aa r X i v : . [ h e p - t h ] A ug RUNHETC-2009-04
Stability and duality in N = 2 supergravity Jan ManschotNHETC, Rutgers UniversityPiscataway, NJ 08854-8019 USA
Abstract
The BPS-spectrum is known to change when moduli cross a wall of marginal stability.This paper tests the compatibility of wall-crossing with S -duality and electric-magneticduality for N = 2 supergravity. To this end, the BPS-spectrum of D4-D2-D0 branes isanalyzed in the large volume limit of Calabi-Yau moduli space. Partition functions arepresented, which capture the stability of BPS-states corresponding to two constituentswith primitive charges and supported on very ample divisors in a compact Calabi-Yau.These functions are “mock modular invariant” and therefore confirm S -duality. Fur-thermore, wall-crossing preserves electric-magnetic duality, but is shown to break the“spectral flow” symmetry of the N = (4 ,
0) CFT, which captures the degrees of freedomof a single constituent. ontents N = 2 supergravity 43 D4-D2-D0 BPS-states 104 Wall-crossing in the large volume limit 165 Conclusion and discussion 22A Two mock Siegel-Narain theta functions 26 The study of BPS-states in physics has been very fruitful. Their invariance under (partof the) supersymmetry transformations of a theory makes them insensitive to variations ofcertain parameters. This allows the calculation of some quantities in a different regime thanthe regime of interest. BPS-states have been specifically useful in testing various dualities,for example S -duality in N = 4 Yang-Mills theory [43] or in string theory [39]. Anothermajor application is the understanding of the spectrum of supersymmetric theories of gravity,leading to the microscopic account of black hole entropy for various supersymmetric blackholes in string theory [41, 32].This article considers the BPS-spectrum of N = 2 supergravity theories in 4 dimensions. N = 2 supersymmetry is the least amount of supersymmetry, which allows massive statesto be BPS. It appears in string theory by compactifying the 10-dimensional space-time ona compact 6-dimensional Calabi-Yau manifold X . A large class of BPS-states are formedby wrapping D-branes around cycles of X , which might correspond to black hole states ifthe number of D-branes is sufficiently large. The Witten index Ω (degeneracy counted with( − F ) is insensitive to perturbations of the string coupling constant g s , and plays therefore acentral role in this paper. It allows to show for certain cases that the magnitude of the indexagrees with black hole entropy: log Ω ∼ S BH . The study of D-branes on X revealed manyconnections to objects in mathematics, like vector bundles, coherent sheaves and derived1ategories, which helps to understand their nature, see for a review Ref. [2]. The index Ωcorresponds from this perspective to the Euler number χ ( M ) of their moduli space M [43],or an analogous but better defined invariant like Donaldson-Thomas invariants [42].An intriguing aspect of BPS-states is their behavior as a function of the moduli of thetheory. The moduli parametrize the Calabi-Yau X and appear in supergravity as scalarfields. Under variations of the moduli, conservation laws allow BPS-states to become stableor unstable at codimension 1 subspaces (walls) of the moduli space. Such changes in thespectrum indeed occur, and were first observed in 4 dimensions by Seiberg and Witten [38].Denef [11] has given an illuminating picture of stability in supergravity as multi black holesolutions whose relative distances depend on the value of the moduli at infinity. At a wall,these distances might diverge or become positive and finite. The changes in the degeneracies∆Ω at a wall show the impact on the spectrum of these processes. Ref. [12] derives formulasfor ∆Ω for n -body semi-primitive decay using arguments from supergravity. The notion ofstability for D-branes is closely related to the notion of stability in mathematics [16, 17].In this context, Kontsevich and Soibelman [30] derive a very general wall-crossing formulafor (generalized) Donaldson-Thomas invariants. Gaiotto et al. [23] shows that this genericformula applied to the indices of 4-dimensional N = 2 quantum field theory, is implied byproperties of the field theory.Much evidence exists for the presence of an S -duality and electric-magnetic duality groupin N = 2 supergravity [7, 44]. S -duality is an SL (2 , Z ) group which exchanges weak andstrong coupling; electric-magnetic duality is the action of a symplectic group on the vectormultiplets. These dualities impose strong constraints on the spectrum of the theory. Thewall-crossing formulas are very generic on the other hand, and the walls form a very intricateweb in the moduli space. It is therefore appropriate to ask: are wall-crossing and dualitycompatible with each other? This paper analyses this question, concentrating on D4-D2-D0 BPS-states or M-theory black holes, in the large volume limit of Calabi-Yau modulispace. The BPS-objects correspond in this limit to coherent sheaves on a Calabi-Yau 3-foldsupported on an ample divisor. The analysis considers the walls, the primitive wall-crossingformula and (part of) the supergravity partition function Z sugra ( τ, C, t ), which enumerates theindices as a function of D2- and D0-brane charges for fixed D4-brane charge. Z sugra ( τ, C, t )captures the changes in the spectrum by wall-crossing. S -duality predicts modularity for this2unction, which is tested in this paper.The degrees of freedom of a single D4-D2-D0 black hole are related via M-theory to a 2-dimensional N = (4 ,
0) superconformal field theory (SCFT) [32]. One of the symmetries of theSCFT spectrum is the “spectral flow symmetry” [4, 21, 31], which are certain transformationsof the charges, which do not change the value of the moduli at infinity. This imposes additionalconstraints on the spectrum to the ones imposed by the supergravity duality groups. Asingle constituent cannot decay any further, and conjectures by [5, 1] indicate that the SCFTdescription of the spectrum (for given charge) might only be valid for a specific value of themoduli. Therefore, interesting dependence of the SCFT spectrum as a function of the moduliat infinity is not expected. This suggests that a natural decomposition for the supergravitypartition function with fixed magnetic charge P might be Z sugra ( τ, C, t ) = Z CFT ( τ, C, t ) + Z wc ( τ, C, t ) , (1.1)where Z CFT ( τ, C, t ) is the well-studied SCFT elliptic genus [4, 21, 31, 33], and all wall-crossing in the moduli space is captured by Z wc ( τ, C, t ). Z CFT ( τ, C, t ) is known to transformas a modular form from arguments of CFT; the modular properties of Z wc ( τ, C, t ) are howeverunknown.This paper considers a small part of Z wc ( τ, C, t ), namely P P P P ample , primitive Z P ↔ P ( τ, C, t ),which enumerates the indices of composite BPS-configurations with two constituents, withample and primitive magnetic charges P and P . An important building block of thesefunctions is the newly introduced “mock Siegel-Narain theta function”. Mock modular formsdo not transform exactly as modular forms, but can be made so by the addition of a relativelysimple correction term [46], which is applied to mock Siegel-Narain theta functions in theappendix. Using its transformation properties, one can show that the corrected partitionfunction transforms precisely as the SCFT elliptic genus, thereby confirming S -duality.From the analysis follows also that electric-magnetic duality remains present in the theory,but the “spectral flow” symmetry of the SCFT is generically not present. This is not quiteunexpected since this is not a symmetry of supergravity. Another indication that the spectralflow symmetry is not present appears in Ref. [1], which explains that the jump in the D4-D2-D0 index by wall-crossing can be larger than the index of a single BPS-object (this effectis known as the entropy enigma [12]).A special property of Z P ↔ P ( τ, C, t ) is that it does not contribute to the index if the mod-3li are chosen at the corresponding attractor point. However, Z P ↔ P ( τ, C, t ) is genericallynot zero, and therefore Z sugra ( τ, C, t ) is nowhere equal to Z CFT ( τ, C, t ) generically. Section 4explains how these observations are in agreement with conjectures of Refs. [5, 1] about theuplift of these BPS-configurations to five dimensions.Although the compatibility with the dualities is expected, it is very interesting to seehow it is realized. The stability condition and primitive wall-crossing formula combine inan almost miraculous way to the mock Siegel-Narain theta function, which gives insights inthe way wall-crossing is captured by N = 2 BPS partition functions for compact Calabi-Yau3-folds. An intriguing property of the corrected partition function is that it is continuous asa function of the K¨ahler moduli t , which is reminiscent of earlier discussions [23, 29].The outline of this paper is as follows. Section 2 reviews briefly the relevant aspectsof N = 2 supergravity. Section 3 describes the BPS-states of interest and the expectedproperties of their partition function. Section 4 is the heart of the paper, it describes thewalls and the partition functions capturing wall-crossing. Section 5 finishes with discussionsand suggestions for further research. The appendix defines two mock Siegel-Narain thetafunctions and gives some of their properties. N = 2 supergravity If IIA string theory is compactified on a compact Calabi-Yau 3-fold X , one obtains N = 2supergravity as the low energy theory in the non-compact dimensions. The most essentialpart of the field content for this article are the b + 1 vector multiplets, which each containa U (1) gauge field F Aµν and complex scalar X A , A = 1 , . . . , b + 1 (with b the second Bettinumber of X ). The gauge fields lead to a vector of conserved charges Γ = ( P , P a , Q a , Q ) T , a = 1 . . . b , which take value in the (2 b + 2)-dimensional lattice L . The magnetic chargesare denoted by P A and electric charges by Q A . The charges arise in IIA string theory aswrapped D-branes on the even homology of X ; the components of Γ represent 6-, 4-, 2- and0-dimensional cycles. A symplectic pairing is defined on the charge lattice h Γ , Γ i = − P Q , + P · Q − P · Q + P Q , . I = − − , where denotes a b × b unit matrix.The scalars X A parametrize the K¨ahler moduli space of the Calabi-Yau X : the complexi-fied K¨ahler moduli are given by t a = B a + iJ a = X a /X . Here, B a and J a are periods of the B -field and the K¨ahler form respectively. The B -field takes values in H ( X, R ). The K¨ahlerforms are restricted to the K¨ahler cone C X , which is defined to be the space of 2-forms suchthat R γ J > R P J > R X J > γ and surface P ∈ X . Anaccurate Lagrangian description of supergravity requires that the volume of X is paramet-rically larger than the Planck length, thus J a → ∞ . This article is mainly concerned withthis parameter regime. Loop and instanton corrections can here be neglected, such that theprepotential simplifies to the cubic expression F ( t ) = 16 d abc t a t b t c , where d abc is the triple intersection number of 4-cycles in X .The supergravity Lagrangian is invariant under the electric-magnetic duality group, whichacts on the vector multiplets and more specifically on the electric-magnetic fields and moduli.This duality group is essentially a gauge redundancy, which appears by working on theuniversal covering space of the moduli space instead of the moduli space itself. The group is Sp (2 b + 2 , Z ): the group of (2 b + 2) × (2 b + 2) matrices K which leave invariant I [44]: K T IK = I . The arguments that the group is Sp (2 b +2 , Z ) are valid in the large volume limit. The correctelectric-magnetic duality group, which is valid for any value of J , is a subgroup of this andgenerated by the monodromies around singularities in the moduli space. These generatorsare generically hard to determine, except for the monodromies in the limit J → ∞ . They are The moduli t a will sometimes be viewed as 2-forms instead of scalars. Similarly, the charges Γ can alsobe viewed as homology cycles or their Poincar´e dual forms. Note that we use here a different notation as in e.g. [44], which is more natural from the point of view ofgeometry. K ( k ) = k a d abc k b k c d abc k c d abc k c k b k c d abc k b k c k a , k ∈ Z b . (2.1)In addition, an SL (2 , Z ) duality group is present, which exchanges the weak and strongcoupling regime. This group acts on the hypermultiplets, and is most manifest in the IIBdescription for large K¨ahler parameters [7].If time is considered as Euclidean and compactified, another SL (2 , Z ) duality group ap-pears. This can be seen from the M-theory viewpoint, where the total geometry is R × T × X ,and T is the product of the time and M-theory circle S × S . A Kaluza-Klein reductionto R , leads to a 3-dimensional N = 4 supergravity theory. Since the physics in R is inde-pendent of large coordinate reparametrizations of T , it should exhibit an SL (2 , Z ) dualitygroup. The duality (complex structure) parameter is given by τ = C + ie − Φ = C + iβ/g s , (2.2)where C ∈ R is the component of the RR-potential 1-form along the time direction. Notethat this SL (2 , Z ) exchanges S and S , which changes the physical interpretation of thestates on both sides of the duality. D2-branes become for example worldsheet instantonsand vice versa. The full BPS-spectrum should however be invariant under these SL (2 , Z )transformations. These transformations also transform the B - and C -fields into each other,which is easily seen from the M-theory perspective: the B - and C -field are reductions ofthe M-theory 3-form over different 2-cycles of the torus. The duality transformations aresummarized by τ → aτ + bcτ + d , C → aC + bB, B → cC + dB, J → | cτ + d | J. (2.3)with (cid:18) a bc d (cid:19) ∈ SL (2 , Z ). Note that this SL (2 , Z ) is not the weak-strong duality of the4-dimensional supergravity. But it is possible to relate this “M-theory” SL (2 , Z ) to the S -duality SL (2 , Z ) of IIB, by a T-duality along the time circle [12]. This transforms C into C and (2.2) becomes the familiar IIB duality parameter. The physical D4-D2-D0 branesof IIA become D3-D1-D-1 instantons of IIB. Therefore, a test of the M-theory SL (2 , Z ) is Note that the upper or lower indices might label either rows or columns in the matrix. S -duality, and in the rest of the paper the M-theory SL (2 , Z ) is referredto as S -duality.The N = 2 supersymmetry algebra contains a central element, the central charge Z (Γ) ∈ C . The central charge of a BPS-state is a linear function of its charge Γ and a non-linearfunction of the K¨ahler or complex structure moduli of X . Only the complexified K¨ahlermoduli t a appear in Z (Γ) for the relevant BPS-states in this article, thus Z (Γ , t ).The mass M of supersymmetric states is determined by the supersymmetry algebra to be M = | Z (Γ , t ) | . In a theory of gravity, a sufficiently massive BPS-state correspond to a blackhole state in the non-compact dimensions. The moduli depend generically on the spatialposition t ( ~x ) in a black hole solution. Their value at the horizon is determined in terms ofthe charge Γ by the attractor mechanism [19], whereas the value at infinity is imposed asboundary condition. The mass M is determined by the moduli at infinity. Following sectionsdeal with the stability of BPS-states, which is determined by these values at infinity. Alsothe SL (2 , Z ) duality group is acting on the complex structure parameter τ of T at infinity.The expression for the central charge as a function of the moduli is generically highlynon-trivial. However in the limit J → ∞ it simplifies to [2] Z (Γ , t ) = − Z X e − t ∧ Γ , where the moduli t and the charge Γ are viewed as forms on X . Alternatively, one can write Z (Γ , t ) = (cid:0) , t a , d abc t b t c , d abc t a t b t c (cid:1) I Γ = Π T I Γ , where we defined the vector of the periods Π.A very intriguing aspect of BPS-states is their stability. The simplest example is the casewith two BPS-objects with primitive charges Γ and Γ . Their total mass is larger than orequal to the mass of a single BPS-object with the same total charge: | Z (Γ , t ) | + | Z (Γ , t ) | ≥| Z (Γ + Γ , t ) | . The equality is generically not saturated, but for special values of the moduli t = t ms , the central charges can align Z (Γ , t ms ) /Z (Γ , t ms ) ∈ R + , and the equality holds.These values form a real codimension 1 subspace of the moduli space, appropriately called the“walls of marginal stability”. They decompose the moduli space into chambers. BPS-statesmight decay or become stable, whenever the moduli cross a wall.Denef [11] has shown how wall-crossing phenomena are manifested in supergravity. Theequations of motions allow for BPS-solutions with multiple black holes. The ones of interest7or the present discussion are solutions with only two black holes. The relative distancebetween the two centers is given by | x − x | = p G h Γ , Γ i | Z (Γ , t ) + Z (Γ , t ) | Im( Z (Γ , t ) ¯ Z (Γ , t )) (cid:12)(cid:12)(cid:12)(cid:12) ∞ , where | ∞ means that the central charges are evaluated at asymptotic infinity in the black holesolution; G is the 4-dimensional Newton constant. In the limit G →
0, or equivalently g s →
0, the distance between the centers also approaches 0. This is the regime, where amicroscopic analysis is typically carried out, it is the D-brane regime as opposed to the blackhole regime.Since distances must be positive, the solution can only exist for h Γ , Γ i Im( Z (Γ , t ) ¯ Z (Γ , t )) > . (2.4)Importantly, | x − x | depends on the moduli: if t approaches a wall of marginal stabilityIm( Z (Γ , t ) ¯ Z (Γ , t )) = 0 , (2.5) | x − x | → ∞ and the 2-center solution decays. An implication of the mechanism for stabilityin supergravity is that single center black holes cannot decay into BPS-configurations withmultiple constituents. If the moduli are chosen at the attractor point at infinity, and are thusconstant throughout the black hole solution, 2-center solutions cannot exist. Moreover, themoduli flow in a 2-center solution from a stable chamber at infinity, to unstable chambers atthe attractor points.In the following, we will analyze wall-crossing between two chambers C A and C B . To avoidambiguities, one can choose Γ and Γ such that Im( Z (Γ , t ) ¯ Z (Γ , t )) < C B , which isequivalent to the convention in the mathematical literature, see for example [45]. This meansthat a stable object with charge Γ satisfiesIm( Z (Γ , t ))Re( Z (Γ , t )) < Im( Z (Γ , t ))Re( Z (Γ , t )) , with h Γ , Γ i > J → ∞ [2]. The charge Γ of the BPS-state is determined by the Chern character of G is the 4-dimensional Newton constant, and is given in terms of IIA and M-theory parameters by G = g α ′ ( α ′ ) V CY and G = ℓ ℓ P πR ℓ V CY , respectively. E and of the ˆ A genus of the Calabi-Yau [35]Γ = ch( i ! E ) q ˆ A ( T X ) , (2.6)where i : P ֒ → X is the inclusion map of the divisor into the Calabi-Yau.Of central interest are the degeneracies of BPS-states with charge Γ. Most useful isactually the index Ω(Γ; t ) = 12 Tr H (Γ; t ) (2 J ) ( − J , (2.7)where J is a generator of the rotation group Spin(3). Ω(Γ; t ) is a protected quantity againstvariations of g s . The degeneracies are only constant in chambers of the moduli space, but jumpif a wall is crossed. This is easily understood from the mechanism for decay in supergravity:the constituents separate, leading to a factorization of the Hilbert spaces, and consequentlya loss of the number of states. The change in the index is [12]:∆Ω(Γ; t s → t u ) = Ω(Γ; t u ) − Ω(Γ; t s ) = − ( − h Γ , Γ i− |h Γ , Γ i| Ω(Γ ; t ms ) Ω(Γ ; t ms ) . Of course, in crossing a wall towards stability one gains states. Therefore the change of theindex is in this case∆Ω(Γ; t u → t s ) = ( − h Γ , Γ i− |h Γ , Γ i| Ω(Γ ; t ms ) Ω(Γ ; t ms ) . Wall-crossing occurs more generally between two chambers C A and C B . If Γ and Γ arechosen such that Im( Z (Γ , t B ) ¯ Z (Γ , t B )) < C B , the change of the index between the twochambers is ∆Ω(Γ; t A → t B ) = ( − h Γ , Γ i h Γ , Γ i Ω(Γ ; t ms ) Ω(Γ ; t ms ) . (2.8)This is consistent with jumps of the invariants in mathematics at walls of marginal stability.We can of course choose the points t A and t B more generally and allow them to lie in thesame chamber. Then the change in the index is∆Ω(Γ; t A → t B ) = ( − h Γ , Γ i h Γ , Γ i Ω(Γ ; t ) Ω(Γ ; t ) (2.9) × (cid:0) sgn(Im( Z (Γ ; t A ) ¯ Z (Γ ; t A ))) − sgn(Im( Z (Γ ; t B ) ¯ Z (Γ ; t B ))) (cid:1) , where sgn( z ) is defined as sgn( z ) = 1 for z >
0, 0 for z = 0, and − z <
0. Note that∆Ω(Γ; t A → t B ) satisfies a cocycle relation [AC] = [AB] + [BC].9ompatibility of the earlier described dualities with wall-crossing is non-trivial. Considerhere the compatibility of electric-magnetic duality. As a gauge redundancy, Sp (2 b + 2 , Z ) (orthe relevant subgroup) leaves invariant the central charge: Z (Γ; t ) = Z ( K Γ; K t ) ( K t denotesthe transformed vector of moduli), and the indices:Ω(Γ; t ) = Ω( K Γ; K t ) , (2.10)for every Γ ∈ L . Since the walls are determined by the central charges, and h Γ , Γ i = h K Γ , K Γ i , it is clear that wall-crossing does not obstruct the electric-magnetic dualitygroup. Note that generically Ω(Γ; t ) = Ω( K Γ; t ), and that no symmetry exists in supergravitywhich relates these two indices. Section 3 comes back to this point.The SL (2 , Z )-duality group also implies non-trivial constraints for the degeneracies andtheir wall-crossing. The test of this duality is however much more involved and the subjectof Section 4, after general aspects of D4-D2-D0 BPS-states and their partition functions areexplained in the next section. This section specializes the general considerations of the previous section to the set of stateswith charge Γ = (0 , P, Q, Q ), and discusses the supergravity partition functions for thisclass of charges. These BPS-states correspond to D4-branes wrapping a divisor in X , withhomology class P ∈ H ( X, Z ). This class of BPS-states is well-described in the literature,see for example [32, 36, 4, 21], therefore the review here will only include the most essentialparts for the discussion.The divisor is also denoted by P and taken to be very ample, which means among othersthat it has non-zero positive components in all 4-dimensional homology classes. The intersec-tion form on P leads to a quadratic form D ab = d abc P c for magnetic charges k ∈ H ( X, Z ),the signature of D ab is (1 , b − Q takesits value in Λ ∗ + P/ Q − P/ ∗ / Λ is denoted by µ . Ifnecessary, the dependence of D ab on P will be made explicit, like P · J , otherwise simply J is used. 10he real and imaginary part of the central charge Z (( P, Q, Q ) , t ) of these states areRe( Z (Γ , t )) = 12 P · ( J − B ) + Q · B − Q , Im( Z (Γ , t )) = ( Q − BP ) · J. The mass | Z (Γ; t ) | of BPS-states in the regime P · J ≫ | ( Q − B ) · B − Q | , | ( Q − BP ) · J | is: | Z (Γ , t ) | = 12 P · J + ( Q − BP ) · B − Q + (( Q − BP ) · J ) P · J + O ( J − ) . (3.1)All but the first term are homogeneous of degree 0 in J , and thus invariant under rescalings.The combination (( Q − BP ) · J ) P · J is positive definite: ( Q − B ) . J has thus a natural interpreta-tion as a point of the Grassmannian which parametrizes 1-dimensional subspaces on which D ab is positive definite. It therefore determines a decomposition of Λ ⊗ R into a 1-dimensionalpositive definite subspace and a ( b − P = 0, P = 0, the transformations (2.1) act on the charges and moduli as Q → Q + k · Q + 12 d abc k a k b P c ,Q a → Q a + d abc k b P c ,t a → t a + k a , with k a ∈ Λ.As mentioned in the introduction, the microscopic explanation for the macroscopic en-tropy S BH = π | Z | of a single center D4-D2-D0 black hole was given by Ref. [32] usingM-theory. The black hole degrees of freedom are in this case those of an M5-brane whichwraps the divisor in X times the torus T . The microscopic counting relied on a 2-dimensional N = (4 ,
0) CFT, which can be obtained as the reduction of the M5-brane worldvolume the-ory to T . The magnetic charge P determines mainly the field content of the CFT, whereasthe electric charges Q and Q are charges of states within the CFT. The BPS-indices of thesingle center black hole are the Fourier coefficients of the SCFT elliptic genus Z CFT ( τ, C, B )[4, 21, 31].To test the compatibility of S -duality in supergravity with wall-crossing, one needs toconsider the full supergravity partition function Z ( τ, C, t ), which captures the stability of The subscript “sugra” used in the introduction will be omitted. t . Properties of Z ( τ, C, t ) are now briefly reviewed, tailored forthe present discussion. It is defined by Z ( τ, C, t ) = X Q , Q Tr H ( P,Q,Q ; t ) 12 (2 J ) ( − J + P · Q exp (cid:0) − πτ | Z (Γ , t ) | + 2 πiτ ( Q − Q · B + B /
2) + 2 πiC · ( Q − B/ (cid:1) , with τ = βg s ∈ R + , τ = C ∈ R , t = B + iJ ∈ Λ ⊗ C and B, C ∈ Λ ⊗ R . This functionsums over Hilbert spaces with fixed magnetic charge and varying electric charges. This isin agreement with a microcanonical ensemble for magnetic charge and a canonical ensemblefor electric charges, which is natural in the statistical physics of BPS black holes [37]. Afterinsertion of (3.1) one finds Z ( τ, C, t ) = exp( − πτ J ) X Q , Q Tr H ( P,Q,Q ; t ) 12 (2 J ) ( − J + P · Q × e (cid:16) − ¯ τ ˆ Q ¯0 + τ ( Q − B ) / τ ( Q − B ) − / C · ( Q − B/ (cid:17) , with ˆ Q ¯0 = Q ¯0 + Q , Q ¯0 = − Q and e ( x ) = exp(2 πix ). The modular invariant prefactorexp( − πτ J ) is in the following omitted. The partition function has an expansion Z ( τ, C, t ) = X Q , Q Ω( P, Q, Q ; t ) ( − P · Q × e (cid:16) − ¯ τ ˆ Q ¯0 + τ ( Q − B ) / τ ( Q − B ) − / C · ( Q − B/ (cid:17) . Note that the partition function depends in various ways on the K¨ahler moduli t : they appearin Ω( P, Q, Q ; t ), moreover B shifts the electric charges and J determines the decompositionof the lattice into a positive and negative definite subspace of Λ ⊗ R . The sum over Q and Q isunrestricted and might at some point invalidate the estimate used for (3.1), even in the limit J → ∞ . To verify that this does not invalidate the analysis, we compute the term O ( J − ).It is given by − Q − B ) ( ˆ Q ¯0 − ( Q − B ) − ) /P · J . It follows from the CFT analysis thatˆ Q ¯0 is bounded below for a single constituent, therefore ˆ Q ¯0 − ( Q − B ) − is as well. Moreover,the next section shows that ˆ Q ¯0 − ( Q − B ) − is also bounded below for stable bound statesof 2 constituents. If ( Q − B ) ( ˆ Q ¯0 − ( Q − B ) − ) is O ( J ), then | Z (Γ , t ) | − P · J is at least O ( J ). Contributions to the partition function of states for which the approximations for Eq.(3.1) are not satisfied, are thus highly suppressed compared to the states for which they aresatisfied, which shows that the analysis is not invalidated. Also for any given value of the12harges, one can always increase J to sufficiently large values, such that the approximationsare valid. It is very well possible however, that not the whole partition function has a niceFourier expansion.It is well known that Z ( τ, C, t ) contains a pole for τ → i ∞ and its SL (2 , Z ) images. It isless clear at this point whether poles in B or C can appear in Z ( τ, C, t ). Examples of CFT’swhere such poles appear, are the characters of massless representations of the N = 4 SCFTalgebra [18], and the sigma model with the non-compact target space H +3 [24]. The Fourierexpansion of a partition function with poles depends on the integration contour. This is howthe partition function of dyons in N = 4 supergravity [40] captures wall-crossing phenomena.However, the stability condition (2.5) for D4-branes on ample divisors show that no wall-crossing as function of C is present. Moreover, the partition functions for bound states oftwo constituents, derived in the next section, are not directly suggestive for “wall-crossingby poles”. Therefore, in the following is assumed that no poles in B or C are present in Z ( τ, C, t ).The translations K ( k ) of the electric-magnetic duality group imply a symmetry for thepartition function. Using (2.10) and assuming the Fourier expansion, one verifies easily that Z ( τ, C, t ) −→ ( − P · k e ( C · k/ Z ( τ, C, t ) , under transformations by K ( k ). Also using (2.10) one can show a quasi-periodicity in B : Z ( τ, C, t + k ) = ( − P · k e ( C · k/ Z ( τ, C, t ) . Additionally, Z ( τ, C, t ) satisfies a quasi-periodicity in C : Z ( τ, C + k, t ) = ( − P · k e ( − B · k/ Z ( τ, C, t ) . (3.2)These translations are large gauge transformations of C . A theta function decomposition isnot implied by the two periodicities since the Fourier coefficients Ω(Γ; t ) explicitly depend on B , and generically Ω( K ( k )Γ; t ) = Ω(Γ; t ).A distinguishing property of the partition function for this class of BPS-states is thatcharges multiply either τ or ¯ τ , in contrast to for example D2- or D6-brane partition functions.Additionally, space-time S -duality suggests that the function transforms as a modular form,such that techniques of the theory modular forms can be usefully applied. Refs. [21, 22]present some coefficients Ω((0 , , Q, Q ); t ) for several Calabi-Yau 3-folds with b = 1. These13oefficients determine the whole partition function, and confirm modularity in a non-trivialway. However, stability phenomena do not occur in the limit J → ∞ for these Calabi-Yau’s,since b = 1. The next section tests modularity, if wall-crossing is present.The arguments from CFT for modularity are very robust. Refs. [4, 21, 33] derive thatthe action of the generators of SL (2 , Z ) on Z CFT ( τ, C, t ) is given by: S : Z ( − /τ, − B, C + i | τ | J ) = τ ¯ τ − ε ( S ) Z ( τ, C, t ) , (3.3) T : Z ( τ + 1 , C + B, t ) = ε ( T ) Z ( τ, C, t ) , where ε ( T ) = e ( − c ( X ) · P/
24) and ε ( S ) = ε ( T ) − [12, 33]. Here the analysis of [4, 21] isadapted to the supergravity point of view following [12]. The next section gives evidencethat the same transformation properties continue to hold for the full supergravity partitionfunction. Note that S -duality is consistent with the two periodicities mentioned above. Theperiodicities and the SL (2 , Z ) form together a Jacobi group SL (2 , Z ) ⋉ ( Z b ) .The partition function for single constituents can be decomposed in a vector-valued mod-ular form and a theta function by arguments from CFT. The indices of the CFT are indepentof the moduli at infinity: Ω CFT (Γ; t ) = Ω CFT (Γ) = Ω(Γ), and obey the “spectral flow sym-metry” Ω(Γ) = Ω( K ( k )Γ). To see this, recall that the D2-brane charges appear in the CFTin a U (1) b current algebra, which can be factored out of the total CFT by the Sugawaraconstruction, which implies that the indices satisfy Ω CFT (Γ) = Ω
CFT ( K ( k )Γ) [4, 21, 31]. Thename “spectral flow” comes originally from the SCFT of superstrings. In the current context,one could see the flow as a flow of the B -field. As mentioned already after Eq. (2.10), noevidence exists that this is a symmetry of the full spectrum of 4-dimensional supergravity. Infact, Section 4 shows that wall-crossing is incompatible with this symmetry at generic pointsof the moduli space.Since the spectral flow symmetry is present in the spectrum of a single D4-D2-D0 blackhole, the theta function decomposition is reviewed here. We define the functions h P,Q − P ( τ ) = X Q ¯0 Ω( P, Q, Q ) q Q ¯0 + Q , (3.4) Evidence exists that Z ( τ, C, t ) does only transform as (3.3) under the full group SL (2 , Z ) if P is prime.Otherwise it transforms as a modular form of a congruence subgroup, whose level is determined by the divisorsof P . Consequently, the rest of the article assumes implicitly that P is prime, although it nowhere explicitlyenters the calculations. K ( k )Γ), one can show that the invariants Ω( P, Q, Q ) depend onlyon ˆ Q ¯0 and the conjugacy class µ of Q ∈ Λ ∗ , thus Ω( P, Q, Q ) = Ω µ ( ˆ Q ¯0 ). Therefore, h P,Q − P ( τ ) = h P,Q − P + k ( τ ) with k ∈ Λ. This allows a decomposition of Z CFT ( τ, C, t )into a vector-valued modular form h P,µ ( τ ) and a Siegel-Narain theta function Θ µ ( τ, C, B ): Z CFT ( τ, C, t ) = X µ ∈ Λ ∗ / Λ h P,µ ( τ ) Θ µ ( τ, C, B ) , (3.5)withΘ µ ( τ, C, B ) = X Q ∈ Λ+ P/ µ ( − P · Q e (cid:0) τ ( Q − B ) / τ ( Q − B ) − / C · ( Q − B/ (cid:1) . (3.6)The dependence of Θ µ ( τ, C, B ) on the K¨ahler moduli J is not made explicit. The transfor-mation properties of Θ µ ( τ, C, B ) are S : Θ µ ( − /τ, − B, C ) = 1 p | Λ ∗ / Λ | ( − iτ ) b +2 / ( i ¯ τ ) b − / e ( − P / X ν e ( − µ · ν )Θ ν ( τ, C, B ) ,T : Θ µ ( τ + 1 , C + B, B ) = e (cid:0) ( µ + P/ / (cid:1) Θ µ ( τ, B, C ) . They satisfy in addition two periodicity relations for B and C with k ∈ Λ:Θ µ ( τ, C, B + k ) = ( − k · P e ( C · k/
2) Θ µ ( τ, C, B ) , Θ µ ( τ, C + k, B ) = ( − k · P e ( − B · k/
2) Θ µ ( τ, C, B ) . All the dependence on τ and the “explicit” dependence on B , C and J of Z CFT ( τ, C, t ) iscaptured by the Θ µ ( τ, C, B ). Note that the Θ µ ( τ, C, B ) are annihilated by D = ∂ τ + i π ∂ C + + B + · ∂ C + − πiB . Z ( τ, C, t ) is also annihilated by D , if holomorphic anomalies in h P,µ ( τ )are ignored; these are known to arise in similar partition functions for 4-dimensional gaugetheory [43].The transformation properties of Θ µ ( τ, C, B ) imply that h P,µ ( τ ) transforms as a vector-valued modular form: S : h P,µ ( − /τ ) = − p | Λ ∗ / Λ | ( − iτ ) − b / − ε ( S ) ∗ e (cid:0) − P / (cid:1) × X δ ∈ Λ ∗ / Λ e ( − δ · µ ) h P,δ ( τ ) ,T : h P,µ ( τ + 1) = ε ( T ) ∗ e (cid:0) ( µ + P/ / (cid:1) h P,µ ( τ ) . S BH = π q ( P + c ( X ) · P ) ˆ Q ¯0 for ˆ Q ¯0 ≫ P + c ( X ) · P . As explained in Section 3, the partition function is expected to exhibit the modular symmetryand electric-magnetic duality in the large volume limit J → ∞ . This section constructs thecontribution Z P ↔ P ( τ, C, t ) of bound states of two primitive constituents with primitiveD4-brane charges P and P = ~ Z ( τ, C, t ), and tests its modular properties. I takethe following Ansatz for the contribution to the index of a bound state of two primitiveconstituents at a point t in the moduli space:Ω Γ ↔ Γ (Γ; t ) = (cid:0) sgn(Im( Z (Γ , t ) ¯ Z (Γ , t ))) + sgn( h Γ , Γ i ) (cid:1) (4.1) × ( − h Γ , Γ i− h Γ , Γ i Ω(Γ )Ω(Γ ) . The first term of the first line ensures that this Ansatz reproduces the wall-crossing formula(2.9). The non-trivial part of the Ansatz is thus the term sgn( h Γ , Γ i ). This section ex-plains that this is also in agreement with other important physical requirements. Based onthe Ansatz, the generating function of the contribution to the index of the bound states isdetermined in Eq. (4.7). A study of the generating function leads to the following results:- the generating function (4.7) is convergent,- the generating function does not exhibit the modular properties of Z CFT ( τ, C, t ) (thepartition function of a single center black hole with magnetic charge P + P ), butit can be made so by the addition of a “modular completion” using techniques ofmock modular forms. The “completed” generating function (4.10) is proposed as thecontribution Z P ↔ P ( τ, C, t ) of 2-center bound states, which is thus compatible with S -duality.- Z P ↔ P ( τ, C, t ) has the unexpected property that it is continuous as function of themoduli, which is reminiscent of earlier work on wall-crossing [23, 29]. The generatingfunction is by construction a discontinuous function of the moduli.The combination of the first and second property is essentially a unique consequence of theAnsatz. The agreement of the Ansatz with the supergravity picture is discussed later.16e continue now by taking a closer look at the walls of marginal stability. SpecializingEq. (2.5), gives for the walls at J → ∞ (without 1 /J corrections) P · J ( Q − BP ) · J − P · J ( Q − BP ) · J = 0 . (4.2)Note that this wall is independent of the D0-brane charges Q ,i . And so states decay at thiswall, independent of their D0-charge and of their distribution between the constituents. Thecondition for stability for this class of states is P · J ( Q − BP ) · J − P · J ( Q − BP ) · J < , if h Γ , Γ i >
0. This stability condition is a natural generalization of slope stability for sheavesor bundles on surfaces [15], since P · J replaces the notion of rank. It can be derived from thestability for sheaves [28]. When 1 /J corrections are included, one finds that actually manyphysical walls merge with each other in the limit J → ∞ [13]. We define I ( Q , Q ; t ) = P · J ( Q − BP ) · J − P · J ( Q − BP ) · J √ P · J P · J P · J , (4.3)which is invariant under rescalings of J .It is instructive to look at the symmetries of the wall (4.2). Clearly, it is invariant underthe translations K ( k ) (2.1), if it acts both on the charges and the moduli. However, the wallis not invariant in general if only the charges are transformed. This is only the case for veryspecial situations like P || P . The change in the index is therefore not consistent with thespectral flow symmetry. Indeed, already in Section 2 we argued that this symmetry is notnatural from the supergravity perspective. The fact that the symmetry is broken has majorimplications for supergravity partition functions, since the decomposition into a vector-valuedmodular form and theta functions is not valid.We can now see that Eq. (4.1) is in agreement with the supergravity picture. As men-tioned before, the picture of stability in supergravity shows that only the single center solutionexists if the moduli are chosen at the corresponding attractor point t (Γ). Therefore, the in-dex should equal the CFT-index at this point: Ω(Γ; t (Γ)) = Ω CFT (Γ), which is consistentwith the account of black hole entropy [32]. More evidence for this idea comes from theconjectures in Refs. [5, 1], which suggest a one to one correspondence between connectedcomponents of the solution space of multi-centered asymptotic AdS × S solutions and IIAattractor flow trees starting at t (Γ) = lim λ →∞ D − Q + iλP . Note that Z ( τ, C, t ) does not17epend on λ in the limit J → ∞ . By the AdS /CFT correspondence, this also suggeststhat Ω(Γ , t (Γ)) = Ω CFT (Γ). If this is correct, (4.1) should not contribute to Ω(Γ; t (Γ)).Indeed, computation of I ( Q , Q ; t (Γ)) gives q P P P P P ( P · Q − P · Q ), and thereforesgn( I ( Q , Q ; t (Γ))) − sgn( P · Q ) = 0, such that there is never a contribution from boundstates at the attractor point using this Ansatz. On the other hand, bound states with twoconstituents for charges ˜Γ = Γ might exist at t (Γ), and consequently Ω(˜Γ , t (Γ)) = Ω CFT (˜Γ).Therefore, these considerations of BPS-configurations with two constituents show that gener-ically Z ( τ, C, t ) equals nowhere in the moduli space Z CFT ( τ, C, t ).A special choice of charges is P = ~
0, i.e. Γ = (0 , , Q , Q , ). If one does not move themoduli outside the K¨ahler cone, then walls for this choice can not be crossed. To see this,recall that Q represents now the support of a coherent sheaf and must therefore representa holomorphically embedded D2-brane. Therefore, Q · J > J ∈ C X . The stabilitycondition for ( P, Q, Q , ) → (0 , Q , Q , ) + ( P, Q , Q , ) is given by P · Q Q · J < , (4.4)which is independent of the B -field. Eq. (4.4) may or may not be satisfied for given charges.However, because Q · J cannot change its sign for J ∈ C X , no walls of marginal stabilityare present in the large volume limit. It is thus consistent to consider only bound states ofconstituents with non-zero D4-brane charge.To construct the generating function, it is covenient to introduce some notation. Forconstituent i = 1 , i , the corresponding quadratic form is denoted by ( Q i ) i and the conjugacy class of Q i in Λ ∗ i / Λ i is µ i . h Γ , Γ i can be written as an innerproduct of 2vectors in Λ ⊕ Λ ⊗ R . Define to this end the unit vector P = ( − P ,P ) √ P P P ∈ Λ ⊕ Λ ⊗ R , then( Q , Q ) · P = Q · P = h Γ , Γ i / √ P P P . In the appendix, also I ( Q , Q ; t ) is written as aninnerproduct.Since the wall is independent of the D0-brane charge, the index Ω( P, Q, Q ; t ) jumpsirrespective of the D0-brane charge. For the partition function, we only want to keep track ofthe magnetic charge of the two constituents and sum over all the electric charge. Therefore,the contribution to the index Ω( P, Q, Q ; t ) from bound states of constituents whose D4-brane18harges are P and P includes a sum over D0- and D2-brane charge:Ω P ↔ P ( P, Q, Q ; t ) = X ( Q ,Q , )+( Q ,Q , )=( Q,Q ) 12 (sgn( I ( Q , Q ; t )) − sgn( h Γ , Γ i )) × ( − h Γ , Γ i ( P · Q − P · Q ) Ω( P , Q , Q , )Ω( P , Q , Q , ) . The generating function of Ω P ↔ P ( P, Q, Q ; t ) analogous to (3.4) is h P ↔ P ,Q − P ( τ ) = P Q Ω P ↔ P ( P, Q, Q ; t ) q − Q + Q . This can be expressed in terms of the vector-valuedmodular forms of the last section: h P ↔ P ,Q − P ( τ ) q − Q = X ( Q ,Q , Q ,Q , Q,Q Q ( − P · Q − P · Q ( P · Q − P · Q ) Ω(Γ ) Ω(Γ ) × ( sgn( I ( Q , Q ; t )) − sgn( h Γ , Γ i ) ) q Q ¯0 , + Q ¯0 , (4.5)= X Q + Q = Q ( sgn( I ( Q , Q ; t )) − sgn( h Γ , Γ i ) ) ( − P · Q − P · Q × ( P · Q − P · Q ) h P ,µ ( τ ) h P ,µ ( τ ) q − ( Q ) − ( Q ) . Note that the spectral flow symmetry is used here to write h P i ,µ i ( τ ) instead of h P i ,Q i − P i / ( τ ).Eq. (4.5) can be seen as a major generalization of a similar formula for rank 2 sheaves on arational surface [27].To obtain the full generating function, we have to multiply h P ↔ P ,Q − P ( τ ) by( − P · Q e (cid:0) τ ( Q − B ) / τ ( Q − B ) − / C · ( Q − B/ (cid:1) , (4.6)and sum over Q ∈ Λ ∗ . The various quadratic forms in the exponent combine to e (cid:0) τ ( Q − B ) / τ (cid:0) ( Q − B ) ⊕ − ( Q − B ) (cid:1) / C · ( Q − B/ (cid:1) , where Q ⊕ = ( Q ) + ( Q ) . See the appendix for more explanation of the notation. Theterm ( Q − B ) ⊕ − Q − B ) , which multiplies πτ in the exponent is not negative definite,but has signature (1 , b − Q , Q ) ∈ Λ ⊕ Λ is thereforeclearly divergent. However, the presence of sgn( I ( Q , Q ; t )) − sgn( P · Q ) ensures that thefunction is convergent, which follows from Proposition 1 in the appendix. Thus the stabilitycondition implies that the quadratic form is negative definite, if evaluated for stable bound19tates. Performing the sum over Q , one obtains the generating series: X µ ⊕ ∈ Λ ∗ ⊕ / Λ ⊕ h P ,µ ( τ ) h P ,µ ( τ ) Ψ µ ⊕ ( τ, C, B ) , (4.7)where Λ ⊕ = Λ ⊕ Λ , µ ⊕ = ( µ , µ ) ∈ Λ ∗ ⊕ / Λ ⊕ andΨ µ ⊕ ( τ, C, B ) = X Q ∈ Λ1+ µ P / Q ∈ Λ2+ µ P / ( P · Q − P · Q ) ( − P · Q + P · Q ( sgn( I ( Q , Q ; t )) − sgn( P · Q ) ) (4.8) × e (cid:0) τ ( Q − B ) / τ (cid:0) ( Q − B ) ⊕ − ( Q − B ) (cid:1) / C · ( Q − B/ (cid:1) , with P = ( − P ,P ) √ P P P ∈ Λ ⊕ ⊗ R .The test of S -duality is now reduced to testing modularity for (4.8). Since Ψ µ ⊕ ( τ, C, B )is not a sum over the total lattice Λ ⊕ , it does not have the nice modular properties of thefamiliar theta functions. However, Ref. [46] explains that a real-analytic term can be addedto a sum over a positive definite cone in an indefinite lattice with signature ( n − , µ ⊕ ( τ, C, B ), and explains in detail how it can be completed to a functionΨ ∗ µ ⊕ ( τ, C, B ), which transforms as a Siegel-Narain theta function. The essential idea ofthis procedure is to make the replacementsgn( z ) −→ Z √ τ z e − πu du, (4.9)which interpolates monotonically and continuously between − z = −∞ and 1 at z = + ∞ .It approaches sgn( z ) in the limit τ → ∞ . To complete Ψ µ ⊕ ( τ, C, B ) to a modular function,one also needs to replace z sgn( z ) by an appriopriate continuous function as explained in theappendix. Indefinite theta functions are prominent in the work on mock modular forms [46];Ψ µ ⊕ ( τ, C, B ) is therefore appropriately called a “mock Siegel-Narain theta function”.By replacing Ψ µ ⊕ ( τ, C, B ) with Ψ ∗ µ ⊕ ( τ, C, B ) in Eq. (4.7), we obtain our final proposalof the contribution of 2-center bound states Z P ↔ P ( τ, C, t ) to Z ( τ, C, t ): Z P ↔ P ( τ, C, t ) = X µ ⊕ ∈ Λ ∗ ⊕ / Λ ⊕ h P ,µ ( τ ) h P ,µ ( τ ) Ψ ∗ µ ⊕ ( τ, C, B ) . (4.10) Note that the Fourier expansion (3.2) is thus not modular. Z P ↔ P ( τ, C, t ) trans-forms precisely as the CFT partition function Z CFT ( τ, C, t ) of the single constituent withD4-brane charge P + P (3.3)! To see that the weight agrees, note that the weight ofΨ ∗ µ ⊕ ( τ, C, B ) is (1 , b + 1) = (1 , b −
1) + (0 , (1 , b −
1) is due to the latticesum and (0 ,
1) is due to the insertion of P · Q − P · Q . Combining this with 2 · (0 , − b − h P i ,µ i ( τ ), one precisely finds the weight ( , − ) for Z P ↔ P ( τ, C, t ). A crucial detail is the grading by ( − P · Q : ( − ( P + P ) · ( Q + Q )+( P · Q − P · Q ) =( − P · Q + P · Q , such that Ψ ∗ µ ⊕ ( τ, C, B ) does transform conjugately to h P ,µ ( τ ) h P ,µ ( τ ).Moreover, as was already mentioned above, coexistence of convergence and modularity isessentially a unique consequence of the Ansatz. In particular, the fact that P is independentof the moduli and satisfies P · ( J, J ) =
P · ( B, B ) = 0 is essential. We thus observe thatall factors in (4.1) combine in a neat way such that Z P ↔ P ( τ, C, t ) has the same modularproperties as Z P + P ( τ, C, t ).One could of course object to correcting the partition function by hand and argue that ananomaly appeared for S -duality. However, the correcting factor could also arise automaticallyin a more physical derivation, for example by perturbative contributions. It is also not sosurprising that corrections to the Fourier expansion (3.2) are necessary, since it was derivedby assuming that the charges are finite and J → ∞ , which is clearly not the case everywherein the Hilbert space. Note that a physical derivation might lead to a slightly different modularcompletion of the generating function, since one could always add a real-analytic function withthe same transformation properties. This would however not change the crucial propertieswe have established.Besides S -duality, there is another very appealing aspect in favor of the correction term.Eq. (4.8) is not continuous as a function of the moduli B and J because of the termssgn( I ( Q , Q ; t )). As discussed above, the correction term is essentially a replacement ofthe discontinuous functions sgn( z ) and z sgn( z ) by real analytic functions (which approachthe original expression in the limit | z | → ∞ ). The modular invariant partition function istherefore continuous in B and J . This might not be such a coincidence as it seems at firstsight. Ref. [29] proposed a continuous and holomorphic generating function for Donaldson-Thomas invariants (or an extension thereof), which captures wall-crossing. Moreover, Ref.[23] describes that continuity of the metric g of the target manifold of a 3-dimensional sigma21odel, essentially implies the Kontsevich-Soibelman wall-crossing formula. Continuity of Z ( τ, C, t ) is very intriguing from this perspective, and it would be interesting to investigatewhether it plays here an as fundamental role as in these references.The contribution of all 2-constituent BPS-states with primitive, ample charges is easilyincluded in Z ( τ, C, t ) by the sum P P P P ample , primitive Z P ↔ P ( τ, C, t ). The above analyses givessome evidence that modularity is also preserved if one of the charges is not ample. The consistency of wall-crossing with S -duality and electric-magnetic duality is tested byanalyzing the BPS-spectrum of D4-D2-D0 branes on a compact Calabi-Yau 3-fold X . Thestability of composite BPS-states with two primitive constituents is considered, in the largevolume limit of the K¨ahler moduli space. The consistency of electric-magnetic duality withwall-crossing follows rather straightforwardly from the structure of the walls and the primitivewall-crossing formula. From the equations for the walls in the moduli space can also be seenthat wall-crossing is not compatible with the spectral flow symmetry, which appears in themicroscopic description of a single D4-D2-D0 object by a CFT [32]. S -duality is tested by theconstruction of a partition function (4.10) for two constituents, which captures the changesof the spectrum if walls of marginal stability are crossed. The essential building block is a“mock Siegel-Narain theta function”, which might be of independent mathematical interest.The stability condition and the BPS-degeneracies combine in a very intricate way in orderto preserve modularity, which is a confirmation of S -duality.The results of this paper are applicable to various problems, for example those related toentropy enigmas [12]. With these are meant BPS-configurations with multiple constituents,whose number of degeneracies is larger than the number of degeneracies of a single constituentwith the same charge. Originally, the common thought was that wall-crossing would onlyhave a subleading effect on the degeneracies. Ref. [1] has shown that enigmatic changesin the spectrum can also happen from D4-D2-D0 configurations with 2 constituents, whichare considered in this paper. The present work shows that these enigmatic phenomena, canbe captured by modular invariant partition functions. This might proof useful in futurestudies on the entropy enigma. For example Eq. (4.10) shows that the leading entropy of twoconstituents (if their bound state exists) is π q ( P + P + c · P ) (cid:0) Q ¯0 + ( Q ) + ( Q ) (cid:1) Q and Q , under the constraint Q + Q = Q . This shouldbe compared with the single constituent entropy π q ( P + c · P )( Q ¯0 + Q ). Based onthese equations, one can show the existence of enigmatic configurations, even in the regime q ˆ Q ¯0 P ≫
1, or large topological string coupling. This shows that Z wc ( τ, C, t ) is not necessarilya small correction to Z CFT ( τ, C, t ) in (1.1). A detailed analysis of the conditions for the firstentropy to be larger than the second would be very instructive. This raises the question ofthe relation of the discussed partition functions in this paper and the OSV-conjecture, whichrelates the black hole partition function and the one of topological strings [37].The D4-D2-D0 BPS-degeneracies are also related to mathematically defined invariants.In the large volume limit, the D4-D2-D0 index correspond to the Euler number (or a variantthereof) of the moduli space of coherent sheaves with support on the divisor of the Calabi-Yau. An explicit calculation of these Euler numbers is currently not feasible, but would bemagnificent. It would for example provide a more rigorous test of modularity of the partitionfunctions. A more tractable possibility for future work is to replace the index Ω(Γ; t ) by a morerefined quantity [14] by including the spin dependence Ω(Γ; t, y ) = Tr H (Γ; t ) ( − y ) J . This isnot a protected quantity, but is nevertheless of interest. The corresponding partition functionmight still exhibit modular properties, and wall-crossing formulas do exist in the literaturefor Ω(Γ; t, y ) in the context of surfaces [27, 45] and also physics [13]. A generalization ofSection 4 to include these refined invariants should therefore be possible. Another suggestionis to move away from the limit J → ∞ by including finite size corrections. This would alsoleave the description of the BPS-states as coherent sheaves, and the relations with dualitiesprobably become probably more intricate.A limitation of this work is that it considers only primitive wall-crossing. One might con-tinue in a similar fashion as Section 4 to construct partition functions for BPS-configurationswith more constituents, and test the compatibility of the semi-primitive wall-crossing formula[12] and S -duality in this way. Much more appealing would be a closed expression for thepartition function, which does not sum over all possible decays. Such an expression mightultimately allow for a test of the generic Kontsevich-Soibelman wall-crossing formula withrespect to S -duality. Or even explain the KS-formula in N = 2 supergravity from physicalconsiderations, as was done for N = 2 field theory [23]. Although this paper took in somesense an opposite approach, some lessons might still be learned.23he requirement of the dualities implies non-trivial constraints for the indices and wall-crossing formulas. These do not seem constraining enough to deduce the KS-formula. Forexample, the appearance of mock modular forms instead of normal modular forms was a pri-ori unknown. This can of course be seen as an anomaly for S -duality. On the other hand, itis really pretty close to modularity, and the functions can be made modular by a simple mod-ification as explained in the appendix. These modifications might appear in a more physicalderivation of the partition function in order to preserve S -duality. The correction terms mightbe determined by a differential equation, similar to the holomorphic anomaly equation of topo-logical strings [3]. Proposition 5 gives the action of D , defined in Section 3, on Ψ ∗ µ ⊕ ( τ, C, B ).This shows that DZ P ↔ P ( τ, C, t ) includes a term Z CFT ,P ( τ, C, B ) Z CFT ,P ( τ, C, B ), which issuggestive and reminiscent of earlier work on holomorphic anomaly equations, see for exam-ple Ref. [34]. Another consequence of the correction terms is that they make the functioncontinuous as a function of the moduli, although it captures the changes of the spectrumunder variations of the moduli. This is quite intriguing, since “continuity” was essential inthe field theory derivation of the KS-formula in Ref. [23], more precisely the continuity ofthe metric of the target space of a 3-dimensional sigma model. The appearance of a continu-ous partition function in this paper suggests that continuity might be fundamental here too.More investigation is clearly necessary to find out to what extent continuity and the duali-ties can imply the generic wall-crossing formula [30] for BPS-invariants. Ref. [29] suggestedearlier a continuous, holomorphic generating function for Donaldson-Thomas invariants, andits discussion resembles in some respects Ref. [23]. However, Z P ↔ P ( τ, C, t ) does not seemto be holomorphic in t .Note that the way Z P ↔ P ( τ, C, t ) captures stability is quite different from how the par-tition function of -BPS states (or dyons) of N = 4 supergravity captures stability. Thatfunction captures wall-crossing in a very appealing way by poles [40] and a proper choice ofthe integration contour [8] to obtain Fourier coefficients. In this way, mock modular formsarise via meromorphic Jacobi forms [9].Section 4 shows that the supergravity partition function is nowhere in moduli space equalto the CFT partition function (except for special cases like a Calabi-Yau with b = 1). Anatural question is: is the supergravity partition function related to the partition function ofa lower dimensional theory, just as the spectrum of a single constituent is captured by the24 = (4 ,
0) SCFT? Ref. [5] (see also [6]) proposes that such a theory might be classically a2-dimensional sigma model into the moduli space of supersymmetric divisors in the Calabi-Yau, whose “beta function does not vanish for Y different from the attractor point andthe Y undergo renormalization group flow till they reach the attractor point, an IR fixedpoint. Along the flow, the constituents of M5-M5 bound states decouple from each other;each of them has its own IR fixed point corresponding to an AdS × S .” The structure ofthe partition function (4.10) shows the decoupled constituents. It is also in agreement withthe suggestion that the theory is not a CFT, since the spectral flow symmetry is not present.On the other hand, Z sugra ( τ, C, t ) does not equal Z CFT ( τ, C, t ) at attractor points, whichindicates that the microscopic theory (if it exists) is not a CFT, not even at these points. Abetter understanding of these issues is clearly desired. Another alternative for a microscopictheory is quiver quantum mechanics [10], which arises in the limit g s →
0, and is knownto capture bound states in 4 dimensions. A connection between this theory, the D4-D2-D0bound states and their partition functions might lead to interesting insights.An intriguing implication of the proposed function is wall-crossing as a function of the C -field for the BPS-states one obtains after S -duality. A D4-D2-D0 BPS-state becomes a D3-D1-D-1 instantonic BPS-state after performing a T-duality along the time circle. This doesnot yet change anything fundamental, stability of this configuration is still captured by B and J . However, S -duality transforms such a configuration to one with instanton D3-branesand fundamental string instantons. Moreover, B and C are interchanged, which implies thatthe degeneracies of these BPS-states jump as a function of C and J . This is quite interestingsince the C -field is generically not considered as a stability parameter, and gives also evidencethat B and C should be considered on a more equal footing. The K-theoretic description ofthe C -fields is however very different in nature than the description of the B -field. Acknowledgements
I would like to thank Dieter van den Bleeken, Wu-yen Chuang, Atish Dabholkar, EmanuelDiaconescu, Davide Gaiotto, Lothar G¨ottsche and Gregory Moore for fruitful discussions.I owe special thanks to Gregory Moore for his comments on the manuscript. This work issupported by the DOE under grant DE-FG02-96ER40949. Y is the vector of normalized 5-dimensional K¨ahler moduli, which is proportional to J . Two mock Siegel-Narain theta functions
This appendix computes the transformation properties of the Siegel-Narain mock theta func-tion which appears in Section 4. The proofs are similar to those given in [46]. The dependenceon the Grassmannian, which parametrizes 1-dimensional positive definite subspaces in the lat-tice Λ, however complicates the discussion. First, properties of a simpler mock Siegel-Naraintheta function are analyzed before those of Ψ ∗ µ ⊕ ( τ, C, B ).Let Λ, Λ and Λ be three lattices with signature (1 , b − d abc : respectively d abc P c , d abc P c and d abc P c . Thevectors P ( i ) are characteristic vectors of the lattices and positive: P i ) >
0. They are relatedby P = P + P . The projection of a vector x ∈ Λ ⊗ R on the positive definite subspace isdetermined by the vector J ∈ Λ ⊗ R : x + = ( x · J/P · J ) J , x − = x − x + , and x = x + x − . Thepositive definite combination x − x − is called the majorant associated to J . It is sufficientfor this appendix that J lies in the space C Λ := n J ∈ Λ ⊗ R : P ( i ) · J , P i ) · J > , i = 1 , o .J is thus positive in all three lattices.The direct sum Λ ⊕ Λ is denoted by Λ ⊕ with quadratic form Q ⊕ = ( Q ) + ( Q ) for Q = ( Q , Q ) ∈ Λ ∗ ⊕ . Vectors in Λ ⊕ are sometimes given the subscript 1 ⊕
2, and in Λ i thesubscript i . For example, P ⊕ = P + P ∈ Λ ⊕ . Similarly, µ ⊕ = µ + µ ∈ Λ ∗ ⊕ / Λ ⊕ ,and µ = µ + µ ∈ Λ ∗ / Λ with µ i ∈ Λ ∗ i / Λ i . With a slight abuse of notation Q denotes(( Q + Q ) · J ) /P · J .Define I ( Q , Q ; t ) as in the main text by I ( Q , Q ; t ) = P · J ( Q − P B ) · J − P · J ( Q − P B ) · J √ P · J P · J P · J . (A.1)Define additionally the vector P = ( − P , P ) √ P P P ∈ Λ ⊕ ⊗ R , (A.2)which satisfies P = 1. Definition 1.
Let t = B + iJ , with B ∈ Λ ⊗ R , and J ∈ C Λ . Then Φ ∗ µ ⊕ ( τ, C, B ) is defined26y:Φ ∗ µ ⊕ ( τ, C, B ) = P Q ∈ Λ ⊕ + µ ⊕ + P ⊕ / ( − P · Q + P · Q (cid:0) E (cid:0) I ( Q , Q ; t ) √ τ (cid:1) − E (cid:0) P · Q √ τ (cid:1) (cid:1) (A.3) × e (cid:0) τ ( Q − B ) / τ (( Q − B ) ⊕ − ( Q − B ) ) / Q − B/ · C (cid:1) , with E ( z ) = 2 Z z e − πu du = sgn( z ) (cid:0) − β ( z ) (cid:1) , where β ( x ) = Z ∞ x u − e − πu du, x ∈ R ≥ . The moduli in the exponent of (A.3) are determined by t . The “ * ” of Φ ∗ µ ⊕ ( τ, C, B ) dis-tinguishes this function from Φ µ ⊕ ( τ, C, B ), which would be defined by replacing E ( z ) bysgn( z ) in the definition. Proposition 1. Φ ∗ µ ⊕ ( τ, C, B ) is convergent for J ∈ C Λ and B, C ∈ Λ ⊗ R . Proof.
First consider the case B = C = 0. The term which multiplies τ in the exponent,and thus determines the absolute value of the exponential is Q J := Q ⊕ − Q + Q ) · J ) P · J = Q ⊕ − Q . (A.4)The signature of this quadratic form is (1 , b −
1) which is problematic for convergence.To show convergence, note that 0 ≤ β ( x ) ≤ e − πx for all R ≥ and that therefore theterms involving β ( x ) in (A.3) are convergent. Consider next the terms with sgn( P · Q ) − sgn( I ( Q , Q ; iJ )). There are essentially two possibilities: sgn( P · Q ) sgn( I ( Q , Q ; iJ )) < >
0. Define the vector s ( J ) = ( − P · J J, P · J J ) √ P · J P · J P · J ∈ Λ ⊕ ⊗ R , such that Q · s ( J ) = I ( Q , Q ; iJ ) and s ( J ) = 1.One can show that P · s ( J ) = q P · J ( P P J ) P P P P · J P · J > P + = s ( J ) + = 0. The spacespan( P , s ( J )) has signature (1 ,
1) in Λ ⊕ with inner product Q J . Therefore (cid:12)(cid:12)(cid:12)(cid:12) P · s ( J ) P · s ( J ) 1 (cid:12)(cid:12)(cid:12)(cid:12) = 1 − ( P · s ( J )) < . Q ∈ Λ ⊕ , which is linearly independent of P and s ( J ), then span( Q, P , s ( J ))is a space with signature (1 , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q J Q · P Q · s ( J ) Q · P P · s ( J ) Q · s ( J ) P · s ( J ) 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > . From this follows directly Q J + 2 P · s ( J )1 − ( P · s ( J )) Q · P Q · s ( J ) < ( Q · P ) + ( Q · s ( J )) − ( P · s ( J )) < . (A.5)Therefore, if sgn( P · Q ) sgn( I ( Q , Q ; iJ )) < Q J <
0. If Q is a linear combination of P and s ( J ), the determinant is zero. From this follows that Q J = 0 only for Q = 0, andotherwise Q J <
0. The sum for sgn( Q · P ) sgn( Q · J ) < >
0. Then all the terms vanish identically, and therefore the wholesum is convergent. Inclusion of B and C does not alter the final conclusion. Proposition 2. Φ ∗ µ ⊕ ( τ, C, B ) transforms under the generators S and T of SL (2 , Z ) as: S : Φ ∗ µ ⊕ ( − /τ, − B, C ) = − i ( − iτ ) / ( i ¯ τ ) b − / p | Λ ∗ / Λ || Λ ∗ / Λ | e ( − P ⊕ / X ν ⊕ ∈ Λ ∗ ⊕ / Λ ⊕ e ( − µ ⊕ · ν ⊕ ) Φ ∗ ν ⊕ ( τ, C, B ) ,T : Φ ∗ µ ⊕ ( τ + 1 , B + C, B ) = e (( µ ⊕ + P ⊕ / ⊕ /
2) Φ ∗ µ ⊕ ( τ, C, B ) , Proof.
The S -transformation is proven using P k ∈ Λ f ( k ) = P k ∈ Λ ∗ ˆ f ( k ), with ˆ f ( k ) the Fouriertransform of f ( k ). Therefore, one needs to determine the following Fourier transform: Z Λ ⊕ ⊗ R d b x E (cid:16) I ( x , x ; iJ ) p − /τ ) (cid:17) × exp (cid:0) πi Re( − / ¯ τ ) x ⊕ + π Im( − / ¯ τ )( x ⊕ − x ) + 2 πi x · y (cid:1) (A.6)= Z Λ ⊕ ⊗ R d b x E (cid:16) I ( x , x ; iJ ) p − /τ ) (cid:17) e (cid:0) − x / τ − ( x ⊕ − x ) / τ + x · y (cid:1) , and the one with I ( x , x ; iJ ) replaced by Q · P . The following concentrates on the case with I ( x , x ; iJ ), the derivation for Q · P is completely analogous.Let Q · s ( J ) = I ( Q , Q ; iJ ) as in Proposition 1, then the following definite quadraticforms can be defined: Q ⊕ = Q + ( Q · s ( J )) , Q ⊕ − = Q ⊕ − Q ⊕ , J, J ) · s ( J ) = 0. Using these quadratic forms, we write e (cid:0) − x / τ − ( x ⊕ − x ) / τ (cid:1) = e (cid:0) − x / τ − ( x · s ( J )) / τ − x ⊕ − / τ (cid:1) The Fourier transform can be written in the form= e (cid:0) τ y / τ I ( y , y ; iJ ) / τ y ⊕ − (cid:1) × Z Λ ⊕ ⊗ R d b x E (cid:16) I ( x , x ; iJ ) p − /τ ) (cid:17) × e (cid:0) − ( x − yτ ) / τ − I ( x − y ¯ τ , x − y ¯ τ ; iJ ) / τ − ( x − y ¯ τ ) ⊕ − / τ (cid:1) . To proceed, one calculates the derivative of the integral ∂∂ I ( y , y ; iJ ) Z Λ ⊕ ⊗ R d b x E (cid:16) I ( x , x ; iJ ) p − /τ ) (cid:17) × e (cid:0) − ( x − yτ ) / τ − I ( x − y ¯ τ , x − y ¯ τ ; iJ ) / τ − ( x − y ¯ τ ) ⊕ − / τ (cid:1) = − i ( − iτ ) / ( i ¯ τ ) b − / p | Λ ∗ / Λ || Λ ∗ / Λ | ∂E (cid:0) I ( y , y ; iJ ) √ τ (cid:1) ∂ I ( y , y ; iJ ) . This is shown by replacing the derivative by − ¯ τ ∂ I ( x ,x ; iJ ) , acting only on the exponent; andperforming a partial integration. The equality is then easily established. Since (A.6) is anodd function of y , the integration constant is 0. Therefore (A.6) is equal to − i ( − iτ ) / ( i ¯ τ ) b − / p | Λ ∗ / Λ || Λ ∗ / Λ | E (cid:0) I ( y , y ; iJ ) √ τ (cid:1) (A.7) × e (cid:0) τ y / τ I ( Q , Q ; iJ ) / τ y ⊕ − (cid:1) Using the standard techniques to include B - and C -field dependence etc., one finds the posedtransformation law. Note that P · ( Q − BP , Q − BP ) = P · ( Q , Q ) = P · Q . The proofof the T -transformation is standard. Proposition 3.
Define D = ∂ τ + i π ∂ C + + B + · ∂ C + − πiB , then τ / D Φ µ ⊕ ( τ, C, B )is a modular form of weight (2 , b − Proof.
The action of D on the exponents vanishes, and therefore only the derivative to τ onthe functions E ( z √ τ ) remains. The proposition follows easily from here.29 efinition 2. With the same input as for Definition 1:Ψ ∗ µ ⊕ ( τ, C, B ) = π √ τ (cid:18)q P · J ( P P J ) P · J P · J Θ µ ( τ, C, B ) Θ µ ( τ, C, B ) − √ P P P Θ µ ⊕ ( τ, C, B, P ) (cid:19) + P Q ∈ Λ ⊕ + µ ⊕ + P ⊕ / ( − P · Q + P · Q ( P · Q − P · Q ) (A.8) × (cid:0) E (cid:0) I ( Q , Q ; t ) √ τ (cid:1) − E (cid:0) P · Q √ τ (cid:1) (cid:1) × e (cid:0) τ ( Q − B ) / τ (( Q − B ) ⊕ − ( Q − B ) ) / Q − B/ · C (cid:1) with Θ µ i ( τ, C, B ) as defined by Eq. (3.6), summing over Λ i . Θ µ ⊕ ( τ, C, B, P ) is defined byΘ µ ⊕ ( τ, C, B, P ) = X Q ∈ Λ ⊕ + P ⊕ / µ ⊕ ( − P ⊕ · Q × e (cid:0) τ ( Q − B ) / τ ( P · Q ) / τ ( Q − B ) ⊕ − / C · ( Q − B/ (cid:1) . In the limit τ → ∞ , Ψ ∗ µ ⊕ ( τ, C, B ) approaches Ψ µ ⊕ ( τ, C, B ), which is defined in Eq. (4.8).This series is convergent because Φ ∗ µ ⊕ ( τ, C, B ) is convergent. Proposition 4. Ψ ∗ µ ⊕ ( τ, C, B ) transforms under the generators S and T of SL (2 , Z ) as: S : Ψ ∗ µ ⊕ ( − /τ, − B, C ) = − ( − iτ ) / ( i ¯ τ ) b +1 / p | Λ ∗ / Λ || Λ ∗ / Λ | e ( − P ⊕ / X ν ⊕ ∈ Λ ∗ ⊕ / Λ ⊕ e ( − µ ⊕ · ν ⊕ ) Ψ ∗ ν ⊕ ( τ, C, B ) ,T : Ψ ∗ µ ⊕ ( τ + 1 , B + C, B ) = e (( µ ⊕ + P ⊕ / ⊕ /
2) Ψ ∗ µ ⊕ ( τ, C, B ) , Proof.
This is a continuation of the proof of Proposition 2. The following Fourier transformneeds to be calculated: Z Λ ⊕ ⊗ R d b x ( P · x − P · x ) E (cid:16) I ( x , x ; iJ ) p − /τ ) (cid:17) (A.9) × e (cid:0) − ( x ⊕ − x ) / τ − x / τ + x · y (cid:1) , and the one with I ( x , x ; iJ ) replaced by P · Q . We again concentrate on the case with I ( x , x ; iJ ). It is instructive to write P · x − P · x as ( − P , P ) · x T with x = ( x , x ).The inner product ( − P , P ) · x + with x + = x · J J/P · J vanishes. Therefore,( − P , P ) · x T = ( − P , P ) · x T − + ( − P , P ) · s ( J ) T x · s ( J )= ( − P , P ) · x T − + q P · J ( P P J ) P · J P · J I ( x , x ; iJ ) , s ( J ) ∈ Λ ⊕ as in the proof of Proposition 1. This shows that the factor P · x − P · x can be replaced by (2 πi ) − (cid:18) ( − P , P ) · ∂ y − + q P · J ( P P J ) P · J P · J ∂ I ( y ,y ; iJ ) (cid:19) . Using Proposition2, one finds that (A.9) equals − ( − iτ ) / ( i ¯ τ ) b / √ | Λ ∗ / Λ || Λ ∗ / Λ | (cid:2) ( P · y − P · y ) E (cid:0) I ( y , y ; iJ ) √ τ (cid:1) e (cid:0) τ y / τ ( y ⊕ − y ) / (cid:1) + √ τ πi ¯ τ q P · J ( P P J ) P · J P · J e (cid:0) τ y / τ I ( y , y ; iJ ) / τ y ⊕ − / (cid:1) (cid:3) . Clearly, this Fourier transform leads to a shift in the modular transformation properties.This can be cured if one recalls the transformation properties of the second Eisenstein series: E ( − /τ ) = τ ( E ( τ ) − iπτ ). A correction term can be added to E ( τ ): E ∗ ( τ ) = E ( τ ) − πτ which transforms as a modular form of weight 2. This leads precisely to the term withtheta functions in the definition. This means that the discontinuous function z sgn( z ), whichappears in (4.8), is replaced in Ψ ∗ µ ⊕ ( τ, C, B ) by the real analytic function F ( z ) = z E ( z ) + π e − πz . F ( z ) approaches z sgn( z ) for | z | → ∞ . Proposition 5.
With D as in Proposition 3 D Ψ ∗ µ ⊕ ( τ, C, B ) = − i √ τ q P · J ( P P J ) P · J P · J Υ µ ⊕ ( τ, C, B )+ i π (2 τ ) / (cid:0) Θ µ ( τ, C, B )Θ µ ( τ, C, B ) − Θ µ ⊕ ( τ, C, B, P ) (cid:1) , with Υ µ ⊕ ( τ, C, B ) = X Q ∈ Λ ⊕ + P ⊕ / µ ⊕ ( − P ⊕ · Q ( − P , P ) · Q − I ( Q , Q ; t ) × e (cid:0) τ ( Q − B ) / τ I ( Q , Q ; t ) / τ ( Q − B ) ⊕ − / C · ( Q − B/ (cid:1) Proof.
The proof is straightforward. Note that Θ µ i ( τ, C, B ) and Υ µ ⊕ ( τ, C, B ) are not mockmodular forms. The weights are respectively (1 , b −
1) and (2 , b ), such that the weight of D Ψ ∗ µ ⊕ ( τ, C, B ) is (5 / , (2 b + 1) /
2) as expected.
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