TTASI lectures on complex structures
Frederik Denef
Center for the Fundamental Laws of Nature, Harvard University,17 Oxford Street, Cambridge, MA 02138.Simons Center for Geometry and Physics,Stony Brook, NY 11794-3636.Institute for Theoretical Physics, University of Leuven,Celestijnenlaan 200D, B-3001 Heverlee, Belgium. denef physics.harvard.edu
Abstract
These lecture notes give an introduction to a number of ideas and methods thathave been useful in the study of complex systems ranging from spin glasses to D-braneson Calabi-Yau manifolds. Topics include the replica formalism, Parisi’s solution of theSherrington-Kirkpatrick model, overlap order parameters, supersymmetric quantum me-chanics, D-brane landscapes and their black hole duals. a r X i v : . [ h e p - t h ] A p r ontents Q ab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.3 Replica symmetric solution . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.4 Replica symmetry breaking and Parisi matrices . . . . . . . . . . . . . 252.3.5 n → D-brane landscapes 54 N = 2 supergravity . . . . . . . . . . . . . . . . . . . . . . . 845.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.4 Landscape structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922 Introduction
If you are a string theorist, chances are that you think of a complex structure as somethingsquaring to minus one, rather than as something consisting of many intricately interactingdegrees of freedom. These lecture notes, however, are concerned with the latter.There is a deeply rooted belief in the natural sciences that the more fundamental a theorygets, the less important complexity becomes. In particular there has traditionally been analmost unquestioned assumption that physics at subatomic scales must be intrinsically simple,and that complexity is only relevant in the macroscopic or living world. Certainly there isa decrease in complexity when going from cells to proteins and from there to amino acids,atoms and finally elementary particles. It would seem logical that any step further down thisreductive tree — for example explaining the origin of elementary particles and their properties,accounting for the entropy of black holes or understanding the initial conditions of the universe— should involve only structures and concepts of the utmost simplicity and elegance.By now considerable theoretical and experimental evidence has accumulated suggestingthis is wrong. In particular, whenever quantum mechanics and gravity pair up — as in cos-mological dynamics on the largest space and time scales, the determination of effective lowenergy parameters in theories with compact extra dimensions, and the stringy microphysics ofblack holes — complexity appears not only unavoidable, but seems to some extent essential.On the theoretical side I am referring here more specifically to the highly complex, fractal-likeiterated structures arising in eternal inflation, the perplexing complexity of the string theorylandscape, and the closely related complexity of wrapped D-brane systems that, among manyother applications, have given us the first quantitative explanation of the entropy of blackholes. On the experimental side, the most influential development has been the gathering ofabundant cosmological precision data: detailed cosmic microwave spectroscopy providing con-vincing evidence for the slow roll inflation hypothesis, and the measurement of an impossiblytiny yet nonzero vacuum energy density, of just the right magnitude to allow self-reproducingresonances to crawl out of gravitationally collapsed dust at precisely the right time to see theonset of cosmic acceleration. Whether we like it or not, these developments, and the absenceof plausible alternative explanations for these and other fine-tuning conundrums, have added Except for very specific items, references will be given at the end of this section and in subsequent sections. CP , , , , , , which boasts about three hundred thousand deformationmoduli, up to seventy-five thousand D3-branes, and a flux-induced superpotential specifiedby a choice of almost two million integers multiplying an equal amount of independent periodintegrals [1, 2]. According to simple estimates, this compactification, assuming all moduli canbe stabilized, gives rise to more than 10 intricately interconnected flux compactifications.Similarly, a macroscopic extremal Reissner-Nordstrom black hole looks considerably simplerthan the corresponding weakly coupled D-brane systems one is led to consider in microscopiccomputations of the black hole entropy. For example a wrapped D4 producing a modestextremal black hole of say the mass of the sun easily involves D4-branes with 10 modulisubject to superpotentials specified by 10 flux quanta. Finally, even in simple toy modellandscapes, the large scale structure generated by eternal inflation is infinitely more complexthan what we observe in our universe.Although the importance and urgency of a better understanding of these matters is clearto many, there has been widespread reluctance to face this kind of stringy complexity directly.Instead, the dominant approach has been to try to isolate particular phenomena of interest, forexample by studying simplified compactification models like local Calabi-Yau spaces, effectivefield theory models, non-disordered toy models of the landscape, D-brane models for blackholes restricted to charge regimes where the dominant contributions to the entropy are highly4tructured by symmetry (Cardy regime), or coarse grained asymptotic parameter distributionsin ensembles of compactifications allowing similar complexity-minimizing limits. There are ofcourse obvious excellent reasons to follow this reductionist approach. However, there are alsoexcellent reasons to try to probe complexity itself:1. In contrast to supersymmetric AdS compactifications, a specific choice of compactifica-tion data does in general not correspond to a superselection sector in cosmologies witha positive vacuum energy, as quantum and thermal fluctuations will force changes in ge-ometric moduli, fluxes and topology. Thus, any complete, nonperturbative descriptionof a sector of string theory rich enough to describe our own universe should be able toencode not just one compactification, but the full space of internal space configurationsthat can be dynamically reached in one way or another. Since various topological tran-sitions between for example Calabi-Yau manifolds are known to be perfectly sensiblephysical processes in string theory, the space of such interconnected configurations islikely to be huge. If we imagine for a moment that there exists a complete holographicdescription of eternally inflating cosmologies in string theory, say in the form of a fieldtheory living at future infinity, then this field theory would somehow have to encodethe full complexity of those googols of geometries — it would literally be a theory ofeverything, and it couldn’t be anything less.2. There is often striking organization, universality and elegance emerging in disorderedsystems, as has become clear over the past decades in studies of spin and structuralglasses, neural networks and other complex systems. Conventional notions of symmetryare largely irrelevant in such systems, but other, equally powerful structures and theirassociated order parameters appear in their place. This includes hierarchical clusterorganization of the state space, replica symmetry breaking and overlap order parameters.Uncovering these led to highly nontrivial exact solutions of various models of complex,disordered systems. Much like symmetries in ordered systems, these structures alsodetermine to a large extent the dynamics and other physical properties. They have awide range of applicability in fields as diverse as condensed matter physics, neuroscience,biology and computer science, and have led to practical applications such as new efficientalgorithms for optimization, data mining and artificial intelligence. Thus, rather thanan annoyance to be avoided, complexity can be the essence, and the key feature to focus It is not necessary that such configurations also support metastable “vacua”. N limit is alsothe limit one is a priori interested in here. Concrete examples are given by supersym-metric branes wrapped on compact cycles in Calabi-Yau manifolds, which exhibit manyof the typical characteristics of mean field models of glasses. In these lecture notes, I will give an introduction to some of the concepts and techniques whichhave been useful in studies of complex systems of many degrees of freedom, with intricate,disordered interactions. Although I will discuss ideas and techniques developed in the theoryof glasses as well as ideas and techniques developed in string theory, there will be little or nodiscussion of applications of one to the other. For this I refer to the work that will appear in[3, 4, 5], and in which any new idea that might be present in these notes originated. The focuswill be on the basics, meaning material usually assumed to be known in most of the recentspecialized literature. Rather than to give a comprehensive review, I will treat a number ofspecific topics in a more or less self-contained and hopefully pedagogical way, to avoid variantsof step 7 of [6], and to allow the reader to learn how to actually compute a number of thingsrather than to just get a flavor of the ideas. This comes at a price of having to leave out manyinteresting and important topics (including some that were discussed to a certain extent in myactual lectures at TASI). In particular, despite its importance in the motivation given above,I will say almost nothing directly here about the landscape of string compactifications, andfocus instead on space localized D-brane systems arising as the weak coupling description ofcharged black holes. This has three additional reasons: ( i ) I have already written extensivelecture notes on string vacua [7], and wanted to avoid overlap altogether. ( ii ) The internalspace geometry of the space-localized D-brane systems describing black holes is identical tothat of space-filling D-brane systems describing compactifications, and the techniques used toanalyze the former can directly be transported to the latter. The advantages of looking firstat space-localized branes are furthermore numerous: They are conceptually and technicallyeasier, allow arbitrarily large charges and hence a proper thermodynamic limit, and have6olographic dual descriptions as black holes, providing effective “experimental measurements”of thermodynamic quantities like the entropy. ( iii ) It prevents, for the time being, potentialdiffusion of confusion from partially unresolved conceptual problems of quantum gravity in acosmological setting (as reviewed by Tom Banks at this school [8]) into what should be anexposition of well-understood and diversely applicable methods.More specifically, I will cover the following topics:1. An introduction to the theory of spin glasses, in particular Parisi’s solution of theSherrington-Kirkpatrick model, using the replica formalism. This was the first non-trivial energy landscape to be studied and understood in detail in physics. Specialattention is given to Parisi’s overlap order parameters, which allow to detect a non-trivial equilibrium state space structure without having to be able to explicitly knowthose equilibrium states. A generalization of this order parameter for arbitrary quan-tum systems is proposed at the end of the section. Some other complex systems such asthe Hopfield model for memory and learning are briefly discussed. Other, non-replicaapproaches such as the cavity method and Langevin dynamics are important but nottreated in any detail in these notes.2. An introduction to supersymmetric quantum mechanics, where simple but powerfulconcepts such as the Witten index make it possible to compute exact quantum groundstate degeneracies of highly complex systems. The computation of nonperturbativelifting effects due to landscape barrier tunneling as well as relations to Morse theoryare discussed in some detail. Finally, the map between nonsupersymmetric Langevindynamics and supersymmetric quantum mechanics is introduced, and its appearance inglass theory is outlined.3. An introduction to the low energy quantum mechanics description of D-branes wrappedon compact cycles in Calabi-Yau manifolds. These are the prime examples of complexsystems in string theory, and excellent models to introduce techniques ubiquitous inthe field of string compactification. Again I start from scratch, assuming only somebackground knowledge in elementary differential geometry and the basics of D-branesin string theory. I describe in detail the low energy reduction of the D4-brane wrappedon a high degree 4-cycle. This results in a supersymmetric quantum mechanics of highcomplexity, which nevertheless is manageable thanks to the underlying supersymmetric7eometry. If the D4 charge is of order N , the brane allows of order N worldvolume mag-netic fluxes, inducing an extremely complicated superpotential for its order N moduli,and leading to an enormous energy landscape with exponentially many minima. Never-theless it is possible to explicitly construct vast numbers of exact supersymmetric criticalpoints, by giving the critical point condition the interpretation of “capturing” holomor-phic 2-cycles. At large D0-charge, this method becomes inadequate, but at the same timeit becomes easier to give good estimates for the number of ground states. A formula forthe index is derived by directly applying the machinery of Lagrangian supersymmetricquantum mechanics, treating the worldvolume fluxes as momenta canonically conjugateto angular variables. This reproduces the well known continuum approximation at lead-ing order, but also provides all corrections in terms of differential euler characteristicsof critical point loci. Finally, bound states with mobile D0-branes are introduced andcounted in a rudimentary fashion, a microscopic formula for the D4-D0 is derived, andsome open ends and generalizations are summed up to conclude.4. An overview of the construction of the zoo of multicentered supersymmetric black holebound state solutions that are the holographic duals of the D-brane energy landscapesdiscussed in the previous section. For the topics covered in these lectures, I will give references along the way. For the topicsmentioned above but which will not be treated in any detail in what follows, here are a fewgeneral referencesthat may be useful: [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,24, 25, 26, 27, 28, 29, 30, 22, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 7, 41, 42].
A spin glass [43, 44, 45] is a system of localized spins with disordered interactions. An exam-ple is copper sprinkled with manganese atoms at random positions. The sign of the spin-spininteraction potential between the manganese atoms oscillates as a function of distance and isrelatively long range, so we get effectively randomized mixed ferromagnetic and antiferromag-8etic interactions. Some spin pairs will want to align, while others will wish to anti-align. Ifamong three given spins, two pairs want to align and one pair wants to anti-align, or if allthree pairs want to anti-align, one pair will not get what it wants and the triangle is calledfrustrated. The presence of many frustrated triangles typically leads to exponentially manylocal minima of the energy, and thus to a landscape.This landscape is highly complex; the problem of finding the ground state in even thesimplest spin glass models is effectively intractable. Nevertheless, as we shall see, some modelsof spin glasses have exactly solvable thermodynamics, thanks to a remarkable underlyingmathematical structure which greatly simplifies the analysis but nevertheless leads to a veryrich phase space structure. Its most striking consequence is ultrametricity of the state space.An ultrametric space is a metric space with a distance function d that satisfies somethingstronger than the standard triangle inequality, namely d ( a, b ) ≤ max { d ( b, c ) , d ( c, a ) } . (2.1)What this means is that all triangles are isosceles, with the unequal side the shortest of thethree. Ultrametric spaces appear in various branches of mathematics; for example the p -adicdistance between rational numbers is ultrametric. For nice introductions to the mathematicsof ultrametric spaces, see chapter 9 of [46], and [47]. Ultrametric spaces are also familiar inbiology: If we define the distance between two current species as how far back in time onehas to go to find a common ancestor, this distance is trivially ultrametric. Indeed, picking aturkey, a monkey and a donkey for example, the distance turkey-donkey and turkey-monkeyis the same, while the distance monkey-donkey is the shorter one of the three. Alternatively,one can define a species distance as the degree of difference of DNA. Since this will be roughlyproportional to how far back a common ancestor is found, this distance will again be essentiallyultrametric. The same is true for any representative set of proteins. This is used with greatsuccess to reconstruct evolutionary trees (for a recent account see e.g. [48]). The lesson herefor us is that ultrametricity is equivalent to the possibility of clustering objects and organizingthem in a hierarchical tree , similar to the evolution tree of species.In the case of spin glasses, the points in the metric space are the different equilibrium states,that is the different ergodic components or superselection sectors in which the standard Gibbsmeasure breaks up in the glass phase. We will describe this in detail in the subsequent sections, The p -adic norm of a rational number is | q | p = p − k when q can be written as q = p k m/n where m and n contain no powers of p . The p -adic distance is then d p ( a, b ) ≡ | a − b | p . An ultrametric tree. Distances between the points in the top layer are set by the vertical distanceto the first common ancestor node. here we give a qualitative picture. The distance between different equilibrium states is givenby some measure of microscopic dissimilarity, for example the sum of local magnetizationdifferences squared. Remarkably, in mean-field models such as the Sherrington-Kirkpatrickmodel [49], this space turns out to be ultrametric [50, 51, 52]: In the thermodynamic limit,with probability 1, triples of magnetization distances are isosceles. Equilibrium states gethierarchically organized in clusters. The cluster division is independent of the distance measureused, as long as it is statistically representative, meaning zero distance implies equal states.Figure 2 shows the results of a simulation illustrating this phenomenon.What is remarkable is that such a nontrivial tree structure appears in a purely staticsetting, without any apparent underlying evolution process creating it. Instead, the role ofevolutionary time is played by temperature or more generally energy scale: At high tempera-tures there is only one ergodic component of phase space. When the temperature is loweredbelow the spin glass phase transition, the phase space starts breaking up into distinct partsthat no longer talk to each other, separated by effectively unsurpassable free energy barriers.Initially these valleys in the free energy landscape will all still look very similar, but uponfurther cooling, their mutual magnetization distance grows, and eventually they in turn willstart breaking up in components separated by free energy barriers. This branching processproceeds all the way down to zero temperature, generating an ultrametric tree of states. Anypair of ground states defines a “time” on this tree, proportional to their ultrametric distance. To be precise, the remarkable thing is the existence of a nontrivial ultrametric structure, with multiplebranchings. Trivial ultrametricity, meaning all triangles are equilateral, is very easy to realize in infinitedimensional spaces. For example all pairs of randomly chosen points on an infinite dimensional sphere arewith probability 1 at the same distance from each other. [5] Dendrogram plots and overlap matrices for the SK model with N = 800 spins. We used paralleltempering Monte Carlo [53, 54] to reach thermal equilibrium, with 50 replicas at equally spaced temperaturesbetween T = 0 . T = 1 .
2. Based on overlaps of 100 configurations sampled with separation of 100 sweeps,clustered using Mathematica. Results are shown for
T /T c = 1 . , . , . , .
12. Red = maximal positiveoverlap q = +1 (i.e. minimal distance), white = overlap q = 0, dark blue = maximal negative overlap q = − As we will see, ultrametricity plays an important role in the exact solution of mean fieldspin glass models, and determines many of their static and dynamic properties. It plays inthis sense a role similar to that of symmetries and symmetry breaking in physics. In fact, itsemergence is closely related to the breaking of an auxiliary symmetry, called replica symmetry,which naturally arises in the description of disordered systems. The order parameter (or ratherthe order function ) capturing the symmetry breaking pattern exhibits ultrametricity as oneof its characteristic features. We will discuss this extensively in what follows.For more realistic, local (short range) spin glass models in three dimensions, for which themean field approximation is poor, there is no consensus on the physical presence or relevance ofultrametricity, and different schools exist with different favorite models and results, the mostwidespread being the droplet model of [55]. On the other hand, in systems where little or nolocality is present, such as neural networks or combinatorial optimization problems, mean fieldmethods and ultrametricity are ubiquitous [43]. A recent numerical study of ultrametricity insimple Ising type models of different degrees of locality can be found in [56].For a review of ultrametricity in physics, see [57]. General criteria for the appearanceultrametricity were formulated in [58]. Introductory texts on spin glass theory include [43,59, 60, 61] 11n the following we will make the above qualitative description more precise, introducethe new order parameters and explain the mysterious but powerful replica trick and thephenomenon of replica symmetry breaking associated with ultrametricity.
The most basic models for spin glasses are formally similar to the classical Ising model, with N spins s i = ± H = (cid:88) ij J ij s i s j . (2.2)In the case of the Ising model, we have J ij = − J for nearest neighbors ( ij ) and zero otherwise.At high temperatures the magnetization M ≡ N (cid:80) i m i is zero ( m i is the average value of s i ).In dimension 2 or higher, below a critical temperature T c , the magnetization acquires a nonzerovalue and the Z spin flip symmetry is spontaneously broken. In the mean field approximation, T c = J and M is a solution of the mean field consistency equation M = tanh( M J/T ).In 1975, to model spin glasses, Edwards and Anderson [62] proposed to study the aboveHamiltonian, still with nearest neighbor couplings J ij , but now drawn randomly and indepen-dently from a Gaussian distribution with zero mean and standard deviation J . Although atfinite N everything in this model will depend on the actual values of J ij , in the thermody-namic limit N → ∞ , intensive quantities such as the (spatially averaged) magnetization orfree energy density are self-averaging: they become equal to their J ij -averaged value with unitprobability. This makes computations possible. Due to the disorder, the magnetization M vanishes at all temperatures in this model. Nevertheless, there exists a critical temperaturebelow which the spins freeze, in the sense that they acquire locally preferred directions. Thisdisordered frozen phase is the spin glass phase. The magnetization is clearly not a good orderparameter to detect this situation. Instead Edwards and Anderson introduced a new orderparameter: q EA ≡ N (cid:88) i m i . (2.3) In the statistical mechanics literature, the common convention is to put a minus sign in front of the sumover the spins, so ferromagnetic couplings are positive. We will not do this in these lecture notes. m i are the local magnetizations in the equilibrium state the system finds itself in,which in an actual physical setting can be thought of as the time averaged value of the spin s i .For the Ising model m i is independent of i , so q EA = M . The Edwards-Anderson parameteris related to the experimentally measured magnetic susceptibility: χ = (1 − q EA ) /T .Though the meaning of time averaging is physically clear, we have not specified any dy-namics for this model, so we need a purely static definition for m i . This is subtle. It is not thecanonical ensemble average (cid:104) s i (cid:105) obtained from the Gibbs probability measure p ( s ) ∝ e − βH ( s ) ,as this average is trivially zero due to the Z symmetry of H . The same issue arises alreadyfor the Ising model, but there it is clear what is going on: below T c the Gibbs probabilitymeasure splits in two “superselection” sectors, characterized by opposite values of the mag-netization. The two sectors can be separated by switching on a small background magneticfield h , which shifts H → H + h (cid:80) i s i , lifting the Z degeneracy and eliminating one of thetwo sectors, depending on the sign of h . We can then define the magnetization for each sectoras m i ± ≡ lim h → ± lim N →∞ (cid:104) s i (cid:105) h .For spin glasses this does not work; since M = 0, switching on a constant h will notlift the degeneracy. Switching on an inhomogeneous field h i tailored to the m i of the frozenequilibrium state we want to single out would of course do the job, but it is impossible toknow in advance what the required profile is going to be; it could be anything, and it willbe different for different values of J ij . To make things worse, there may be many distinctequilibrium states, all with different values of the m i , and again there is no way of tellingwithout actually solving the system. There is no standard symmetry and no order apparentin the system at any temperature. What “order” parameter could possibly distinguish betweenthose phases?These questions were considerably sharpened and then answered through the study of aneven more simplified spin glass model, proposed by Sherrington and Kirkpatrick [49] in 1975.The Hamiltonian is as above except that now the couplings J ij are nonzero for all pairs ofspins, not just nearest neighbors; they are all independent Gaussian random variables withzero mean and variance J / N , J ij J kl = J N δ ik δ jl . (2.4)Overlines will denote averages over the J ij throughout these notes. The model is completely The variance is scaled with N such that the typical size of the interaction potential at each lattice siteremains finite in the thermodynamic limit: ( (cid:80) j J ij s j ) = J /
2. The factor of 2 is added for consistency withstandard conventions, in which usually J ij = J ji is imposed, which we will not do. T = 0, where it predicted for instance negative entropy densities. Afterconsiderable effort, the correct solution was finally found in 1979, in a seminal breakthroughby Parisi [50, 51]. (It took another 25 years before the solution was rigorously proven to becorrect [63, 64].) In the course of the process, Parisi uncovered the proper order parametersto fully describe the spin glass phase, as well as a hidden “statistical” symmetry group whosebreaking they parametrize. We will describe the solution in section 2.3, but first we describethe physical meaning of the order parameter. To do this we must introduce the notion ofthermodynamic pure states.
The probability measure characterizing the Gibbs state is p G ( s ) = 1 Z e − βH ( s ) , Z = (cid:88) s e − βH ( s ) . (2.5)For a given probability measure p , we denote the expectation value of an observable A ( s )by (cid:104) A (cid:105) p . For the Gibbs state we usually just write (cid:104) A (cid:105) G = (cid:104) A (cid:105) . We say p satisfies clusterdecomposition if correlation functions of local observables factorize in the thermodynamiclimit for almost all points. That is, for any finite r , in the limit N → ∞ , (cid:104) A i B i · · · C i r (cid:105) p = (cid:104) A i (cid:105) p (cid:104) B i (cid:105) p · · · (cid:104) C i r (cid:105) p + R i i ··· i r , (2.6)where the remainder R is negligible on average: lim N →∞ N r (cid:80) i ··· i r | R i ··· i r | = 0. Here A i , B i ,. . . , C i are local observables, like for example A i = s i or B i = s i − s i s i +3 . In particular theclustering property implies that intensive quantities like the free energy density have definite,non-fluctuating values. In local theories (2.6) is equivalent to the property that correlationfunctions factorize in the limit of infinite spatial separation.The Gibbs measure for the Ising model satisfies cluster decomposition above the criticaltemperature, but not below: at high temperatures we have (cid:104) s i s j (cid:105) = 0 = (cid:104) s i (cid:105)(cid:104) s j (cid:105) when | i − j | →∞ , but at low temperatures we have instead (cid:104) s i s j (cid:105) = M (cid:54) = (cid:104) s i (cid:105)(cid:104) s j (cid:105) = 0 ·
0. However we can Historically it was first obtained as a formal mathematical object in the replica formalism. The physicalinterpretation was given later in [65]. p G ( s ) = p + ( s ) + p − ( s ) , (2.7)where p ± ( s ) = lim h →± lim N →∞ Z e − β ( H ( s )+ h (cid:80) i s i ) . Sets of finite measure according to p + havezero measure according to p − and vice versa. The measures p ± do satisfy cluster decomposi-tion: (cid:104) s i s j (cid:105) + = M = (cid:104) s i (cid:105) + (cid:104) s j (cid:105) + . The superselection sectors described by p ± are called “purestates” in the statistical mechanics literature.Rigorously defining pure states for general systems is subtle. For a discussion in thecontext of spin glasses we refer to page 89 (appendix 1) of [66]. What is known rigorously isthat any probability measure p G defined for a system of infinite size, which locally behaveslike the Gibbs measure (i.e. it gives relative probabilities proportional to e − β ∆ H for finite sizefluctuations), can always be uniquely decomposed into pure states p α as p G ( s ) = (cid:88) w α p α ( s ) . (2.8)Here w α > α . The pure states satisfy theclustering property and cannot be further decomposed. This gives them an implicit definition. As mentioned earlier, for disordered systems, it is in general impossible to explicitly findthe actual decomposition into pure states. But granting the decomposition exists, one canformally define the overlap q αβ between pure states as q αβ ≡ N (cid:88) i (cid:104) s i (cid:105) α (cid:104) s i (cid:105) β . (2.9)As a special case, notice that q αα is nothing but the Edwards-Andersen order parameter q EA defined in (2.3). Although not at all obvious, it turns out [67] that q αα is independent ofthe pure state α (and of the disorder realization J ij ), so q EA is actually an invariant of the More intuitively, in physical systems, pure states correspond to distinct ergodic components, individuallyinvariant under time evolution and not further decomposable into smaller time invariant components. Inergodic components the ergodic theorem implies that the time average of any observable equals its ensemble(phase space) average.
T /J . Using this, we see that the overlap is closely related to theEuclidean distance between states: d ( α, β ) = 1 N (cid:88) i ( m iα − m iβ ) = 2( q EA − q αβ ) . (2.10)At this point the overlap matrix may still seem like an abstract, incomputable quantity.However, consider the overlap probability distribution : P ( q ) = (cid:88) αβ w α w β δ ( q − q αβ ) . (2.11)This is the probability of finding an overlap q αβ = q when one samples the Gibbs state. Thewonderful thing is that this quantity is actually computable without any knowledge of theactual decomposition into pure states. It can be rewritten purely in terms of the Gibbs stateas the overlap distribution for two identical replicas of the system, with spins s (1) and s (2) : P ( q ) = (cid:68) δ (cid:16) q − N (cid:80) i s (1) i s (2) i (cid:17)(cid:69) n =2 , (2.12)where n = 2 means we are considering two replicas: (cid:10) A ( s (1) , s (2) ) (cid:11) n =2 ≡ Z (cid:88) s e − βH ( s (1) ) − βH ( s (2) ) A ( s (1) , s (2) ) . (2.13)To prove this, one shows the moments are equal. Consider for example its second moment: (cid:104) q (cid:105) = (cid:90) dq P ( q ) q = (cid:88) αβ w α w β N (cid:88) i (cid:104) s i (cid:105) α (cid:104) s i (cid:105) β N (cid:88) j (cid:104) s j (cid:105) α (cid:104) s j (cid:105) β = 1 N (cid:88) αβ w α w β (cid:88) ij (cid:104) s i s j (cid:105) α (cid:104) s i s j (cid:105) β = 1 N (cid:88) ij (cid:104) s i s j (cid:105)(cid:104) s i s j (cid:105) = 1 N (cid:42)(cid:32)(cid:88) i s (1) i s (2) i (cid:33) (cid:43) n =2 . Since the SK model is completely nonlocal and does not a priori distinguish any site j relative to a givensite i , we can think of the permutation symmetry acting on the site indices i as a gauge symmetry, similarto the diffeomorphism group in gravity (where J ij is the analog of the spacetime metric). Then the localmagnetization distribution P ( m ) ≡ N (cid:80) i δ ( m − m iα ) = δ ( m − m iα ) encodes all gauge invariant informationbased on the “vevs” m iα . It is shown in [67] that again this quantity is independent of the state α . So we cansay that there is no gauge invariant distinction between different pure states based on just the magnetizations;all equilibrium states look the same as far as non-fluctuating quantities are concerned. From [68]. Disorder-averaged overlap distribution P ( q ) for the SK model at various temperatures,from T = 0 .
95 on right to T = 0 .
30 on left in steps of 0.05 ( T c = 1). The dotted lines represent delta-functions,localized at q αα = q EA . To get to the third line we used the clustering property (2.6), and for the fourth line we used(2.8) in reverse to express everything in terms of Gibbs state expectation values. The highermoments are treated analogously.Equation (2.12) in principle allows us to compute or approximate P ( q ) by standard meth-ods, for example by using Monte Carlo simulations sampling the Boltzmann-Gibbs distribu-tion. It captures the presence and properties of pure states very well and is therefore a goodorder parameter for the spin glass phase. If only one pure state is present, namely the Gibbsstate itself, as is the case at high temperatures, then q = q EA = 0 due to the Z symme-try, and P ( q ) = δ ( q ). If the system freezes and splits into one Z pair of pure states, withlocal magnetizations m i and − m i , then for both states the self-overlap is q EA = N (cid:80) i m i and P ( q ) = δ ( q − q EA ) + δ ( q + q EA ). If there are many pure states, there will be manydelta-functions, and if there is an infinite number of them, P ( q ) becomes continuous.Computing P ( q ) analytically for a given coupling matrix J ij is not possible. It is howeverpossible to compute its average P ( q ), by a variety of methods that can also be used to computemore basic thermodynamic quantities; the most prominent ones are the replica, cavity (TAP),Langevin dynamics and supersymmetry methods [43, 59, 69, 70, 71, 72]. In these lecturenotes we will focus on the replica method. To already get an idea of where all this is heading,a result of such computations is shown in fig. 3, displaying the disorder-averaged P ( q ) forvarious temperatures below T c . The function is smooth except for a delta-function peak at17he highest value of q , which is the self-overlap q αα = q EA . Its presence is a consequence ofthe state and disorder independence of q EA , leading to a term ( (cid:80) α w α ) δ ( q − q EA ) in (2.11).The nonvanishing of (cid:80) α w α in turn indicates the presence of states of finite weight w α . Thecontinuity of P ( q ) shows that there are in general multiple pure states, but it does not imply P J ( q ) is continuous at fixed J ; in fact because some w α are finite, P J ( q ) must have multipledelta peaks, but with J -dependent locations, so integrating over J smooths them out. Indeed,unlike the magnetization or the free energy, P J ( q ) is not a self-averaging quantity; it fluctuatesbetween different realizations of the disorder. One can also define more refined overlap distributions, such as the overlap triangle distribution P ( q , q , q ) ≡ (cid:88) αβγ w α w β w γ δ ( q − q βγ ) δ ( q − q γα ) δ ( q − q αβ ) , (2.14)which similar to (2.12) can be written in terms of the overlap distribution of three independentreplicas in the Gibbs state. Restricting to triples with positive overlaps, one finds [52] P ( q , q , q ) = 12 (cid:90) q dq (cid:48) P ( q (cid:48) ) P ( q ) δ ( q − q ) δ ( q − q )+ 12 (cid:18) P ( q ) P ( q ) θ ( q − q ) δ ( q − q ) + permutations (cid:19) , (2.15)where θ is the step function and P ( q ) is as in (2.11). This manifestly exhibits ultrametric-ity, with the first term encoding equilateral and the last three terms more general isoscelestriangles. We now turn to the actual computations in the SK model, first using the replica formalism[62, 49, 50, 51]. The Hamiltonian is, as stated earlier: H J [ s ] = (cid:88) i (cid:54) = j J ij s i s j , (2.16) This is necessary because without this restriction, the Z symmetry trivially destroys the ultrametricstructure: A sign flip of the spins in state α will flip the sign of two out of three overlaps of an ( αβγ ) triangle,and clearly this does not preserve isoscelesness. This also explains the structure in the blue regions of fig. 2. i, j = 1 , . . . , N , s i = ± J ij is drawn randomly out of a Gaussian distribution withmean 0 and variance J N , i.e. p ( J ij ) ∝ e − NJ J ij . (2.17)Other distributions would give equivalent results; only the first two moments matter. It isimpossible to solve this system exactly for arbitrary given couplings J ij — just finding theabsolute minimum energy configuration is already an NP-hard problem [73]. Fortunatelythe usual intensive thermodynamic quantities of interest are self-averaging , that is they areindependent of the random matrix J ij with probability going to 1 in the thermodynamic limit.Thus we can compute them by computing averages. For example the free energy density is F = F J ≡ (cid:90) (cid:89) ij dJ ij p ( J ij ) F J , F J ≡ − βN log Z J , Z J ≡ (cid:88) s ∈{± } N e − βH J [ s ] . (2.18)Crucially, the average log Z J has to be computed after taking the logarithm. Such an aver-age is called quenched , the disorder represented by the J ij is called quenched disorder, etc.Computing the average first, i.e. on the partition function itself, is called annealed averaging.Physically, this corresponds to a situation in which the couplings themselves are fluctuatingvariables. The annealed average is much easier to compute than the quenched average, but itis not the situation we are in; in a real spin glass for example the couplings are determined bythe positions of the impurity atoms in the host crystal, which vary randomly over space butdo not fluctuate in time, or if they do, on much longer time scales than the spins fluctuate.To deal with the quenched average, one can use the replica trick. This is based on theobservation that log Z = lim n → Z n − n = ∂∂n Z n | n =0 , (2.19)together with the fact that for positive integers n , the average of Z n is just an annealed averageof the disorder coupled to n replicas of the original system, and therefore easy to compute.Of course, the subtle part is the “analytic continuation” to n = 0, which is hard to makerigorous. However, its effective power in a large range of applications is undeniable, so the A perhaps somewhat more precise version starts from the observation that Z n is an entire function of n ,with Taylor expansion Z n = e n log Z = (cid:80) ∞ k =0 n k k ! (log Z ) k valid for all n , so Z n = 1 + n log Z + · · · for all n . Theidea is then that the term linear in n can be extracted by considering arbitrary integers n and computing Z n as an expansion in n . However this is in principle ambiguous: for example 1 + sin( πn ) is entire and evaluatesto 1 for all integers, yet it has a nonvanishing first order term in its Taylor expansion. For a deeper analysissee e.g. [74, 75]. a, b = 1 , . . . , n , the coupling-averaged n -fold replicated partitionfunction is, putting J = 1, Z n = (cid:90) dJ ij e − NJ ij (cid:88) s ∈{± } nN e − βJ ij s ia s ja . (2.20)Repeated indices are summed over and we absorb the Gaussian normalization factors in themeasure dJ ij . At this point, the replicas do not interact with each other. However, becausethey all couple to the same J ij , integrating out the disorder induces effective interactionsbetween the replicas: Z n = (cid:88) s ∈{± } nN e β N s ia s ja s ib s jb . (2.21)The positive sign in the exponent means the interaction is attractive. Now, reverse-mimickinghow replicas got coupled by integrating out the site coupling matrix J ij , we de couple thelattice sites by integrating in a “replica coupling matrix” Q ab : Z n = (cid:88) s ∈{± } nN (cid:90) dQ e − N β ( Q ab ) e β Q ab s ia s ib = (cid:90) dQ e − N β ( Q ab ) (cid:18) (cid:88) S ∈{± } n e β Q ab S a S b (cid:19) N . (2.22)The S -sum here runs over the n replica copies only; lattice indices no longer occur. The sumover the lattice sites has simply produced an overall power of N . This is awesome becauseit means that in the thermodynamic limit N → ∞ , we can evaluate the integral over Q in asaddle point approximation. It all boils down now to finding the critical points of F ( Q ) ≡ β Q ab − β log Z ( Q ) , Z ( Q ) ≡ (cid:88) S ∈{± } n e β Q ab S a S b . (2.23)Denoting the dominant critical point(s) of F ( Q ) by Q (cid:63) , we thus get Z n = e − βN F ( Q (cid:63) ) . If wecan find the saddle points for general n , we are done. Using the trick (2.19) and noting thatconsistency with Z = 1 requires F ( Q (cid:63) ) | n =0 = 0, we obtain the free energy density: F = ∂∂n F ( Q (cid:63) ) | n =0 . (2.24)20o summarize, what we have done is trade summing over the lattice for summing over replicas,which allows us to do a saddle point computation of the quenched average. Before proceedingto find the solution, we pause to ponder the meaning of the matrix Q ab that apparentlycaptures the large N behavior of the model. Q ab So far the physical meaning of the matrix Q ab is obscure, although the conspicuous notationalsimilarity with the overlap matrix q αβ introduced in (2.9) suggests the two are related. Wewill now show that this is indeed the case, in that its structure in the limit n → P ( q ) defined in (2.11), or more precisely P ( q ). To see this,first observe that the saddle point equations F (cid:48) ( Q (cid:63) ) = 0 can be written as the self-consistencyequation Q (cid:63)ab = (cid:104) S a S b (cid:105) Q (cid:63) ≡ Z ( Q (cid:63) ) (cid:88) S ∈{± } n S a S b e β Q (cid:63)cd S c S d . (2.25)Hence Q ab equals the overlap in replica space . Now consider the expression (2.12) for P ( q ): P ( q ) = (cid:88) s ,s ∈{± } N e − β ( H [ s ]+ H [ s ]) Z δ ( q − N N (cid:88) i =1 s i s i ) . (2.26)This depends on the random couplings J ij . We want to average it but are facing a problemsimilar to the problem we had when we wanted to compute the average of log Z : now theproblem is the explicit appearance of J in the factors 1 /Z . We deploy again the replica trick,this time based on P ( q ) = lim n → (cid:88) s ∈{± } nN e − β (cid:80) na =1 H [ s a ] δ ( q − N N (cid:88) i =1 s i s i ) , (2.27)where we have singled out the first two replicas to appear in the delta function. Indeed, theright hand side can for n ≥ Z n − (cid:88) s ,s ∈{± } N e − β ( H [ s ]+ H [ s ]) δ ( q − N N (cid:88) i =1 s i s i ) , (2.28)formally reducing to (2.26) when continued to n →
0. Since in (2.27) there are no longer anydenominators, we can compute the disorder average by simple Gaussian integration.21sing (2.27) and manipulations similar to those leading up to (2.22), one computes thefinite moments (cid:104) q k (cid:105) ≡ (cid:82) dq P ( q ) q k . Dropping 1 /N suppressed terms: (cid:104) q k (cid:105) = lim n → (cid:90) dQ e − β N ( Q ab ) (cid:18) (cid:88) S ∈{± } n e β Q ab S a S b (cid:19) N − k (cid:18) (cid:88) S ∈{± } n e β Q ab S a S b S S (cid:19) k = lim n → (cid:90) dQ e − βN F ( Q ) (cid:18) (cid:88) S ∈{± } n e β Q ab S a S b Z ( Q ) S S (cid:19) k , (2.29)where we have absorbed in the measure dQ a normalization factor 1 / (cid:82) dQ e − βN F ( Q ) , whichensures (cid:104) q (cid:105) = 1. The functions Z ( Q ) and F ( Q ) were defined in (2.23). We may replace S S by S a S b with a (cid:54) = b and average over replicas, since there is no distinction between replicas.The resulting expression has the advantage of being the same at distinct saddle points relatedby the permutation symmetry of the replicas. In view of (2.25), the saddle point evaluationthen gives (cid:104) q k (cid:105) = lim n → (cid:88) [ Q (cid:63) ] W Q (cid:63) n ( n − (cid:88) a (cid:54) = b Q k(cid:63)ab = − lim n → (cid:88) [ Q (cid:63) ] W Q (cid:63) (cid:88) b (cid:54) =1 Q k(cid:63) b . (2.30)The sum over Q (cid:63) is a sum over distinct replica permutation symmetry orbits, i.e. it is non-trivial only if there are different saddle points not related by the permutation symmetry ofthe replicas, as may be the case when there are additional symmetries beyond the replicapermutation symmetry. For each such orbit Q (cid:63) , W Q (cid:63) denotes its relative weight. In the caseat hand, we do have an additional Z spin flip symmetry in the system. If this is the onlyother source of saddle degeneracy, we have W Q (cid:63) = . In general we conclude: P ( q ) = − lim n → (cid:88) [ Q (cid:63) ] W Q (cid:63) (cid:88) b (cid:54) =1 δ ( q − Q (cid:63) b ) , (2.31)and if there is no degeneracy besides the Z one (as one would expect generically), this becomes P ( q ) = − lim n → (cid:88) b (cid:54) =1 12 δ ( q − Q (cid:63) b ) + δ ( q + Q (cid:63) b ) . (2.32)Thus the pure state overlap distribution will be entirely determined by the saddle pointsolution Q (cid:63) continued to n →
0. To obtain the probability to find an overlap q , all we need todo is compute the fraction of entries Q (cid:63) b that are equal to q . The weird looking minus signis not a typo: it must be there because (cid:80) b (cid:54) =1 n − → − n →
0. Still, countingthe number of entries Q b in a 0 × × Q ab in fact has infinitely many degrees of freedom!22 .3.3 Replica symmetric solution Finding the critical points of F ( Q ) for general n is still a nontrivial task. Some obvious conse-quences of (2.25) are Q (cid:63)aa = 1 and the fact that it is a positive definite symmetric matrix, butto make further progress, one must make an ansatz for the form of the solution. The simplestone, which was the one used by Sherrington and Kirkpatrick [49], is the replica symmetric(RS) ansatz, which is the unique ansatz leaving the permutation symmetry unbroken (we willdrop the explicit (cid:63) subscripts from here on): Q ab = uδ ab + q (1 − δ ab ) . (2.33)So Q ab = u if a = b and Q ab = q if a (cid:54) = b . The relation (2.32) then gives P ( q (cid:48) ) = 12 (cid:0) δ ( q (cid:48) − q ) + δ ( q (cid:48) + q ) (cid:1) . (2.34)Thus the RS ansatz is equivalent to assuming not more than one Z pair of pure states. If q = 0, there is just one pure state and the system is in the paramagnetic phase.We wish to extremize F ( Q ) = β Q ab − β log (cid:80) e β Q ab S a S b . From the ansatz (2.33) we obtain Q ab = nu + n ( n − q , e β Q ab S a S b = e β ( n ( u − q )+ q ( (cid:80) a S a ) ) . (2.35)Assuming without loss of generality q ≥
0, we linearize the term involving S a in the exponentwith the transformation (cid:88) S ∈{± } n e β q ( (cid:80) a S a ) = 1 √ πq (cid:90) dz e − z q (cid:88) S ∈{± } n e zβ (cid:80) a S a (2.36)= 1 √ πq (cid:90) dz e − z q [2 cosh( βz )] n (2.37)= 1 + n √ πq (cid:90) dz e − z q log [2 cosh( βz )] + O ( n ) , (2.38)whence ∂ n F | n =0 = β u − q ) − β u − q ) − β √ πq (cid:90) dz e − z q log [2 cosh( βz )] . (2.39)This is to be extremized with respect to u and q . Extremizing u is trivial: u = 1 , (2.40)23eproducing the result Q aa = 1 we arrived at earlier directly from (2.25). We set u = 1 inwhat follows. There is no simple closed form solution for the saddle point value of q , so let usconsider limiting cases. In the high temperature limit β →
0, we expand in powers of β : ∂ n F | n =0 = − β (cid:0) q (cid:1) − log 2 β + O ( β ) . (2.41)The extremum is at q = 0: As expected for high temperatures the system is in its paramagneticphase. The free energy and entropy densities at q = 0 are, from (2.24): F = − log 2 β − β , S = β ∂ β F = log 2 − β . (2.42)The log 2 corresponds the two-fold spin degeneracy at each lattice site.Below a critical temperature T c , q = 0 ceases to be the thermodynamically stable saddlepoint. To find this transition point, we expand (2.39) for small q∂ n F | n =0 = − log 2 β − β β ( β − q − β q + O ( q ) . (2.43)The coefficient of q changes sign when β = 1, and a new saddle point with q ≈ β − > This signals the spin glass phase transition.Recalling we chose units such that the coupling variance parameter J = 1, we conclude T c = J . (2.44)This seems all fine, and is consistent with Monte Carlo simulations. However, in the lowtemperature limit β → ∞ something awkward happens. We have ∂ n F | n =0 = − (1 − q ) β − (cid:114) qπ + O ( β − ) . (2.45)The relevant saddle point is at q = 1 − (cid:113) π β . This means the self-overlap q EA of the purestates approaches 1, as expected. But the free energy and entropy density are, to leading What “stable” means in the limit n → maxima in q -space, whereas ordinarily in physicalparameter spaces, stable equilibria are those that minimize the free energy. This “inverted” rule finds itsorigin in the formally negative dimension of the fluctuation modes of Q ab in the limit n = 0. For example,the expression U = n (cid:80) a,b Q ab is manifestly positive definite for positive integers n . It has a positive definiteHessian (the n × n unit matrix) for all n and the Q = 0 extremum of U is stable. Nevertheless, inserting theRS ansatz (2.33) gives U ( q ) = 1 − q . It is in general nontrivial to do a full stability analysis in the replicaformalism. For more discussion see [76, 59]. Tree and matrix representation of a Parisi matrix Q ab . Different colors and different tree connectionheights correspond to different values of Q ab , with darker red representing larger values. In this example, n = 90, K = 3, { m , m , m , m , m } = { , , , , } , { u, q , q , q , q , q } = { , . , . , . , . , } . order F = − (cid:114) π + 12 πβ ≈ − . , S = − π ≈ − . . (2.46)The free energy disagrees with numerical simulations, which indicate F ≈ − .
76 at T = 0.More dramatically, the negative entropy clearly does not make any sense. Therefore thesolution must be wrong at low temperatures. The replica symmetric ansatz (2.33) is apparentlytoo restrictive. The replica symmetry must be broken. We want to relax the replica symmetric ansatz (2.33), which was Q ab = δ ab + q (1 − δ ab ). The effective replica Hamiltonian H R = − βN (cid:80) ab (cid:0) N (cid:80) i s ia s ib (cid:1) appearing in (2.21) de-scribes an attractive interaction between the replicas: It is energetically favorable if the repli-cas line up. Competing with this is the fact that lining up means less phase space. Below T c ,the energy gain is more important than the entropy loss and we get a nonzero overlap expec-tation value between the replicas. In the RS ansatz the degree of correlation is assumed to bethe same for all pairs of replicas. Relaxing this assumption means considering the possibilitythat groups of replicas form clusters that have larger overlaps amongst themselves than withother clusters. The maximally symmetric situation in such a scenario corresponds to the case We put the diagonal part of Q ab equal to δ ab here and it what follows. This is justified by observing thatthe diagonal part of Q ab decouples from the off-diagonal part in (2.25), and that for any saddle point Q aa = 1.
25n which the clusters are indistinguishable from each other, in particular equal in size andmutual overlaps. In this case there is also no absolute distinction between the replicas, onlyrelative to some fixed other replica.To find the thermodynamically relevant critical points of F ( Q ) (defined in equation (2.23)),it is therefore natural to consider [50, 51] a minimal permutation symmetry breaking scheme S n → S n/m × S m , by splitting up the n replicas in clusters of size m , and assuming theoverlap between distinct replicas within one cluster to be q and the overlap between replicasin different clusters to be q < q . Refining the replica labeling by a = a a where a =1 , . . . , n/m labels the clusters and a = 1 , . . . , m the replicas inside each cluster, this translatesto Q a a ,b b = δ a b δ a b + q δ a b (cid:15) a b + q (cid:15) a b , (cid:15) ab ≡ − δ ab (2.47)(so (cid:15) ab = 1 if a (cid:54) = b and 0 if a = b ). The idea is then to substitute this new ansatz in F ( Q ),extract the O ( n ) term, and look for saddle points by varying q , q and m .The clustering process can be iterated, by breaking up the clusters into smaller clustersand then those in turn into even smaller ones. When there are K distinct nontrivial clustersizes, this is referred to as replica symmetry breaking at level K ( K -RSB).To write things out explicitly, put m ≡ m , the next larger one by m and so on, up to m K +1 = n . Let q i be the overlapbetween replicas within a cluster of size m i +1 (excluding those contained in an even smallercluster). Labeling replicas by a = a K a K − · · · a a where a i = 1 , . . . , m i +1 m i labels the clustersof size m i , the ansatz can be written as Q a K ··· a ,b K ··· b = δ ab + K (cid:88) i =0 q i δ a K b K · · · δ a i +1 b i +1 (cid:15) a i b i (2.48)= (cid:88) i ∆ i δ a K b K · · · δ a i b i , ∆ i ≡ q i − − q i > , (2.49)where q i ≡ i < q i ≡ i > K . For example for K = 2 this becomes Q a a a ,b b b = δ a b δ a b δ a b + q δ a b δ a b (cid:15) a b + q δ a b (cid:15) a b + q (cid:15) a b = (1 − q ) δ a b δ a b δ a b + ( q − q ) δ a b δ a b + ( q − q ) δ a b + q . When m does not divide n , it is of course not possible to split the replicas in equal blocks of size m . Theidea is to consider values of n and m for which it is possible, and then to analytically continue the resultingexpressions, treating m , like n , as a continuous parameter. In this section we only consider the integer n case.The continuation will be discussed in section 2.3.5. K = 3 is shown in fig. 4.The hierarchical block structure is equivalent to having an ultrametric structure in theoverlap matrix. In other words, it can be organized as a tree, as illustrated in fig. 4. Definingan ultrametric reference distance r ab between two replicas to be, say, the size m of the smallestcluster to which they both belong, we can reformulate the Parisi ansatz simply as the statementthat the overlap only depends on the distance: Q ab = q ( r ab ) , (2.50)where q ( r ) is an arbitrary function (encoding the q i ).We want to evaluate the replica free energy F ( Q ) = β Q ab − β log (cid:80) S e β Q ab S a S b defined in(2.23). The first term is straightforward: (cid:88) ab Q ab = n (cid:18) K (cid:88) i =0 q i ( m i +1 − m i ) (cid:19) . (2.51)The interpretation of the coefficient ( m i +1 − m i ) multiplying each q i is clear: it is the numberof replicas having overlap q i with some fixed reference replica. The second term in F ( Q ),proportional to log Z ( Q ), where Z ( Q ) ≡ (cid:80) S e β Q ab S a S b , is more interesting. Using (2.49), wehave Z = (cid:88) S exp (cid:20) β (cid:88) i ∆ i (cid:88) a K ··· a i (cid:18) (cid:88) a i − ··· a S a K ··· a (cid:19) (cid:21) . (2.52)The squares in the exponential can be linearized by Gaussian transforms similar to (2.36).The resulting expression initially involves Gaussian integrals over many variables z ( i ) a K ··· a i , butthey can be evaluated iteratively in clusters starting from the smallest cluster and integratingup to larger and larger distance scales (i.e. larger i / larger m i / smaller q i ). The result ofthis little exercise is conveniently and suggestively expressed in terms of convolutions with theGreen’s function of the heat equation, G q ( z ) ≡ √ πq exp (cid:2) − z q (cid:3) , (2.53)27 q q q q m m m m m q q q q m m m m m q m ! q " Figure 5:
The cluster size function m ( q ) for the K = 3 example of fig. 4, and the corresponding tree. Thearrow indicates the direction of the RG flow. producing Z as the outcome of the following recursion: Z ( z ) ≡ (cid:88) s e − βzs = 2 cosh( βz ) , (2.54) Z i +1 ( z ) ≡ (cid:18)(cid:90) dz (cid:48) G | q i − q i − | (cid:0) z − z (cid:48) (cid:1) Z i ( z (cid:48) ) (cid:19) k i , k i = m i +1 m i , (2.55) Z = (cid:90) dz G q K (cid:0) z (cid:1) Z K +1 ( z ) . (2.56)The powers k i arise from identical copies of Gaussian integrals, the copies corresponding todifferent values of the sub-index a i = 1 , . . . , k i . The first step in the recursion is easy enough: Z ( z ) = (cid:16) e β (1 − q ) (2 cosh( βz )) (cid:17) m , but past this point one has to resort to expansions or nu-merical evaluation. As a simple check of the above result, notice that when we remove the dis-tinction between the clusters, i.e. q K = · · · = q = q (but keeping q − ≡ q K +1 ≡ G | q i − q i − | ( z − z (cid:48) ) = δ ( z − z (cid:48) ) for 1 ≤ i ≤ K , so Z = (cid:82) dz G q ( z ) (cid:16) e β (1 − q ) (2 cosh( βz )) (cid:17) m K +1 .Recalling m K +1 = n , we see this correctly reproduces the replica symmetric formulae of section2.3.3.The recursion (2.55) can be thought of as an exact Wilsonian renormalization group action,evolving from the UV (small clusters) to the IR (large clusters), rescaling the number of degreesof freedom in jumps by factors k i = m i +1 /m i . We can make the equations more familiarlooking by defining a function Z ( q, z ) for all q ∈ [0 ,
1] such that Z ( q i − (cid:15), z ) = Z i +1 ( z ). The28quations (2.55) and the identification of G as the Green’s function of the heat equation showthat we can take this function Z ( q, z ) to be the single spin sum 2 cosh( βz ) at q = 1, and thenevolve it down in q according to the heat equation ∂ q Z = − ∂ z Z , with a jump Z → Z k i whenever q crosses a q i . As the latter transformation is nonlinear, it is more convenient towork with g ( q, z ) = log Z ( q, z ), since on g it acts linearly: g → k i g . (The price to pay forthis is that the heat equation becomes nonlinear.) To implement this without reference to theindices i , we define a function m ( q ) (illustrated in fig. 5) as m ( q ) = size of clusters with overlap > q . (2.57)So in particular m ( q i ) = m i and ∂ q m ( q ) = (cid:80) i ( m i − m i +1 ) δ ( q − q i ). Then g ( q, z ) = log Z ( q, z )is the solution to D q g = − (cid:20)(cid:18) ∂g∂z (cid:19) + ∂ g∂z (cid:21) , D q ≡ ∂∂q − m ∂m∂q , (2.58)evolving down from q = 1 to q = 0 with initial conditionlim q (cid:37) g ( q, z ) = log (2 cosh( βz )) , (2.59)and final identification log Z = lim q (cid:38) g ( q, . (2.60)The “gauge connection” A q = ( ∂ q m ) /m keeps track of the increase in the number of degrees offreedom, while the nonlinear equation in g is equivalent to the heat equation. The connectionmay be gauged away by redefining f ≡ m g , so ∂ q f = m D q g , for which ∂f∂q = − (cid:20) m ( q ) (cid:18) ∂f∂z (cid:19) + ∂ f∂z (cid:21) . (2.61)We thus see that the iterative system is nothing but a particularly simple example of the exactrenormalization group equations, with q the analog of the scale and f ( q, z ) the analog of theWilsonian action.For some purposes it is more convenient to take the scale variable to be some arbitraryauxiliary parameter λ , parametrizing the ( q, m ) staircase (fig. 5) by continuous functions q ( λ )and m ( λ ). Letting λ evolve from λ = 0 to λ = 1, with( q, m ) | λ =0 = (1 , , ( q, m ) λ =1 = (0 , n ) , (2.62)29nd denoting λ -derivatives by a dot, the RG equation becomes˙ f = − ˙ q (cid:20) m ( λ ) (cid:18) ∂f∂z (cid:19) + ∂ f∂z (cid:21) , f ( z ) | λ =0 = log (cid:0) βz ) (cid:1) . (2.63)Obviously this introduces a reparametrization gauge symmetry. Putting everything together,we get 1 n F [ q, m ] = β (cid:18) (cid:90) λ =0 dm q (cid:19) − β f ( z = 0) | λ =1 . (2.64)The remaining tasks are ( i ) make sense of the analytic continuation to n = 0 and ( ii ) extremizethe functional F [ q, m ]. n → with RSB Given that 1 = m < m < m < · · · < m K < m K +1 = n in the Parisi matrix construction,it is not obvious how to properly continue to n = 0 <
1, to say the least. To get an idea ofwhat could constitute a sensible continuation, consider the overlap distribution (2.32), whichin K -RSB becomes, taking into account the crucial overall sign of (2.32), P ( q ) = lim n → K (cid:88) i =0 ( m i − m i +1 ) δ ( q − q i ) = lim n → ∂ q m ( q ) . (2.65)For this to make sense as a probability density we need 1 = m > m > m > · · · > m K +1 = n = 0, i.e. the above cluster size inequalities must be inverted, and the m i obviously canno longer be integers; we will allow them to be arbitrary real numbers. Equivalently, m ( q )must become an increasing function from the unit interval to the unit interval, to make ∂ q m a proper probability density. This also inverts the interpretation of m ( q ) from (2.57) tolim n → m ( q ) = probability of finding an overlap ≤ q . (2.66)In general the function m ( q ) may have continuously increasing parts, which can be thoughtof as the K → ∞ limit of the discretized construction. This is illustrated in fig. 6. Recallingthe discussion of section 2.2.3, getting a smooth function should indeed not surprise us: evenif individual realizations of the disorder J ij produce a discrete overlap structure like the onedepicted on the right of the figure, if this structure itself is sensitive to the disorder, averagingover the J ij will smooth out the discreteness. To avoid having to duplicate everything, we restrict again to positive overlaps here, i.e. we condition theprobabilities on q ≥ .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 q m q EA state space Figure 6: (Fictitious) example of ( m, q )-line after continuation to n = 0. The arrow again indicates thedirection of the RG flow. The diagram on the right is a representation of the ultrametric state space treefor some choice of the disorder J ij . The vertical lines are separating the branches of the tree. Differentcompartments at the top (the leaves) correspond to different pure states α , and their width represents theirprobability weight w α . The value of q where two branches separate sets their mutual overlap (so the taller thewall, the smaller the overlap). Thus, going up along the q -axis, m ( q ) increases in discrete jumps. But differentchoices of J ij lead to different barrier structures, and averaging produces the smooth overlap distribution onthe left. (The residual step up is due to the disorder independence of the self-overlap q EA .) This suggests what we should aim for. Looking back at the equations (2.62)-(2.64), wesee that in fact, the required structure is very naturally obtained by simply taking n → m ( λ ) becomes now a decreasingfunction while q ( λ ) remains decreasing. Everything else stays the same. In particular theintrinsic ultrametric structure is preserved. Notice that now the quadratic terms (cid:82) λ =0 dm q in (2.64) become negative definite, with q = 0 being the maximum. This is not an indicationof thermodynamic instability, but rather the converse, as mentioned already in footnote 12.In the limit n →
0, thermodynamically stable states correspond to maxima of the free energy F ( Q ) instead of minima.Admittedly, the above is not a particularly solid justification for the proposed continuationto n = 0. One can probably do better, but I will not try to do this here. The continuation procedure is somewhat reminiscent of continuation in the p -adic numbers: A p -adicintegral over a region with p -adic norm between 1 and n is a finite sum with p n terms, and it naturallycontinues to a p -adic integral over a region with p -adic norm between n = 0 and 1, which is an infiniteseries. Similarities with p -adic and adelic cumbers were explored in chapter 9 of [46] and [77, 78]. However toreproduce the above one still needs to do a formal continuation p → .3.6 The Parisi solution To summarize, the Parisi solution of the SK model is (going back to the gauge λ = 1 − q ),in all its glory: F ( β ) = max m ( q ) (cid:20) β (cid:18) − (cid:90) q =0 dm q (cid:19) − β f m ( z = 0) | q =0 (cid:21) , (2.67)where m ( q ) is an non-decreasing function on the unit interval satisfying the boundary condi-tions m (0) = 0 , m (1) = 1 , (2.68)and f m ( q, z ) is the solution to the following flow equation running from q = 1 to q = 0: ∂f∂q = − (cid:20) m ( q ) (cid:18) ∂f∂z (cid:19) + ∂ f∂z (cid:21) , f ( z ) | q =1 = log (cid:0) βz ) (cid:1) . (2.69)Although this may still look somewhat unwieldy, in practice a simple trial family of m ( q )functions (for example stepwise constant functions with just a few steps K ) can be treatedeasily and already lead to excellent approximations of the exact result. For example [51],for K = 1 one gets for the ground state free energy F = − .
765 and entropy S = − . K = 2 this improves further to F = − . S = − . S = 0. Detailed computations at various temperatures can be found in [68]. A recentstability analysis of the low T solution is given in [79]. This is as far as we will go in describing the explicit solution of the Sherrington-Kirkpatrickmodel. We have left many topics untouched, in particular the various other ways in whichthe model can be solved (for which we refer to [43, 59] as a starting point). We hope we havemade it clear that the hierarchical organization of the state space was the central structure inthe story. This appears to be typical for disordered mean field (i.e. nonlocal) models. Otherpopular examples of such models include the Ising and spherical p -spin models [80, 81, 61]. The To compare to [51]: x , h there equals m resp. z here, and the apparent extra terms there are generated bythe initial flow from q = 1 to q = q EA during which m ( q ) = 1, which is included here but not there. This part ofthe flow is easy to solve because it is basically exactly the heat equation: f ( q, z ) = β (1 − q ) + log(2 cosh( βz )),as is easily checked. s i constrained to lie on an N -dimensional sphere (cid:80) i s i = N ,with a Hamiltonian H = (cid:80) i ··· i p J i ··· i p s i · · · s i p , where the J are random couplings. It isknown [61] that when p = 2, the replica symmetric solution is exact, while for p ≥
3, the1RSB solution is exact. In the limit p → ∞ , the model becomes equivalent to the “randomenergy model” [80] and can be solved exactly and explicitly [81]. One can interpolate betweenspherical spin models and Ising spin models such as SK by switching on an additional non-random potential V = (cid:80) i λ ( s i − , where p = 2, λ → ∞ reproduces the SK model. Variousgeneralizations to higher spins, SU ( N ) spins, rotors, quantum spins etc have been studiedtoo, for example in [44, 82, 83, 84].As mentioned in section 2.2.1, such mean field models are not expected to be a goodapproximations for physical, short range spin glasses in spaces with a moderate number ofdimensions (including three), and it is not clear to what extent the hierarchical organizationof the mean field models is a useful zeroth order expansion point to understand the physics ofsuch systems. But there are other systems of interest, in physics and beyond, where nonlocalinteractions occur naturally, and where the conceptual framework developed for the meanfield theory of spin glasses has turned out to be very useful. This includes the theory ofneural networks, either as models for the brain or for various forms of machine learning, likethe way Google figures out all the time what you really mean. One of the oldest and mostfamous such models is the Hopfield model [85], which is effectively described by a Hamiltonian H = (cid:80) ij J ij s i s j , where the couplings J ij encode synaptic strengths and s i = ± i -th neuron. Stable firing states of the neurons correspond tominima of H . N -component patterns ξ i with ξ i = ± J ij → J ij − ξ i ξ j (or variants thereof), which energetically favors the stability of neuron states s i = ξ i . After M patterns are learned, the couplings are thus J ij = J (0) ij − (cid:80) Mµ =1 ξ µi ξ µj . However,as is well known, learning does not equal remembering. Memory retrieval dynamics in thiscontext is modeled by a standard Monte Carlo type relaxation process, which tries to minimizethe energy. Remembering a learned pattern ξ µ then means that the change in couplingeffectively has given rise to a stable minimum s (cid:63) at or very near the pattern ξ µ . Clearly thismodel shares many characteristics with mean field spin glass models, and mean field spin glasstheory has indeed been very useful in the analysis of these models. For example in [86] it wasshown that with the above learning rule, the memory undergoes a phase transition when ittries to learn too much: for M < M c with M c ≈ . N it remembers practically everything,33hile for M > M c it gets completely confused and remembers practically nothing. In thisframework, the phase in which learned patterns are remembered is similar to the spin glassphase (many highly metastable states), hierarchical clustering in state space gets interpretedas categorization and organization of memories in concepts, and so on. A collection of someof the original papers along these lines can be found in [43].Another important branch of human activity where the ideas of spin glass theory havefound fruitful applications is optimization algorithms for NP-hard problems. Canonical ex-amples of such problems are the satisfiability problem and the traveling salesman problem,which find incarnations in practical problems going from optimal chip design to the problemof finding string vacua satisfying certain conditions [40]. For an early review we refer againto [43]. Recent developments include for example the relation of clustering phase transitionsto effective hardness of optimization problem instances and ways to exploit these insights indevising new efficient algorithms [89, 90]. The statistical mechanics community is fond of Ising spins, and for a good reason: there is alot in the real world that can be naturally modeled by large collections of interacting binarydegrees of freedom. String theorists on the other hand face the sad reality that the complexsystems that are natural in their world are rarely as simple. On the other hand, Parisi’sidea of using the overlap distribution to characterize the emergence of different pure states /ergodic components / superselection sectors in an intrinsic, quantitative and computable wayis an attractive one. Thus, when pondering for example the state space structure of complexD-brane systems [5] or de Sitter space [4], one is led to the question what the analog is of thepure state spin overlap q αβ ≡ N (cid:80) i (cid:104) s i (cid:105) α (cid:104) s i (cid:105) β defined in (2.9) for general quantum systems. Forthe more general systems of interest the degrees of freedom could be fluxes, scalars, fermions,KK modes, or whatever. Since typically there is no canonical scalar product between suchdegrees of freedom, a naive direct generalization of the overlap will not work.A simple construction that is fully quantum and makes sense in general, and that isequivalent to the standard spin overlap in the Ising spin case, is the following. Since wewant to consider systems for which we can at least formally take a thermodynamic limit, It turns out that the key to retain the ability to remember new things is forgetting old things [87]. As measured by the 2 × citations of e.g. [88].
34e assume we can randomly sample N independent “representative” degrees of freedom x i , i = 1 , . . . , N of the system, with the thermodynamic limit of infinite system size correspondingto N → ∞ . For example consider a scalar field theory living in a large box of volume V .We can subdivide the large box into smaller identical boxes of some fixed volume v , and thenuniformly sample at some fixed sampling density a subset of N of the smaller boxes, which welabel by i = 1 , . . . , N . Finally, in each of the sampled boxes i , we compute the average valueof the scalar field, and we call this x i . Another example would be a U ( M ) matrix quantummechanics describing a wrapped D-brane system. Here we may sample a uniform subset of N of the M matrix degrees of freedom. A final example is again an Ising spin glass, for whichwe sample N spins.The state of the complete system is assumed to be described by some density matrix ρ .Expectation values are computed as usual as (cid:104) A (cid:105) ρ ≡ Tr ( ρA ). For each degree of freedom i there is a corresponding reduced density matrix ρ i obtained by tracing out all other degreesof freedom, so Tr H i (cid:0) ρ i A i (cid:1) = Tr ( ρA i ) . (2.70)Here H i is the Hilbert space associated to the degree of freedom x i (for the scalar field examplethis would be H i = L ( R ), whereas for (quantum) Ising spins it is H i = C ), and A i is anobservable referring to H i alone (so for the scalar this could for example be A i = x i , or A i = − i∂ x i ).A natural definition of overlap between two states ρ α , ρ β is then Q αβ ≡ N N (cid:88) i =1 Tr H i ρ iα ρ iβ . (2.71)When the system consists of Ising spins, this reduces to the standard magnetization (or spin)overlap, up to a rescaling and a shift. For indeed, the general density matrix for a spin 1/2degree of freedom can be written as ρ = ( + (cid:126)m · (cid:126)s ), where (cid:126)m = (cid:104) (cid:126)s (cid:105) ρ is the magnetization, Sampling is not necessary for the construction, but it spares us the trouble of having to know and workwith the complete set of degrees of freedom, which may lead to UV divergences or may not even be known.Moreover the overlap q αβ and corresponding spin glass order parameter P ( q ) could equally well have beendefined by sampling say every other spin instead of all spins; ultrametricity implies that the results wouldhave been identical. In general this need not be true of course, so the notion of overlap may depend on thechoice of degrees of freedom that are sampled. Q αβ = N (cid:80) i (1 + (cid:126)m iα · (cid:126)m iβ ). For continuous degrees of freedom this can also bewritten in terms of the associated Wigner densities on phase space: Q αβ = 1 N N (cid:88) i =1 π (cid:126) (cid:90) drdp W iα ( r, p ) W iβ ( r, p ) , (2.72)where W iα ( r, p ) ≡ π (cid:126) (cid:82) ds e ips/ (cid:126) (cid:104) r − s/ | ρ iα | r + s/ (cid:105) . In the classical limit this becomes (aftersuitable coarse graining to kill off highly oscillatory modes) the overlap of ordinary phasespace probability densities, and we can think of Q αβ as the fraction of degrees of freedom( x i , p i ) found in the same elementary phase space cell (of size 2 π (cid:126) ) when randomly samplingfrom two states α and β .We can mimic (2.9) even more closely, by writing the reduced density matrix ρ i as theexpectation value (cid:104) E i (cid:105) of the operator E i defined by E irs ≡ P † x i = r P x i = s = | x i = r (cid:105)(cid:104) x i = s | , (2.73)where P x i = r is the projection operator on the eigenspace x i = r . Then we get Q αβ = 1 N N (cid:88) i =1 (cid:88) rs (cid:104) E irs (cid:105) α (cid:104) E isr (cid:105) β . (2.74)Finally, for some states ρ , an analog of the clustering property (2.6) will hold, for othersnot. We define this analog to be the property that correlation functions of “local” observablesfactorize in the thermodynamic limit for almost all evaluation points. That is, for any finite r lim N →∞ N r (cid:88) i , ··· ,i r (cid:12)(cid:12)(cid:12)(cid:12) (cid:104) A i B i · · · C i r (cid:105) ρ − (cid:104) A i (cid:105) ρ (cid:104) B i (cid:105) ρ · · · (cid:104) C i r (cid:105) ρ (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (2.75)By “local” we mean here observables A i referring to H i alone (we could relax this definitionby allowing dependence on other, “nearby” degrees of freedom, but this will be good enoughfor our purposes).With these definitions set up, we can proceed in complete parallel with the Ising spinglass case, and define “pure states” ρ α , decompositions ρ = w α ρ α of non-pure states (suchas ρ = e − βH for glassy systems at low temperatures), and their overlap distributions P ( q ) = (cid:80) αβ w α w β δ ( q − Q αβ ). Whenever ρ fails to satisfy cluster decomposition (signaling a nontrivial This is valid for quantum Ising spins. Classical Ising spins are trivially obtained from this by restrictingto the up/down eigenstates of s z , yielding Q αβ = (1 + q αβ ) with q αβ as defined in (2.9). P ( q ) becomes nontrivial. For example for an isotropic quantum ferromagnet,the low temperature pure states will be labeled by a vector (cid:126)m such that for all i , ρ i(cid:126)m = (1 + (cid:126)m · (cid:126)s ), and the overlap distribution P ( q ) is uniform on the interval [ − m , m ].Moreover P ( q ) will again be computable in principle for any known state ρ , even if we donot know how to describe its decomposition into pure states. Explicitly, the analog of (2.12)that allows this is P ( q ) = (cid:10) δ (cid:0) q − N (cid:80) irs E i rs E i sr (cid:1)(cid:11) n =2 , (2.76)where again the subscript ‘ n = 2’ means we consider two replicas of the system, with densitymatrix ρ ⊗ ρ , and E depends on the first replica’s degrees of freedom while E depends onthe second replica’s degrees of freedom. In the classical limit, P ( q ) becomes the probabilitydistribution for finding a fraction q of the degrees of freedom ( x i , p i ) of the two replicas in thesame elementary phase space cell.The operator inside the delta-function has a more concrete interpretation. For a given i ,let X i ≡ (cid:88) rs E i rs E i sr = (cid:88) rs | x i = r, x i = s (cid:105)(cid:104) x i = s, x i = r | . (2.77)Notice the r ↔ s flip on the right hand side, which we can equivalently think of as a x i ↔ x i flip. Thus, what this operator X i does for a given i (when acting on a wave function say) isexchange the replica variables x i , x i . The first moment of P ( q ) is then (cid:104) q (cid:105) = 1 N (cid:88) i Tr( ρ X i ρ ) = 1 N (cid:88) i Tr H i ( ρ i ) = 1 N (cid:88) i e − S i (2) , (2.78)where S i (2) is the second Renyi entropy [91] of ρ i . If ρ = | Ψ (cid:105)(cid:104) Ψ | , this gives information onhow entangled the x i are in the wave function Ψ; if Ψ factorizes (no entanglement), then (cid:104) q (cid:105) = 1. For the isotropic ferromagnet (strong entanglement), we have (cid:104) q (cid:105) = . The secondmoment is (cid:104) q (cid:105) = 1 N (cid:88) ij Tr( ρ X i X j ρ ) = 1 N (cid:88) i (cid:54) = j Tr H ij ( ρ ij ) + 1 N = 1 N (cid:88) i (cid:54) = j e − S ij (2) , (2.79)where ρ ij is the density matrix obtained by tracing out all degrees of freedom except x i , x j , andin the last expression we dropped the 1 /N term since it is understood that the thermodynamiclimit N → ∞ is taken. The ratio (cid:104) q (cid:105) / (cid:104) q (cid:105) gives information about the degree of long rangecorrelation (and failure of cluster decomposition) in the system. For a thermodynamic pure37tate it equals 1, for the isotropic ferromagnet it equals 1 + m . Moments of order k areobtained analogously by fixing k degrees of freedom.Thus, for rather general systems, we have arrived at an infinite set of in principle com-putable order parameters that detect the emergence of a nontrivial state space landscape.Applications are discussed in [4, 5]. Complex systems are complex. As a result, it is hard in general to quantitatively determineeven their basic features. When it can be done, such as for the Sherrington-Kirkpatrick model,the results often have a striking richness. Complex systems arising in string theory, such as thegeometrically highly complex wrapped D-branes reviewed in section 4, are often significantlymore intricate than simple spin models. This makes straightforward analysis more daunting.Although in principle the probability overlaps defined in section 2.5 define an intrinsic orderparameter that could be used to analyze those stringy systems along the lines of the SK model,in practice this may still be difficult. This is even true for much simpler quantities like theentropy.On the other hand, the existence of dualities in string theory and the presence of supersym-metry often tremendously simplify things. In this section we will focus on supersymmetry.The main simplification that arises due to this is that many quantities of interest can becomputed exactly in a semiclassical limit. The primary example of this is the Witten index[92], which counts ground state degeneracies in supersymmetric quantum mechanics, and isgenerically invariant under continuous deformations of the theory. Famously this allowed Stro-minger and Vafa [93] to microscopically compute the entropy of certain classes of extremalblack holes arising in string theory, by tuning the string coupling constant from the stronglyself-gravitating black hole regime to a weakly coupled, non-self-gravitating regime. In thisregime the system has a well-understood D-brane description, and the Witten index can becomputed in a relatively straightforward and precise way.The ground state degeneracy is not the only thing supersymmetry allows us to control.Through its relation with Morse theory, the structure of supersymmetric quantum mechanicsgives useful insights into the saddle point structure of energy landscapes and the way they38re dynamically connected, and makes it possible to systematically compute nonperturbativelevel splittings and their associated exponentially slow relaxation rates. Clearly this is ofpotential interest in the study of complex systems. In fact, through a formal map betweennonsupersymmetric classical statistical mechanics and supersymmetric quantum mechanics,this has ramifications beyond the realm of supersymmetric systems (see section 3.8).We will review these fundamental aspects of supersymmetric quantum mechanics in whatfollows. I will not assume any familiarity with the topic, and start from scratch, building upthe topic along the lines of chapter 10 of [94], an excellent introduction to supersymmetricquantum mechanics and its interplay with geometry. A review focusing more on relations toexactly solvable systems is [95], and finally Witten’s original [92] is a classic.Specific applications to string theory are left to section 4.
By definition, the Hamiltonian of a supersymmetric quantum mechanical system can be writ-ten as H = 12 { Q, Q † } , (3.1)where Q , the supercharge, is an operator satisfying Q = (cid:0) Q † (cid:1) = 0 . (3.2)The Hilbert space is Z graded, splitting up in a even or “bosonic” part, and an odd or“fermionic” part: H = H B ⊕ H F . (3.3)The operator ( − F is defined as +1 on H B and − H F . The operator Q is odd/fermionic,i.e. it maps from H B into H F and from H F into H B :[( − F , Q ] = − Q . (3.4)Some immediate consequences are • [( − F , H ] = 0, [ Q, H ] = 0, i.e. ( − F and Q are symmetries. At this level “bosonic” and “fermionic” are just conventional names. They do not necessarily mean thatthe quantum mechanical states represent bosonic or fermionic particles in the usual sense of the word, althoughin specific situations the notion may be correlated. H ≥
0, all energies are positive or zero. • H | α (cid:105) = 0 ⇔ Q | α (cid:105) = 0 = Q † | α (cid:105) , zero energy states are supersymmetric. • For each positive energy level H E = H E,B ⊕ H
E,F , E >
0, the fermionic and bosonicsubspaces are isomorphic . The isomorphism is given by the Hermitian supercharge Q ≡ Q + Q † , (3.5)which satisfies Q = 2 E on H E and is therefore invertible when E (cid:54) = 0.The latter property is particularly important. It means that all positive energy eigenstatescome in boson-fermion pairs. This is not necessarily true for zero energy ground states. TheWitten index Ω quantifies the degree to which this is not true:Ω ≡ dim H ,B − dim H ,F . (3.6)Since all positive energy states come in even-odd pairs, this can equally well be written in thefollowing alternative ways: Ω = Tr ( − F (3.7)= Tr ( − F e − βH (3.8)= (cid:90) D ( · · · ) e − S ( ··· ) . (3.9)The last line represents the usual Euclidean path integral representation of the partitionfunction, except that now both fermionic and bosonic variables satisfy periodic boundaryconditions, instead of the usual antiperiodic boundary conditions on the fermions. This is dueto the insertion of ( − F , which flips the sign of the fermionic variables. (The example belowmay make this more clear.)The most important property of the Witten index is that it is invariant under genericcontinuous deformations. This is clear from its definition: although the total number ofsupersymmetric ground states may vary when we vary the parameters of the model, as statesmove in and out of the E = 0 level, all such arrivals and departures must come in bose-fermipairs, and therefore the index will not be affected. Figure 7:
1d susy QM with h ( x ) = x − x + 7. Blue: h ( x ), red: V ( x ) = h (cid:48) ( x ) /
2, green: Φ ( x ) ∝ e − h ( x ) . A simple example is obtained by considering a bosonic (i.e. commuting) variable x and afermionic (i.e. anticommuting) variable ¯ ψ , wave functionsΦ( x, ¯ ψ ) = Φ B ( x ) + Φ F ( x ) ¯ ψ , (3.10)and canonical commutation relations [ x, p ] = i , { ¯ ψ, ψ } = 1, where ψ ≡ ¯ ψ † . So acting on wavefunctions we have p = − i∂ x , ψ = ∂ ¯ ψ . (3.11)Then we consider the supercharge Q ≡ ¯ ψ ( ip + h (cid:48) ( x )) , Q † = ψ ( − ip + h (cid:48) ( x )) . (3.12)Here h ( x ) is an arbitrary function, sometimes called the superpotential. Expressed in compo-nents Φ = (cid:0) Φ B Φ F (cid:1) this becomes Q = (cid:32) ∂ x + h (cid:48) ( x ) 0 (cid:33) , (3.13) Although generically robust, there are important exceptions to this argument. In particular, when thegap we have implicitly assumed between the zero and first excited energy levels vanishes in the course of thedeformation, the Witten index may jump. This can happen for example when the deformation passes througha degeneration allowing a ground state wave function to spread out all the way to infinite distance, becomingnon-normalizable and therefore no longer a proper state in the Hilbert space, but rather part of the continuumof scattering states. H = 12 { Q, Q † } = 12 (cid:0) − ∂ x + h (cid:48) ( x ) + h (cid:48)(cid:48) ( x )[ ¯ ψ, ψ ] (cid:1) (3.14)= 12 (cid:32) − ∂ x + h (cid:48) ( x ) − h (cid:48)(cid:48) ( x ) 00 ∂ x + h (cid:48) ( x ) + h (cid:48)(cid:48) ( x ) (cid:33) . We can think of this as the Hamiltonian of a spin 1/2 particle of unit mass in a potential V ( x ) = h (cid:48) ( x ) and a magnetic field h (cid:48)(cid:48) ( x ). In the case h ( x ) = ω x , this becomes theharmonic oscillator potential V ( x ) = ω x plus a constant magnetic field ω . Assuming ω > F = 0) energy levels are then E B = 0 , ω, ω, ω, . . . and the fermionic energylevels E F = ω, ω, ω, . . . . We explicitly see the pairing of E > E = 0, captured by the Witten index Ω = +1. When on the other hand ω <
0, we get E B = | ω | , | ω | , . . . and E F = 0 , | ω | , | ω | , . . . , so Ω = − Q Φ = Q † Φ = 0:Φ = A B e − h ( x ) + A F e + h ( x ) ¯ ψ . (3.15)For these to be true supersymmetric ground states, Φ has to be normalizable. When lim | x |→∞ h ( x ) =+ ∞ , this requires A F = 0, and thus Ω = +1. When lim | x |→∞ h ( x ) = −∞ , we get instead A B = 0 and Ω = −
1. In all other cases, none of the solutions are normalizable, and Ω = 0.An example is shown in fig. 7.
We can easily generalize this to multiple degrees of freedom x I and ¯ ψ I , I = 1 , . . . , N , leadingto 2 N -component wave functionsΦ( ¯ ψ, x ) = (cid:88) b , ··· ,b N =0 , Φ b ··· b N ( x ) ( ¯ ψ ) b · · · ( ¯ ψ N ) b N , (3.16)and a fermion number quantum number F counting the number of ¯ ψ I . We take the super-charge to be Q = (cid:88) I ¯ ψ I ( ip I + ∂ I h ( x )) , (3.17) Here ¯ ψ is, as in (3.10), a Grassman variable. Equivalently we could write Φ = A B e − h ( x ) | (cid:105) + A F e + h ( x ) ¯ ψ | (cid:105) , with | (cid:105) the “fermionic vacuum state”, i.e. ψ | (cid:105) = 0. Q = 0. The Hamiltonian is then H = (cid:88) I p I + 12 ( ∂ I h ) + 12 (cid:88) IJ [ ¯ ψ I , ψ J ] ∂ I ∂ J h . (3.18)It is now no longer possible in general to find explicit solutions to Q Φ = Q † Φ = 0, but when h ( x ) is a nondegenerate quadratic function, say h ( x ) = 12 N (cid:88) I =1 ω I ( x I ) , (3.19)then the problem just factorizes into N copies of the 1d harmonic oscillator situation we solvedabove, and thus the unique normalizable supersymmetric ground state isΦ ( x ) = e − (cid:80) I | ω I | ( x I ) (cid:89) I : ω I < ¯ ψ I . (3.20)This has fermion number F equal to the number of negative eigenvalues of the Hessian of h at the critical point x = 0. This number µ is called the Morse index of the critical point. TheWitten index is Ω = ( − µ .Using this, we can compute the Witten index for any h ( x ) that has only isolated, non-degenerate critical points. Indeed, since the Witten index is invariant under continuousdeformations that do not change the asymptotics of h ( x ), we can consider the deformation h ( x ) → λh ( x ) and compute Ω in the limit λ → ∞ . This limit corresponds to a semiclassicallimit. The classical supersymmetric ground states are given by the critical points of h ( x ).The corresponding perturbative semiclassical wave functions are to lowest order Gaussianscentered at the critical points, of the form (3.20). All of these remain supersymmetric toall orders in perturbation theory. (Roughly this is because perturbation theory around onecritical point does not know about the presence of other critical points, and if there is onlyone critical point, the Witten index is 1.) However, nonperturbative tunneling effects may liftthis degeneracy, just like tunneling splits up the doubly degenerate perturbative ground stateenergy level of a double well potential in ordinary quantum mechanics. But because all lifting Such a function is called a Morse function, and it corresponds to the generic case. We will always assume h is a Morse functions in this section. More precisely, the perturbation theory for each critical point x (cid:63) of h is a series in inverse powers of λ and is obtained by changing variables from x to y defined by x I = x I(cid:63) + y I / √ λ so H = λ ( H + λ − / H + · · · )where H is the harmonic oscillator part of the Hamiltonian, obtained by truncating h at quadratic order in y , and H the part corresponding to the higher order terms. Figure 8:
Critical points of the vertical height function on a pretzel, with fermion number (Morse index)indicated. The Witten index is Ω = 3 − − must occur in bose-fermi pairs, it will not affect the total Witten index, so we can computethe index simply by adding the contributions of all critical points x (cid:63) of h :Ω = (cid:88) x (cid:63) : h (cid:48) ( x (cid:63) )=0 ( − µ ( x (cid:63) ) . (3.21)Checking this for the 1d example shown in fig. 7, we see that indeed there h has two minimaand one maximum, so Ω = +1 + 1 − h is a harmonic function, as is the case when it is for example equal to the realpart of a holomorphic function W ( z ) of variables z a = x a − + ix a , a = 1 , · · · , n = N/
2, allcritical points have Morse index n , and the Witten index is equal to ( − n times the absolutenumber of critical points of W . This situation occurs naturally in the description of wrappedD-branes in Calabi-Yau compactifications of type II string theory. On a compact N -dimensional Riemannian manifold M , we can naturally define a supersym-metric quantum mechanics with Hilbert space given by the space of complex valued differ-ential forms Φ = Φ + Φ I dx I + Φ IJ dx I ∧ dx J + · · · on M . The elementary 1-forms dx I getidentified with the Grassmann coordinates ¯ ψ I . The inner product of two differential formsΦ ( a ) = (cid:80) NF =0 Φ I ··· I F dx I ∧ · · · ∧ dx I F , a = 1 , (cid:104) Φ (1) | Φ (2) (cid:105) = (cid:90) Φ (1) ∧ (cid:63) Φ (2) = N (cid:88) F =0 (cid:90) d N x √ g g I J · · · g I F J F Φ (1) I ··· I F Φ (2) J ··· J F , (3.22)44here g IJ is the metric on M . The fermion number F is the form degree and the superchargein the absence of a potential is the exterior derivative Q = d = dx I ∇ I , (3.23)where ∇ I is the covariant derivative. The Hamiltonian is the Laplace-Beltrami operator∆ = 12 (cid:0) dd † + d † d (cid:1) . (3.24)Comparing with our previous notation, we have, at the level of operators, the identifications¯ ψ I = dx I ∧ , ψ I = g IJ ∂∂ ¯ ψ J = g IJ ι ∂ I , (3.25)implying in particular the anticommutation relations { ψ I , ¯ ψ J } = g IJ . More explicitly, usingthe relation between curvature and covariant derivative commutators, the Hamiltonian canbe written as H = − g IJ ∇ I ∇ J + 12 R IJKL ψ I ¯ ψ J ψ K ¯ ψ L . (3.26)The supersymmetric ground states are the harmonic differential forms, and the Witten indexis the alternating sum of the number of harmonic forms at each form degree, which is nothingbut the Euler characteristic of M :Ω = N (cid:88) F =0 ( − p b p ( M ) = χ ( M ) . (3.27)The Betti number b p is also equal to the number of homologically independent ( N − p ) − cycles .On a compact manifold we have moreover b p = b N − p . An immediate consequence of this isthat Ω = 0 when N is odd. For a Riemann surface of genus g , we have b = b = 1 and b = 2 g , so χ = 2 − g .Adding a potential can be implemented by putting Q h = e − h de h = d + dh . (3.28)This gives a potential term V = g IJ ∂ I h∂ J h in the Hamiltonian. Switching on a potential willnot change the Witten index, since the spectrum is guaranteed to be discrete on a compactspace. In fact, it will not even change the absolute number of supersymmetric ground statesin each fermion number. To prove this one uses the very useful general fact that the space This is not hard to show. First, for positive energy states, we have that if Q | α (cid:105) = 0, then | α (cid:105) = E ( Q † Q + QQ † ) | α (cid:105) = Q (cid:16) Q † E | α (cid:105) (cid:17) , so all positive energy states are trivial in cohomology. For zero energystates on the other hand we have Q | α (cid:105) = 0 = Q † | α (cid:105) , so if | α (cid:105) = Q | β (cid:105) , then (cid:104) α | α (cid:105) = (cid:104) β | Q † | α (cid:105) = 0, i.e. | α (cid:105) = 0.So each zero energy state is nontrivial in cohomology.
45f supersymmetric ground states is isomorphic to the Q -cohomology: H susy (cid:39) ker Q im Q . (3.29)This is combined with the observation that there is an isomorphism between the cohomologiesof Q = d and Q h = e − h de h provided by the map | α (cid:105) → e − h | α (cid:105) .An example of a Riemann surface, embedded in R is shown in fig. 8. This Riemannsurface is known as a pretzel and acquires a natural metric from its embedding in R . For thefunction h we take the height along the vertical axis. There are 3 minima, 9 saddle pointsand 2 maxima, each of which gives rise in the large pretzel limit to a perturbative susy groundstate, here of fermion number 0, 1 and 2 respectively. We know we can compute the Wittenindex in perturbation theory, yielding Ω = 3 − −
4, so the Riemann surface is ofgenus 3, which is confirmed by visual inspection. We know also that b = 1 = b , and thus b = 6. Therefore, since the Betti numbers count the number of supersymmetric groundstates at given fermion number, several of the perturbative ground states must be lifted bytunneling effects: only one linear combination of the F = 0 and the F = 2 states can remainsupersymmetric, and six linear combinations of the F = 1. The remaining three F = 1 statespair up with two F = 0 and one F = 2 state and get lifted. The lifting will be exponentiallysmall in the pretzel size (which sets the scale of h ). So far we have described everything in a Hamiltonian framework. Semiclassical tunnelingamplitudes responsible for lifting perturbative ground states are however most easily computedin a Lagrangian framework. The Lagrangian framework and its associated path integral arealso most convenient to efficiently obtain a differential geometric expression for the Wittenindex, as the integral over M of a differential form made out of the curvature, namely theEuler density.The Lagrangian is obtained from the Hamiltonian in the usual way. We are primarilyinterested in the Euclidean Lagrangian. Starting from (3.26) with the addition of the potentialterm, this is L = 12 g IJ ˙ x I ˙ x J + 12 g IJ ∂ I h∂ J h + g IJ ¯ ψ I D τ ψ J + 12 R IJKL ψ I ¯ ψ J ψ K ¯ ψ L + ∇ I ∂ J h ¯ ψ I ψ J . (3.30)46here τ is the Euclidean time conjugate to H and ˙ x = dxdτ , D τ ψ I = ∂ t ψ I + Γ IJK ∂ t x J ψ K , ∇ I V J = ∂ I V J − Γ KIJ V K . The supersymmetries act on the fields as δ ( · · · ) = [ (cid:15)Q + ¯ (cid:15)Q † , · · · ]: δx I = (cid:15) ¯ ψ I − ¯ (cid:15)ψ I (3.31) δψ I = (cid:15) ( − ˙ x I − Γ IJK ¯ ψ J ψ K + g IJ ∂ J h ) (3.32) δ ¯ ψ I = ¯ (cid:15) ( ˙ x I − Γ IJK ¯ ψ J ψ K + g IJ ∂ J h ) , (3.33)leaving the action invariant.As an application we derive a differential geometric expression for the Witten index, whichis essentially the generalized Gauss-Bonnet formula [96]. The path integral representation ofthe Witten index is Ω = Tr ( − F e − βH = (cid:90) D x D ¯ ψ D ψ e − (cid:82) β L dt , (3.34)with periodic boundary conditions on all fields: φ ( β ) = φ (0) for φ = x, ψ, ¯ ψ . In the limit β →
0, only the constant trajectories contribute to the path integral. Thus, taking h = 0 andworking in an orthonormal frame, the path integral reduces to the finite dimensional integralΩ = 1(2 πβ ) N/ (cid:90) d N x d N ¯ ψ d N ψ e − β R IJKL ψ I ¯ ψ J ψ K ¯ ψ L , (3.35)where the normalization can be fixed for example by comparing path integral and canonicalexpressions for the free propagator in flat space (cid:104) x | e − βH | x (cid:105) . Using the symmetries of theRiemann tensor and the Grassmann integral representation of the Pfaffian of a matrix M ,Pf M = (cid:82) dψ · · · dψ N e − ψ i M ij ψ j (with (Pf M ) = det M ), this becomesΩ = (cid:90) M Pf R , R I J ≡ π R I JKL dx K ∧ dx L . (3.36)This expression, which gives the Euler characteristic of M in terms of the integrated Eulerdensity, is known as the generalized Gauss-Bonnet theorem. By completing the squares, the bosonic part of the action (3.30) can be brought into the form S B = s (cid:0) h ( x f ) − h ( x i ) (cid:1) + (cid:90) T/ − T/ dτ g IJ (cid:0) ˙ x I − sg IK ∂ K h (cid:1) (cid:0) ˙ x J − sg JL ∂ L h (cid:1) , (3.37)where s = ±
1. The first term comes from integrating a total derivative, and x ( − T /
2) = x i and x ( T /
2) = x f are the start and end points of the trajectory. This implies the lower bound47 A DEC F GH
Figure 9:
Steepest ascent gradient flow lines for the function h ( x, y ) = λ sin(2 πx ) sin(2 πy ) on the 2-torusdefined by the identifications x (cid:39) x + 1, y (cid:39) y + 1. The critical points A, B are minima,
G, H are maxima,
C, D, E, F are saddle points. Big green arrows are paths which give a positive contribution to Q , big purplearrows give negative contributions. Little red arrows indicate wave function orientations. S B ≥ s (cid:0) h ( x f ) − h ( x i ) (cid:1) for s = ±
1, hence S B ≥ | h ( x f ) − h ( x i ) | , which is saturated for steepestdescent or ascent trajectories:˙ x I = sg IJ ∂ J h ( x ) , s = sign (cid:0) h ( x f ) − h ( x i ) (cid:1) . (3.38)These gradient flow trajectories (with the fermions put to zero) are automatically solutionsto the Euclidean equations of motion, and moreover from the supersymmetry variations(3.32)-(3.33) it can be seen that ascending ( s = +1) trajectories are annihilated by the Q -supersymmetry, while descending trajectories are annihilated by the Q † -supersymmetry. Suchminimal action trajectories, in the limit T → ∞ , are called instantons.Instantons play a key role in the computation of nonperturbative lifting of ground states.For concreteness consider the example shown in fig. 9, a two dimensional flat unit torus withsuperpotential h ( x, y ) = λ sin(2 πx ) sin(2 πy ). There are 8 critical points: 2 minima, 4 saddlepoints and 2 maxima, giving Ω = 2 − χ ( T ). The Betti numbers for T are b = 1, b = 2, b = 1, so again some of the perturbative ground states must be lifted by tunnelingeffects. The relevant tunneling trajectories responsible for this are precisely the instantonsintroduced above.The general story goes as follows. Denoting the perturbative ground states by | i (cid:105) , lifting48eans that the matrix elements of the supercharge reduced to the perturbative zero energylevel, ˆ Q ij ≡ (cid:104) i | Q | j (cid:105) , (3.39)although identically zero in perturbation theory, is not identically zero when nonperturbativeeffects are taken into account. Notice that since Q has fermion number 1, Q ij can only benonzero if the fermion numbers of | i (cid:105) and | j (cid:105) statisfy the selection rule F i = F j + 1. Thematrix elements Q ij can be given a path integral representation. It can be shown that thanksto supersymmetry, this path integral (generically) only gets contributions from instantontrajectories annihilated by Q , i.e. paths of steepest ascent, interpolating between criticalpoints j → i with F i = F j + 1. Moreover the Gaussian approximation to this contribution isexact, resulting in the very simple expressionˆ Q ij = (cid:88) γ : j → i n γ e − h ij , (3.40)where the sum is over steepest ascent flows from j to i satisfying F i − F j = 1. The prefactor n γ = ± h ij is the instanton action, h ij = | h ( i ) − h ( j ) | . This simplification in the pathintegral is referred to as localization. We will not derive this here; see [94] chapter 10 for adetailed pedagogical account.The sign of n γ is determined by orientation considerations. This is best explained bygoing back to the torus example. First we need to choose an orientation for the perturbativeground states | i (cid:105) . In the figure the critical points are denoted by the letters A to G , andwe will denote the perturbative ground states by these letters too. Then | A (cid:105) and | B (cid:105) are0-forms, | C (cid:105) , | D (cid:105) , | E (cid:105) , | F (cid:105) are 1-forms, and | G (cid:105) , | H (cid:105) are 2-forms. For example, to lowest orderin perturbation theory, | A (cid:105) = α e − λ (2 π ) ( u + v ) , α = √ πλ , u = x − , v = y − , (3.41) | C (cid:105) = α e − λ (2 π ) ( u + v ) du , u = x + y − , v = x − y + , (3.42) | G (cid:105) = α e − λ (2 π ) ( u + v ) du ∧ dv , u = x − , v = y − . (3.43)Note that whereas the top and bottom forms can be given a natural orientation inherited fromthe orientation of the torus itself, this is not so for the 1-forms. We choose them such that atthe critical point, the form is positive in the positive x -direction, i.e. the local coordinate u that is such that h ∼ − u + v near the critical point is taken to increase when going to theright. In the figure, the orientation of the local u -coordinate is indicated by the red arrows.49he arrow is a local frame for the unstable direction. More generally, the choice of orderedstandard local coordinates near each critical point determines a local frame for the unstabledirections of h at each critical point. On the other hand when moving along an instantontrajectory, we can transport this frame from the initial point to the final point, and then addto it a new vector pointing in the direction of the path, to obtain a new frame for the unstabledirections around the final point (which indeed generically has one more unstable directionthan the initial point, in the direction of the arriving path). If this new frame has the sameorientation as the frame we already had there, then n γ = +1, and otherwise n γ = − h = λ for successive critical points,we get for example ˆ Q | A (cid:105) = e − λ ( | D (cid:105) + | F (cid:105) − | C (cid:105) − | E (cid:105) ), ˆ Q | C (cid:105) = e − λ ( | H (cid:105) − | G (cid:105) ), ˆ Q | G (cid:105) = 0.In matrix form, we thus compute ˆ Q , ˆ Q † and ˆ H = { ˆ Q, ˆ Q † } (writing zeros as blanks):ˆ Q = e − λ − − − − − − − − , ˆ H = 2 e − λ − − − − − − − − The space of true supersymmetric ground states is isomorphic to the cohomology ker ˆ Q/ im ˆ Q ofthe matrix ˆ Q , which is relatively easily computed. This is called the Morse-Witten cohomology.Alternatively, we can find the actual supersymmetric ground states (within the approximationof working within the finite dimensional space of perturbative supersymmetric ground states,i.e. to leading order in e − λ ), as the joint null space of ˆ Q and ˆ Q † , or equivalently the null spaceof ˆ H . The reduced Hamiltonian ˆ H furthermore gives the dynamical transition rates betweendifferent perturbative ground states, and diagonalizing it gives the energy level splittings.The conclusion in our example is that the supersymmetric ground states in this approxi-mation are given by the symmetric combinations | A (cid:105) + | B (cid:105) , | C (cid:105) + | F (cid:105) , | D (cid:105) + | E (cid:105) , | G (cid:105) + | H (cid:105) ,consistent with the Betti numbers (1 , , e − λ .Of course in more general examples, the splitting pattern and linear combinations will beless simple. But in principle at least there is a general algorithm to compute it in terms ofsimple, generically isolated, “atomic” (∆ F = 1) gradient flows, which is quite remarkable; innonsupersymmetric quantum mechanics in dimensions greater than 1, doing something similaris typically very hard if not impossible. 50or complex energy landscapes h ( x ) in high dimensional spaces, it will typically be moreappropriate to compute statistical, thermodynamic quantities, but again the structure ofsupersymmetry will lead to significant simplifications.As a check on the above results, and also to get further insight into the nature of theapproximation, observe that it is actually easy to write down the exact bottom and topfermion number ground state wave functions; they are: | F = 0 (cid:105) = e − h , | F = dim (cid:105) = e + h (cid:81) I ¯ ψ I . (3.44)The approximation we made replaces this to lowest order in perturbation theory by an evensuperposition of Gaussians centered at the critical points. This is indeed an excellent approx-imation at large λ .Note that in this limit perturbative ground states are exponentially close to being realground states, and physically the number of perturbative ground states would therefore be abetter measure for the ground state degeneracy than the Witten index. However, at generic,order 1 couplings, this is no longer the case, and one may reasonably expect all states that arenot protected to be lifted to finite energies, making the Witten index an accurate, physicallymeaningful measure for the number of true ground states. On the other hand, as manyexamples in string theory have taught us, one must of course be careful that in the apparentstrong coupling regime there isn’t a dual weak coupling description hiding, which would againpotentially produce many new near-exact ground states, not captured by the Witten index.This is the case in particular in thermodynamic, “large N ” limits. For example the SKmodel and other glassy systems develop many pseudo ground states in the thermodynamiclimit. Although not supersymmetric, there is an interesting map from classical statisticalmechanics systems to supersymmetric quantum mechanics systems, showing this is not justan analogy, but really the same thing. To this we turn next. Supersymmetric quantum mechanics can be interpreted as a BRST description of a classicalstochastic system [70, 97, 98, 71, 72]. This goes as follows. Consider a stochastic dynamicalLangevin equation for a particle with position x ( t ) in a potential h ( x ): dxdt = − ∂ x h ( x ) + η ( t ) , (3.45)51here η is a Gaussian random noise function with variance α , P ( η ) ∝ e − η / α (the noise canbe thought of as thermal fluctuations with temperature proportional to α ). Starting fromsome initial point x i at time t = 0, the probability density of finding the configuration at apoint x f at time t is given by the path integral P ( x f , t | x i ,
0) = (cid:90) D x D η e − (cid:82) t α η δ (cid:0) dxdt + h (cid:48) ( x ) − η (cid:1) | det (cid:0) ddt + h (cid:48)(cid:48) ( x ) (cid:1) | , (3.46)appropriately normalized and with boundary conditions x (0) = x i , x ( t ) = x f . The combina-tion of delta function and determinant ensures that the path integration over x simply picksout with unit weight the unique solution to the first order linear differential equation (3.45) forgiven η . It is the functional analog of the finite dimensional formula (cid:82) dx δ ( A · x + B ) | det A | = 1where A is a constant matrix and B is a constant vector. Ignoring for the time being the abso-lute value signs, we can write the determinant as a fermionic path integral, so that amusingly,after also integrating out η , we obtain exactly the Euclidean propagator of supersymmetricquantum mechanics with superpotential h ( x ): P ( x f , t | x i ,
0) = (cid:90) D x D ¯ ψ D ψ e − (cid:82) t α ( ˙ x + h (cid:48) ( x )) + ¯ ψ ˙ ψ + ¯ ψh (cid:48)(cid:48) ( x ) ψ (3.47)= e ( h ( x i ) − h ( x f )) /α (cid:90) D x D ¯ ψ D ψ e − (cid:82) t α ( ˙ x + h (cid:48) ( x ) )+ ¯ ψ ˙ ψ + ¯ ψh (cid:48)(cid:48) ( x ) ψ (3.48)= e ( h ( x i ) − h ( x f )) /α (cid:104) x f | e − tH susy /α | x i (cid:105) . (3.49)The last line is the propagator between states in the zero fermion number sector, with elapsedEuclidean time t and Planck’s constant (cid:126) = α (before we were setting (cid:126) ≡ F = 0) energy eigenstates | n (cid:105) , we have (cid:104) x f | e − tH susy /α | x i (cid:105) = (cid:88) n (cid:104) x f | n (cid:105)(cid:104) n | x i (cid:105) e − tE n /α . (3.50)In the long time limit t → ∞ , only the ground state survives. As we have seen, when Z ≡ (cid:82) dx e − h ( x ) /α < ∞ , the Witten index is 1 and the unique supersymmetric ground stateis (cid:104) x | (cid:105) = √ Z e − h ( x ) /α . We conclude P ( x f , ∞| x i ,
0) = 1
Z e − β h ( x f ) , β ≡ α , (3.51)with Z = (cid:82) dx e − βh ( x ) . This is indeed the expected Boltzmann equilibrium distribution withinverse temperature β , which alternatively can be derived from the Fokker-Planck equationassociated to the Langevin equation. In fact the Fokker-Planck equation can easily be seen52o be equivalent to the Schr¨odinger equation in the zero fermion number sector (identifying P ( x ) ∝ e h ( x ) ψ ( x )), providing an alternative derivation of the above identifications.Note that when the Witten index is −
1, i.e. e + h ( x ) is normalizable, there is no futureequilibrium state, but there is a past equilibrium state. When it is zero, there is no futureand no past equilibrium. If there are multiple perturbative supersymmetric ground stateswith fermion number zero (i.e. local minima of h ( x )), then in the small α (i.e. large β ) limit,there will be highly metastable states. According to (3.50), the time scale for decay of thesemetastable states is set by the nonperturbatively lifted energy, which as we have seen in theprevious section can be computed from simple instantons and the Morse-Witten complex.All this generalizes in a straightforward way to higher dimensional and curved spaces.If h ( x ) is the Hamiltonian of some classical statistical mechanical system, then in the ther-modynamic limit N → ∞ there may be degenerations that change the number of exactsupersymmetric ground states at a given fermion number and even the Witten index. This isbecause degeneracy lifting instanton effects may vanish altogether, thus turning perturbativeground states into exact ground states. When this happens at fermion number zero, ergod-icity is broken and we get multiple exact equilibrium states. Whether or not this happenswill depend on the distribution of energy barriers, i.e. the distribution of supersymmetricground states with low fermion number, and how they are connected by instantons, i.e. theMorse-Witten complex. This should capture, if present, the ultrametric structure of the statespace.Some discussion of spin glasses in the language of supersymmetric quantum mechanicscan be found in e.g. [71, 99, 72, 59]. Recent considerations in string theory involving theabove connection between stochastic equations and supersymmetry include [100, 101]. Anice example of the relation between transition rate distributions and universal propertiesof relaxation dynamics can be found in [102], where the typical log(1 + t w /t ) behavior ofreturn to equilibrium after a perturbation of duration t w of a glass was derived from generalarguments based on the form of a transition matrix with exponentially suppressed off-diagonalmatrix elements. Such a transition matrix and Hamiltonian are typical for systems with awide distribution of relevant instanton actions.53 D-brane landscapes
We now move on to the description of complex systems that appear in the context of stringtheory, and review some of the tools that have been used to describe their basic features, suchas ground state degeneracies and distributions. One of the interesting things string theoryadds to conventional analysis is the power of holographic duality [103], which in favorablecircumstances can reduce a quantitative understanding of the strong coupling physics of themodel under consideration to simple computations in gravity. This works particularly well forthermodynamics, as black holes are the holographic duals to thermodynamic states, capturingthe strong coupling thermodynamics of the model with often rather astonishing simplicity.More specifically, we will have a look at the complex configuration spaces that arise whenwrapping D-branes around various cycles of compactification manifolds, with various world-volume fields turned on. Such brane configurations arise in type II constructions of stringvacua, where they are among other things responsible for the particle physics content of thecompactification. They also arise in the description of charged black hole microstates, extrap-olated to weak coupling. In the weak coupling limit (large volumes, small string coupling),the only difference between the two as far as their perturbative description is concerned isthe fact that in the former case, branes are filling the observable, noncompact space, whilein the latter, they are localized at a point. But in particular the geometric internal spacedescription of the configuration is mathematically identical in the two cases. Of course theactual dynamics and other physics of the two systems will be very different, one describingthe universe we may live in, the other an object we may look at, but the fact that they arerelated at the level of classical configuration spaces is one of the beautiful incarnations of theunifying nature of string theory. Thus, in particular, the largeness of the landscape and thelargeness of the entropy of a black hole are intimately connected.Despite this map, the fact that branes fill all of space has one dramatic consequence, andthat is that the charges (brane wrapping numbers etc) are tightly constrained by so-calledtadpole cancellation conditions. What this means is simply that whenever we have chargedobjects (such as a branes) with compact transversal dimensions (as is the case for space-fillingbranes but not space-localized ones), the sum of all charges must necessarily be zero. This isa consequence of Gauss’ law, or more colloquially, the fact that flux lines otherwise have noplace to go to; when there’s a source, there must be a sink. Combined with the requirement of54tability (typically realized by staying close to a supersymmetric configuration), this imposesstrict bounds on the allowed charges. On the other hand, for space localized branes, no suchcharge restrictions arise, and in particular we can take the charges to infinity and considerthermodynamic limits, which can be expected to have universal properties. This motivates usto consider localized brane systems.At finite coupling another difference between the two creeps in: the presence of gravity.This has a very different effect in the two settings. In the space-filling case, “vacuum” solutionswith slightly different internal brane configurations can give rise to very different asymptoticgeometries. As emphasized by Tom Banks at this school [8], for various reasons including thefact that our most successful descriptions of quantum gravity use fixed spacetime asymptoticsin an essential way, this makes it unclear if we can even think of these different solutions asbeing part of the same theory, or, as he would put it, it makes it clear that these solutionsare not part of the same theory. This does not mean that the presence of a space of braneconfigurations is of no physical relevance here of course; they do all exist after all, and it iscertainly of interest to classify the set of all possibilities. Furthermore local fluctuations aswell as larger fluctuations in the form of bubble nucleations can certainly probe — albeit in alimited sense — the configuration space. But it does mean that the situation is conceptuallymuch more subtle than, say, a collection of manganese atoms in copper. In contrast, forspace-localized branes, these issues do not arise. Gravitational backreaction just turns theweakly coupled brane system into a black hole, leaving the asymptotics unaffected. Evenbetter, thanks to the effects of gravity, we get a host of thermodynamic parameters for free,courtesy of the Bekenstein-Hawking entropy formula. This gives another motivation to studybrane systems associated to black holes.Besides their intrinsic interest as models of complex, glassy systems with intricate land-scapes and known holographic dual descriptions, the study of such localized brane systemscan also teach us things about the space-filling systems. Given their close physical and math- These bounds are set by the background curvature or orientifold contributions to the charge, which inturn are set by topological invariants such as the Euler number of the compactification manifold. Althoughthese numbers can get fairly high, for example there are known [2] elliptic CY 4-folds suitable for F-theorycompactifications allowing up to O (10 ) transversal D3-branes, constructions get increasingly sparse at thehigher end, leading many to suspect the set is finite. For reasons we do not understand any deeper thanthrough case by case examination of various conspiracies, the string theory landscape is uncannily “boxedin”, making it for example very hard, and plausibly impossible [38], to find infinite families of parametricallycontrolled solutions with positive cosmological constant. In this section we derive in detail the quantum mechanics describing the low energy dynamicsof a D4-brane wrapped on a 4-cycle, in the limit of small string coupling and large volume.Besides being an interesting and accessible example of a complex system in string theory withapplications to black hole physics, mathematics and string model building, it also providesa clean example illustrating various useful Kaluza-Klein reduction techniques, in particularthe relation between fluxes and superpotentials. (The idea to consider such an open stringlandscape was proposed in [106].)
Consider type IIA string theory on R × X , where X is some compact 6-dimensional manifold.The part of the IIA bulk action relevant for our purposes is, in units with the string scale (cid:96) s = 2 π √ α (cid:48) ≡ S IIA = S IIA , NSNS + S IIA , RR (4.1) S IIA , NSNS = 2 π (cid:90) R × X d x √− g (cid:18) g s R − g s H MNP H MNP (cid:19) (4.2) S IIA , RR = − π (cid:90) R × X d x √− g (cid:18) F (2) MN F (2) MN + 12 F (4) MNP Q F (4) MNP Q (cid:19) , (4.3)where H = dB is the NSNS 3-form field strength, F ( p +2) = dC ( p +1) are the RR field strengthsand g s is the string coupling constant. The theory contains D p -branes with p even. In56articular a single D4 wrapped around a 4-cycle Σ in the compact manifold X has, in thelimit of large volume and weak string coupling, an effective worldvolume action S D = S D4 , NSNS + S D4 , RR (4.4) S D4 , NSNS = − πg s (cid:90) R × Σ d x √− h (cid:18) F IJ F IJ (cid:19) (4.5) S D4 , RR = 2 π (cid:90) R × Σ F ∧ C (3) + (cid:18) F ∧ F + 124 e ( R ) (cid:19) ∧ C (1) , (4.6)where h IJ is the induced brane metric, F = dA + B , A is the U (1) gauge connection livingon the brane and e ( R ) is a quadratic polynomial in the tangent and normal bundle curvatureforms. When X is a Calabi-Yau manifold, this is the Euler density e ( R ): e ( R ) = Pf R , R I J ≡ π R I JKL dx K ∧ dx L , (4.7)which we encountered earlier in (3.36).The couplings to the RR fields given by S D4 , RR are topological in the sense that they aremetric-independent. They specify the gauge flux and curvature induced D2- and D0-chargeson the brane.We have ignored curvature corrections to S D4 , NSNS , but in the end we will fix this bymatching the energy required by supersymmetry for a given charge. A slightly more preciseversion of this action and references to derivations in the literature can be found in section 2of [107].
The low energy dynamics of the wrapped D4 system is obtained by Kaluza-Klein reduction.This is done most easily, and leads to the richest structure, when there is residual supersym-metry. An N = 1 theory in four dimensions does not have pointlike objects that preservepart of the supersymmetry, so one has to consider at least N = 2, i.e. compactifications witheight preserved supercharges, with the wrapped D-brane breaking half of those. When no bulkmagnetic fluxes are turned on, this requires the compactification manifold X to be Calabi-Yau, i.e. Ricci-flat and K¨ahler. Concretely this means that there exist complex coordinates y m , m = 1 , , θ m = θ mn dy n , such that themetric is of the form ds = | θ | + | θ | + | θ | , (4.8)57ith associated K¨ahler form J and covariantly constant holomorphic 3-form Ω given by J = i (cid:0) θ ∧ ¯ θ + θ ∧ ¯ θ + θ ∧ ¯ θ (cid:1) , Ω = θ ∧ θ ∧ θ . (4.9)The Calabi-Yau condition means that J and Ω are globally well-defined. The K¨ahler formhas one holomorphic and one anti-holomorphic index and is therefore called a (1 , , i (cid:90) X Ω ∧ Ω = 8 (cid:90) X J V X , (4.10)where V X is the volume of X .As a concrete example we can take X to be the quintic Calabi-Yau, described by a homo-geneous degree 5 polynomial equation in CP . The four complex dimensional projective space CP is the set of all nonzero ( x , x , x , x , x ) ∈ C , modulo the equivalence relation x (cid:39) λx , λ ∈ C ∗ . The simplest choice of polynomial is of Fermat form: X : Q ( x ) ≡ x + x + x + x + x = 0 . (4.11)Deformations of the metric preserving Calabi-Yauness are called (geometric) moduli. The fourdimensional low energy effective field theory will contain these moduli as massless scalars. Ingeneral it is not possible to write down explicit expressions for the Ricci flat Calabi-Yau metric,but Yau’s theorem states that for a given K¨ahler class (i.e. the cohomology class of the K¨ahlerform) and complex structure (i.e. a choice of complex coordinates), there is a unique Ricci flatK¨ahler metric. Thus, there are two kinds of moduli, those specifying the cohomology classof the K¨ahler form and those specifying the complex structure. In the case at hand there isjust one K¨ahler modulus, which parametrizes the overall scale of the metric, i.e. the volumeof X — the second cohomology of X is one-dimensional. The different complex structuresare simply parametrized by the choice of defining polynomial Q ( x ), modulo linear coordinatetransformations of the x i . This space is 101-dimensional.More generally, the dimension of the second cohomology of X will be b ( X )-dimensional,i.e. there are b ( X ) independent harmonic 2-forms on X . Equivalently, because of Poincar´eduality, there are b ( X ) independent homology 4-cycles in X . Let D A , A = 1 , . . . , B ≡ Poincar´e duality is a natural isomorphism between p -cycle homology classes and ( n − p )-form cohomologyclasses in an n -dimensional space. For a p -cycle C locally given by n − p equations f i ( x ) = 0, the associated dualcohomology class can be represented by the closed delta-function ( n − p )-form ˆ C = δ ( f ) df ∧· · ·∧ δ ( f n − p ) df n − p .Note that for two cycles C and C (cid:48) with dimension adding up to n , we can thus write (cid:82) ˆ C ∧ ˆ C (cid:48) = (cid:82) C ˆ C (cid:48) = C ∩ C (cid:48) ), where the last expression counts intersection points for generic representatives, with signs accordingto orientations. Usually we will drop the hat in the dual to avoid cluttering. ( X ) be an integral basis of harmonic 2-forms. Then J = J A D A , (4.12)and the J A are the K¨ahler moduli. We choose the signs of the 2-forms D A such that J A > C (3) = φ A dt ∧ D A + · · · , C (1) = φ dt + · · · . (4.13) In the weak coupling limit g s →
0, the backreaction of the wrapped D4 on the bulk geometryis negligible, due to the lower power in 1 /g s appearing in the D4-action. We can thereforeconsider the D-brane to be a probe in a background geometry specified by arbitrary, fixed bulkmoduli. For the wrapped brane to be supersymmetric, it must wrap a minimal volume 4-cycle,or somewhat stronger even, it must be holomorphic [108]. In the case of the quintic this isequivalent to saying it can be described by a degree N homogeneous polynomial equationΣ : P N ( x , x , x , x , x ) = 0 . (4.14)The degree N can be identified with the multiplicity of the D4-charge. For example for N = 1, the most general polynomial is P ( x ) = a x + · · · + a x . The complex coefficients c are deformation moduli of Σ. Since overall rescaling of the coefficients does not changethe zero set, we have a four complex dimensional deformation moduli space, topologically CP . An example of a degree N polynomial is ( P ( x )) N , which corresponds to N coincidentbranes wrapping P ( x ) = 0, with gauge group enhanced to U ( N ). The most general degree N homogeneous polynomial on the other hand gives a smooth singly wrapped brane with gaugegroup U (1). In the case at hand it has d ( N ) = N + N − z a , as can be verified by direct monomial counting, taking into account that when N ≥ q N − ( x ) Q ( x ) to P N ( x ), with Q as in (4.11), does not change the zero set.More generally one can deduce the number of deformations using standard algebraic geometrytechniques [109]. For a D4-brane wrapping a homology class [Σ] = N A D A with N A > d = 16 D ABC N A N B N C + 112 c ,A N A − . (4.15) Here the D A are the Poincar´e duals to the D A introduced in (4.12), for which as mentioned in footnote29 we use the same notation. D ABC are defined as D ABC ≡ (cid:90) X D A ∧ D B ∧ D C = D A ∩ D B ∩ D C ) (4.16)and the c ,A are topological numbers (the second Chern class of X ) depending on X only.For the quintic we see by comparing that D = 5 and c , = 50. The same techniques allowstraightforward computation of the generic Euler characteristic of Σ, namely χ = D ABC N A N B N C + c ,A N A , (4.17)which for the quintic leads to χ = 5 N + 50 N . The second Betti number, i.e. the number ofindependent 2-cycles or 2-forms on Σ is obtained from this as b (Σ) = χ − − b − b . When N A > b = b of Σ are inherited from X , so they vanish except in thehigher susy cases when X is T ( b = 6) or T × K b = 2). In any case, b grows as N , andtherefore, crucially, we will get a huge magnetic flux degeneracy on these branes in the large N limit. Indeed we can turn on harmonic gauge field strengths F = dA on Σ characterizedby b integers S i , the flux quanta: F = b (Σ) (cid:88) i =1 S i σ i , (4.18)where the σ i form a basis of integrally quantized harmonic 2-forms on Σ. In terms of Poincar´edual 2-cycles, we can equivalently write S i = η ij (cid:90) F ∧ σ j = η ij (cid:90) σ j F , η ij ≡ (cid:90) Σ σ i ∧ σ j = σ i ∩ σ j ) , (4.19)with η ij the inverse of η ij . The matrix η ij is called the intersection form of Σ. It is integraland unimodular, but in general not positive definite — in fact as we will see later at large N it has signature ( b +2 , b − ) ∝ ( , ) × N .The flux and curvature induced D2 and D0 brane charge can be read off from (4.6) withthe reduction (4.13): S D4 , RR = 2 π (cid:90) dt (cid:0) − q φ + q A φ A (cid:1) (4.20) q = − χ (Σ) − (cid:90) Σ F ∧ F = − χ − η ij S i S j (4.21) q A = (cid:90) Σ F ∧ D A = D iA S i , (4.22)60 ( z ) Σ( z + δz ) v m δz Figure 10:
Infinitesimal variation of Σ along its moduli space. where the integers D iA are given by D A,i = (cid:90) Σ σ i ∧ D A = σ i ∩ D A ) . (4.23)Kaluza-Klein reduction of the D4 produces a supersymmetric quantum mechanics with d (Σ) complex continuous degrees of freedom z a and b (Σ) discrete flux degrees of freedom S i . The moduli z a parametrize a moduli space M . The discrete fluxes can be thoughtof as quantized momenta dual to periodic coordinates that do not explicitly appear in theHamiltonian. The wrapped D-brane with F = 0 preserves 4 supercharges; it is the dimensionalreduction from 4 to 1 of a four dimensional N = 1 theory. In a sector with fixed flux wetherefore expect the low energy effective action for the z a to be of the general chiral multipletform [110] S = 2 πg s (cid:90) dt (cid:16) g a ¯ b ˙ z a ˙¯ z b − g a ¯ b ∂ a W ( z ) ¯ ∂ ¯ b ¯ W (¯ z ) (cid:17) + · · · , (4.24)where the + · · · consists of terms independent of z , W ( z ) is a holomorphic superpotential(absent when F = 0 but generically nonzero when F (cid:54) = 0), and g a ¯ b a K¨ahler metric: g a ¯ b = ∂ a ¯ ∂ ¯ b K ( z, ¯ z ) . (4.25)We have explicitly retained the overall prefactor πg s instead of absorbing it in the metric andsuperpotential. In this way g s can be thought of as Planck’s constant in the supersymmetricquantum mechanics. Note that this is of the form (3.30) (in its Lorentzian version) with h = ( W + ¯ W ). The special form of h is due to the presence of four supersymmetries Q α , Q † α , α = 1 ,
2, rather than the generic minimal amountof 2 supercharges.
61o identify the metric, we should take the brane to be slowly moving along its modulispace and expand the first term in (4.5) to second order in the velocities. An infinitesimaldisplacement δz a along the moduli space causes an infinitesimal normal displacement δy m = v ma δz a of Σ inside X , where the v ma depend holomorphically on the coordinates of Σ. This isillustrated in fig. 10. Thus, when slowly moving along the moduli space with velocity ˙ z a , thefirst term in (4.5) becomes − (cid:90) R × Σ d x √− h = (cid:90) dt (cid:90) Σ dV (cid:18) − g m ¯ n v ma ¯ v ¯ n ¯ b ˙ z a ˙¯ z ¯ b (cid:19) , (4.26)where dV is the volume element on Σ. Choosing the orthonormal frame θ appearing in (4.9)such that θ and θ lie along Σ (i.e. they span T ∗ Σ) while θ is normal to it, we can write dV = (cid:0) i θ ∧ ¯ θ (cid:1) ∧ (cid:0) i θ ∧ ¯ θ (cid:1) so the kinetic term becomes (cid:18) (cid:90) Σ θ ∧ θ v a ∧ ¯ θ ∧ ¯ θ ¯ v b (cid:19) ˙ z a ˙¯ z ¯ b = (cid:18) (cid:90) Σ ω a ∧ ¯ ω ¯ b (cid:19) ˙ z a ˙¯ z ¯ b , (4.27)where ω a = Ω · v a = θ ∧ θ v a (4.28)is the contraction of Ω with the vector field v a . It is a well-defined holomorphic (2 ,
0) form onΣ. This map is an isomorphism between deformations of Σ and harmonic (2 , g a ¯ b = 14 (cid:90) Σ ω a ∧ ¯ ω ¯ b . (4.29)To show that this is K¨ahler, we first expand the ω a in the integral basis introduced in (4.18): ω a = ω ai σ i , ω ai = η ij (cid:90) σ j ω a . (4.30)Now fix an arbitrary reference point z a ≡ M parametrizing the holo-morphic cycles Σ( z ). If we move away from this point, the 2-cycles σ i will sweep out 3-chainsΓ i ( z ). This is illustrated in figure 11. We can integrate the holomorphic 3-form over these3-chains, producing the holomorphic 3-chain periodsΠ i ( z ) = (cid:90) Γ i ( z ) Ω . (4.31)Because Σ is holomorphic this is independent of the choice of representative of σ i inside Σ( z ).By the definition of the ω a , we then have ∂ a Π i ( z ) = (cid:90) σ i ( z ) Ω · v a ( z ) = (cid:90) σ i ω a , (4.32)62igure 11: The inner (yellow) hyperboloid represents Σ(0), the outer (blue) one is Σ( z ), the green horizontaldisk stretched between the two represents the 3-chain Γ( z ), and its boundary in Σ( z ) is σ ( z ). and combining this with (4.30) and (4.29) gives us g a ¯ b = 14 ∂ a Π i η ij ¯ ∂ ¯ b ¯Π j = ∂ a ¯ ∂ ¯ b K ( z, ¯ z ) , K = 14 Π i η ij ¯Π j . (4.33)This makes it manifest that the metric is K¨ahler.To identify the superpotential, we consider the second term in (4.5), which for a givenmagnetic flux F gives a potential energy (cid:90) Σ F ∧ (cid:63)F = (cid:90) Σ
14 ( F + (cid:63)F ) ∧ ( F + (cid:63)F ) − (cid:90) Σ F ∧ F , (4.34)where (cid:63) is the Hodge star operator on Σ and we used (cid:63) = 1. The second term is topological— it does not depend on the z a , and (4.21) shows it is nothing but the D0-charge inducedby the fluxes. Thus, the magnetic potential energy is bounded below by the flux inducedD0-charge, with the bound saturated by anti-self-dual flux F = − (cid:63) F . In the following wewill show that the first term is (almost) the | ∂W | term of (4.24).The (cid:63) operator commutes with the decomposition of 2-forms according to their number of(holomorphic, antiholomorphic) indices. It acts as +1 on (2 ,
0) forms: (cid:63)ω (2 , = ω (2 , . Thisis easily checked by considering a holomorphic orthonormal frame of Σ: (cid:63) ( θ ∧ θ ) = + θ ∧ θ .The complex conjugate (0 , , This is true because Σ is a K¨ahler manifold, being a complex submanifold of the K¨ahler manifold X . , ,
1) formsorthogonal to that are anti-selfdual, i.e. (cid:63) = −
1. This is checked by considering θ ∧ ¯ θ + θ ∧ ¯ θ resp. θ ∧ ¯ θ − θ ∧ ¯ θ .Accordingly the first term in (4.34) is (cid:90) Σ
14 ( F + (cid:63)F ) ∧ ( F + (cid:63)F ) = (cid:90) Σ F (2 , ∧ F (0 , + F (1 , ∧ F (1 , . (4.35)The component F (1 , is obtained by projection onto the K¨ahler form J = J A D A (pulled backto Σ): F (1 , = f + J , f + = 1 (cid:82) Σ J (cid:90) Σ F ∧ J = 1 (cid:82) Σ J q A J A , (cid:90) Σ J = D ABC N A J B J C , (4.36)where we used (4.22). This expression is again independent of the D4 moduli z a . The secondterm in (4.35) thus equals (cid:82) Σ J ( q A J A ) . Note this is independent of the overall scale of J .The remaining term does depend on the moduli z a and we will see it can be identified with | ∂W | . Expand F (2 , in the basis of (2 , F (2 , = f a ω a , f a = 14 g a ¯ b (cid:90) Σ F ∧ ¯ ω ¯ b , (4.37)with g a ¯ b defined in (4.29) and g a ¯ b its inverse. Using the decomposition (4.18) and the relationto chain period derivatives (4.32), we write (cid:90) Σ F ∧ ¯ ω ¯ b = S i (cid:90) σ i ¯ ω ¯ b = ¯ ∂ ¯ b (cid:0) S i ¯Π i (¯ z ) (cid:1) , (4.38)and the first term in (4.35) becomes g a ¯ b ∂ a ( S i Π i ) ¯ ∂ ¯ b ( S j ¯Π j ). Comparing to (4.24), we identifythe flux induced superpotential W ( z ) = S i Π i ( z ) . (4.39)We put everything together below. To summarize, the low energy degrees of freedom at weak coupling and large volume of ageneric D4-brane wrapped on a 4-cycle Σ = N A D A in a generic Calabi-Yau 3-fold with Generic means here that the brane worldvolume is smooth and the gauge group is U (1). Coincident braneswill give rise to an enhanced nonabelian gauge symmetry and intersecting branes will give rise to additionallight bifundamentals. We also do not include pointlike D0-branes yet. We will return to this in section 4.4. D ABC and K¨ahler form J = J A D A ( J A , N A >
0) are d = b − z a and b = χ − S i parametrizing the U (1) flux F = S i σ i , with b = D ABC N A N B N C + c ,A N A and χ = D ABC N A N B N C + c ,A N A . Upto a constant energy term the bosonic part of the action is S = S dyn + S top (4.40) S dyn = 2 πg s (cid:90) dt (cid:20) g a ¯ b ˙ z a ˙¯ z b − g a ¯ b ∂ a W ¯ ∂ ¯ b ¯ W (cid:21) (4.41) S top = − πg s (cid:90) dt (cid:104)(cid:16) q + ( q A ˜ J A ) (cid:17) + g s (cid:0) q φ − q A φ A (cid:1)(cid:105) , (4.42)with superpotential and K¨ahler metric given by W ( z ) = S i Π i ( z ) , Π i ( z ) = (cid:90) Γ i ( z ) Ω (4.43) g a ¯ b = ∂ a ¯ ∂ ¯ b (cid:0)
14 Π i η ij Π j (cid:1) , (4.44)flux-dependent D0- and D2-charges q = − χ − η ij S i S j = − χ − (cid:90) Σ F ∧ F (4.45) q A = D iA S i = (cid:90) Σ D A ∧ F , (4.46)and unit K¨ahler form ˜ J = 1( D ABC N A J B J C ) / J = 1( (cid:82) Σ J ) / J . (4.47)The matrix η ij = σ i ∩ σ j ) is the intersection form of Σ and D iA = D A ∩ σ i ). The 3-chainsΓ i ( z ) are 3-chains swept out by the 2-cycles σ i on Σ when moving from a fixed reference pointto z in the moduli space. We recall that for the quintic D = 5, ˜ J = √ N , c , = 50.This is a reliable description at weak coupling and low energies. What exactly does weakcoupling mean? We recall from our general discussion of supersymmetric quantum mechanicsin section 3 that weak coupling corresponds to a limit in which the superpotential h is scaledup as h → λh with λ → ∞ , while keeping the metric constant. To see how this relatesto other scalings, consider the simplest case of a single variable with bosonic Lagrangian L = A ˙ x + B ( dh/dx ) . Then we can redefine x = y/ √ A such that L = ˙ y + AB ( dh/dy ) .So weak coupling means AB → ∞ . Applying this to the case at hand and remembering that65ecause of (4.10) the metric and periods scale as g a ¯ b ∝ V X , Π i ∝ √ V X with the volume of X ,we have effectively A ∝ V X /g s and B ∝ /g s . Thus weak coupling means V X g s → ∞ . (4.48)The left hand side is essentially the ratio of the string length squared over the 4d Planck lengthsquared, (cid:96) s /(cid:96) . Hence the weak coupling regime is the regime in which the string length ismuch larger than the 4d Planck length, and the low energy supergravity description breaksdown.We see from (4.41) that switching on the flux quanta S i generates a highly complex po-tential energy function for the z a , leading at fixed q and q A (and large q ) to a vast energylandscape with exponentially many minima, parametrized by the values of S i and z a for which ∂W S ( z ) = 0. This is kinematically very similar to the landscape of flux vacua in string theory,except of course that here we have a quantum mechanical system rather than a universe.The potential is in general not a single valued function on the moduli space M , due topossible monodromies σ i → M ij σ j of the 2-cycle basis when going around noncontractibleloops in M . It does become single valued on the covering space (cid:102) M , also known as Teichm¨ullerspace, but then one has to quotient the theory by the covering group, which acts as a discretegauge symmetry on z a and S i . The metric g a ¯ b and the charges q and q A on the other handare single valued, since they can be expressed without reference to the basis σ i .The different zero energy classical minima of the potential can mix quantum mechanicallythrough tunneling instantons. By completing the squares in the Euclidean version of theaction one finds a bound S E dyn ≥ πg s Re( e − iα ∆ W ) for any real α , saturated when˙ z a = − e − iα g a ¯ b ¯ ∂ ¯ b ¯ W (¯ z ) . (4.49)The strongest bound is obtained with α = arg(∆ W ), so this is what we should take α to bein order to be able to find instanton solutions. The instanton action is then S E dyn = 2 πg s | ∆ W | , (4.50)and the trajectories are straight lines when projected to the W -plane. Before turning to the supersymmetric completion of the model, it may be useful to quicklyreview some basic formulae in K¨ahler geometry as well as more specialized expressions ap-plicable to the model under consideration. When a metric is K¨ahler, i.e. g a ¯ b = ∂ a ¯ ∂ ¯ b K , all66hristoffel symbols Γ KIJ = g KL ( ∂ I g LJ + ∂ J g IL − ∂ L g IJ ) vanish except when all indices areholomorphic or all indices are anti-holomorphic. When all indices are holomorphic, we haveΓ abc = g a ¯ d ∂ b g c ¯ d = ¯ ∂ a ¯Π i ∂ b ∂ c Π i = (cid:90) X ¯ ω a ∧ ∂ b ω c . (4.51)We are lowering and raising indices with g a ¯ b and η ij . The first expression is true for generalK¨ahler manifolds, the remainder specializes to (4.44). Equivalently, (cid:82) ¯ ω ¯ a ∧ ∇ b ω c = 0. TheRiemann curvature, defined by [ ∇ I , ∇ J ] X K = R IJLK X L , also simplifies: R a ¯ bc ¯ d = g e ¯ d ¯ ∂ ¯ b Γ eac = ¯ ∇ ¯ b ¯ ∂ ¯ d ¯Π i ∇ a ∂ c Π i = (cid:90) X ¯ ∇ ¯ b ¯ ω ¯ d ∧ ∇ a ω c . (4.52)Again, the first equation is valid for general K¨ahler manifolds. Components of the curvaturetensor not related by symmetries to the above ones (such as R abc ¯ d ) all vanish. Finally, inaddition to the usual curvature symmetries R IJKL = − R JIKL = − R IJLK = R KLIJ and R I [ JKL ] = 0, a K¨ahler manifold also satisfies R a ¯ bc ¯ d = R c ¯ ba ¯ d .Geometrically, the fact that (cid:82) ¯ ω ¯ a ∧ ∇ b ω c = 0 means that ∇ b ω c is of type (1 , , , , ,
0) form from Σ( z ) to Σ( z + δz ) will at most produce one extra antiholomorphicleg to first order in δz ). The same is then true for ∇ b ω c , and (cid:82) ¯ ω ¯ a ∧ ∇ b ω c = 0 further impliesthat the (2 ,
0) part is actually zero, leaving only a (1 ,
1) part.Keeping in mind the ( p, q )-type of various forms, the following orthogonality propertieshold: ∂ a Π i ∂ b Π i = 0 , ∂ a Π i ∇ b ∂ c Π i = 0 , ∂ a Π i D Ai = 0 . (4.53)Further orthogonality properties can be derived from these by differentiation, for example ∇ b ∂ a Π i D Ai = 0. To find the supersymmetric completion of this model, we could either reduce the fermionicpart of the full D4-action, or we can infer it from the structure of the bosonic part. We willgo for the latter route. To do so, we first write down the bosonic Hamiltonian derived fromthe action given earlier. This is H (0) = H (0)dyn + H (0)top with H (0)dyn = g s π g a ¯ b p a ¯ p ¯ b + πg s g a ¯ b ∂ a Π i S i ¯ ∂ ¯ b ¯Π j S j , (4.54) H (0)top = πg s (cid:0) q + ( ˜ J A q A ) (cid:1) . (4.55)67ere p a = πg s g a ¯ b ˙¯ z a and we dropped the “chemical potential terms” proportional to φ and φ A here. If we consider the S i as constants, this is of the form of a 4d N = 1 theory ofchiral superfields reduced to 1d, and the supersymmetric completion would be immediate.However the S i are not dynamically conserved in this system, since motion from a point z inthe moduli space M to itself along a noncontractible loop will in general change S i → M ij S j due to monodromy. On the other hand q , q A and H (0)top are monodromy invariant, and hencethey are conserved. In particular for an instanton, the change ∆ W that appears in (4.50) isdue to a change ∆ z with constant S i on Teichmuller space , but it will in general lead to achange ∆ S when reduced back to the moduli space. Therefore we should consider the S i tobe dynamical. We can think of the S i as quantized momenta conjugate to angular coordinates ϑ i that do not appear explicitly in the Hamiltonian, i.e. S i = − i∂ ϑ i . The ϑ i can then beviewed as the potentials obtained by KK reduction of the 2-form potential on the D4 that isdual to the U (1) gauge field. The appropriate metric for these coordinates can be read offfrom the above expression for the Hamiltonian, which can be cast in the form H (0) = g s π g a ¯ b p a ¯ p ¯ b + πg s g ij S i S j , (4.56)where we have defined a metric on flux space g ij ≡ (cid:16) Π ia ¯Π ja + ¯Π i ¯ a Π j ¯ a + ˜ J i ˜ J j (cid:17) − η ij , Π ia ≡ ∂ a Π i , ˜ J i ≡ ˜ J A D iA . (4.57)In this expression we use η ij and g a ¯ b for index raising and lowering. The three terms insidethe brackets are projectors onto (2 , , ,
1) direction parallel tothe K¨ahler form; together they project to the space of self-dual forms. That these terms areproperly normalized projectors follows from the relation (4.29) and from the definition (4.47).In fact this metric is nothing but the the Hodge (cid:63) -product, and the decomposition intoprojections is just (4.34) with (4.35) again. Consistent with this, we have that g ij g jk = δ ik ,which is the property (cid:63) = 1. This also shows that the inverse of g ij is g ij = η ik η jl g kl .The above Hamiltonian is just that of a free supersymmetric quantum mechanics in acurved space. Accordingly we could construct the usual supercharges Q = d and Q † = d † and take H = { Q, Q † } , i.e. the Laplacian. However then the Q = 0 supersymmetric Incidentally, an open D2-instanton wrapping a special Lagrangian 3-chain Γ in X with boundary ∂ Γ = σ + − σ − on Σ has an action S D2 = πg s | (cid:82) Γ Ω | = πg s | ∆ W | , where ∆ W is the change in superpotential when F → F + σ + − σ − . This is exactly the same as for monodromy instantons, suggesting a possible identificationof the two. H dyn = H top = 0, which corresponds to D-branestates without any flux. This is too restrictive. We only need H dyn = 0 for a state to besupersymmetric, so we need to find supercharges that square to H dyn only. These are basicallythe dimensionally reduced 4d N = 1 supersymmetry generators, except that we interpret the S i now as dynamical momenta: S i = − i∂ i . Introducing the fermionic operators ¯ ψ a = dz a ∧ ,¯ ψ ¯ a = d ¯ z ¯ a ∧ and their conjugates ¯ ψ ¯ a = ( ψ a ) † , ¯ ψ a = ( ψ ¯ a ) † , we define the supercharges Q − = (cid:113) g s π ¯ ψ a ∇ a + (cid:113) πg s ¯ ψ ¯ a ¯ ∂ ¯ a ¯Π i ∂ i , ¯ Q + = (cid:113) g s π ¯ ψ ¯ a ¯ ∇ ¯ a + (cid:113) πg s ¯ ψ a ∂ a Π i ∂ i . (4.58)and their conjugates ¯ Q − = ( Q − ) † , Q + = ( ¯ Q + ) † . Equivalently Q − = (cid:113) g s π ∂ + (cid:113) πg s ¯ ∂ ¯Π i ∂ i , ¯ Q + = (cid:113) g s π ¯ ∂ + (cid:113) πg s ∂ Π i ∂ i . (4.59)The supercharges satisfy the extended supersymmetry algebra { Q α , ¯ Q β } = δ αβ H dyn , { Q α , Q β } = 0 = { ¯ Q α , ¯ Q β } , (4.60)where H dyn reduces to (4.54) in the zero fermion number sector. This defines the summetriccompletion of H (0)dyn . Working out e.g. { Q − , Q †− } we get explicitly, acting on wave functionsΦ( ¯ ψ, x ) (i.e. differential forms): H dyn = − g s π ¯ ∇ a ∇ a + g s π R a ¯ bc ¯ d ¯ ψ a ¯ ψ ¯ b ψ c ψ ¯ d − πg s ∂ a Π i ¯ ∂ a ¯Π j ∂ i ∂ j − ¯ ψ a ψ b ∇ a ∂ b Π i ∂ i + ¯ ψ ¯ b ψ ¯ a ¯ ∇ ¯ a ¯ ∂ ¯ b ¯Π i ∂ i . Here, again acting on differential forms, ∇ a = ∂ a + Γ bac ¯ ψ c ψ b , and p a = − i∂ a , S i = − i∂ i .Evidently neither the ϑ i nor their fermionic superpartners (i.e. something like dϑ i ) appearexplicitly in the Hamiltonian. Nevertheless they do not completely decouple from the z -degrees of freedom, again because of global monodromies — the ϑ -torus fibration over themoduli space M has nontrivial identifications when going around nontrivial loops.To summarize, the full supersymmetric Hamiltonian including the RR potential terms is,in terms of the conserved charges: H = H dyn + πg s (cid:0) q + ( ˜ J A q A ) (cid:1) + 2 π ( φ q + φ A q A ) . (4.61) Because a 4d N = 1 theory has four supersymmetries, the system has twice as much supersymmetry asthe minimal susy QM case studied in section 3. As “the” Q -operator (satisfying Q = 0, { Q, Q † } = H dyn ), wecan therefore choose from a family of linear combinations of the supercharges. The phase α appearing in theinstanton flow equation (4.49) determines which supersymmetry is preserved by the instanton, and thereforewhich supersymmetry gets corrected. ϑ momenta S i explicit in the topological part of the Hamiltonian, this becomes H = H dyn − π (cid:0) g s + φ (cid:1)(cid:0) χ + η ij S i S j (cid:1) + πg s ( ˜ J i S i ) + 2 πφ i S i . (4.62)Unpacking the whole thing gives us H = − g s π ¯ ∇ a ∇ a + g s π R a ¯ bc ¯ d ¯ ψ a ¯ ψ ¯ b ψ c ψ ¯ d − π ( g s + φ ) χ + π (cid:0) g s g ij − φ η ij (cid:1) S i S j + 2 πL i S i . (4.63)Here φ i = φ A D iA , g ij is the metric (4.57), and L i = φ i − i π ¯ ψ a ψ b ∇ a ∂ b Π i + i π ¯ ψ ¯ b ψ ¯ a ¯ ∇ ¯ a ¯ ∂ ¯ b ¯Π i . (4.64)It is often convenient to split quantities like S i and L i into self-dual and antiself-dual parts,as this leads to more transparent “left-moving” and “right-moving” expressions. Thus, forinstance (cid:0) g s g ij − φ η ij (cid:1) S i S j = ( g s − φ ) S − ( g s + φ ) S − , (4.65)where squares denote contraction with η , i.e. X = X i X i = η ij X i X j (so X > X − < X + and hence also for X − = X − X + . Legendre transforming the Hamiltonian, with ˙ z a = ∂H∂p a , ˙ ϑ i = ∂H∂S i , we get the correspondingLagrangian L = πg s g a ¯ b ˙ z a ˙¯ z ¯ b + i ¯ ψ a D t ψ a + i ¯ ψ ¯ b D t ψ ¯ b − g s π R a ¯ bc ¯ d ¯ ψ a ¯ ψ ¯ b ψ c ψ ¯ d + 2 π ( g s + φ ) χ
24+ 14 π ( g s − φ ) (cid:0) ˙ ϑ + − πL + (cid:1) − π ( g s + φ ) (cid:0) ˙ ϑ − − πL − (cid:1) , (4.66)where D t ψ a = ˙ ψ a + ˙ z b Γ abc ψ c . Note that L + = φ + as ( ∇ a ∂ b Π) + = 0. We have constructed the supersymmetric quantum mechanics describing the low energy physicsof smooth D4-branes with U (1) gauge symmetry and abelian fluxes F turned on. In the ther-modynamic limit N → ∞ we get O ( N ) discrete flux degrees of freedom S i interacting with70 .0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 Figure 12:
Analog in 2d of the problem of fitting a holomorphic 4-cycle to contain a given collection of rigidholomorphic 2-spheres in a 6-dimensional space: here we are fitting a 1-cycle given by a degree 5 polynomialin x and y to contain 36 randomly generated points, thus fixing all moduli of the 1-cycle. each other in a complicated, nonlocal way through coupling to O ( N ) continuous degreesof freedom z a . The effective Hamiltonian (4.56) is somewhat reminiscent of models for spinglasses or the Hopfield model for neural networks, but more intricate, and a priori withoutquenched disorder, making it more like a structural glass than a spin glass. In any case thissuggests the presence of many effectively stable states. We will now make this quantitative,ignoring for the time being the possibility of brane degenerations and nonabelian gauge groupenhancement.Classical supersymmetric brane configurations are those for which Π ia S i = 0, i.e. ∂W F ( z ) =0. As the 3-chain periods are very hard to compute explicitly, it might seem hopeless to tryto construct any of those configurations explicitly. But a little thought shows otherwise. Thecondition ∂W F = 0 is equivalent to F (0 , = F (2 , = 0 . (4.67)In other words if we switch on a flux F = S i ω i , the brane moduli must adapt themselvesto make the flux of type (1 , A spin glass has quenched disorder built into the Hamiltonian. A structural glass has a simple Hamiltonian,and spontaneously generates its own disorder. This includes the material normal people refer to when theyuse the word glass. σ is clearly of type (1 ,
1) since for every (2 , ω we have (cid:82) σ ∧ ω = (cid:82) σ ω = 0. The last equation is true because σ is the image of some map y m ( u )with u a single complex coordinate, and so the pullback of a (2 ,
0) form will be proportionalto du ∧ du = 0. Thus, any linear combination (with arbitrary signs) of fluxes Poincar´e dualto holomorphic 2-cycle will be of type (1 , ,
1) is sometimes called the (generalized) Noether-Lefschetz locus inalgebraic geometry.This interpretation gives a relatively simple way of explicitly constructing classical super-symmetric ground states. Let us illustrate this with an example. Consider again the quinticCalabi-Yau Q ( x ) = 0 and let us assume that we have chosen the polynomial Q such that itcontains the following holomorphic 2-cycle, parametrized by ( u, v ) ∈ CP : γ : ( x , x , x , x , x ) = ( u, − u, v, − v, . (4.68)The Fermat quintic (cid:80) i x i = 0 for example contains γ . The topology of γ is that of CP ,i.e. it is a 2-sphere. Such “degree 1 rational curves” are generically isolated in a Calabi-Yau.This can be checked in the case at hand by simple counting: the most general linear map( u, v ) → ( x , · · · , x ), has 10 coefficients, 4 of those can be eliminated by linear coordinatetransformations of ( u, v ), and requiring Q ( x ( u, v )) = 0 for all ( u, v ) produces 6 equations onthe remaining coefficients (namely the coefficients of u , u v , . . . , v after expanding out),so we generically expect a discrete solution set. Indeed in the generic quintic there are 2875isolated degree 1 rational curves.Now consider our single wrapped D4 ( N = 1), described by a linear equationΣ : (cid:88) i a i x i = 0 . (4.69)Requiring Σ to contain γ produces the equations a = a , a = a . Therefore, if we switchon a flux Poincar´e dual to γ on Σ, rigidity of the rational curve means that in order topreserve supersymmetry, these equations must remain satisfied when deforming Σ. Hencethe D4 moduli space is reduced from CP to CP . If we also require say the rational curve γ (cid:48) : x = (0 , u, − u, v, − v ) to lie on Σ, enforced by switching on F = γ (cid:48) , we end up with a = a = a = a = a , i.e. all moduli have been lifted. A simple 2d analog of this isillustrated in fig. 12. By combining different pairs of the 2875 degree 1 rational curves we can72hus explicitly construct × = 4131375 different isolated supersymmetric configurations(assuming there are no duplicates or degenerates). In fact for each choice of pair ( γ, γ (cid:48) ) thereis an infinite number of different flux states, obtained by switching on a more general linearcombination of the harmonic forms γ and γ (cid:48) : F = kγ + k (cid:48) γ (cid:48) , (4.70)where k, k (cid:48) ∈ Z . Of course these are not all degenerate in total energy. They all have H dyn = 0but they will have different H top . To compute this we should find the D2- and D0-charges.They are q ( F ) = (cid:90) Σ D ∧ F = k + k (cid:48) , q ( F ) = − χ − (cid:90) Σ F ∧ F = − χ
24 + 32 ( k + k (cid:48) ) , (4.71)with χ = 55. Here we used the fact that the self-intersection number of γ inside Σ equals − γ inside Σ (necessarily nonholomorphi-cally) and counting intersections, or by using some algebraic geometry (as in section 4.3 of[107]). The total energy H = H top is given by g s π H = q + q − χ
24 + 32 ( k + k (cid:48) ) + 15 ( k + k (cid:48) ) . (4.72)Thus we get a nondegenerate lattice of vacua for each generic pair of degree 1 rational curves.This can be generalized in various ways. Instead of degree 1 rational curves, we can considerdegree d rational curves. The same simple counting argument as before indicates again thatthey are generically isolated: there are 5( d +1) − d + 1 equations, which happens to be the same number. The number of rationalcurves grows exponentially with the degree. There are for example 609250 degree 2 curves and704288164978454686113488249750 degree 10 curves. Furthermore, we can consider arbitraryD4-charge N . Requiring such a degree N d N d + 1moduli, out of N +25 N . So at large N the 4-cycle can “store” up to N d degree d rationalcurves by switching on the appropriate fluxes. If d is sufficiently large so that the numberof degree d curves N d ∼ e κd is much larger than this number, then this leads, naively atleast, to up to ∼ e κ N susy configurations. Switching on one degree d flux quantum increases q by dN + 1, and q by ± d depending on the sign of the flux (see section 4.3 of [107] for Actually we should shift F by the “half-flux” D , because for N = 1, Σ is not spin [111, 112]. Thisensures proper charge quantization. We will ignore this here.
73 derivation). So at large N the energy in the maximal amount of N d stored curves is g s π H ≈ q ≈ − χ + N d × dN ≈ − N + N = N . Although the above discussion is for weak coupling, there is strong evidence that some ofthis structure carries over the strong coupling black hole regime, in the form of the structureof multicentered black hole bound states (cf section 5). We refer to [107] for more discussionon this.
Although the above construction is explicit, and the number of ground state configurationsthat can be built in this way is huge, it is still only a small subset, especially at large q .In this regime the generic ground state configuration is isolated, and their total number anddistribution over the moduli space can be computed using the statistical methods to countflux vacua developed in [36, 113, 114] and reviewed in section 6 of [7]. Those methods mapthe classical critical point counting problem, after making a continuum approximation, tothe computation a supersymmetric finite dimensional integral, essentially a finite dimensionalversion of what we did is section 3.8. In the present case we already have a supersymmet-ric quantum mechanics to start from. Not surprisingly, it is closely related to the effectivesupersymmetric quantum mechanics used in the statistical approach.In what follows we will consider the problem by applying the general machinery of su-persymmetric quantum mechanics reviewed is section 3. This clarifies and complements theresults obtained by the methods reviewed in [7] (and applied to the case at hand in appendixG of [107]). In particular it allows us to go beyond the continuum approximation. The com-putations will get a little technical, for which I apologize. I chose to include them here becauseit may be useful for some readers to see a nontrivial example worked out, and because as faras I know this has not been done in the literature (the treatment has some overlap with [107],but we will not use S-duality as an input, but rather derive it directly from the susy quantummechanics).From our general consideration of supersymmetric quantum mechanics in the previoussection, it is not hard to come up with a formula for the Witten index Ω( q ) as a weighted Notice in particular that the energy becomes positive before all moduli are frozen. If what we werestudying were not D4-branes but D7-branes in IIB orientifolds, this would basically mean that we cannot fixall moduli in this way while respecting tadpole cancelation, going against common genericity arguments. Thisis not conclusive though since this construction certainly does not exhaust all possibilities.
Artist impression of a smooth abelian D4 configuration with U (1) magnetic fluxes turned on(flux density represented by colors). sum of Euler characteristics of critical point loci M S of the superpotential W S , summed overfluxes/momenta S with the given total charge q :Ω( q ) = (cid:88) S ( − d − d S χ ( M S ) . (4.73)The sign ( − d − d S appears because m transversal dimensions contribute m (the Morse index)to the fermion number. This is a useful formula for supersymmetric configurations with smallD0-charge, but not for large D0-charge. We will try to extract the large charge asymptoticsfrom the partition function.The supersymmetric partition function generating the Witten indices Ω( q ) in each con-served charge sector is, with H as in (4.61)-(4.63): Z ( β, g s , φ ) = Tr ( − F e − βH ( g s ,φ ) = (cid:88) q Ω( q ) e − πβgs ( q +( ˜ J A q A ) ) − πβ ( φ q + φ A q A ) . (4.74)Introducing rescaled RR potentials ϕ ≡ βφ , (4.75)we obtain the following expression for the Witten index Ω( q ):Ω( q ) = lim β → (cid:90) dϕ e πϕ · q Z ( β, ϕ ) , (4.76)where the ϕ integrals in principle are over a unit imaginary interval, e.g. from − i/ i/
2. Forcomputational purposes it is more convenient to work with integrals over the entire imaginaryaxis, because it allows easy implementation of contour integration techniques. Extendingthe integration ranges in the integrals above only produces trivial additional delta-functionfactors, which in the end can be stripped off manually.75e will now try to compute the Witten indices from the Euclidean path integral repre-sentation of Z in the limit β →
0, analogous to the derivation leading to (3.36): Z = (cid:90) D z D ψ D ϑ e − S [ z,ψ,ϑ ] , (4.77)with periodic boundary conditions on all fields: z a ( τ + β ) = z a ( τ ), ψ a ( τ + β ) = ψ a ( τ ), ϑ i ( τ + β ) = ϑ i ( τ ) + 2 πn i , with n i ∈ Z . By Wick rotating (4.66) to rescaled Euclidean time t = − iβτ , (4.78)we obtain the Euclidean action S = (cid:90) dτ (cid:18) πg s β | ˙ z | + ¯ ψD t ψ + g s βπ R ¯ ψ ¯ ψψψ − πu χ
24 + 14 πv (cid:0) ˙ ϑ + 2 πi(cid:96) (cid:1) − πu (cid:0) ˙ ϑ + 2 πi(cid:96) (cid:1) − (cid:19) , (4.79)where we suppressed indices for clarity, denoted (cid:96) ≡ L/β , i.e. (cid:96) i ≡ ϕ i − iβ π ¯ ψψ ∇ ∂ Π i + iβ π ¯ ψψ ¯ ∇ ¯ ∂ ¯Π i , (4.80)and introduced “light cone” coordinates u ≡ βg s + ϕ , v ≡ βg s − ϕ . (4.81)As in (3.35), in the limit β →
0, the path integral localizes to constant paths ( z, ψ ), with thefluctuation determinants for z and ψ canceling each other. The path integral over ϑ does notlocalize to constant paths, since the ϑ kinetic terms remain finite when β → ϕ (cid:54) = 0).The ϑ path integral is essentially the partition function of a free particle on a b -dimensionaltorus. Making the sum over different winding sectors ϑ i ( τ + 1) = ϑ i ( τ ) + 2 πn i explicit, this is (cid:88) n e − πv ( n + i(cid:96) ) + πu ( n + i(cid:96) ) − (cid:90) D ϑ e − π (cid:82) dτ ( v ˙ ϑ − u ˙ ϑ − ) . (4.82)The first factor corresponds to the action weights for the classical straight line trajectories ineach sector, and the remaining path integral corresponds to the fluctuations from the classicalpaths and thus has strictly periodic boundary conditions ϑ i ( τ + 1) = ϑ i ( τ ). This is just thestandard Euclidean free particle propagator integrated over the ϑ -torus: (cid:90) D ϑ e − π (cid:82) dτ [ v ˙ ϑ − u ˙ ϑ − ] = u − b − / v − b +2 / . (4.83)By making use of (4.52) and the various orthogonality and (anti-)self-duality properties dis-cussed there, as well as the projectors introduced in (4.57) to write explicit expressions for76 + and n − = n − n + , the sum in (4.82) together with the 4-fermion curvature term in (4.79)reduces after some tender and care to (cid:88) n e − πv ( n + iϕ ) J + πu ( n + iϕ ) ⊥ J e − β ( πgsuv | n Π (cid:48) | − u ( ¯ ψψ n Π (cid:48)(cid:48) +cc) − gsvπu R ¯ ψ ¯ ψψψ ) , (4.84)where we separated out a factor in the summand that is independent of the dynamical variables z, ψ , in which we also introduced the notation X iJ ≡ ˜ J i X i for the projection of X in thedirection of J , and X ⊥ J for its orthogonal complement (w.r.t. η ij ). Now consider the z, ψ integral of the other factor: (cid:90) d d z d d ψ d d ¯ ψ e − β ( πgsuv | n Π (cid:48) | − u ( ¯ ψψ n Π (cid:48)(cid:48) +cc) − gsvπu R ¯ ψ ¯ ψψψ ) . (4.85)This integral has the typical zero dimensional supersymmetric form with superpotential W n ( z ) = n i Π i ( z ). It localizes on the critical point locus M n of W n , meaning the Gaussian approxi-mation to the integral is exact. The Gaussian integral for quadratic fluctuations normal to M n produces a factor g s uvβ for each complex normal direction, while the corresponding normalfermions produce a factor β u . Together this gives a factor βvu for each normal direction. Thetangential directions to M n remain integrated over, and the corresponding tangent fermionintegral gives for each complex tangent dimension a factor − g s βvπu R . Finally, there will bean overall path integral normalization factor (2 πg s β ) − d as in (3.35). Altogether the aboveintegral thus reduces to ( − d n e ( M n ) v d u − d , where d n ≡ dim M n and e ( M n ) is the Eulerdensity on M n , as in (3.36). The sign is physically meaningful; we expand on this below.Putting everything together, we find the generating function: Z = ( u/v ) / u − b / e πu χ (cid:88) n ( − d n χ n e − πv ( n + iϕ ) J + πu ( n + iϕ ) ⊥ J , (4.86)where we defined the differential geometric Euler characteristics χ n ≡ (cid:90) M n e ( R ) , (4.87)and we used d = b − . Note that since by definition Ω n is nonzero only if ( n ⊥ J ) + can bemade to vanish, all terms in this series have n ⊥ J ≤ Actually, because n can transform by a monodromy when going around a loop in moduli space, this onlyholds up to boundary terms for a given n , but these cancel between different values of n . To see localizationmore directly, note that since the Witten index is invariant under a change of g s , we can make the integrandas sharply peaked on M n as we want, with vanishing boundary terms when g s → q A = D Ai n i , ˜ q = − χ − n , we can also express this as Z = ( uv ) / u − b / e π χ ( u − u ) − πu ϕ − + πv ϕ ( − d (cid:88) ˜ q Ω(˜ q ) e − πu ˜ q − π ( u + v )˜ q + πiu ˜ q − ˜ ϕ − − πiv ˜ q + ˜ ϕ + , (4.88)with as in (4.73) Ω(˜ q ) = (cid:80) n ( − d n − d χ n , summing over the values of n with the given charges˜ q . Comparing this to the original (4.74), which can be written as Z = (cid:88) q Ω( q ) e − πuq − π ( u + v ) q − πqϕ , (4.89)we see that what we have shown is roughly that Z is a modular (Jacobi) form under themodular transformation u → /u , v → /v , transforming similar to an ordinary theta function(although what we have here is much more nontrivial than a theta function, due to thenontrivial moduli dynamics). The duality exchanges winding and momentum modes, andthere are many ways of understanding it: electromagnetic duality of the D-brane theory, S-duality of the parent string theory, modularity of the parent CFT in M-theory, T-duality ofthe torus, a version of Poisson resummation, etc. The modular transformation relates thelarge and small flux / D0-charge regimes.In any case, we can now extract the large q asymptotics of Ω( q ) from Z . For simplicitywe will consider here the case q A = 0, but this is easily extended. Performing the Gaussianintegrals over the ϕ A in (4.76) is easy. We get (cid:90) i ∞− i ∞ dϕ · · · dϕ B Z ( u, v, ϕ ) = c u − k e πu χ (cid:88) n ( − d n χ n e πu n ⊥ . (4.90)Here k ≡ b − B with b = dim H ( Σ), B = dim H ( X ), and n ⊥ is the component of n orthogonal to all ϕ = ϕ A D A , i.e. the component of n orthogonal to the pullback of H ( X )in H (Σ). Explicitly n i ⊥ = n i − ( n j D Aj ) D iA , where D A = D AB D B , D AB being the inverse of D AB ≡ D ABC N C , with D ABC defined in (4.16). The constant c is the Gaussian determinantfactor: c = (det D AB ) − / . For the example of the quintic we have k = (5 N + 50 N − / D = 5 N , c = √ N . The sum over n is trivially divergent because adding an arbitrary m = m A D A ∈ H ( X ) to n does not alter n ⊥ . From the point of view of the integral over ϕ A this is due to the redundancy φ A → φ A − im A , n → n + m A D A , which we introducedourselves a little earlier when we extended the integration domain from an interval to the fullimaginary axis. The upshot is that we can take this into account simply by restricting thesum over n ∈ H (Σ , Z ) to a sum over the quotient n ∈ H (Σ , Z ) /H (Σ , Z ).78he q → ∞ asymptotics of Ω will be captured by the u → Z . In this regimeall n (cid:54) = 0 exponential corrections can be dropped. The ϕ integral can then be evaluatedexactly by closing the contour. There is a pole of order k at u = 0 i.e. ϕ = − β <
0. If q (cid:48) ≡ q + χ < q (cid:48) > q , ≈ π | c | ( − d χ ( M ) (2 πq (cid:48) ) k − ( k − ∼ (cid:18) πeq (cid:48) k (cid:19) k . (4.91)The last approximation is valid for k (cid:29)
1. It is obtained as a Stirling approximation, orequivalently as the saddle point approximation of the integral. The saddle point lies at ϕ =2 πq (cid:48) /k . This implies we need at least q (cid:48) (cid:29) k to justify dropping the n (cid:54) = 0 terms, orequivalently q (cid:29) χ ∼ N .This expresses the number of supersymmetric flux configurations as the volume of a 2 k -dimensional shell of radius squared 2 q (cid:48) , which loosely (but not literally because η ij is indefinite)can be thought of as the shell in flux space for which the D0-charge − χ − η ij S i S j equals q and the D2-charge is constrained to be zero. This reproduces the large q asymptoticsfor the number of flux vacua found in various contexts in [36, 113, 114, 7, 107], based on anapproximation in which fluxes were taken to be continuous.Note that χ ( M ) = (cid:82) M e ( R ) is the Witten index for the pure D4 without the flux degreesof freedom. Since this space is topologically CP d , one may expect χ ( M ) = χ ( CP d ) = d + 1 ∼ N /
6. This is not obviously correct because of possible singularities in M , but argumentswere given in [107] that this is nevertheless the correct identification. The sign factor ( − d has a physical meaning: In the large q limit, almost all supersymmetric configurations willbe isolated critical points of the superpotential W S ( z ). Isolated critical points lead to susystates with fermion number equal to their Morse index, which here always equals d becausethe superpotential is holomorphic. This explains the ( − d . The sign ( − d n in the termswith n (cid:54) = 0 further suggests (but does not prove) that these correspond to the contributionsfrom non-isolated configurations, with d − d n residual moduli.It is of course easy to get exponentially large numbers of flux configurations out of theseestimates, even for modest charges. For example for the quintic with, say, N = 5, q =365, we get Ω ∼ , widely considered to be a very large number. (Similar estimates inthe context of mathematically very similar constructions of type IIB string vacua with D7-branes form the basis for suspicions that there exists a staggeringly huge landscape of stringvacua sweeping out a for all practical purposes dense set in parameter space, giving a simple79olution to the cosmological constant problem but obliterating hopes of top-down predictivity[31, 32, 33, 34, 36, 113, 114, 39, 7].)On the other hand note that when 2 πeq (cid:48) < k , the above estimates give exponentially small estimates. Clearly then, in this regime, it must be that the n (cid:54) = 0 terms dominate thedegeneracy. If it is indeed true, as suggested above, that n (cid:54) = 0 contributions can be identifiedwith non-isolated configurations, this means that in this regime, non-isolated configurationsbecome entropically dominant.For q (cid:48) (cid:28) k we could try to build up the spectrum along the lines of section 4.4, and thiscould in turn be used to infer the corrections in the regime q (cid:48) (cid:29) k . However in the intermediateregime q (cid:48) ∼ k , the system is extremely complex. Trying to count degeneracies in this regimemay be like trying to compute the boiling point of water from first principles. In the caseof water we can just measure the boiling point and be done with it. We let nature do thecomputation for us. In the case of D-branes, this would not appear to be an option. However,there is something analogous, provided we enlarge our task to the problem of counting all
D4-D2-D0 bound states, not just those corresponding to abelian flux configurations of the D4.In that case, we can construct the corresponding BPS black hole solutions, and simply readoff their Bekenstein-Hawking entropy S = log Ω( q ). We let gravity do the computation forus! To include all entropically relevant D4-D2-D0 bound states we need in particular considerbound states with localized D0-branes. We turn to this next. So far we have only considered D4-brane and U (1) worldvolume flux degrees of freedom. Asimple but entropically important extension is to include bound states with pointlike D0-branes. For a D4 wrapped on a fixed smooth Σ, the number of such bound states is easy tocompute [115]. We can think of the D0-branes as a gas of noninteracting particles. D0-branesform bound states among themselves of arbitrary D0-charge k (the easiest way to see this isto consider their uplift to M-theory, where they are KK modes along the M-theory circle, k being the KK momentum), so each particle in the D4-D0 gas is characterized by a chargequantum number k >
0, as well as by the supersymmetric 1-particle state | pα (cid:105) it occupies. This happens to be the relevant regime for applications to type IIB string theory vacua [7, 105]. In thiscontext space-localized D4-branes are replaced with space-filling D7-branes, carrying U (1) fluxes inducingD5- and D3-charges. The D3-tadpole cancelation condition dictates that the total D3-charge vanishes, hence q (cid:48) = χ ≈ k and πeq (cid:48) k ≈ πe ≈ .
42, barely above the threshold.
Artist impression of a smooth abelian D4 configuration with U (1) magnetic fluxes and mobileD0-branes bound to it. As we have seen in section 3.5, these 1-particle states are given by the harmonic differentialforms on Σ. The label p = 0 , , , , | pα (cid:105) indicates the fermion number (form degree)and α = 1 , . . . , b p . By the usual Fock space construction, an arbitrary multi-particle statecan be represented by specifying occupation numbers n k,p,α , which take values in Z + if p iseven and in { , } is p is odd. In the absence of fluxes, the total D0-charge of the D4-D0state is Q = − χ + (cid:80) k,p,α k n kpα and the fermion number is F = (cid:80) k,p,α p n kpα . Here we arestill ignoring the moduli degrees of freedom of Σ, considering those frozen for the time being.The generating function for the degeneracies d QF of supersymmetric ground states of totalD0-charge Q and fermion number F is then (cid:88) QF d QF y F t Q = t − χ (cid:88) { n kα } y (cid:80) kpα p n kpα t (cid:80) kpα k n kpα = t − χ (cid:89) kpα (cid:88) n y np t nk = t − χ (cid:89) k (cid:81) p odd (1 + y p t k ) b p (cid:81) p even (1 − y p t k ) b p . (4.92)A generating function for the Witten indices Ω D0 ( Q ) = (cid:80) F ( − F d QF is obtained by setting y = − (cid:88) Q Ω D0 ( Q ) t Q = t − χ (cid:89) k (1 − t k ) − χ = η ( t ) − χ , (4.93)where χ = (cid:80) p ( − p b p is the Euler characteristic of Σ, and η ( t ) is the Dedekind eta-function.If we ignore the fact that there can be nontrivial interplay between the D4 and D0 moduli, Ignoring this is actually not a good idea at small q , but at large q it is a good approximation since mostD4 flux configurations will be isolated anyway in this regime. Z in (4.76) by e − πu χ η ( e − πu ) − χ . Using the modularity property of the eta function, wefurthermore have η ( e − πu ) − χ = u χ/ η ( e − πu ) − χ . (4.94)Combining this with the D4 partition function (4.86), using χ = b + 2, and for notationalsimplicity at the cost of slight formality setting v = − u = − ϕ in (4.86), we get, within ourno D4-D0 interplay approximation: Z D4+D0 = η ( e − πϕ ) − χ (cid:88) n ( − d − d n χ n e πϕ n − πϕ n (4.95)= ϕ e πϕ ( χ − ϕ ) (cid:89) k (1 − e − πkϕ ) − χ (cid:88) n ( − d n χ n e πϕ ( n + inϕ ) . (4.96)Again we see that Z transforms as a modular form under ϕ → /ϕ . Notice in particularthat the large positive and negative weights of the D4 resp. the D0-part have canceled.To extract the large q asymptotics, we retaining only the leading terms at small positive ϕ . After integrating out ϕ :Ω( q ) ≈ (cid:90) dϕ ϕ B +10 e π ˆ q ϕ + πχ ϕ ( − d χ ( M ) . (4.97)where ˆ q ≡ q + D AB q A q B . The saddle point of this integral is at ϕ = (cid:113) χ
24 ˆ q , with valueΩ( q ) ≈ e π (cid:113) ˆ q χ . (4.98)We recall that χ = D ABC N A N B N C + c ,A N A , so this is an explicit formula for the index as afunction of the charges. The logarithm of this expression can therefore directly be comparedto the corresponding black hole entropy.This is reliable when the saddle point value of ϕ is small, i.e. ˆ q (cid:29) χ . When the latter isnot the case, the saddle point value of ϕ is not small, and so, as in the case without D0-branes,even within the model limitations we have made (such as ignoring singular or nonabelian D-branes), the above approximation for Ω( q ) is not necessarily reliable. We will see in the nextsection that the agreement with the black hole entropy is excellent when ˆ q (cid:29) χ . In factfor the leading order matching in this regime, we do not even need to include the fluxes andmoduli degrees of freedom, and a much simpler derivation is possible [93, 109, 115]. Thematch does get better at subleading order when the D4 degrees of freedom are included. Butwhen ˆ q (cid:46) χ , the correct entropy is not anywhere near (4.98). In this regime, the flux andD4-moduli degrees of freedom dominate the entropy.82 .7 Some extensions With the goal of introducing in detail a complex, glassy system that occurs naturally in stringtheory and has a precise geometric description and well-controlled holographic counterparts,we have given a fully explicit construction of the supersymmetric quantum mechanics de-scribing the ground state sector of wrapped D4-branes bound to D0-branes, and discussedexplicit constructions of supersymmetric ground states and their counting in some detail.Some of this was a review of parts of [107] and the ideas and results used there, with somesimplifications and some points worked out more explicitly. Our discussion was necessarilyincomplete in scope. Rather than try to give an overview of the huge related literature, letme mention a few immediate extensions that could have directly followed this part, with asmall, non-representative, biased sample of possible starting points for further reading:1. The lift to M-theory as an M5 wrapped on S × Σ, reducing to a (0 ,
4) CFT. This is insome ways a more natural framework as it unifies D4 and D0 degrees of freedom andallows for a holographic dual description beyond the ground state sector [109, 116, 117].2. Quiver quantum mechanics: these are 1d gauged linear sigma models, providing a simplebut very rich class of models describing complex D-brane systems. The field content isrepresented by quiver diagrams, nodes being partonic branes and arrows open stringsbetween them [118]. Quantization allows explicit interpolation between the weakly cou-pled geometric regime and the strongly coupled black hole bound state regimes [119](see also below).3. A more refined description of D4-D0 bound states that takes into account the interplaybetween D4 and D0 moduli, nonabelian degrees of freedom, and so on. One conceptuallysimple but efficient approach is the brane-anti-brane tachyon condensation picture [120,121, 122, 123, 124, 107, 125].4. Counting of ground states away from the regime ˆ q (cid:29) χ . Relation to Gromov-Witten,Gopakumar-Vafa and Donaldson-Thomas invariants and the OSV conjecture [104, 126,127, 128, 129, 130, 107, 131].5. Wall crossing phenomena: e.g. the possibility for a D4 to split into a D6 and an anti-D6away from the strict large volume limit [132, 121, 133, 119, 107, 134, 135].6. Applications to global aspects of IIB model building [125, 136]83t would certainly be useful to have a fully explicit example in which in particular the periodvectors can be computed exactly. This may be possible along the lines of [137].We end with some references to key papers in the history of the subject. The idea thatbranes can be wrapped on nontrivial compact cycles to obtain charged particles in four dimen-sions appeared first in [138] (and shown to be necessary for the consistency of string theory in[139]). The discovery of the perturbative string description of D-branes [140] made it possibleto quantize and count states at weak coupling, leading to the computation of the Bekenstein-Hawking entropy in [93]. Many of the ideas in the more systematic development of the physicsand mathematics of wrapped D-branes in Calabi-Yau manifolds, including stability and wallcrossing, originated in [121, 141], with precursors in [142, 132]. In the appropriate regime, D-branes wrapped on compact cycles manifest themselves as blackholes in the low energy effective field theory. It turns out that an intricate zoo of stationarysupersymmetric bound states of such black holes exists, like giant molecules, all of whichcollapse to a D-brane localized at a single point in space in the limit g s → N = 2 supergravity The four dimensional effective theory of type IIA string theory compactified on a Calabi-Yaumanifold X is N = 2 supergravity coupled to B = b ( X ) abelian N = 2 vector multiplets.There are B + 1 gauge fields A Λ , Λ = 0 , , · · · , B , obtained as in (4.13) by KK reduction ofRR potentials: C (1) = A , C (3) = A A ∧ D A . The B vector multiplets furthermore each containa complex scalar, obtained by KK reduction of the complexified K¨ahler form: B + iJ = t A D A .They also contain spin 1/2 fermions, but we don’t need those. Finally, the theory containsmassless hypermultiplets too, but they do not affect the solutions of interest to us and can be84onsistently put to constant values. We will work in this section in units with the 4d Newtonconstant G N ≡ L of magnetic-electric charges Γ carries a fundamen-tal symplectic product, which in a symplectic basis has the canonical form (cid:104) Γ , ˜Γ (cid:105) ≡ Γ Λ ˜Γ Λ − Γ Λ ˜Γ Λ . (5.1)Upper indices denote magnetic and lower indices electric components. In the IIA case athand, the electric charges are the D0 and D2 charges Q and Q A , and the dual magneticcharges are the D6 and D4 charges N and N A . In terms of these, the symplectic product is (cid:104) ( N, Q ) , ( ˜ N , ˜ Q ) (cid:105) = N ˜ Q + N A ˜ Q A − Q ˜ N − Q A ˜ N A . Integrality of this product is equivalentto Dirac quantization.In the weak string coupling regime (4.48), the wrapped D-branes are well described as pointparticles moving in flat R , interacting with each other primarily through the lightest stretchedopen string modes. The coupling g ≡ g s / √ V X ∝ (cid:96) p /(cid:96) s is in a hypermultiplet and cantherefore be tuned at will. When it gets larger, excited open string modes become important,until eventually the interactions are better described by massless closed string exchange, i.e.graviton, photon and scalar exchange. This is the regime in which the supergravity descriptionbecomes valid. When the charges are large, the wrapped D-brane states manifest themselvesas weakly curved black hole solutions.A single centered BPS (i.e. supersymmetric) solution to the equations of motion is neces-sarily static and spherically symmetric, with a metric of the form ds = − e U dt + e − U d(cid:126)x ,and all fields functions of r = | (cid:126)x | only. The BPS equations of motion take the first ordergradient flow form typical for supersymmetric solutions [143, 144]:˙ U = − e U | Z | , (5.2)˙ t A = − e U g A ¯ B ∂ ¯ B | Z | , (5.3)where g A ¯ B is the metric on the vector multiplet moduli space, the dot denotes derivation withrespect to τ ≡ /r , and Z (Γ , t ) is the central charge of the magnetic-electric charge Γ in a Usually these are denoted by P and P A , but to be consistent with the previous sections we use N and N A instead. t A . It is a complex function on the vector multiplet moduli space,holomorphic up to a normalization factor, and linear in the charge vector Γ: Z (Γ; t ) = 1 |(cid:104) V, ¯ V (cid:105)| / (cid:104) Γ , V (cid:105) , (5.4)where V Λ = X Λ , V Λ = ∂F∂X Λ , X A = t A X , A = 1 , . . . , B . (5.5)Here F ( X ) is the prepotential of the N = 2 theory, which determines all couplings and metricsin the 4d action. In general it is a locally defined holomorphic function, homogeneous of degree2. For our type IIA theory, it takes the form F ( X ) = − D ABC X A X B X C + · · · (5.6)where D ABC was defined in (4.16) and the ellipsis denote string worldsheet instanton cor-rections, which are exponentially suppressed in the K¨ahler moduli J A = Im t A and thereforenegligible when the Calabi-Yau is large. Dropping those, (5.4) boils down to Z = 1 (cid:16) D ABC J A J B J C (cid:17) / (cid:18) N D ABC t A t B t C − N A D ABC t B t C + Q A t A + Q (cid:19) . (5.7)The central charge gets its name because it appears as a charge commuting with everythingin the N = 2 supersymmetry algebra. Its absolute value equals the lowest mass a particle ofcharge Γ can possibly have in a background specified by t . This bound is saturated for BPSstates. Its phase determines the supercharges preserved by the BPS state. Two BPS objectsare mutually supersymmetric if their central charge phases line up.The gradient flow equations (5.2) drive | Z (Γ , t ) | to its minimal value | Z (cid:63) | . If this minimalvalue is zero at a nonsingular point in moduli space, no solution exists. If it is nonzero, weget a black hole with near horizon solution t = const . = t (cid:63) , e − U ( r ) = | Z (cid:63) | r , as can be checkeddirectly from the above BPS flow equations. This describes AdS × S with S horizon area A = 4 π | Z (cid:63) | and therefore Bekenstein-Hawking entropy S (Γ) = π | Z (cid:63) (Γ) | . (5.8) Although all regular critical points of | Z | are isolated local minima [145], in the presence of singularitiesat finite distance in the moduli space, there can be multiple basins of attraction, but we will ignore this here,as it does not occur in the large volume approximation. Warp factor e U for some three centered D6 − D6 − D4 bound state, for a choice of charges thatdoesn’t really matter because this figure is primarily ornamental.
When the D6-charge N is zero, the minimization of | Z | is straightforward and there is asimple closed form expression for S (Γ) (still in the large volume approximation) [146]: S ( N, Q ) | N =0 = 2 π (cid:115) (cid:98) Q D ABC N A N B N C , (cid:98) Q = Q + 12 D AB Q A Q B , (5.9)where D AB = D ABC N C and D AB is its inverse. Note that to leading order in the large chargelimit, this is exactly the same as the microscopic result (4.97)! The subleading term can bereproduced macroscopically as well, from the Wald entropy in the presence of an R term[147]. When N is not zero there is no general closed form solution for S (Γ), except when Q A is chosen to be proportional to D ABC N B N C . In that case we define for some fixed K A > n , n, q, q ) by ( N , N A , Q A , Q ) = ( n , n K A , q ( K ) A , q K ), where K is a short for D ABC K A K B K C and ( K ) A for D ABC K B K C . Then we can write S = π K (cid:113) n q − n q + 6 n q − n n q q − n q . (5.10)As we will see below, once the entropy function S (Γ) is known, it is easy to write down fullyexplicit expressions for all fields at all points in space.Quite remarkably, these theories also have multicentered, supersymmetric, stationary blackhole bound states , like giant molecules. These are genuine bound states in the sense that87he centers are constrained by nontrivial potentials, generated by scalar, electomagnetic andgravitational forces [133, 148, 149, 150, 117, 107]. They have a metric of the form ds = − e U ( dt + ω i dx i ) + e − U d(cid:126)x , (5.11)where U and ω depend on (cid:126)x , and they are fully characterized by harmonic functions H Λ , H Λ with sources at the positions (cid:126)x i of the charges Γ i : H = (cid:88) i Γ i | (cid:126)x − (cid:126)x i | + h , (5.12)where the constant term h is determined by the total charge Γ and the asymptotic moduli: h = − (cid:0) e − iα ˜ V (cid:1) | r = ∞ , α = arg Z (Γ) , ˜ V = V / |(cid:104) V, ¯ V (cid:105)| / , (5.13)and V is as in (5.5).As mentioned earlier, once the entropy function S (Γ) is known on charge space, the com-plete solution is known [150], simply by substituting H ( (cid:126)x ) for Γ in S and its derivatives: e − U = 1 π S ( H ) , A Λ = 1 π ∂ log S ( H ) ∂H Λ ( dt + ω ) + A Λmon , t A = H A − iπ ∂S∂H A H − iπ ∂S∂H . (5.14)The one-form A Λmon is the vector potential for a system of Dirac magnetic monopoles of charge N Λ i located at the positions (cid:126)x i . The off-diagonal components ω of the metric are the solutionsto ∇ × ω = (cid:104)∇ H, H (cid:105) , (5.15)where ∇ is the flat space gradient. This equation implies an important integrability condition: ∇ · ( ∇ × ω ) = 0 ⇒ (cid:104)∇ H, H (cid:105) = 0, from which, using ∇ | (cid:126)x | = − πδ ( x ), we get for everycenter i a condition: (cid:88) j (cid:104) Γ i , Γ j (cid:105)| (cid:126)x i − (cid:126)x j | = −(cid:104) Γ i , h (cid:105) = 2 Im (cid:0) e − iα Z (Γ i , t ) (cid:1)(cid:12)(cid:12) r = ∞ . (5.16)This imposes constraints on the positions (cid:126)x i ; solutions are the BPS equilibrium positions ofthe black holes in their mutual force fields. For the 2-centered case we have in particular | (cid:126)x − (cid:126)x | = (cid:104) Γ , Γ (cid:105) | Z + Z | Im( Z ¯ Z ) , (5.17)where Z i = Z (Γ i , t ) | r = ∞ . When the right hand side goes from positive to negative through awall where Z and Z line up, the equilibrium separation diverges and the bound state decays.This is the supergravity incarnation of the wall crossing phenomenon.88igure 16: (from [107]) Attractor flow tree in the t -plane for a particular bound state of 7 flux-carryingD6-branes and 7 flux-carrying anti-D6-branes. The flow starts at the yellow dot at the top and goes down.The iterated structure can be thought of as representing clutering of individual centers. In the case at hand allattractor points lie on the boundary of moduli space, which is due the special choice of charges. The symmetricform is also special to our particular choice. The main motivation to pick those charges was esthetics anddramatic effect. More generic examples can be found in [107]. For completeness we also give the solution to (5.15) [150]. For a 2-centered bound statewith centers at (0 , , L ) and (0 , , − L ), up to residual diffeomorphism gauge transformations t → t + f ( (cid:126)x ), ω → ω − df : ω = (cid:104) Γ , Γ (cid:105) L (cid:18) L − r ( L + r − L r cos 2 θ ) / + 1 − cos θ + cos θ (cid:19) dφ . (5.18)Here ( r, φ, θ ) are standard spherical coordinates centered at the origin and θ , θ are anglesof spherical coordinates centered at the two particle positions. The integrability condition isequivalent to the absence of physical singularities along the z -axis. Because (5.15) is linear,the solution for more centers is obtained by superposition.The solutions are generically stationary but not static: they have intrinsic angular mo-mentum, given by (cid:126)J = 12 (cid:88) i
I would like to thank the organizers of TASI 2010, Tom Banks, Michael Dine and SubirSachdev, for giving me the opportunity to teach on these topics. I am very much indebtedto Dionysios Anninos, Tarek Anous, Jacob Barandes, Marcus Benna, Hyeyoun Chung, MikeDouglas, Bram Gaasbeek, Hajar Ebrahim, Greg Moore and Andy Strominger, whose insights,collaborations and discussions were crucial in the genesis of these notes. Many of the ideas mo-tivating the theme of these lectures were developed over the past year together with DionysiosAnninos, and we would like to thank Charlies Kitchen for hospitality. I’m grateful to A.P.Young and Subir Sachdev for clarifying conversations about spin glasses. Finally, specialthanks to Dionysios Anninos, Tarek Anous, Jacob Barandes and Marcus Benna for a carefulreading of parts of these notes and their useful suggestions for improvements. This work wassupported in part by DOE grant DE-FG02-91ER40654.92 eferences [1] M. Lynker, Landau-Ginzburg vacua of string, M- and F-theory at c=12,
Nuclear PhysicsB . (1-2), 123–150 (June, 1999). ISSN 05503213. doi: 10.1016/S0550-3213(99)00204-7. URL http://arxiv.org/abs/hep-th/9812195 .[2] M. Kreuzer. Calabi-Yau data. URL http://tph16.tuwien.ac.at/%7ekreuzer/CY .[3] J. Barandes, H. Chung, F. Denef, H. Ebrahim, and P. Petrov, Tunneling transitionsbetween black hole bound states, to appear .[4] D. Anninos and F. Denef, Replica order parameter for de Sitter dynamics, to appear .[5] D. Anninos, T. Anous, J. Barandes, and F. Denef, String Glasses, to appear .[6] A. Goose. Prerequisites. URL http://abstrusegoose.com/272 .[7] F. Denef, Les Houches lectures on constructing string vacua, arXiv 0803.1194 . (2008).URL http://arxiv.org/pdf/0803.1194 .[8] T. Banks, TASI Lectures on Holographic Space-Time, SUSY and Gravitational Effec-tive Field Theory, arXiv:1007.4001 (July. 2010). URL http://arxiv.org/abs/arXiv:1007.4001 .[9] A. Starobinsky, Dynamics of phase transition in the new inflationary universe scenarioand generation of perturbations, Physics Letters B . (3-4), 175–178 (Nov., 1982).ISSN 03702693. doi: 10.1016/0370-2693(82)90541-X. URL http://dx.doi.org/10.1016/0370-2693(82)90541-X .[10] A. Linde, Chaotic inflation, Physics Letters B . (3-4), 177–181 (Sept., 1983). ISSN03702693. doi: 10.1016/0370-2693(83)90837-7. URL http://dx.doi.org/10.1016/0370-2693(83)90837-7 .[11] A. Linde, D. Linde, and A. Mezhlumian, From the Big Bang theory to the theory of astationary universe, Physical Review D . (4), 1783, (1994). URL http://arxiv.org/abs/gr-qc/9306035 .[12] S. Winitzki, Eternal fractal in the universe, Physical Review D . (8), 083506, (2002).URL http://prd.aps.org/abstract/PRD/v65/i8/e083506 .9313] A. Strominger, The dS/CFT correspondence, Journal of High Energy Physics . (10), 034–034 (Oct., 2001). ISSN 1029-8479. doi: 10.1088/1126-6708/2001/10/034.URL http://arxiv.org/abs/hep-th/0106113 .[14] A. Strominger, Inflation and the dS/CFT Correspondence, Journal of High EnergyPhysics . (11), 049–049 (Nov., 2001). ISSN 1029-8479. doi: 10.1088/1126-6708/2001/11/049. URL http://arxiv.org/abs/hep-th/0110087 .[15] M. Spradlin and A. Strominger, Les Houches lectures on de Sitter space, Arxiv preprinthep-th/0110007 (Sept. 2001). URL http://arxiv.org/abs/hep-th/0110007 .[16] E. Witten, Quantum Gravity In De Sitter Space, arXiv: hep-th/0106109 (June. 2001).URL http://arxiv.org/abs/hep-th/0106109 .[17] J. Maldacena, Non-gaussian features of primordial fluctuations in single field inflationarymodels,
Journal of High Energy Physics . (05), 013–013 (May, 2003). ISSN 1029-8479. doi: 10.1088/1126-6708/2003/05/013. URL http://arxiv.org/abs/astro-ph/0210603 .[18] D. Anninos and G. Ng, Asymptotic Symmetries and Charges in De Sitter Space, Arxivpreprint arXiv:1009.4730 (Sept. 2010). URL http://arxiv.org/abs/1009.4730 .[19] R. Bousso, B. Freivogel, and S. Leichenauer, Geometric origin of coincidences and hi-erarchies in the landscape,
Arxiv preprint arXiv:1012.2869 (Dec. 2010). URL http://arxiv.org/abs/1012.2869 .[20] R. Bousso, B. Freivogel, S. Leichenauer, and V. Rosenhaus, Boundary definition of amultiverse measure,
Physical Review D . (12), 39 (Dec., 2010). ISSN 1550-7998. doi:10.1103/PhysRevD.82.125032. URL http://arxiv.org/abs/1005.2783 .[21] R. Bousso, B. Freivogel, Y. Sekino, S. Shenker, L. Susskind, I.-S. Yang, and C.-P. Yeh,Future foam: Nontrivial topology from bubble collisions in eternal inflation, PhysicalReview D . (6), 23 (Sept., 2008). ISSN 1550-7998. doi: 10.1103/PhysRevD.78.063538.URL http://arxiv.org/abs/0807.1947 .[22] R. Bousso, The cosmological constant, General Relativity and Gravitation . (2-3),607–637 (Dec., 2007). ISSN 0001-7701. doi: 10.1007/s10714-007-0557-5. URL http://arxiv.org/abs/0708.4231 . 9423] J. Garriga and A. Vilenkin, Holographic multiverse, Journal of Cosmology and Astropar-ticle Physics . , 021, (2009). URL http://arxiv.org/abs/0809.4257 .[24] A. Vilenkin, Holographic multiverse and the measure problem, Arxiv preprintarXiv:1103.1132 (Mar. 2011). URL http://arxiv.org/abs/1103.1132 .[25] Wikipedia. Multiverse. URL http://en.wikipedia.org/wiki/Multiverse .[26] H. Everett, ”Relative State” Formulation of Quantum Mechanics,
Reviews of ModernPhysics . (3), 454–462 (July, 1957). ISSN 0034-6861. doi: 10.1103/RevModPhys.29.454. URL http://rmp.aps.org/abstract/RMP/v29/i3/p454_1 .[27] S. Giddings, Axion-induced topology change in quantum gravity and string theory, Nuclear Physics B . (4), 890–907 (Sept., 1988). ISSN 05503213. doi: 10.1016/0550-3213(88)90446-4. URL http://dx.doi.org/10.1016/0550-3213(88)90446-4 .[28] S. Coleman, Black holes as red herrings: topological fluctuations and the loss of quantumcoherence, Nuclear Physics B . (4), 867–882, (1988). ISSN 0550-3213. URL http://linkinghub.elsevier.com/retrieve/pii/0550321388901101 .[29] N. Arkani-Hamed, J. Orgera, and J. Polchinski, Euclidean wormholes in string theory, Journal of High Energy Physics . , 018, (2007). URL http://iopscience.iop.org/1126-6708/2007/12/018 .[30] S. Weinberg, Anthropic Bound on the Cosmological Constant, Physical Review Letters . (22), 2607–2610 (Nov., 1987). ISSN 0031-9007. doi: 10.1103/PhysRevLett.59.2607.URL .[31] A. Strominger, Superstrings with torsion, Nuclear Physics B . (2), 253–284 (Sept.,1986). ISSN 05503213. doi: 10.1016/0550-3213(86)90286-5. URL http://dx.doi.org/10.1016/0550-3213(86)90286-5 .[32] R. Bousso and J. Polchinski, Quantization of Four-form Fluxes and Dynamical Neu-tralization of the Cosmological Constant, JHEP . , 0006, (2000). URL http://arxiv.org/abs/hep-th/0004134 .[33] S. Kachru, R. Kallosh, A. Linde, and S. Trivedi, de Sitter vacua in string theory, PhysicalReview D . (4) (Aug., 2003). ISSN 0556-2821. doi: 10.1103/PhysRevD.68.046005. URL http://arxiv.org/abs/hep-th/0301240 .9534] L. Susskind, The anthropic landscape of string theory, Universe or multiverse . pp. 247–66 (Feb., 2007). URL http://arxiv.org/abs/hep-th/0302219 .[35] A. N. Schellekens, The Landscape ”avant la lettre”,
ArXiv: physics/0604134 (Apr.2006). URL http://arxiv.org/abs/physics/0604134 .[36] M. R. Douglas, The statistics of string/M theory vacua,
Journal of High Energy Physics . (05), 046–046 (May, 2003). ISSN 1029-8479. doi: 10.1088/1126-6708/2003/05/046.URL http://arxiv.org/abs/hep-th/0303194 .[37] M. Grana, Flux compactifications in string theory: A comprehensive review, PhysicsReports . (3), 91–158 (Jan., 2006). ISSN 03701573. doi: 10.1016/j.physrep.2005.10.008. URL http://arxiv.org/abs/hep-th/0509003 .[38] B. S. Acharya and M. R. Douglas, A Finite Landscape?, hep-th/0606212 (June. 2006).URL http://arxiv.org/abs/hep-th/0606212 .[39] M. Douglas and S. Kachru, Flux compactification, Reviews of Modern Physics . (2),733–796 (May, 2007). ISSN 0034-6861. doi: 10.1103/RevModPhys.79.733. URL http://rmp.aps.org/abstract/RMP/v79/i2/p733_1 .[40] F. Denef and M. R. Douglas, Computational complexity of the landscape: Part I, Annalsof Physics . (5), 1096–1142 (May, 2007). ISSN 00034916. doi: 10.1016/j.aop.2006.07.013. URL http://arxiv.org/abs/hep-th/0602072 .[41] M. R. Douglas, Effective potential and warp factor dynamics, Journal of High EnergyPhysics . (3), 1–32–32 (Mar., 2010). ISSN 1029-8479. doi: 10.1007/JHEP03(2010)071. URL http://arxiv.org/abs/0911.3378 .[42] A. Nabutovsky and S. Weinberger, The fractal nature of Riem/Diff I, Geometriae Ded-icata . (1), 1–54, (2003). ISSN 0046-5755. URL .[43] M. M´ezard, G. Parisi, and M. A. Virasoro, Spin Glass Theory and Beyond . vol. 9,
Lecture Notes in Physics , (World Scientific, 1987). ISBN 9971501155. URL http://books.google.com/books?id=ZIF9QgAACAAJ .9644] K. Binder and A. Young, Spin glasses: Experimental facts, theoretical concepts, andopen questions,
Reviews of Modern physics . (4), 801–976, (1986). ISSN 1539-0756.URL http://link.aps.org/doi/10.1103/RevModPhys.58.801 .[45] K. H. Fisher and J. A. Hertz, Spin Glasses, American Journal of Physics . (5), 478,(1992). ISSN 00029505. doi: 10.1119/1.16856. URL http://books.google.com/books?id=zXqel8dS-rIC .[46] P. Cartier, J. B. Bost, H. Cohen, D. Zagier, R. Gergondey, H. M. Stark, E. Reyssat,F. Beukers, G. Christol, M. Senechal, A. Katz, J. Bellissard, P. Cvitanovic, and J. C.Yoccoz, From Number Theory to Physics . (Springer, 1992). ISBN 3540533427. URL http://books.google.com/books?id=UUFIQgAACAAJ .[47] L. Brekke, P. Freund, M. Olson, and E. Witten, Non-archimedean string dynamics,
Nuclear Physics B . (3), 365–402 (June, 1988). ISSN 0550-3213. doi: 10.1016/0550-3213(88)90207-6. URL http://dx.doi.org/10.1016/0550-3213(88)90207-6 .[48] S. Roch, Toward extracting all phylogenetic information from matrices of evolutionarydistances., Science (New York, N.Y.) . (5971), 1376–9 (Mar., 2010). ISSN 1095-9203. doi: 10.1126/science.1182300. URL .[49] D. Sherrington and S. Kirkpatrick, Solvable Model of a Spin-Glass, Physical ReviewLetters . (26), 1792–1796 (Dec., 1975). ISSN 0031-9007. doi: 10.1103/PhysRevLett.35.1792. URL http://prl.aps.org/abstract/PRL/v35/i26/p1792_1 .[50] G. Parisi, Infinite number of order parameters for spin-glasses, Physical Review Letters . (23), 1754–1756, (1979). ISSN 1079-7114. URL http://link.aps.org/doi/10.1103/PhysRevLett.43.1754 .[51] G. Parisi, A sequence of approximated solutions to the SK model for spin glasses, Journalof Physics A: Mathematical and General . , L115, (1980). URL http://iopscience.iop.org/0305-4470/13/4/009 .[52] M. M´ezard, G. Parisi, N. Sourlas, G. Toulouse, and M. Virasoro, Nature of the spin-glass phase, Physical Review Letters . (13), 1156–1159, (1984). ISSN 1079-7114. URL http://link.aps.org/doi/10.1103/PhysRevLett.52.1156 .9753] E. Marinari, Optimized monte carlo methods, Advances in Computer Simulation . pp.50–81, (1998). URL http://dx.doi.org/10.1007/BFb0105459 .[54] D. J. Earl and M. W. Deem, Parallel tempering: Theory, applications, and new perspec-tives,
Physical Chemistry Chemical Physics . (23), 3910 (Sept., 2005). ISSN 1463-9076.doi: 10.1039/b509983h. URL http://pubs.rsc.org/en/Content/ArticleHTML/2005/CP/B509983H .[55] D. Fisher and D. Huse, Ordered phase of short-range Ising spin-glasses, Physical ReviewLetters . (15), 1601–1604, (1986). ISSN 1079-7114. URL http://link.aps.org/doi/10.1103/PhysRevLett.56.1601 .[56] H. G. Katzgraber and A. K. Hartmann, Ultrametricity and clustering of States in spinglasses: a one-dimensional view., Physical review letters . (3), 037207 (Jan., 2009).ISSN 0031-9007. URL http://dx.doi.org/10.1103/PhysRevLett.102.037207 .[57] R. Rammal, G. Toulouse, and M. Virasoro, Ultrametricity for physicists, Reviews ofModern Physics . (3), 765–788, (1986). ISSN 1539-0756. URL http://link.aps.org/doi/10.1103/RevModPhys.58.765 .[58] G. Parisi and F. Ricci-Tersenghi, On the origin of ultrametricity, Journal of PhysicsA: Mathematical and General . , 113, (2000). URL http://iopscience.iop.org/0305-4470/33/1/307 .[59] C. De Dominicis and I. Giardina, Random fields and spin glasses: a field theory approach .(Cambridge Univ Press, 2006). ISBN 0521847834. URL http://books.google.com/books?id=58iM6EFKx9cC .[60] T. Castellani and A. Cavagna, Spin-glass theory for pedestrians,
Journal of StatisticalMechanics: Theory and Experiment . , P05012, (2005). URL http://arxiv.org/abs/cond-mat/0505032 .[61] A. Crisanti and H. J. Sommers, The spherical p-spin interaction spin glass model: thestatics, Zeitschrift fur Physik B Condensed Matter . (3), 341–354 (Oct., 1992). ISSN0722-3277. doi: 10.1007/BF01309287. URL . 9862] S. Edwards and P. Anderson, Theory of spin glasses, Journal of Physics F: MetalPhysics . , 965, (1975). URL http://iopscience.iop.org/0305-4608/5/5/017 .[63] F. Guerra, Broken replica symmetry bounds in the mean field spin glass model, Com-munications in Mathematical Physics . (1), 1–12, (2003). ISSN 0010-3616. URL http://dx.doi.org/10.1007/s00220-002-0773-5 .[64] M. Talagrand, The Parisi formula, Annals of Mathematics-Second Series . (1), 221–264, (2006). ISSN 0003-486X. URL http://annals.princeton.edu/annals/2006/163-1/annals-v163-n1-p04-s.pdf .[65] G. Parisi, Order parameter for spin-glasses, Physical Review Letters . (24), 1946–1948,(1983). ISSN 1079-7114. URL http://link.aps.org/doi/10.1103/PhysRevLett.50.1946 .[66] A. P. Young, Ed., Spin glasses and random fields . (World Scientific Pub Co Inc, 1998).ISBN 9810232403. URL http://books.google.com/books?id=9hsCr-i6b5wC .[67] M. M´ezard and M. Virasoro, The microstructure of ultrametricity,
Journal de Physique . (8), 1293–1307, (1985). ISSN 0302-0738. URL http://dx.doi.org/10.1051/jphys:019850046080129300 .[68] A. Crisanti and T. Rizzo, Analysis of the-replica symmetry breaking solution of theSherrington-Kirkpatrick model, Physical Review E . (4), 46137, (2002). ISSN 1550-2376. URL http://link.aps.org/doi/10.1103/PhysRevE.65.046137 .[69] D. J. Thouless, P. W. Anderson, and R. G. Palmer, Solution of ’Solvable model ofa spin glass’, Philosophical Magazine . (3), 593–601 (Mar., 1977). ISSN 1478-6435.doi: 10.1080/14786437708235992. URL http://adsabs.harvard.edu/abs/1977PMag...35..593T .[70] G. Parisi and N. Sourlas, Random Magnetic Fields, Supersymmetry, and Negative Di-mensions, Physical Review Letters . (11), 744–745, (1979). ISSN 1079-7114. URL http://link.aps.org/doi/10.1103/PhysRevLett.43.744 .[71] J. Kurchan, Supersymmetry in spin glass dynamics, Journal de Physique I . (7), 1333–1352, (1992). URL http://hal.archives-ouvertes.fr/jpa-00246625/ .9972] J. Kurchan, Supersymmetry, replica and dynamic treatments of disordered systems: aparallel presentation, Arxiv preprint cond-mat/0209399 . pp. 1–19, (2002). URL http://arxiv.org/pdf/cond-mat/0209399 .[73] F. Barahona, On the computational complexity of Ising spin glass models,
Journal ofPhysics A: Mathematical and General . , 3241, (1982). URL http://iopscience.iop.org/0305-4470/15/10/028 .[74] I. Kondor, Parisi’s mean-field solution for spin glasses as an analytic continuation inthe replica number, Journal of Physics A: Mathematical and General . , L127, (1983).URL http://iopscience.iop.org/0305-4470/16/4/006 .[75] J. Verbaarschot and M. Zirnbauer, Critique of the replica trick, Journal of PhysicsA: Mathematical and General . , 1093, (1985). URL http://iopscience.iop.org/0305-4470/18/7/018 .[76] J. Almeida and D. Thouless, Stability of the Sherrington-Kirkpatrick solution of a spinglass model, Journal of Physics A: Mathematical and General . , 983, (1978). URL http://iopscience.iop.org/0305-4470/11/5/028 .[77] V. Avetisov, A. Bikulov, and S. Kozyrev, Application of p-adic analysis to models ofbreaking of replica symmetry, Journal of Physics A: Mathematical and General . (50),8785, (1999). ISSN 0305-4470. doi: http://dx.doi.org/10.1088/0305-4470/32/50/301.URL http://iopscience.iop.org/0305-4470/32/50/301 .[78] G. Parisi and N. Sourlas, P-adic numbers and replica symmetry breaking, The Eu-ropean Physical Journal B . (3), 535–542 (Mar., 2000). ISSN 1434-6028. doi:10.1007/s100510051063. URL http://dx.doi.org/10.1007/s100510051063 .[79] A. Crisanti and C. De Dominicis, Stability of the Parisi Solution for the Sherrington-Kirkpatrick model near T= 0, Arxiv preprint arXiv:1101.5233 . (2011). URL http://arxiv.org/abs/1101.5233 .[80] B. Derrida, Random-energy model: Limit of a family of disordered models,
PhysicalReview Letters . (2), 79–82, (1980). ISSN 1079-7114. URL http://link.aps.org/doi/10.1103/PhysRevLett.45.79 . 10081] D. Gross and M. M´ezard, The simplest spin glass, Nuclear Physics B . (4), 431–452,(1984). ISSN 0550-3213. URL http://dx.doi.org/10.1209/0295-5075/20/3/002 .[82] A. Bray and M. Moore, Replica theory of quantum spin glasses, Journal of Physics C:Solid State Physics . , L655, (1980). URL http://iopscience.iop.org/0022-3719/13/24/005 .[83] J. Ye, S. Sachdev, and N. Read, Solvable spin glass of quantum rotors, Physical reviewletters . (25), 4011–4014, (1993). ISSN 1079-7114. URL http://link.aps.org/doi/10.1103/PhysRevLett.70.4011 .[84] L. Cugliandolo, D. Grempel, and C. da Silva Santos, Imaginary-time replica formalismstudy of a quantum spherical p-spin-glass model, Physical Review B . (1), 14403,(2001). ISSN 1550-235X. URL http://link.aps.org/doi/10.1103/PhysRevB.64.014403 .[85] J. Hopfield, Neural networks and physical systems with emergent collective computa-tional abilities Biophysics, Proceedings of the National Academy of Sciences of the UnitedStates of America . (8), 2554, (1982). URL .[86] D. Amit, H. Gutfreund, and H. Sompolinsky, Spin-glass models of neural networks, Physical Review A . (2), 1007–1018, (1985). ISSN 1094-1622. URL http://link.aps.org/doi/10.1103/PhysRevA.32.1007 .[87] G. Parisi, A memory which forgets, Journal of Physics A: Mathematical and General . (M), L617, (1986). URL http://iopscience.iop.org/0305-4470/19/10/011 .[88] S. Kirkpatrick, Optimization by simulated annealing: Quantitative studies, Journal ofStatistical Physics . (5-6), 975–986 (Mar., 1984). ISSN 0022-4715. doi: 10.1007/BF01009452. URL http://dx.doi.org/10.1007/BF01009452 .[89] M. M´ezard, T. Mora, and R. Zecchina, Clustering of Solutions in the Random Satis-fiability Problem, Physical Review Letters . (19) (May, 2005). ISSN 0031-9007. doi:10.1103/PhysRevLett.94.197205. URL http://prl.aps.org/abstract/PRL/v94/i19/e197205 . 10190] M. M´ezard and R. Zecchina, Random K-satisfiability problem: From an analytic so-lution to an efficient algorithm, Physical Review E . (5) (Nov., 2002). ISSN 1063-651X. doi: 10.1103/PhysRevE.66.056126. URL http://pre.aps.org/abstract/PRE/v66/i5/e056126 .[91] A. Renyi. On measures of entropy and information. In Fourth Berkeley Symposiumon Mathematical Statistics and Probability , vol. 547, pp. 547–561, (1961). URL http://l.academicdirect.org/Horticulture/GAs/Refs/Renyi_1961.pdf .[92] E. Witten, Constraints on supersymmetry breaking,
Nuclear Physics B . (2), 253–316 (July, 1982). ISSN 05503213. doi: 10.1016/0550-3213(82)90071-2. URL http://dx.doi.org/10.1016/0550-3213(82)90071-2 .[93] A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Physics Letters B . (1-4), 99–104 (June, 1996). ISSN 03702693. doi: 10.1016/0370-2693(96)00345-0. URL http://arxiv.org/abs/hep-th/9601029 .[94] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, andE. Zaslow, Mirror symmetry . (Clay mathematics monographs, 2003). ISBN 0821829556.URL http://books.google.com/books?id=uGYRaAhFGx0C&pgis=1 .[95] F. Cooper, A. Khare, and U. Sukhatme, Supersymmetry and quantum mechanics,
Physics Reports . (5-6), 267–385 (Jan., 1995). ISSN 03701573. doi: 10.1016/0370-1573(94)00080-M. URL http://dx.doi.org/10.1016/0370-1573(94)00080-M .[96] L. Alvarez-Gaume, Supersymmetry and the Atiyah-Singer index theorem, Communi-cations in Mathematical Physics . (2), 161–173 (June, 1983). ISSN 0010-3616. doi:10.1007/BF01205500. URL http://dx.doi.org/10.1007/BF01205500 .[97] S. Cecotti and L. Girardello, Stochastic and Parastochastic Aspects of SupersymmetricFunctional Measures : A New Approach to Supersymmetry, Annals of Physics . ,81–99, (1983). URL http://dx.doi.org/10.1016/0003-4916(83)90172-0 .[98] P. Damgaard and H. Huffel, Stochastic quantization, Physics Reports . (5-6), 227–398 (Aug., 1987). ISSN 03701573. doi: 10.1016/0370-1573(87)90144-X. URL http://dx.doi.org/10.1016/0370-1573(87)90144-X .10299] S. Franz and J. Kurchan, On the Computation of Static Expectation Values from Dy-namics in Spin Glasses, Europhysics Letters (EPL) . (3), 197–203 (Oct., 1992). ISSN0295-5075. doi: 10.1209/0295-5075/20/3/002. URL http://dx.doi.org/10.1209/0295-5075/20/3/002 .[100] S. Cecotti and C. Vafa, 2d Wall-Crossing, R-twisting, and a Supersymmetric Index, Arxiv preprint arXiv:1002.3638 (Feb. 2010). URL http://arxiv.org/abs/1002.3638 .[101] R. Dijkgraaf, D. Orlando, and S. Reffert, Relating field theories via stochastic quan-tization,
Nuclear Physics B . (3), 365–386 (Jan., 2010). ISSN 05503213. doi:10.1016/j.nuclphysb.2009.07.018. URL http://arxiv.org/abs/0903.0732 .[102] A. Amir, Y. Oreg, and Y. Imry, Mean-field model for electron-glass dynamics, PhysicalReview B . (16), 1–8 (Apr., 2008). ISSN 1098-0121. doi: 10.1103/PhysRevB.77.165207.URL http://link.aps.org/doi/10.1103/PhysRevB.77.165207 .[103] O. Aharony, S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, Large N field theories,string theory and gravity, Physics Reports . (3-4), 183–386 (Jan., 2000). ISSN 0370-1573. doi: 10.1016/S0370-1573(99)00083-6. URL http://arxiv.org/abs/hep-th/9905111 .[104] H. Ooguri, A. Strominger, and C. Vafa, Black hole attractors and the topological string, Physical Review D . (10), 106007, (2004). ISSN 1550-2368. URL http://link.aps.org/doi/10.1103/PhysRevD.70.106007 .[105] F. Denef, M. Esole, and M. Padi, Orientiholes, Journal of High Energy Physics . , 045(Mar., 2010). ISSN 1029-8479. doi: 10.1007/JHEP03(2010)045. URL http://arxiv.org/abs/0901.2540 .[106] J. Gomis, D. Mateos, and F. Marchesano, An open string landscape, Journal of High En-ergy Physics . (11), 021–021 (Nov., 2005). ISSN 1029-8479. doi: 10.1088/1126-6708/2005/11/021. URL http://arxiv.org/abs/hep-th/0506179 .[107] F. Denef and G. W. Moore, Split States , Entropy Enigmas , Holes and Halos, Arxivpreprint hep-th/0702146 . (2007). URL http://arxiv.org/abs/hep-th/0702146 .[108] M. Mari˜no, R. Minasian, G. W. Moore, and A. Strominger, Nonlinear instantons fromsupersymmetric p -branes,
Journal of High Energy Physics . (01), 005–005 (Jan.,103000). ISSN 1029-8479. doi: 10.1088/1126-6708/2000/01/005. URL http://arxiv.org/abs/hep-th/9911206 .[109] J. Maldacena, A. Strominger, and E. Witten, Black hole entropy in M-Theory, Journalof High Energy Physics . (12), 002–002 (Dec., 1997). ISSN 1029-8479. doi: 10.1088/1126-6708/1997/12/002. URL http://arxiv.org/abs/hep-th/9711053 .[110] J. Wess and J. Bagger, Supersymmetry and supergravity . (Princeton UniversityPress, 1992). ISBN 0691025304. URL http://books.google.com/books?id=4QrQZ_Rjq4UC&pgis=1 .[111] D. Freed, Anomalies in string theory with D-branes,
Arxiv preprint hep-th/9907189 (July. 1999). URL http://arxiv.org/abs/hep-th/9907189 .[112] R. Minasian and G. W. Moore, K-theory and Ramond-Ramond charge,
Journal ofHigh Energy Physics . (11), 002–002 (Nov., 1997). ISSN 1029-8479. doi: 10.1088/1126-6708/1997/11/002. URL http://arxiv.org/abs/hep-th/9710230 .[113] S. K. Ashok and M. R. Douglas, Counting Flux Vacua, Journal of High Energy Physics . (01), 060–060 (Jan., 2004). ISSN 1029-8479. doi: 10.1088/1126-6708/2004/01/060.URL http://arxiv.org/abs/hep-th/0307049 .[114] F. Denef and M. R. Douglas, Distributions of flux vacua, Journal of High Energy Physics . (05), 072–072 (May, 2004). ISSN 1029-8479. doi: 10.1088/1126-6708/2004/05/072.URL http://arxiv.org/abs/hep-th/0404116 .[115] C. Vafa, Black Holes and Calabi-Yau Threefolds, Adv.Theor.Math.Phys. , 207 (Nov.,1998). URL http://arxiv.org/abs/hep-th/9711067 .[116] R. Minasian, G. Moore, and D. Tsimpis, Calabi-Yau black holes and (0, 4) sigma models, Communications in Mathematical Physics . (2), 325–352 (Apr., 2000). ISSN 0010-3616. URL http://arxiv.org/abs/hep-th/9904217 .[117] J. de Boer, F. Denef, S. El-Showk, I. Messamah, and D. V. den Bleeken, Black holebound states in AdS 3 S 2, Journal of High Energy Physics . (11), 050–050 (Nov.,2008). ISSN 1029-8479. doi: 10.1088/1126-6708/2008/11/050. URL http://stacks.iop.org/1126-6708/2008/i=11/a=050 .104118] M. Douglas and G. Moore, D-branes, Quivers, and ALE Instantons, Arxiv preprinthep-th/9603167 (Mar. 1996). URL http://arxiv.org/abs/hep-th/9603167 .[119] F. Denef, Quantum quivers and Hall/hole halos,
Journal of High Energy Physics . ,023, (2002). URL http://arxiv.org/abs/hep-th/0206072 .[120] E. Witten, D-branes and K-theory, Journal of High Energy Physics . (12), 019–019 (Dec., 1998). ISSN 1029-8479. doi: 10.1088/1126-6708/1998/12/019. URL http://arxiv.org/abs/hep-th/9810188 .[121] I. Brunner, M. R. Douglas, A. Lawrence, and C. R¨omelsberger, D-branes on the quintic, Journal of High Energy Physics . (08), 015–015 (Aug., 2000). ISSN 1029-8479. doi:10.1088/1126-6708/2000/08/015. URL http://arxiv.org/abs/hep-th/9906200 .[122] Y. Oz, D. Waldram, and T. Pantev, Brane-antibrane systems on Calabi-Yau spaces, Journal of High Energy Physics . (02), 045–045 (Feb., 2001). ISSN 1029-8479. doi:10.1088/1126-6708/2001/02/045. URL http://arxiv.org/abs/hep-th/0009112 .[123] D. Gaiotto, M. Guica, L. Huang, A. Simons, A. Strominger, and X. Yin, D4-D0 braneson the quintic, Journal of High Energy Physics . , 19 (Sept., 2006). URL http://arxiv.org/abs/hep-th/0509168 .[124] D. Gaiotto and L. Huang, D4-branes on complete intersection in toric variety, hep-th/0612295 (Dec. 2006). URL http://arxiv.org/abs/hep-th/0612295 .[125] A. Collinucci, F. Denef, and M. Esole, D-brane deconstructions in IIB orientifolds, Journal of High Energy Physics . (02), 005–005 (Feb., 2009). ISSN 1029-8479. doi:10.1088/1126-6708/2009/02/005. URL http://arxiv.org/abs/0805.1573 .[126] M. Guica and A. Strominger, Carg`ese Lectures on String Theory with Eight Super-charges, Nuclear Physics B - Proceedings Supplements . , 39–68 (Sept., 2007). ISSN09205632. doi: 10.1016/j.nuclphysbps.2007.06.007. URL http://arxiv.org/abs/0704.3295 .[127] D. Gaiotto, A. Strominger, and X. Yin, The M5-brane elliptic genus: modularityand BPS states, Journal of High Energy Physics . (08), 070–070 (Aug., 2007).ISSN 1029-8479. doi: 10.1088/1126-6708/2007/08/070. URL http://arxiv.org/abs/hep-th/0607010 . 105128] C. Beasley, D. Gaiotto, M. Guica, L. Huang, A. Strominger, and X. Yin, Why ZBH =Ztopˆ2, hep-th/0608021 (Aug. 2006). URL http://arxiv.org/abs/hep-th/0608021 .[129] D. Gaiotto, A. Strominger, and X. Yin, From AdS 3 /CFT 2 to black holes/topologicalstrings, Journal of High Energy Physics . (09), 050–050 (Sept., 2007). ISSN 1029-8479. doi: 10.1088/1126-6708/2007/09/050. URL http://arxiv.org/abs/hep-th/0602046 .[130] J. de Boer, M. C. Cheng, R. Dijkgraaf, J. Manschot, and E. Verlinde, A Farey tail forattractor black holes, Journal of High Energy Physics . (11), 024–024 (Nov., 2006).ISSN 1029-8479. doi: 10.1088/1126-6708/2006/11/024. URL http://arxiv.org/abs/hep-th/0608059 .[131] A. Collinucci and T. Wyder, The elliptic genus from split flows and Donaldson-Thomasinvariants, Journal of High Energy Physics . (5), 1–37 (Oct., 2010). URL http://arxiv.org/abs/0810.4301 .[132] S. Kachru and J. McGreevy, Supersymmetric three-cycles and (super)symmetry break-ing, Physical Review D . (2), 10 (Dec., 1999). ISSN 0556-2821. doi: 10.1103/PhysRevD.61.026001. URL http://arxiv.org/abs/hep-th/9908135 .[133] F. Denef, Supergravity flows and D-brane stability, Journal of High Energy Physics . , 050, (2000). URL http://arxiv.org/abs/hep-th/0005049 .[134] B. Pioline, Four ways across the wall, Arxiv preprint arXiv:1103.0261 (Mar. 2011). URL http://arxiv.org/abs/1103.0261 .[135] J. Manschot, B. Pioline, and A. Sen, Wall-Crossing from Boltzmann Black Hole Halos, arXiv 1011.1258 (Nov. 2010). URL http://arxiv.org/abs/1011.1258 .[136] S. Cecotti, C. Cordova, J. J. Heckman, and C. Vafa, T-Branes and Monodromy, arXiv:1010.5780 (Oct. 2010). URL http://arxiv.org/abs/1010.5780 .[137] M. Aganagic and C. Beem, The geometry of D-brane superpotentials,
Arxiv preprintarXiv:0909.2245 (Sept. 2009). URL http://arxiv.org/abs/0909.2245 .[138] A. Strominger, Heterotic solitons,
Nuclear Physics B . (1), 167–184 (Oct., 1990).ISSN 05503213. doi: 10.1016/0550-3213(90)90599-9. URL http://dx.doi.org/10.1016/0550-3213(90)90599-9 . 106139] A. Strominger, Massless black holes and conifolds in string theory, Nuclear Physics B . (1-2), 96–108 (Sept., 1995). ISSN 05503213. doi: 10.1016/0550-3213(95)00287-3.URL http://arxiv.org/abs/hep-th/9504090 .[140] J. Polchinski, Dirichlet Branes and Ramond-Ramond Charges, Physical Review Letters . (26), 4724–4727 (Dec., 1995). ISSN 0031-9007. doi: 10.1103/PhysRevLett.75.4724.URL http://arxiv.org/abs/hep-th/9510017 .[141] M. R. Douglas, D-branes, categories and N=1 supersymmetry, Journal of MathematicalPhysics . (7), 2818 (Nov., 2001). ISSN 00222488. doi: 10.1063/1.1374448. URL http://arxiv.org/abs/hep-th/0011017 .[142] M. Berkooz, M. Douglas, and R. Leigh, Branes intersecting at angles, Nuclear Physics B . (1-2), 265–278 (Nov., 1996). ISSN 0550-3213. doi: 10.1016/S0550-3213(96)00452-X.URL http://arxiv.org/abs/hep-th/9606139 .[143] S. Ferrara, R. Kallosh, and A. Strominger, N=2 extremal black holes, Physical Review D . (10), R5412–R5416 (Nov., 1995). ISSN 0556-2821. doi: 10.1103/PhysRevD.52.R5412.URL http://arxiv.org/abs/hep-th/9508072 .[144] S. Ferrara, Black holes and critical points in moduli space, Nuclear Physics B . (1-3), 75–93 (Sept., 1997). ISSN 05503213. doi: 10.1016/S0550-3213(97)00324-6. URL http://arxiv.org/abs/hep-th/9702103 .[145] G. Moore, Arithmetic and Attractors, hep-th/9807087 (July. 1998). URL http://arxiv.org/abs/hep-th/9807087 .[146] M. Shmakova, Calabi-Yau Black Holes, Phys. Rev.
D56 , 540–544 (Dec., 1997). doi:10.1103/PhysRevD.56.540. URL http://arxiv.org/abs/hep-th/9612076 .[147] G. L. Cardoso, B. de Wit, and T. Mohaupt, Corrections to macroscopic supersymmetricblack-hole entropy,
Phys. Lett.
B451 , 309–316 (Dec., 1999). URL http://arxiv.org/abs/hep-th/9812082 .[148] G. L. Cardoso, B. de Wit, J. K¨appeli, and T. Mohaupt, Stationary BPS Solutions inN=2 Supergravity with Rˆ2-Interactions,
Journal of High Energy Physics . (12),19 (Sept., 2000). URL http://arxiv.org/abs/hep-th/0009234 .107149] K. Behrndt, D. Lust, and W. Sabra, Stationary solutions of N= 2 supergrav-ity, Nuclear Physics B . (1-2), 264–288 (Jan., 1998). ISSN 0550-3213. doi:10.1016/S0550-3213(97)00633-0. URL http://linkinghub.elsevier.com/retrieve/pii/S0550321398810146 .[150] B. Bates and F. Denef, Exact solutions for supersymmetric stationary black hole com-posites, hep-th/0304094 (Apr. 2003). URL http://arxiv.org/abs/hep-th/0304094 .[151] F. Denef and G. Moore, How many black holes fit on the head of a pin?, Gen-eral Relativity and Gravitation . (10), 1539–1544 (May, 2007). ISSN 0001-7701.doi: 10.1007/s10714-007-0469-410.1142/S0218271808012437. URL http://arxiv.org/abs/arXiv:0705.2564http://arxiv.org/abs/arXiv:0705.2564