Adding Flavor to the Narain Ensemble
Shouvik Datta, Sarthak Duary, Per Kraus, Pronobesh Maity, Alexander Maloney
CCERN-TH-2021-020
Adding Flavor to the Narain Ensemble
Shouvik Datta , Sarthak Duary , Per Kraus , Pronobesh Maity & Alexander Maloney Department of Theoretical Physics, CERN,1 Esplanade des Particules, Geneva 23, CH-1211, Switzerland. International Centre for Theoretical Sciences-TIFR,Shivakote, Hesaraghatta Hobli, Bengaluru North 560 089, India. Mani L. Bhaumik Institute for Theoretical Physics,Department of Physics and Astronomy,University of California, Los Angeles, CA 90095, USA. Department of Physics, McGill University,Montr´eal, QC, H3A 2T8, Canada.
Abstract
We revisit the proposal that the ensemble average over free boson CFTs intwo dimensions — parameterized by Narain’s moduli space — is dual to an exotictheory of gravity in three dimensions dubbed U (1) gravity. We consider flavoredpartition functions, where the usual genus g partition function is weighted byWilson lines coupled to the conserved U (1) currents of these theories. Theseflavored partition functions obey a heat equation which relates deformations ofthe Riemann surface moduli to those of the chemical potentials which measurethese U (1) charges. This allows us to derive a Siegel-Weil formula which computesthe average of these flavored partition functions. The result takes the form of a“sum over geometries,” albeit with modifications relative to the unflavored case. a r X i v : . [ h e p - t h ] F e b ontents τ → One of the most striking observations in the study of quantum gravity is that certain simplegravitational theories – primarily those in a low number of space-time dimensions – appearto be described not by a single quantum theory, but rather by an ensemble average of manytheories. This phenomenon was initially described for Jackiw-Teitelboim gravity in AdS ,which is dual to a random matrix theory [1]. This is a prototypical example of an AdS /CFT duality. In order to understand higher dimensional versions of this phenomenon, one wouldlike to understand ensembles of random conformal field theories which are dual to putativetheories of gravity in Anti-de Sitter space. At first sight, constructing a random conformalfield theory seems quite difficult, as it would involve an ensemble average over the space ofconformal field theories, a space which is itself quite poorly understood. For this reason,1ecent work in this direction [2, 3] has focused on CFTs with enhanced symmetry algebraswhere the space of CFTs can be understood precisely (related works in this direction include[4–8]).The natural starting point is perhaps the simplest possible family of two dimensionalCFTs: unitary, compact CFTs with U (1) D × U (1) D current algebra and central charge c = D . These are simply theories of D free compact bosons, and the data which defines sucha theory is an even, self-dual lattice of signature ( D, D ). The moduli space of such theoriesis the homogeneous space [9, 10] M D = O ( D, D, Z ) \ O ( D, D ) /O ( D ) × O ( D ) . (1.1)This space has finite volume, and a unique homogeneous metric which can be used to define aprobability distribution on the associated space of CFTs. The work of [2, 3] argued that thisensemble average is dual to an exotic three dimensional theory of gravity in AdS dubbed“ U (1) gravity.” This theory of gravity includes as its perturbative degrees of freedom a U (1) D Chern-Simons theory describing the gauge dynamics dual to the U (1) D global symmetryin the boundary. The non-perturbative structure of the theory is defined by a sum overgeometries in the bulk. Together these ingredients were shown to reproduce the ensembleaverage of the genus g partition function, which was computed using the Siegel-Weil formulain terms of a real analytic Eisenstein series [11–13].The genus g partition function, however, is not the most general observable of the theory.The theory contains global U (1) charges, so one can in addition consider “flavored” partitionfunctions which include fugacities that couple to these global U (1) charges. For example, onthe torus one can consider the flavored partition function Z (cid:0) τ, ¯ τ , z IL , z IR (cid:1) = Tr (cid:104) e πiτ ( L − c ) e − πi ( L − c ) e πiz IL J I e − πiz IR ¯ J I (cid:105) , (1.2)which depends on both the conformal structure parameter τ as well as a D -component vector( z IL , z IR ) of chemical potentials. Geometrically, these chemical potentials can be interpretedas background Wilson lines which couple to the global U (1) charges ( Q I , ¯ Q I ) of a state. Athigher genus, one can consider more general flavored partition functions which include Wilsonlines wrapping arbitrary cycles in the boundary surface.The natural question is then: is there a version of the Siegel-Weil formula which allowsone to compute the ensemble average of these more general observables? And second – andperhaps more importantly – does the result yield some insights into the structure of thetheory and its gravity dual beyond the higher genus partition functions considered in [2]?The answer to the first question is, in fact, not difficult. The observation begins with the fact(that we will explain in much more detail below) that the counting function for primaries,2( z L , z R , τ, ¯ τ ), obeys a version of the heat equation: ∂ Θ ∂τ = 14 πi ∇ z L Θ , ∂ Θ ∂τ = − πi ∇ z R Θ . (1.3)This equation follows from the fact that the stress tensor of a free boson theory is Sugawara,and hence a composite operator quadratic in the U (1) currents; this relates variations withrespect to the conformal structure to variations with respect to the U (1) gauge potentials. Byaveraging this equation over Narain moduli space we will completely determine the ensembleaverage of the flavored partition function, a novel (and somewhat less commonly studied)version of the Siegel-Weil formula.Equation (1.3) hints as well towards an answer to our second question, as it allows us totrade conformal structure dependence for dependence on the fugacities. At the level of thetorus partition function this is not particularly interesting, as it simply reflects the fact thatthe dimensions and spins of primary operators are uniquely given by their U (1) charges∆ = Q · Q + ¯ Q · ¯ Q , j = Q · Q − ¯ Q · ¯ Q . (1.4)The situation at higher genus is considerably more interesting since the dependence of thehigher genus partition function on conformal structure encodes not just the dimensions andspins of primary states but also the operator product expansion coefficients. In a free CFT,however, the OPE coefficients are completely determined by charge conservation C Q ,Q ,Q ∝ δ ( Q + Q + Q ) . (1.5)Thus one might expect that all of the data of a higher genus partition function can becompletely packaged into information about the corresponding conserved charges. Indeed,we will see that this is the case by writing down a higher genus version of the heat equation(1.3). An interesting feature of this result is that it is possible to go to the boundary ofmoduli space where a higher genus surface degenerates into a disjoint union of tori. Theresult is that all of the data contained in a genus g partition function can be repackaged intothe data of the g th moment of the (flavored) torus partition function: (cid:104) Z ( τ , z ) Z ( τ , z ) · · · Z ( τ g , z g ) (cid:105) ↔ (cid:104) Z g ( τ ) (cid:105) . (1.6)The averages of these quantities are given by appropriate Eisenstein series, just as in theunflavored case. In a sense, therefore, this perspective allows us to completely dispensewith the higher genus partition functions and consider only statistical properties of the toruspartition function. An additional interesting feature of our result is that it allows us to This may provide an interesting perspective on the analogy between sphere packing and the modularbootstrap described in [14, 15]. The natural question following [15] is: what is the sphere packing analogue ofthe conformal bootstrap constraints which go beyond torus modular invariance, such as higher genus modularsymmetry or the crossing symmetry of local correlation functions? Our considerations suggest the followinganswer: modular properties of higher moments of the theta series appearing in the sphere packing problem. (cid:104) ρ (∆ , j, Q I ) (cid:105) and thetwo point function (cid:104) ρ (∆ , j , Q I ) ρ (∆ , j , Q J ) (cid:105) ; it turns out that by including dependence oncharge, one finds expressions which are considerably simpler than those which have previouslyappeared in the literature.Turning to the holographic interpretation, we show that the statement [2, 3] that theaveraged partition function can be naturally reproduced in terms of U (1) D × U (1) D Chern-Simons theory generalizes to the flavored case. The chemical potentials appearing in theflavored partition function map to a choice of boundary conditions in the Chern-Simonstheory, in a manner which enforces the proper behavior under modular transformations.This paper is structured as follows. In Section 2 we begin with a few remarks on theNarain moduli space and define the averaging procedure for partition functions. The flavoredpartition function on the torus is evaluated in Section 3 using a generalization of the Laplaceequation, as well as via a heat equation. The analysis for the partition function is generalizedto higher genus in Section 4. Section 5 reproduces the flavored partition function from U (1) D × U (1) D Chern-Simons theory in AdS . In this section we recall aspects of free CFTs in two dimensions, with emphasis on theirsymmetries and moduli spaces.We consider the theory of D real compact bosons X I , I = 1 , , . . . D , and its associated U (1) D × U (1) D current algebra. Current algebra primaries are given by the vertex operators V l = e il L · X L + il R · X R . (2.1)The momentum vectors l ≡ ( l IL , l IR ) live in a lattice Γ, which has a signature ( D, D ) innerproduct: l ◦ l ≡ l L · l L − l R · l R . (2.2)The choice of lattice Γ labels different possible CFTs, i.e. different compactifications of thefree bosons. This choice is constrained by modular invariance of the torus partition function.First, invariance under τ → τ + 1 (i.e. the quantization of spin) implies that Γ is even, i.e.that the vectors ( l IL , l IR ) obey l ◦ l ∈ Z . Second, invariance under τ → − /τ implies that Γis self-dual, i.e. that Γ ∗ = Γ, where the dual lattice Γ ∗ consists of all vectors with integer ◦ product with all elements of Γ. An even, self-dual lattice of signature ( D, D ) is known as aNarain lattice. In string theory language we are setting α (cid:48) = 2. L , ˜ L ) are L = 12 l L + N , ˜ L = 12 l R + ˜ N , (2.3)where N, ˜ N ∈ Z are the integer valued oscillator levels. The (unflavored) partition functionis Z Γ ( τ ) = 1 | η ( τ ) | D (cid:88) l ∈ Γ e iπτl L − iπτl R . (2.4)Here the prefactor counts the oscillator states, i.e. the descendants under the U (1) D × U (1) D current algebra, and the lattice sum counts primaries. The partition function is modularinvariant, in the sense that Z Γ ( γτ ) = Z Γ ( τ ) , γ = (cid:18) a bc d (cid:19) ∈ SL (2 , Z ) , γτ ≡ aτ + bcτ + d . (2.5)Given a Narain lattice Γ, one can always apply an O ( D, D ) rotation Λ to produce anotherNarain lattice Γ Λ ≡ ΛΓ, with Λ ∈ O ( D, D ). In fact, it is not hard to show that any
Narainlattice may be obtained by some O ( D, D ) rotation of a fixed reference lattice Γ . However,not all such O ( D, D ) rotations yield distinct CFTs. First, an O ( D ) × O ( D ) ∈ O ( D, D )rotation will act as a symmetry of a particular theory, since its effect can be undone by acompensating O ( D ) × O ( D ) field redefinition of the fields ( X L , X R ). The result is that thespectrum of vertex operators and their OPE coefficients will be unchanged by such a rotation.Second, a subgroup of O ( D, D ) will leave the lattice Γ itself invariant. This subgroup is just O ( D, D, Z ), as can be seen by taking our reference lattice Γ to be the integer lattice in R D,D .The result is that the moduli space of inequivalent Narain theories is the coset M D ≡ O ( D, D, Z ) \ O ( D, D ) /O ( D ) × O ( D ) . (2.6)This moduli space has dimension D .This space of free theories can also be described more explicitly as σ -models, with action S = 14 π (cid:90) d σ (cid:0) √ gg αβ G IJ ∂ α X I ∂ β X J + (cid:15) αβ B IJ ∂ α X I ∂ β X J (cid:1) . (2.7)Here the boson fields have been scaled to have integer periodicities: X I ∼ = X I + 2 πm I , m I ∈ Z , so the choice of theory has been packaged into the target space metric G IJ and B -field B IJ . These are constant symmetric and antisymmetric matrices, respectively, whichcan be combined into a D × D matrix E IJ = G IJ + B IJ . (2.8)5ne can think of E as a coordinate on the moduli space M D .To understand the Narain moduli space in this language, we introduce the O ( D, D )element g that acts on the matrix E as g : E → gE ≡ ( aE + b )( cE + d ) − , (2.9)where g = (cid:32) a bc d (cid:33) , g T J g = J , J = (cid:32) II (cid:33) (2.10)is an element of O ( D, D ). Any matrix E is invariant under some O ( D ) × O ( D ) subgroupof O ( D, D ). This can be seen by first noting that E = I is invariant under the action ofmatrices of the form g = (cid:32) a a (cid:33) , (cid:32) bb (cid:33) , a T a = b T b = I . (2.11)The corresponding statement for general E is obtained by conjugating by the action of O ( D, D ).To write the spectrum, we introduce the matrix M = (cid:32) G − BG − B BG − − G − B G − (cid:33) . (2.12)This is convenient because the O ( D, D ) rotations act equivariantly on M , in the sense that g : M → gM g T . (2.13)Since in the σ -model formulation the fields have integer periodicities, the primary states ofthe theory can be labelled by a vector of integers ( m I , n I ). In terms of these, the spin L − ¯ L and dimension L + ¯ L of a given primary state is l L − l R = 2 m I n I , l L + l R = Z T M Z , Z ≡ (cid:32) m I n I (cid:33) . (2.14)The T-duality group O ( D, D, Z ) is given by those g for which the entries of g T Z are integer.In this case the action (2.13) is the usual Buscher rule for the T-duality transformation ofthe target space metric and B -field.It is important to note that the moduli space M D defined above is a homogeneous spacewhich has a unique Riemannian metric which is invariant under an O ( D, D ) isometry group(generated, in terms of the coset structure, by left multiplication). This coincides with the6sual “Zamolodchikov” metric on the CFT moduli space, and is the natural one to use whenconsidering averages over this space of theories. In particular, we average over moduli spaceby integrating: (cid:104)·(cid:105) = 1
V ol ( M D ) (cid:90) M D ( · ) dµ (2.15)where dµ is the associated invariant measure. We have divided by the volume of M D inorder to properly normalize this measure as a probability distribution. It is important tonote that, although O ( D, D ) has infinite volume, the moduli space M D has finite volumewhen D >
1. This is due to the fact that we have quotiented by the action of the T-dualitygroup; without such a quotient, an interpretation of dµ as a normalizable probability measurewould be impossible.We wish to study the flavored partition function, which is obtained by introducing a setof 2D chemical potentials z ≡ ( z IL , z IR ) that couple to the U (1) D × U (1) D charges of a state.These charges are just the individual components of the lattice vector l = ( l IL , l IR ), so theflavored partition function is Z Γ ( τ, z ) = 1 | η ( τ ) | D (cid:88) l ∈ Γ e iπτl L − iπτl R +2 πiz L · l L − πiz R · l R . (2.16)We note that only the lattice sum has been modified; the prefactor remains the same, becausethe action of the U (1) current algebra will not change the charge of a state.There is one important distinction, which is that the potentials z are not invariant underthe O ( D ) × O ( D ) rotations described above. The reason is easy to understand. Given apoint in moduli space corresponding to a choice of E there is an equivalence class of latticesrelated by O ( D ) × O ( D ) rotations, all corresponding to the same CFT. However, in a givenCFT there are many possible choices of basis for the U (1) D × U (1) D symmetry algebra, whichare related precisely by these O ( D ) × O ( D ) transformations. When we introduce potentials z we have implicitly made a choice of basis. So the flavored partition function should beviewed as a function on the space of Narain lattices O ( D, D, Z ) \ O ( D, D ) rather than on themoduli space of CFTs (2.6). This will be important when we consider the average of flavoredquantities, because we must now integrate over this larger moduli space. In particular, wewill consider averages of the form (cid:104)·(cid:105) = 1Vol( O ( D, D, Z ) \ O ( D, D )) (cid:90) O ( D,D, Z ) \ O ( D,D ) ( · ) dµ (2.17)For quantities which are O ( D ) × O ( D ) invariant (such as unflavored partition functions) thisreduces to the average over M D described above. But this procedure can now be applied toflavored quantities as well. 7 Siegel-Weil formula for flavored partitionfunctions: torus case
In this section we will compute the average of flavored CFT partition functions on the torus.We will do so by showing that it satisfies a set of differential equations, combined withknowledge of its behavior at the boundary of moduli space. We will begin with a review ofthe unflavored case, before describing two differential equations – both a “Laplace equation”and a “heat equation” – obeyed in the flavored case. This latter equation in particular willallow us to easily reduce the computation of the averaged flavored partition function to theunflavored case.
We begin by describing the Laplace equation obeyed by the partition function, which was usedby [2] to derive the Siegel-Weil formula in the unflavored case. We will present a streamlinedderivation of this equation in a form which can be easily adapted to the flavored case.We start by writing the partition function as Z Γ ( τ ) = 1 | η ( τ ) | D Θ Γ ( τ ) , Θ Γ ( τ ) ≡ (cid:88) l ∈ Γ Q ( l, τ ) (3.1)where Q ( l, τ ) ≡ e iπτl L − iπτl R = e iπτ ( l L − l R ) e − πτ ( l L + l R ) . (3.2)with τ = τ + iτ , τ = τ − iτ . We have separated out the theta function Θ Γ ( τ ) which countsprimary states. We denote the Laplacian acting on the modular parameter τ as∆ H = − τ (cid:18) ∂ ∂τ + ∂ ∂τ (cid:19) = − τ ∂ ∂τ ∂τ . (3.3)It is then straightforward to check that∆ H Q ( l, τ ) = − π τ l L l R Q ( l, τ ) . (3.4)We now consider the Laplacian ∆ M acting on the moduli space of Narain lattices. Whilewe could write this operator in terms of the ( G IJ , B IJ ) target space fields, it is simpler to thinkof this Laplacian as an operator on the O ( D, D ) group manifold. Since Q ( l, τ ) is invariantunder O ( D ) × O ( D ) rotations, these two versions of the Laplacian will be proportional toone another. We start by defining O ( D, D ) as the linear transformations which preserve the8uadratic form η AB Z A Z B where A, B = 1 , , . . . , D and η AB = diag(1 D , − D ). Writing the O ( D, D ) generators as J AB = η BC Z A ∂∂Z C − η AC Z B ∂∂Z C (3.5)the quadratic Casimir is J = η AC η BD J AB J CD . (3.6)We now use the fact that the charge vector l = ( l IL , l IR ) transforms as a (contravariant) vectorunder the O ( D, D ) rotations. In particular, we can assemble these charges into an O ( D, D )vector Z A as: l IL = Z I , l IR = Z D + I , I = 1 , , . . . D . (3.7)Since the quadratic Casimir is proportional to the Laplacian, this provides an explicit ex-pression for the Laplacian as a differential operator.Explicitly, when acting on functions of the charge vector the quadratic Casimir takes theform J = L IJ L IJ + R IJ R IJ + 2 T IJ T IJ , (3.8)with L IJ = l IL ∂∂l JL − l JL ∂∂l IL , R IJ = l IR ∂∂l JR − l JR ∂∂l IR , T IJ = l IL ∂∂l JR + l JR ∂∂l IL . (3.9)Since the l L,R are annihilated by L IJ and R IJ , and l L − l R is annihilated by all the generators,we have J Q ( l, τ ) = e iπτ ( l L − l R ) × (cid:104) T IJ T IJ e − πτ ( l L + l R ) (cid:105) . (3.10)An elementary computation yields e iπτ ( l L − l R ) T IJ T IJ e − πτ ( l L + l R ) = 8 (cid:20) τ ∂ ∂τ ∂τ + Dτ ∂∂τ (cid:21) e iπτl L − iπτl R = − (cid:20) ∆ H − Dτ ∂∂τ (cid:21) e iπτl L − iπτl R . (3.11)We will normalize our Laplacian as ∆ M = − J (3.12)9o match [2], so that our result reads (cid:20) ∆ H − Dτ ∂∂τ − ∆ M (cid:21) Q ( l, τ ) = 0 . (3.13)We will rewrite this as[∆ H + s ( s − − ∆ M ] (cid:16) τ D/ Q ( l, τ ) (cid:17) = 0 , s ≡ D/ . (3.14)We can now sum this over Γ to conclude that the theta function obeys the same differentialequation: [∆ H + s ( s − − ∆ M ] (cid:16) τ D/ Θ Γ ( τ ) (cid:17) = 0 . (3.15)In this expression ∆ M is now the Laplacian on the space of Narain lattices Γ. We notethat, since | η ( τ ) | − D and τ D/ have the same modular transformation properties, τ D/ Θ Γ ( τ )is modular invariantWe now integrate this equation over the moduli space M D to obtain an equation for theobject H ( τ ) ≡ τ D/ (cid:104) Θ Γ ( τ ) (cid:105) . (3.16)The crucial observation is that, since ∆ M Θ Γ ( τ ) is a total derivative on M D , its integralvanishes. The result is that H ( τ ) is a modular invariant eigenfunction of the Laplacian onthe upper half plane: [∆ H + s ( s − H ( τ ) = 0 . (3.17)One solution to this equation is the Eisenstein series E D/ ( τ ) ≡ τ D/ (cid:88) ( c,d )=1 | cτ + d | D . (3.18)We can now argue that in fact H = E ( τ ). One way to do so is to note that, sincewe are considering modular invariant functions, we can effectively view this as a Laplaceequation on the fundamental domain H /SL (2 , Z ) which has finite volume. Compactifying To argue that this one must in addition show that the surface terms arising on the boundary of M D vanish. It is easy to see that this occurs when D > To see that this is an eigenfunction of the Laplacian we note that τ D/ is itself an eigenfunction of ∆ H with the correct eigenvalue. The Eisenstein series is the sum of this eigenfunction over the modular group, E ( τ ) = (cid:80) γ ∈ SL (2 , Z ) / Z γτ D/ , which gives a modular invariant eigenfunction with the same eigenvalue. Herethe subgroup Z is the set of matrices (cid:0) n (cid:1) which leave τ invariant. The coset SL (2 , Z ) / Z can then labelledby pairs of coprime integers ( c, d ) which make up the lower row of an SL (2 , Z ) matrix, giving the form of theEisenstein series given in equation (3.18). τ = i ∞ ), we can use the uniquenessof solutions to the Laplace equation with negative eigenvalue. One only has to check that H ( τ ) and E D/ ( τ ) have the same behavior as τ → ∞ . Putting this together gives theSiegel-Weil formula for the torus partition function: (cid:104) Z Γ ( τ ) (cid:105) = τ D/ | η | D (cid:88) ( c,d )=1 | cτ + d | D . (3.19) We now extend this to the flavored partition function, which we write as Z Γ ( τ, z ) = 1 | η ( τ ) | D Θ Γ ( τ, z ) , Θ Γ ( τ, z ) ≡ (cid:88) l ∈ Γ P ( l, z, τ ) , (3.20)where again the function Θ Γ ( τ, z ) counts the contribution of primary states, and P ( l, z, τ ) = e iπτl L − iπτl R +2 πiz L · l L − πiz R · l R . (3.21)The flavored partition function is not modular invariant, but instead transforms covariantlyas Z Γ (cid:18) aτ + bcτ + d , a ¯ τ + bc ¯ τ + d , z IL cτ + d , z IR c ¯ τ + d (cid:19) = exp (cid:20) icπz L cτ + d − icπz R c ¯ τ + d (cid:21) Z Γ ( τ, ¯ τ , z IL , z IR ) . (3.22)We will begin by deriving a version of the Laplace equation. In our derivation of theLaplace equation above, we used the fact that the charge vector l = ( l IL , l IR ) could be packagedinto a contravariant vector under O ( D, D ) transformations. Similarly, the chemical potentials z = ( z IL , z IR ) can be assembled into a covariant vector under O ( D, D ), since the inner product z L · l L − z R · l R is O ( D, D ) invariant. The result is that the O ( D, D ) generators (3.9) will,when acting on functions of both the charge vector l and the chemical potential z , take theform L IJ = l IL ∂∂l JL − l JL ∂∂l IL − z IL ∂∂z JL + z JL ∂∂z IL R IJ = l IR ∂∂l JR − l JR ∂∂l IR − z IR ∂∂z JR + z JR ∂∂z IR T IJ = l IL ∂∂l JR + l IR ∂∂l JL + z JL ∂∂z IR + z JR ∂∂z IL . (3.23)The quadratic Casimir on O ( D, D ) will again take the formˆ J = L IJ L IJ + R IJ R IJ + 2 T IJ T IJ , (3.24)11here the hat indicates that this quadratic Casimir is now understood as a differentialoperator on functions of both l and z .We now follow the same logic as in our derivation of equation (3.14). Of course, thegenerators (4.25) all annihilate the O ( D, D ) invariant combination z L · l L − z R · l R whichappears in our expression for P ( l, z, τ ). The result is that the computation reduces to theone described earlier, and we find[∆ H + s ( s − − ∆ M ] (cid:16) τ D/ P ( l, z, τ ) (cid:17) = 0 , s = D/ , (3.25)where again ∆ M = − ˆ J . As before, we can sum over the lattice Γ to see that:[∆ H + s ( s − − ∆ M ] (cid:16) τ D/ Θ Γ ( τ, z ) (cid:17) = 0 . (3.26)We note that the Laplacian ∆ M appearing here is an O ( D, D ) Laplacian which now acts onboth the space of Narain lattices Γ as well on the vector z = ( z IL , z IR ) of chemical potentials.We now wish to integrate this equation over the space of Narain lattices Γ to obtain anequation for G ( z, τ ) ≡ τ D/ (cid:104) Θ Γ ( τ, l, z ) (cid:105) . (3.27)It is important to remember that – as described in section 2 – the average (cid:104)·(cid:105) should beunderstood as an integral over all of O ( D, D ) rather than just O ( D, D ) /O ( D ) × O ( D ). Thisimplies that the integrated expression G ( z, τ ) will only depend on the potentials through the O ( D ) × O ( D ) invariant combinations z L and z R . Thus in evaluating this integral many ofthe terms appearing in ∆ M = − ˆ J will vanish. In particular, z L and z R are annihilatedby the z -dependent terms in L IJ and R IJ . Therefore, all that survives from ˆ J is the purely z -dependent contribution from 2 T IJ T IJ . The result is that (cid:20) ∆ H + s ( s −
1) + 14 (cid:18) z JL ∂∂z IR + z JR ∂∂z IL (cid:19) (cid:18) z JL ∂∂z IR + z JR ∂∂z IL (cid:19)(cid:21) G ( z, τ ) = 0 . (3.28)This is the version of the Laplace equation which is obeyed by the average of the flavoredpartition function.One obvious solution to this equation can be obtained by generalizing the Eisenstein series(3.18) in order to accommodate the more general modular transformation rule (3.22): G ( z, τ ) = τ D/ (cid:88) ( c,d )=1 e − iπ (cid:18) cz Lcτ + d − cz Rcτ + d (cid:19) | cτ + d | D . (3.29) We have also used the fact that the terms in ˆ J involving only l L,R will lead to boundary terms whichvanish upon integrating over O ( D, D ) when
D >
2, exactly as in the unflavored case. z L − z R . We will therefore give an alternativeargument based on the “heat equation” obeyed by the flavored partition function. The starting observation for our derivation is that equation (2.3) implies that the conformalweights L and ¯ L are determined by the charge vector l = ( l IL , l IR ). This implies that the ( τ, ¯ τ )dependence of the flavored partition function is be determined its z = ( z IL , z IR ) dependence.In particular, we note that P ( l, z, τ ) ≡ e iπτl L − iπτl R +2 πiz L · l L − πiz R · l R , (3.30)obeys ∂∂τ P ( l, z, τ ) = 14 πi ∇ L P ( l, z, τ ) , ∂∂τ P ( l, z, τ ) = − πi ∇ R P ( l, z, τ ) , (3.31)with ∇ L = ∂ ∂z IL ∂z IL , ∇ R = ∂ ∂z IR ∂z IR . (3.32)Summing over lattice points, we conclude that Θ Γ ( τ, z ) will obey the same equation.This heat equation is obeyed by the partition function for every CFT in the Narainensemble. We can therefore integrate over Narain lattices, and conclude that Y ( z L , z R , τ, τ ) ≡ (cid:104) Θ Γ ( τ, z ) (cid:105) (3.33)obeys the same equation. We note that, as in the previous section, the integration over O ( D, D ) implies that the only dependence on potentials is through the O ( D ) × O ( D ) in-variants z L = (cid:112) z IL z IL and z R = (cid:112) z IR z IR . Writing the Laplace operators ∇ L,R in sphericalcoordinates and discarding the angular parts, the heat equation becomes ∂Y∂τ = 14 πi (cid:20) ∂ Y∂z L + D − z L ∂Y∂z L (cid:21) , ∂Y∂τ = − πi (cid:20) ∂ Y∂z R + D − z R ∂Y∂z R (cid:21) . (3.34)These heat equations can be used to fix Y ( z L , z R , τ, τ ). First, from the unflavored analysiswe know that Y ( z L = 0 , z R = 0 , τ, τ ) = (cid:88) ( c,d )=1 | cτ + d | D . (3.35)13rom its definition, we know that it is possible to make the power series expansion Y ( z, τ ) = (cid:80) ∞ m,n =0 Y m,n ( τ, τ ) z mL z nR . The heat equation then provides a set of recursive relations amongthe coefficients in this series expansion. This is easiest to implement by writing Y ( z L , z R , τ, τ ) = (cid:88) ( c,d )=1 e − iπ (cid:18) cz Lcτ + d − cz Rcτ + d (cid:19) | cτ + d | D + ∆ Y ( z L , z R , τ, τ ) . (3.36)The first term on the right hand side obeys the heat equations, so ∆ Y must as well. It willtherefore obey the heat equations and admit the expansion Y ( z, τ ) = (cid:80) ∞ m,n =0 ∆ Y m,n ( τ, τ ) z mL z nR with ∆ Y , = 0. It is simple to see that the heat equation implies recursion relations thatforce all of ∆ Y m,n = 0. We conclude Y ( z L , z R , τ, τ ) = (cid:88) ( c,d )=1 e − iπ (cid:18) cz Lcτ + d − cz Rcτ + d (cid:19) | cτ + d | D . (3.37)We saw in the previous section that this also obeys the flavored Laplace equation, as itshould. The above object is often called the non-holomorphic Jacobi-Eisenstein series; to ourknowledge, holomorphic versions of this quantity were first considered in [16]. Our conclusion is that the average flavored partition function is (cid:104) Z ( τ, z ) (cid:105) = 1 | η ( τ ) | D (cid:88) ( c,d )=1 e − iπ (cid:18) cz Lcτ + d − cz Rc ¯ τ + d (cid:19) | cτ + d | D . (3.38)We wish to extract from this formula the averaged density of states ρ ( j, ∆ , Q I , ¯ Q I ), as afunction of the spin j = ( L − ¯ L ) ∈ Z , dimension ∆ = L + ¯ L and charges Q I , ¯ Q I ofa state. As our partition function depends only on the O ( D ) × O ( D ) invariants z L and z R , the resulting density of states is a function only of the total charges Q = (cid:112) Q I Q I and¯ Q = (cid:112) ¯ Q I ¯ Q I in the left- and right-moving sectors, respectively. As we are interested only inthe density of primary states, we will omit the prefactor | η ( τ ) | − D in what follows.We begin by noting that the term in the sum with ( c, d ) = (0 ,
1) simply describes thecontribution of the ground state. We will therefore concentrate on the terms in the sum with c >
0. To extract the average density of states, we will first perform the Fourier transformwhich takes us from a sector of fixed chemical potentials ( z IL , z IR ) to a sector of fixed charges( Q I , ¯ Q I ), where the contribution to the partition function is: Z ( τ, Q I , ¯ Q I ) = (cid:90) dz IL dz IR e πi ( z IL Q I + z IR ¯ Q I ) (cid:88) ( c,d )=1 e − iπ (cid:18) cz Lcτ + d − cz Rcτ + d (cid:19) | cτ + d | D . (3.39)14he usual unflavored torus partition function can be obtained by integrating this expressionwith measure dQ I d ¯ Q I . Now, the integrals over ( z IL , z IR ) are straightforward D -dimensionalGaussian integrals. These cancel out the factor of | cτ + d | D in the denominator to give: Z ( τ, Q I , ¯ Q I ) = e − πτ ( Q + ¯ Q ) (cid:88) ( c,d )=1 c − D e πi ( Q − ¯ Q ) ( τ + d/c ) . (3.40)We now let d = d ∗ + nc and replace the sum over d with a sum over n ∈ Z and a sum overthe integers 0 ≤ d ∗ < c which are coprime to c . The sum over n is: (cid:88) n ∈ Z e πin ( Q − ¯ Q ) = (cid:88) j ∈ Z δ (cid:0) j − ( Q − ¯ Q ) (cid:1) , (3.41)which gives Z ( τ, Q I , ¯ Q I ) = e − πτ ( Q + ¯ Q ) (cid:88) j ∈ Z δ (cid:0) j − ( Q − ¯ Q ) (cid:1) e πijτ ∞ (cid:88) c =1 c − D (cid:32)(cid:88) d ∗ e πijd ∗ /c (cid:33) . (3.42)We recognize ∆ = Q + ¯ Q and j = Q − ¯ Q as the dimension and spin of a primary state,as expected. The quantity in the parenthesis is known as Ramanujan’s sum, and is usuallydenoted: c c ( j ) ≡ (cid:88) d ∗ e − πijd ∗ /c . (3.43)The sum over c in the expression (3.42) can be computed, and the result is a factor we willcall: κ ( j, D ) ≡ ∞ (cid:88) c =1 c c ( j ) c D = (cid:40) σ D − ( j ) j D − ζ ( D ) , if j (cid:54) = 0 ζ ( D − ζ ( D ) , if j = 0 (3.44)We can now take the inverse Laplace transform in the τ variable to extract the density ofstates: ρ ( j, ∆ , Q I , ¯ Q I ) = κ ( j, D ) δ (cid:0) ∆ − ( Q + ¯ Q ) (cid:1) δ (cid:0) j − ( Q − ¯ Q ) (cid:1) . (3.45)This is our final formula for the averaged density of states. As anticipated, it depends onlyon the total charges Q and ¯ Q , and these total charges are related to dimension and spin inthe usual way.To understand this formula, it is useful to compare this to the expression the total densityof states in the averaged Narain theory. To do so, we simply integrate this over the space ofcharges ( Q I , ¯ Q I ) to give the total density of states: ρ ( j, ∆) = κ ( j, ∆) (cid:90) dQ I d ¯ Q I δ (cid:0) ∆ − ( Q + ¯ Q ) (cid:1) δ (cid:0) j − ( Q − ¯ Q ) (cid:1) = κ ( j, ∆) (cid:32) π D Γ (cid:0) D (cid:1) (cid:33) (∆ − j ) D/ − . (3.46)15he second line involves a Jacobian factor as well as the volumes of D -dimensional spheres incharge space of radius Q = (cid:112) (∆ + j ) / Q = (cid:112) (∆ − j ) /
2, respectively. This expressionmatches precisely the averaged density of states in the Narain theory derived in [3].In retrospect, we could have derived our expressions for the average flavored partitionfunction using a somewhat different logic. In particular, we could have started with theobservation that the average density of states ρ ( j, ∆ , Q I , ¯ Q I ) must be a function only of Q = (cid:112) Q I Q I and ¯ Q = (cid:112) ¯ Q I ¯ Q I , and that these are completely determined by that dimensionand spin using the usual formulas ∆ = Q + ¯ Q and j = Q − ¯ Q . Equation (3.45) is thenthe only possible form of the density of states which is consistent with the known expressionfor the total density of states appearing in [3]. τ → We have emphasized the use of the heat equation in fixing the form of flavored partitionfunctions. In familiar physical systems in which a heat equation arises one is usually interestedin solutions with specified boundary conditions at some initial time. In our context τ playsthe role of time. As a choice of initial time we here consider the case τ →
0, at which theflavored partition function takes a distributional form which can be computed fairly explicitly.Our point in this section simply is to note that the partition function at generic τ can berecovered from this singular limit by using the heat equation.We focus on the factor in the partition function counting primaries, Y D = (cid:88) ( c,d )=1 e − πi (cid:16) cz cτ + d − c ¯ z c ¯ τ + d (cid:17) | cτ + d | D . (3.47)We consider the d = 0 term first. For odd D , we can express this term as a derivative of theDirac delta function as follows Y ( d =0) D, odd = e − πi (cid:16) z τ − ¯ z τ (cid:17) | τ | D = 1 π D − ( ∂ z ∂ ¯ z ) D − e − πi (cid:16) z τ − ¯ z τ (cid:17) | τ | τ → → π D − (cid:104) ( ∂ z ) D − δ ( z ) (cid:105) (cid:104) ( ∂ ¯ z ) D − δ (¯ z ) (cid:105) . (3.48)On the other hand, for even D this is Y ( d =0) D, even = e − πi (cid:16) z τ − ¯ z τ (cid:17) | τ | D = 1 π D − | τ | ( ∂ z ∂ ¯ z ) D − e − πi (cid:16) z τ − ¯ z τ (cid:17) | τ | τ → → π D − | τ | (cid:104) ( ∂ z ) D − δ ( z ) (cid:105) (cid:104) ( ∂ ¯ z ) D − δ (¯ z ) (cid:105) . (3.49) Indeed, our derivation based on the heat equation in the previous subsection – as it is similarly a simpleconsequence of ∆ = Q + ¯ Q and j = Q − ¯ Q –could be considered a different version of this argument. d (cid:54) = 0, we have the following sum over co-primes Y ( d (cid:54) =0) D = (cid:88) ( c,d )=1 (cid:48) e − πi (cid:16) cz cτ + d − c ¯ z c ¯ τ + d (cid:17) | cτ + d | D . (3.50)Here the prime indicates that we consider terms with d (cid:54) = 0. Next, we can write c as j + kd ,with j = 0 , · · · , d − j, d ) = 1. Setting τ = 0, we get Y ( d (cid:54) =0) D (cid:12)(cid:12) τ → = ∞ (cid:88) d =1 ∞ (cid:88) k = −∞ exp (cid:2) − πik ( z − ¯ z ) (cid:3) d − (cid:88) j =0( j,d )=1 exp (cid:20) − πi jd ( z − ¯ z ) (cid:21) d − D . (3.51)The sum over k above gives the Dirac comb, X ( x ) = (cid:80) ∞ a = −∞ δ ( x − a ), while the sum over j is the Ramanujan sum Y ( d (cid:54) =0) D (cid:12)(cid:12) τ → = X ( ¯ z − z ) ∞ (cid:88) d =1 c d ( ¯ z − z ) d − D . (3.52)Performing the sum over d yields the result Y ( d (cid:54) =0) D (cid:12)(cid:12) τ → = σ D − ( ¯ z − z )( ¯ z − z ) D − ζ ( D ) X ( ¯ z − z ) . (3.53)where, σ r ( s ) is the divisor function and ζ ( p ) is the Riemann-zeta function. Hence, the primarycounting partition function at τ → Y D | τ =0 = 1 π D − m | τ | − m (cid:12)(cid:12)(cid:12) ( ∂ z ) D − m δ ( z ) (cid:12)(cid:12)(cid:12) + σ D − ( ¯ z − z )( ¯ z − z ) D − ζ ( D ) X ( ¯ z − z ) . (3.54)where, m = ( D mod 2).A natural question is: why does the contribution to the partition function localize tothe points where ¯ z − z is an even integer? This can be understood by considering theunaveraged D = 1 theory, where a similar phenomenon happens. The primary countingpartition function at τ → Y ( R ) (cid:12)(cid:12) τ → = Tr[ y J ¯ y ¯ J ] = (cid:88) n,w e πiz ( nR + wR ) e − πi ¯ z ( nR − wR )= (cid:88) n e πi ( z − ¯ z ) nR (cid:88) w e πi ( z +¯ z ) Rw = X (cid:18) z − ¯ zR (cid:19) X (cid:18) ( z + ¯ z ) R (cid:19) (3.55)We see that the partition function localises to the points where z − ¯ zR ∈ Z , ( z + ¯ z ) R ∈ Z (3.56)17ombining the above two conditions gives the weaker condition z − ¯ z ∈ Z (3.57)which is independent of R and is the same condition enforced by the Dirac comb appearingin (3.54). The results of the previous section can be generalized in a reasonably straightforward way tothe partition function on higher genus surfaces. In particular, we will show that a flavoredversion of the genus g partition function obeys analogs of the Laplace and heat equationsdescribed above. This will lead to a similar formula for the average flavored partition function.The partition function of a Narain CFT on a Riemann surface Σ of genus g is Z g, Γ ( τ ) = 1Φ( τ ) θ Γ ( τ ) , (4.1)where the Siegel-Narain theta function is θ Γ ( τ ) = (cid:88) l i ∈ Γ e iπτ ij l iL · l jL − iπτ ij l iR · l jR . (4.2)The prefactor Φ( τ ) comes from the integral over oscillator modes, and can be expressedin terms of the one-loop determinant of the scalar Laplacian on Σ. This contribution isindependent of the Narain lattice Γ, so will not be important in what follows. The periodmatrix τ ij is a complex, symmetric g × g matrix with positive imaginary part; i.e. τ ij lives inthe Siegel upper half-space H g . Not every such matrix is actually the period matrix of someRiemann surface, but (4.1) is well defined regardless.The flavored partition function is now obtained by introducing a set of chemical potentials z = ( z ILi , z
IRi ) with i = 1 . . . g and I = 1 . . . D . These measure the charges that propagatearound the various cycles in the Riemann surface, and can be understood as holonomies forbackground U (1) D Wilson lines. The flavored partition function is Z g, Γ ( τ, z ) = 1Φ( τ ) θ Γ ( τ, z ) , (4.3)with θ Γ ( τ, z ) = (cid:88) l i ∈ Γ e iπτ ij l iL · l jL − iπτ ij l iR · l jR +2 πiz Li · l iL − πiz Ri · l iR . (4.4) Φ( τ ) does, however, depend on the central charge and is necessary in order to obtain the correct behaviorunder Weyl transformations.
18n this expression and in what follows we have not written the I indices explicitly. We notethat, as the U (1) D descendants are uncharged, the prefactor Φ( τ ) is exactly the same as inthe unflavored case.The higher genus modular transformations act on the period matrix τ and potential z as τ → γτ ≡ ( Cτ + D ) − ( Aτ + B ) , z L,R → γz L,R ≡ ( Cτ + D ) − z L,R , (4.5)where γ = (cid:0) A ,BC D (cid:1) ∈ Sp (2 g, Z ). The salient point is that C and D are matrices which acton the i = 1 . . . g indices, but not on the I = 1 . . . D flavor indices; therefore much of theanalysis of M D in the previous section will apply in the higher genus case as well. The thetafunction transforms as θ ( γτ, γz ) = exp (cid:2) πiz L C (cid:0) Cτ + D − (cid:1) z L − πiz R C (cid:0) Cτ + D − (cid:1) z R (cid:3) θ ( τ, z ) . (4.6) We begin, as in the genus one case, with the unflavored partition function. The theta functionis a sum over the lattice Γ of Q g ( l, τ ) = e iπτ ij l iL · l jL − iπτ ij l iR · l jR . (4.7)As before, our goal is to write an equation relating the Laplacians on Narain moduli spaceand the Siegel upper half-space.We start by writing the Siegel Laplacian. Decomposing τ ij into its real and imaginaryparts as τ ij = x ij + iy ij , the metric on Siegel upper half-space is ds = y ij y kl ( dy ik dy jl + dx ik dx jl ) . (4.8)Here y ij denotes the inverse of y ij , i.e y ik y kj = δ ij . It is important to note that the lineelement (4.8) should be expressed in terms of unconstrained variables, which we take to be( x ij , y ij ) with i ≤ j . With this in mind, the Laplacian is∆ H g = − √ g g AB ∂ A ( √ gg AB ∂ B ) = − y ik y jl ( ˆ ∂ x ij ˆ ∂ x kl + ˆ ∂ y ij ˆ ∂ y kl ) , (4.9)where ˆ ∂ x ij = (1 + δ ij ) ∂∂x ij and ˆ ∂ y ij = (1 + δ ij ) ∂∂y ij . In (4.9) the index sums each run from 1to g , but ∆ H g should be expressed in terms of the unconstrained variables.We wish to act with the ∆ H g on Q g ( l, τ ), which we now write as Q g ( l, τ ) = e iπx ij ( l iL · l jL − l iR · l jR ) e − πy ij ( l iL · l jL + l iR · l jR ) . (4.10)19 slight inconvenience is the presence of hatted derivatives in ∆ H g . However, these can bedispensed with by the following observation. We are instructed to express Q g ( l, τ ) in termsof the unconstrained quantities ( x ij , y ij ) with i ≤ j and then act with the hatted derivatives( ˆ ∂ x ij , ˆ ∂ y ij ). It is simple to verify that this gives the same result as if we think of all ( x ij , y ij )as being independent, act with ordinary derivatives ( ∂∂x ij , ∂∂y ij ), and then at the end impose( x ji = x ij , y ji = y ij ). This statement relies on the form Q g ( l, τ ) and does not hold for allfunctions. We conclude that when acting on Q g ( l, τ ) we can write∆ g = − y ik y jl (cid:18) ∂∂x ij ∂∂x kl + ∂∂y ij ∂∂y kl (cid:19) , (4.11)and view ( x ji , y ji ) as independent of ( x ij , y ij ) until the end of the computation.This simplification in hand, we can now proceed as we did for g = 1. Defining T IJ = g (cid:88) i =1 (cid:18) l iIL ∂∂l iJR + l iJR ∂∂l iIL (cid:19) , (4.12)the same logic as for g = 1 leads to J Q g ( l, τ ) = e iπx ij ( l iL · l jL − l iR · l jR ) T IJ T IJ e − πy ij ( l iL · l jL + l iR · l jR ) = (cid:2) π y ij y kl l iL · l kL l jR · l lR − πDy ij (cid:0) l iL · l jL + l iR · l jR (cid:1)(cid:3) Q g ( l, τ ) . (4.13)We also have the differential operator relations − y ij y kl ∂∂τ ik ∂∂τ jl Q g ( l, τ ) = − π y ij y kl ( l iL · l kL )( l jR · l lR ) Q g ( l, τ ) , (4.14) y ij ∂∂y ij Q g ( l, τ ) = − πy ij ( l iL · l jL + l iR · l jR ) . (4.15)So that equation (4.13) can be rewritten as J Q g ( l, τ ) = (cid:20) y ij y kl ∂∂τ ik ∂∂τ jl + 8 Dy ij ∂∂y ij (cid:21) Q g ( l, τ ) . (4.16)We can write the Laplacians of the Narain moduli space and the genus- g Riemann surface as∆ M = − J , ∆ H g = − y ij y kl ∂∂τ ik ∂∂τ jl . (4.17)The action of these on Q g ( l, τ ) are related in the following manner (cid:20) ∆ H g − Dy ij ∂∂y ij − ∆ M (cid:21) Q g ( l, τ ) = 0 . (4.18)We now use ∆ H g (det y ) s = − gs (2 s − g − y ) s . (4.19)20his relation is straightforward to derive by using the relationsˆ ∂ y ij det y = y ij , ˆ ∂ y ij y kl = −
12 ( y ki y jl + y kj y jl ) , (4.20)which follow from the definitions.We also have∆ H g ((det y ) s Q g ( l, τ ))) = (cid:2) ∆ H g (det y ) s (cid:3) Q g ( l, τ ) + (det y ) s (cid:2) ∆ H g Q g ( l, τ ) (cid:3) + cross term , (4.21)where the cross term is − y ij y kl (cid:16) ˆ ∂ y ik (det y ) s (cid:17) (cid:16) ˆ ∂ y jl Q g ( l, τ ) (cid:17) = − s (det y ) s y jl ∂∂y jl Q g ( l, τ ) . (4.22)This finally gives (cid:20) ∆ H g − ∆ M + gs (2 s − g − (cid:21) ((det y ) s Q g ( l, τ )) = 0 , (4.23)which is the result quoted in [2].Summing over lattice points, we see that the theta function itself obeys the same differ-ential equation. As in the genus one case, we can sum equation (4 .
23) over Sp (2 g, Z ) imagesto obtain an Eisenstein series which is a modular invariant eigenfunction of the Laplacianwith the same eigenvalue. We will write this Eisenstein series as Y ( z L = 0 , z R = 0 , τ, τ ) = (det y ) s (cid:88) ( C,D )=1 | det( Cτ + D ) | s , (4.24)where the notation “( C, D ) = 1” means that the matrices
C, D together form the lower rowof an Sp (2 g, Z ) matrix; of course when g = 1 this reduces to the usual condition that C and D are coprime integers. It is now straightforward to generalize this to the flavored case, since the M D structuremore or less comes along for the ride. For example, acting on functions of ( l, z ) the generatorsof the O ( D, D ) currents now take the form T IJ = g (cid:88) i =1 l IiL · ∂∂l JiR + l IiR · ∂∂l JiL + z JLi · ∂∂z IRi + z JRi · ∂∂z ILi . (4.25)The argument follows that given above, resulting in the final equation for the averagedpartition function, (cid:20) ∆ H + s ( s −
1) + 14 (cid:18) z JiL ∂∂z
IRi + z JRi ∂∂z
ILi (cid:19) (cid:18) z JLi ∂∂z
IRi + z JRi ∂∂z
ILi (cid:19)(cid:21) (det y ) s (cid:104) Θ Γ ( τ, z ) (cid:105) ) = 0 . (4.26) As in the g = 1 case, this should be regarded not as a sum over Sp (2 g, Z ) but rather a sum over a coset Sp (2 g, Z ) /P where the subgroup P just consists of all transformations which leave det y invariant. .2 Heat equation and the Siegel-Weil formula at higher genus The most natural solution to the differential equation (4.26) is the Eisenstein series (cid:104) Θ Γ ( τ, z ) (cid:105) = (cid:88) ( C,D )=1 e − iπ ( z L ( Cτ + D ) − Cz L ) + iπ ( z R ( Cτ + D ) − Cz R ) | det( Cτ + D ) | D . (4.27)To demonstrate that this is indeed the correct solution, we will utilize a heat equation as inthe g = 1 case. In particular, writing θ Γ ( τ, z ) = (cid:88) l i ∈ Γ P ( l, z, τ ) , (4.28)with P ( l, z, τ ) = e iπτ ij l iL · l jL − iπτ ij l iR l jR +2 πiz iL · l iL − πiz iR · l iR , (4.29)an identical argument as described at g = 1 gives (cid:18) ∂∂τ ij − πi ∂ ∂z iL · ∂z jL (cid:19) P ( l, z, τ ) = 0 = (cid:18) ∂∂τ ij + 14 πi ∂ ∂z iR · ∂z jR (cid:19) P ( l, z, τ ) . (4.30)As before, this reflects the fact that the stress tensor is Sugawara. Summing over latticepoints and integrating over Narain lattices, we find that θ Γ ( τ, z ) and (cid:104) θ Γ ( τ, z ) (cid:105) both obey thedifferential equation (4.30) as well.As at genus one, this provides a relationship between the τ and z dependence of thepartition function which can be used to prove (4.27). The important point is that one canstart with the solution at z = 0 (where (4.27) was proven in [2]), and then expand the heatequation order by order in z to develop recursive relations relating different orders in thisexpansion. It is straightforward to check that (4 .
27) is the unique solution to these recursionrelations which obeys the correct boundary condition at z = 0.There is, however, one important distinction between this case and the simple g = 1case considered earlier. The integral over O ( D, D ) implies that the resulting expressions for (cid:104) θ Γ ( τ, z ) (cid:105) will be invariant under the O ( D ) × O ( D ) symmetries which rotate the chemicalpotential vectors z ILi and z IRi ; these rotations act on the I = 1 . . . D indices, but not onthe i = 1 . . . g index. The result is that the averaged partition function will be a functiononly of the invariants z iL · z jL and z iR · z jR . When g = 1 this includes only the lengths of thechemical potential vectors, which we denoted z L and z R . At g >
1, however, there are nownew invariants which appear with i (cid:54) = j . Loosely speaking, this reflects the fact that whenthe genus g partition function is constructed as a sum over states (corresponding to somechannel decomposition of the genus g surface), charge will flow between different channels.22 .3 A genus 2 example In order to illustrate the utility of (4.27), let us consider in more detail the genus 2 case. Here the period matrix τ ij = (cid:16) τ τ τ τ (cid:17) is two dimensional, and the averaged partition functionwill depend as well on the inner products of the charge vectors z = z I z I , z = z I z I and z · z = z I z I in both the left and right moving sectors. Of particular interest is the pinchinglimit τ →
0, where the genus two surfaces factorizes into disjoint union of two tori. We notethat, generically, the partition function of this pinched Riemann surface does not contain allof the data of our CFT. In the Narain case, however, as long as one keeps z · z non-zero,it is possible to completely reconstruct the genus two partition function in terms of thefactorized torus correlators; by using the heat equation one can determine the dependenceon τ . This holds at higher genus as well, and means that if one wishes one can completelydisregard the genus g partition functions and instead just work with the expectation values (cid:104) Z ( τ , z ) . . . Z ( τ g , z g ) (cid:105) of a product of flavored partition functions; all of the dependence ofthe genus g partition function on the moduli τ ij with i (cid:54) = j can be reconstructed from thefactorized limit by considering the dependence on z i · z j .To understand this in more detail, we can begin by considering the averaged genus 2partition function Z ( τ, Q Ii , ¯ Q iI ) in a sector of fixed charge, rather than fixed potential, whichis given by the Fourier transform: (cid:104) Z ( τ, Q Ii , ¯ Q iI ) (cid:105) = 1Φ( τ ) (cid:90) dz iIL dz iIR e πi ( z iIL Q iI + z iIR ¯ Q iI ) (cid:104) Θ Γ ( τ, z ) (cid:105) , (4.31)where (cid:104) Θ Γ ( τ, z ) (cid:105) = (cid:88) ( C,D )=1 e − iπ ( z L ( Cτ + D ) − Cz L ) + iπ ( z R ( Cτ + D ) − Cz R ) | det( Cτ + D ) | s . (4.32)We will focus on the contribution of the primary states so will drop the prefactor τ ) . Wewill also consider the case τ = 0 where this genus two partition function factorizes intoa product of genus 1 surfaces. The partition function Z ( τ, Q Ii , ¯ Q iI ) can then be used tocompute the two point function of the density of states: (cid:104) ρ (∆ , j , Q I ) ρ (∆ , j , Q I ) (cid:105) .It is possible to unpack the sum over coprime matrices ( C, D ) following [17]. Thistechnique is applied extensively in [18], so we will only summarize a few relevant detailshere. One starts by considering separately the cases where C has rank 0, 1 or 2. The casewith rank 0 just corresponds to ( C, D ) = (0 , Q iI = 0. When rank C = 1, the Fourier transform of Z ( τ, Q Ii , ¯ Q iI ) vanishes unless We are especially grateful to S. Collier for discussions related to the computations appearing in thissubsection. is proportional to Q , and we find (cid:104) ρ (∆ , j , Q I ) ρ (∆ , j , Q I ) (cid:105) rank 1 = (cid:88) ( m,n )=1 κ ( j m,n , D ) δ ( mQ + nQ ) δ (cid:0) ∆ − ( Q + ¯ Q ) (cid:1) δ (cid:0) ∆ − ( Q + ¯ Q ) (cid:1) × δ (cid:0) j − ( Q − ¯ Q ) (cid:1) δ (cid:0) j − ( Q − ¯ Q ) (cid:1) . (4.34)Finally, the set of coprime matrices ( C, D ) with rank( C ) = 2 are parameterized by symmetricmatrices P = C − D with rational entries, leading to (cid:104) Θ Γ ( τ, z ) (cid:105) rank 2 = (cid:88) P ν ( P ) − s e − iπ ( z L ( τ + P ) − z L ) + iπ ( z R ( τ + P ) − z R ) | det( τ + P ) | s (4.35)where ν ( P ) = det( C ) is the product of the elementary divisors of P . The Fourier transformgives a Gaussian integral as in the genus one case: (cid:104) Z ( τ, Q Ii , ¯ Q iI ) (cid:105) rank 2 = e − π ( QyQ + ¯ Qy ¯ Q )+2 πi ( QxQ − ¯ Qx ¯ Q ) (cid:88) P ν ( P ) − s e πi ( QP Q − ¯ QP ¯ Q ) , where we have written τ ij = x ij + iy ij . The sum over P is a version of Siegel’s singular series S s ( Q , Q , Q · Q ) ≡ (cid:88) R ν ( R ) − s e πi ( QRQ − ¯ QR ¯ Q ) , (4.36)where the sum is over rational symmetric matrices with entries between zero and one. Thisseries should be regarded as a generalization of the zeta function relevant for Eisenstein seriesof higher genus. The Fourier transform then gives (cid:104) ρ (∆ , j , Q I ) ρ (∆ , j , Q I ) (cid:105) rank 2 = S s ( Q , Q , Q · Q ) × δ (cid:0) ∆ − ( Q + ¯ Q ) (cid:1) δ (cid:0) ∆ − ( Q + ¯ Q ) (cid:1) × δ (cid:0) j − ( Q − ¯ Q ) (cid:1) δ (cid:0) j − ( Q − ¯ Q ) (cid:1) . (4.37) To see this, we note that the set of coprime matrices with rank( C ) = 1 is parameterized by two pairs ofcoprime integers ( c, d ) = 1 (with c (cid:54) = 0) and ( m, n ) = 1, with C = (cid:18) c
00 0 (cid:19) U T , D = (cid:18) d
10 1 (cid:19) U, (4.33)where U = (cid:16) m pn q (cid:17) is a unimodular matrix. Note that Sp (4 , Z ) invariance implies that the partition function is invariant under τ → τ + N (i.e. x → x + N ) for any symmetric integral matrix N . This implies that QN Q − ¯ QN ¯ Q is an integer. So we canlet P = R + N where N is a symmetric integral matrix and the entries of R are rational numbers between 0and 1, and replace P by R in the sum. Flavored partition function from Chern-Simons the-ory
In this section we show that the averaged flavored partition function can be reproduced in anatural way by summing over a class of geometries weighted by appropriate U (1) D × U (1) D Chern-Simons partition functions. As compared to the unflavored case, the new feature isthat we allow for more general boundary conditions on the gauge fields, corresponding to thepresence of chemical potentials in the partition function. These have to be treated carefullyin order to respect the modular behavior of the partition. In this section we restrict attentionto the genus one CFT flavored partition function, which on the Chern-Simons side meansthat we restrict the class of bulk geometries to be solid tori. Our discussion is similar to [19].We consider U (1) D × U (1) D Chern-Simons theory on a manifold M with boundary ∂M .The action is S = i π (cid:90) M (cid:16) A I ∧ dA I − A I ∧ dA I (cid:17) − π (cid:90) ∂M d x √ gg ab (cid:16) A Ia A Ib + A Ia A Ib (cid:17) , (5.1)where g ab is the metric on ∂M . The boundary term is chosen so that the on-shell variationof the action is δS = i π (cid:90) ∂M d x √ g (cid:16) J aI δA Ia − J aI δA Ia (cid:17) , (5.2)where the currents are J Ia = i (cid:0) A Ia − i(cid:15) ba A Ib (cid:1) , J Ia = i (cid:0) A Ia + i(cid:15) ba A Ib (cid:1) , (5.3)and ( A Ia , A Ia ) function as their conjugate potentials. The stress tensor is defined via thevariation with respect to the metric, δS = 12 (cid:90) ∂M d x √ g T ab δg ab , (5.4)yielding T ab = 18 π (cid:18) A Ia A Ib − A Ic A Ic g ab + A Ia A Ib − A Ic A Ic g ab (cid:19) . (5.5)Choosing the flat metric g ab dx a dx b = dwdw , these formulas read J Iw = i A Iw , J Iw = 0 , J Iw = 0 , J Iw = i A Iw , (5.6) T ww = 18 π (cid:16) A Iw A Iw + A Iw A Iw (cid:17) , T ww = 18 π (cid:16) A Iw A Iw + A Iw A Iw (cid:17) , T ww = T ww = 0 . M be a solid torus. We choose a radial coordinate r such that at fixed r wehave a T on which we choose a complex coordinate w . The w coordinate is taken to haveperiodicities w ∼ = w + 2 π ∼ = w + 2 πτ , τ = τ + iτ . (5.7)The boundary cycle defined by the identification w ∼ = w + 2 π is taken to be contractible whenextended into the solid torus.The flavored partition function is defined by fixing boundary conditions for the connection.We fix (in this section z L ≡ z , z R ≡ ¯ z ) A Iw = iτ z I , A Iw = − iτ z I . (5.8)Note that z I and z I are not related by complex conjugation. Demanding vanishing holonomyaround the contractible circle imposes A Iw = − A Iw , A Iw = − A Iw . (5.9)For flat connections with these boundary values, the full contribution to the classical actioncomes from the boundary term in (5.1), and gives S = − π τ ( z + z ) , (5.10)where we are now writing z = z I z I and z = z I z I . Since the action is quadratic, the 1-loopfluctuation determinant is not affected by the potentials ( z I , z I ). It is equal to the partitionfunction of D free bosons on the torus [2, 20]. Altogether, the path integral for the theoryon the solid torus is Z P I ( τ, z ) = 1 | η ( τ ) | D e π ( z z τ . (5.11)An important point is that the path integral differs from the partition function, where thelatter is defined as Z ( τ, z ) = Tr (cid:104) e πiτ ( L − c/ e − πiτ ( L − c/ e πiz I Q I e − πiz I ¯ Q I (cid:105) . (5.12)In the above, L and L take the Sugawara form, quadratic in the currents. In the presenceof chemical potentials as implemented by our boundary conditions, we noted previously that The notation ( τ, z ) is shorthand for ( τ, τ , z I , z I ). Z ( τ, z ) = e − π ( z z τ Z P I ( τ, z ) . (5.13)The prefactor is responsible for the fact that while the path integral for a CFT with U (1)currents is modular invariant, the partition function with nonzero potentials picks up a mul-tiplicative factor, as written in (3.22). Noting cancellation of the prefactor, the contributionto the partition function is therefore simply Z ( τ, z ) = 1 | η ( τ ) | D . (5.14)The fact that this is independent of the potentials follows from our assumption of trivialholonomy around the contractible cycle; this implies that no charge propagates around thenon-contractible cycle.We now include the sum over bulk manifolds, corresponding to summing over inequivalentchoices for which boundary cycle is contractible in the bulk. We can implement this by writing w = ( cτ + d ) w (cid:48) , with identification w (cid:48) ∼ = w (cid:48) + 2 π ∼ = w (cid:48) + 2 πτ (cid:48) , with τ (cid:48) = ( aτ + b ) / ( cτ + d ).As usual, ad − bc = 1. We now take the contractible cycle to be the one corresponding tothe identification w (cid:48) ∼ = w (cid:48) + 2 π . The classical action is given by the boundary term, which iscoordinate invariant. With τ (cid:48) = τ / | cτ + d | and z (cid:48) I = − iτ (cid:48) A Iw (cid:48) = − i τ cτ + d A Iw = z I cτ + d , (5.15)we obtain S ( τ (cid:48) , z (cid:48) ) = − π τ (cid:48) ( z (cid:48) + z (cid:48) ) = − π τ ( z + z ) + πi (cid:18) cz cτ + d − cz cτ + d (cid:19) . (5.16)Using | η ( τ (cid:48) ) | = | cτ + d || η ( τ ) | , we find that the contribution to the path integral is Z ( c,d ) P I ( τ, z ) = e π ( z z τ | η ( τ ) | D e − πi (cid:16) cz cτ + d − cz cτ + d (cid:17) | cτ + d | D . (5.17)We convert the partition function using (5.13) and, following [21], sum over inequivalentgeometries labelled by relatively prime integers c and d to get Z ( τ, z ) = (cid:88) ( c,d )=1 e − π ( z z τ Z ( c,d ) P I ( τ, z ) = 1 | η ( τ ) | D (cid:88) ( c,d )=1 e − πi (cid:16) cz cτ + d − cz cτ + d (cid:17) | cτ + d | D . (5.18)27his reproduces our previous expression (3.37) for the averaged flavored partition function.As in the unflavored case, we can think of extending this computation to higher genusboundaries. The classical action will again come from boundary terms, with boundaryconditions that fix the holonomy around all boundary cycles that are non-contractible in thebulk. This classical part will reproduce terms in the flavored Siegel-Narain theta function(4.4). The one-loop contribution, denoted as 1 / Φ( τ ), is much more complicated than atgenus one, but we again expect it to be independent of the boundary conditions since theaction is quadratic. Acknowledgements
We are grateful to S. Collier, A. Dymarsky, K. Jensen, A. Shapere and E. Witten for usefulconversations. P.K. is supported in part by the National Science Foundation under researchgrant PHY-1914412. Research of AM is supported in part by the Simons Foundation GrantNo. 385602 and the Natural Sciences and Engineering Research Council of Canada (NSERC),funding reference number SAPIN/00032-2015.
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