Superconformal algebras and mock theta functions 2: Rademacher expansion for K3 surface

Transcription

1 communications in number theory and physics Volume 3, Number 3, 53 55, 009 Superconformal algebras and mock theta functions : Rademacher expansion for K3 surface Tohru Eguchi and Kazuhiro Hikami The elliptic genera of the K3 surfaces, both compact and noncompact cases, are studied by using the theory of mock theta functions We decompose the elliptic genus in terms of the N = superconformal characters at level-, and present an exact formula for the coefficients of the massive non-bps representations using Poincaré Maass series Introduction Studies of an asymptotic behavior of pn, which is the number of partitions of n defined by q n = pnq n, n= were initiated by Hardy and Ramanujan [] The generating function in the left-hand side is essentially the inverse of the Dedekind η function, and it is well-known that its asymptotic behavior is given by n=0 pn n 3 eπ n 3 An exact asymptotic expansion was later derived by Rademacher by use of the circle method eg, [, Chapter ] as π pn = n 3 c= π I 3 n c 6c d mod c d = n+ mod c e d πi 6c, d Here I 3/ denotes the modified Bessel function and we have introduced the Legendre symbol The number of partition pn has received wide interests [], and has played an important role in various areas of mathematics and physics One 53

2 53 Tohru Eguchi and Kazuhiro Hikami of the generalizations of pn is given by the Ramanujan mock theta function [] see also [3, ] for a review, fq =+ =+ q n + q + q + q n αnq n n= n= Asymptotic behavior of αn was written in Ramanujan s last letter to Hardy as αn n e π n 6, n which was proved by Dragonette [0] This asymptotics was later improved by Andrews [], and an exact formula for αn called the Andrews Dragonette identity was conjectured Bringmann and Ono proved the conjecture using the work of Zwegers [] on mock theta functions In our previous paper [], we have shown that the theory of mock theta functions is useful in studying the elliptic genera of hyperkähler manifolds in terms of the representations of N = superconformal algebra We have used the results of Zwegers [7] that mock theta function is a holomorphic part of the harmonic Maass form with weight-/ and has a weight-3/ vector modular form as its shadow see, eg, [39, 6] for reviews As one of the applications of this structure behind the mock theta functions, we employ the method of Bringmann and Ono [,5] see also [6] in this paper to derive an exact formula for the Fourier coefficients of the elliptic genus for the K3 surface, which counts the number of non-bps massive representations Analogous computations of Fourier coefficients of the partition function of three-dimensional gravity using the Poincaré series were discussed in [9, 33, 3, 36] In these papers Jacobi forms are considered instead of mock theta functions We shall also give an exact formula for the number of non-bps representations for the ALE space, which is a degenerate limit of the K3 surface The ALE spaces, or the asymptotically locally Euclidean spaces, are hyperkähler -manifolds, and are constructed from resolutions of the Kleinian singularities

3 Superconformal algebras and mock theta functions 533 We also would like to point out that the non-holomorphic partner of the level- superconformal characters is connected to the Witten Reshetikhin Turaev WRT invariant for 3-manifold associated with the D -type singularity This paper is organized as follows In Section we recall a relationship between the N = superconformal characters and the mock theta functions studied in [] We briefly review the elliptic genus of the K3 surfaces in Section 3 We show how to decompose the elliptic genus in terms of the superconformal characters In Section, we introduce the Poincaré series, whose holomorphic part has the same Fourier coefficients as the number of non-bps representations Following Bringmann and Ono, we compute the Fourier expansion of the Poincaré series, and give an exact asymptotic formula We numerically compute the asymptotic expansion to confirm the validity of our analytic expressions In Section 5 we recall a fact that a limiting value of the non-holomorphic part of the harmonic Maass form is related to the SU WRT invariant for the Seifert manifold M,, The last section is devoted to concluding remarks The N = superconformal algebras and mock theta functions The N = superconformal algebra is generated by the energy-momentum tensor, supercurrents and a triplet of currents that constitute the affine Lie algebra SU k The central charge is quantized to c =6k, and the unitary highest weight state is labeled by the conformal weight h and the isospin l; L 0 Ω = h Ω, T0 3 Ω = l Ω where h k/ and 0 l k/ in the Ramond sector The character of a representation is given by ch k,h,l z; τ =Tr H e πizt 0 3 q L 0 c, where q =e πiτ with τ H, and H denotes the Hilbert space of the representation There exist two types of representations in the N = superconformal algebra [7, 8]: massless BPS and massive non-bps representations In the Ramond sector, their character formulas are given as follows:

4 53 Tohru Eguchi and Kazuhiro Hikami Massless representations h = k, and l =0,,, k, ch R i k, k,lz; τ = θ z; τ [θ 0z; τ] [ητ] 3 ε=± m Z ε eπiεk+m+lz +e πiεz q m qk+m +lm See the Appendix for definitions of the Jacobi theta functions Massive representations h > k and l =,,, k, 3 ch R k,h,l l z; τ =qh k [θ k+ 0 z; τ] [ητ] 3 χ k,l z; τ, where χ k,l z; τ denotes the affine SU character χ k,l z; τ = ϑ k+,l+ ϑ k+, l z; τ, ϑ, ϑ, with the theta series defined by ϑ P,a z; τ = q Pn+a P e πizpn+a n Z Characters in other sectors are related to each other by the spectral flow: R : z + τ R : z + +τ NS : z ÑS : z + Hereafter we study the theory at level k =c = 6 in the R-sector, where the massless character is given by ch R i k=,h=,l=0z; τ = θ z; τ [θ z; τ] [ητ] 3 m Z It is known that this formula may be rewritten as [9] 5 ch R k=,h=,l=0z; τ =[θ z; τ] [ητ] 3 μz; τ, q m e 8πimz +eπiz q m e πiz q m

5 Superconformal algebras and mock theta functions 535 where μz; τ is a Lerch sum defined by 6 μz; τ = ieπiz θ z; τ n q nn+ e πinz q n e πiz n Z Note that the massless characters fulfill an identity ch R k=,h=,l= z; τ+ch R k=,h=,l=0z; τ [θ =q z; τ] 8 [ητ] 3 7 = lim ch R h k=,h,l= z; τ, which shows that the non-bps representation decomposes into a sum of BPS representations at the unitarity bound As shown in [7], we can complete the Lerch sum μz; τ to a Jacobi-like form μz; τ as 8 μz; τ =μz; τ Rτ Here Rτ denotes a non-holomorphic function defined by n + E n + 9 Rτ = [sgn n n Z Iτ ] q n+ Ez denotes the error function given by 0 Ez = z 0 e πu du = erfc πz The function μz; τ transforms like a Jacobi form [0] as follows: i z μz; τ = τ μ τ ;, τ μz; τ +=e πi μz; τ, μz +;τ = μz + τ; τ = μz; τ One can conclude from [7] that the non-holomorphic function 9 is a period integral of the third power of the Dedekind η-function, Rτ = i i τ [ηx] 3 x + τ dx

6 536 Tohru Eguchi and Kazuhiro Hikami Then the function μz; τ satisfies 3 τ μz; τ = i [η τ] 3 Iτ and Iτ 3 Iτ μz; τ =0 τ τ Here the derivatives in τ = u +iv are defined by τ = u i, v τ = u +i v Thus the function μz; τ is a harmonic Maass form with weight / and has [ητ] 3 as its shadow according to the terminology of Zagier [6] Similar q-series was studied in recent work on the Donaldson invariant [3] We note that the above differential equation reduces to 5 Δ μz; τ =0, where Δ k is the hyperbolic Laplacian of weight k, 6 Δ k = v u + v +ikv u +i v 3 Elliptic genus of K3 surface and harmonic Maass form 3 Decomposition of elliptic genus into N = characters The elliptic genus of the hyperkähler manifold X with complex dimension c/3 is identified as [] 3 Z X z; τ =Tr HR H R F e πit 0 3z q L0 c q L 0 c, where F =e πi T0 3 T 3 0, and H R is the Hilbert space on the Ramond sector In the case of K3 surface, it is known that [, 30] 3 Z K3 z; τ =8 [ θ0 z; τ + θ 0 0; τ θ00 z; τ + θ 00 0; τ ] θ0 z; τ θ 0 0; τ

7 Superconformal algebras and mock theta functions 537 To rewrite the elliptic genus in terms of the level- characters we introduce the function Jz; w; τ by 33 Jz; w; τ = [θ z; τ] [ητ] 3 μz; τ μw; τ = [θ z; τ] [ητ] 3 μz; τ μw; τ =ch R k=,h=,l=0z; τ [θ z; τ] [ητ] 3 μw; τ Note that 3 Jz; z; τ =0 It was shown [] that Jz; w; τ behaves like a -variable Jacobi form [0] under the modular transformation: Jz; w; τ =e πi z τ J z τ ; w τ ; τ, 35 Jz +;w; τ =Jz; w; τ +=Jz; w + τ; τ =Jz; w; τ, Jz + τ; w; τ =q e πiz Jz; w; τ These modular properties together with 3 show that Jz; w; τ with w at the half-periods {, +τ, τ } is given by Jacobi theta functions as follows: 36 J z; ; τ θ0 z; τ =, θ 0 0; τ J z; +τ ; τ θ00 z; τ =, θ 00 0; τ J z; τ ; τ θ0 z; τ = θ 0 0; τ

8 538 Tohru Eguchi and Kazuhiro Hikami Note that the Lerch sums introduced in [9] are proportional to μw; τ at w =, +τ, τ, respectively, 37 h τ μ ; τ = ητ h 3 τ μ +τ ; τ = ητ h τ μ τ ; τ = ητ q nn+ ητθ 0 0; τ +q n, n Z ητθ 00 0; τ n Z ητθ 0 0; τ q n 8, +q n n q n 8 q n n Z We thus find that, using 36, the elliptic genus 3 is written as [ ] Z K3 z; τ =ch R k=,h=,l=0z; τ 8 θ z; τ h a τ ητ a=,3, For later convenience let us define 38 Στ 8 w {, +τ, τ } =8ητ a=,3, μw; τ h a τ = q 8 A n q n Here coefficients A n are positive integers, and some of them are computed as follows: 39 n A n Using the massive character 3 k =,l=/ and the identity 7 we conclude that the elliptic genus 3 for K3 surface is decomposed into a sum of superconformal characters as [] 30 n= Z K3 z; τ =0ch R,,0z; τ ch R,, z; τ+ n= A n ch R,n+, z; τ

9 Superconformal algebras and mock theta functions 539 Here the first two terms are isospin l = 0 and / massless representations, and the last term denotes an infinite sum of massive representations 3 Elliptic genus of ALE space The isospin- term in 30 comes from μ ; τ and corresponds to the identity representation in the NS sector It describes the gravity multiplet in string compactification on K3 surface When we decompactify K3 into an ALE space, we decouple gravity Thus we may drop μ ; τ from the elliptic genus and consider [5, 6] 3 Z K3,decompactified z; τ =8 When we define 3 Σ τ 8μ ; τ = q 8 [ θ00 z; τ + θ 00 0; τ A nq n, n= ] θ0 z; τ θ 0 0; τ we have a character decomposition of the elliptic genus as 33 Z K3,decompactified z; τ =6ch R,,0z; τ+ A n A nch R,n+, z; τ Here coefficients A n are integers, and some of them are computed from 6 as follows: 3 n A n It is known that the K3 surface may be decomposed into a sum of 6 A spaces [0], and the elliptic genus of A space is proposed as [5] [ Z A z; τ = θ00 z; τ ] θ0 z; τ + θ 00 0; τ θ 0 0; τ =ch R,,0z; τ+ 35 A n A nch R 6,n+, z; τ n= n= Note that A n A n/6 are all positive integers

10 50 Tohru Eguchi and Kazuhiro Hikami Poincaré Maass series and character decomposition Multiplier system for harmonic Maass forms As the completion of Στ defined in 38, we define Στ =8 μw; τ, which transforms as 3 w {, +τ, τ } Σ τ = τ i Στ, Στ +=e πi Στ One finds in and 3 that the multiplier system for Σ is a complex conjugate to that of [ητ] 3 We recall a transformation formula of the Dedekind η-function eg, [, Chapter 9]: η γτ = i a+d e πi sd,cπi c cτ + dητ a b Here γ = SL; Z with c>0, and sd, c is the Dedekind sum c d defined by sd, c = k kd, c c k mod c where x x, for x R \ Z, x = 0, for x Z We thus conclude that the completion satisfies 5 Σγτ = i 3 e a+d c πi+3sd,cπi cτ + d Στ The Whittaker function and the Poincaré Maass series The Whittaker functions [3], M α,β z and W α,β z, are solutions of the second-order differential equation [ 6 z + + α z + ] β z wz =0

11 Superconformal algebras and mock theta functions 5 We have and W α,β z = Γ β Γ Γβ α βm α,βz+ Γ α + βm α, βz, M α,β z Rz + Γ+β Γ α + z α, βez W α,β z Rz + e z z α Following [6], we define functions M k sv and Ws k v in terms of the Whittaker functions as 7 M k sv = v k M k sgnv,s v, Ws k v = v k W k sgnv,s v Some of them are given as follows: W 3 W 3 v =e v, v = π E M 3 v = π E v π v e v, π e v, where we assume v>0, and the error function Ez is defined in 0 We define the function ϕ k h,s τ for h>0by 8 ϕ k h,s τ =Mk s πhiτ e πihrτ Then it becomes an eigenfunction of the second-order differential equation: [ 9 Δ k ϕ k h,s τ = s s+ k ] k ϕ k h,s τ, where Δ k is the Laplacian defined in 6 Note that at Iτ + ϕ k h,s τ Γs Γ k + sq h

12 5 Tohru Eguchi and Kazuhiro Hikami By using the function ϕ k h,s τ, we introduce the Poincaré Maass series [6] as 0 P s τ = π a b γ= Γ c d \Γ where the multiplier system is chosen to be [χγ] ϕ γτ, cτ + d,s 8 i 3 e a+d πi+3πisd,c c, for c>0, χγ = e b πi, for c = 0 and d = We mean that Γ = SL; Z, and Γ is the stabilizer of, Γ = { } n 0 n Z We see that the Poincaré Maass series P s τ transforms in the same way as Στ, 3 P s γτ = χγ cτ + dp s τ, and that, due to a commutativity of the Laplacian Δ k 6 and the γ action, it is an eigenfunction of the Laplacian [ 3 Δ P s τ = s s 3 ] P s τ 6 At Iτ +, the asymptotic behavior of the M-Whittaker function shows that the Poincaré Maass series behaves exponentially as e πiτ/ Note that P s τ is annihilated by the Laplacian Δ like μz; τ 5 when we set s = 3 ; Δ P 3 τ =0 These facts show that the Poincaré Maass series P 3 τ is the harmonic Maass form with weight /

13 Superconformal algebras and mock theta functions 53 We shall compute the Fourier coefficients of P s τ in the form of Rademacher expansion By using the standard method, we have P s τ = π ϕ 8,sτ+ π Using γτ = a c as c>0 γ Γ \Γ/Γ c 0 n Z γ Γ \Γ/Γ c τ + n+d ϕ 8,s γ ccτ+d n 0 [ χ γ τ ] n 0, the second term up to an overall factor is rewritten [χγ] e [ a πi c χ c n Z M s π Iτ τ + n + d c τ + n + d e c c ] n 0 We then apply a Fourier transformation formula [6, 3] 5 n Z c R τ+n+ d c πi M Iτ s πh τ + n c τ + n e πih n+πi h R c τ+n = n Z a n Iτ e πin h Rτ, where the Fourier coefficients a n v are given as follows: for n>h, a n v = h Γs i πc v Γ s + π W n h,s for n = h, π n h v I s π c n h h, h s a n v = 3 π s+ 3 Γs i s Γ s + Γ s, c s v s 3

14 5 Tohru Eguchi and Kazuhiro Hikami for n<h, a n v = h Γs i πc v Γ s π h n W,s π h n v J s π c Here the modified Bessel function, I α z and J α z, satisfy I α z z 0 Γα + z J α z z 0 Γα + z α, Iα z z α, Jα z z h n h e z, πz z πz cos α π π As a result, we obtain the Fourier expansion of the Poincaré Maass series as follows: Iτ P 3 τ =E q π 8 8n 6 c>0 n Z n π c I 8n c e 3πisd,c+πin d c qn 8 d mod c c,d= [ ] π 8nIτ E n Z 8n n 0 π c J 8n c c>0 d mod c c,d= e 3πisd,c+πi d c n qn 8 Proving the convergence of the above series in c>0 is a somewhat difficult issue Such Poincaré Maass series appeared in the Andrews Dragonette identity, and its convergence was proved by Bringmann and Ono [, Section ] by using properties of the Kloosterman sums and Salié sums In our case, we recall that the sum involved in 6 can be rewritten as 7 d mod c c,d= e 3πisd,c+πi d c n = i c k mod c k = 8n+ mod 8c k e πi k c

15 Superconformal algebras and mock theta functions 55 The sum on the right-hand side can be parameterized by using binary quadratic forms of discriminant 8n +, and the method of Bringmann and Ono shows how this description naturally forces sufficient cancellation to justify convergence In Section we present evidence for the convergence of the series by numerically computing the truncated series at finite values of c Our claim is that the Poincaré Maass series 0 coincides with the completion : 8 Στ =P 3 τ Proof of this statement is analogous to that of the Andrews Dragonette identity by Bringmann and Ono [] We first look at the non-holomorphic part of the Poincaré Maass series P 3 τ From the above expression 6, we obtain i Iτ τ P 8τ = 3 δ π n,0 +8n + π c J 8n + c 9 n=0 d mod c c,d= e 3πisd,c+πi d c n c>0 q8n+ As was proved by Bruinier and Funke [7, Proposition 3], the left-hand side is a weight-3/ cusp form on Γ 0 6 with a trivial character On the other hand, we have from 3 that 0 i Iτ τ Σ8τ = [η 8τ] 3 = [ q 3q 9 +5q 5 7q 9 +9q 8 q +3q 69 ], which is also a weight-3/ cusp form on Γ 0 6 with a trivial character According to the dimension formulas for spaces of half-integral weight modular forms [8] see also [38], the dimension of cusp form on Γ 0 6 is We thus see that Σ8τ is proportional to P 3 8τ Next by comparing the coefficients of q, we see that the holomorphic part of P 3 8τ coincides with that of Σ8τ and hence we have the equality 8

16 56 Tohru Eguchi and Kazuhiro Hikami Finally, we obtain an exact asymptotic expansion for A n 39 as A n = π 8n c= c I π 8n c d mod c c,d= e 3πisd,c+πi d c n The dominating contribution comes from the term c = in the above expression, π π 8n A n I 8n Substituting an explicit form of the Bessel function, I x = πx sinhx, we have the Cardy-type formula 3 log A n π n 8 3 Non-compact case We shall next determine the Fourier coefficients A n 3 which are related to the number of non-bps representations in the decompactified K3 surface We set the completion of Σ τ tobe Σ τ =8 μ ; τ Its modular transformation formulae can be deduced from Recalling the transformation formulae for the Jacobi theta function θ 0 z; τ eg, [, Chapter 0], we see that for γ Γ 0 with c>0 5 Σ γτ = i 3 e a+d πi+3πisd,c c cτ + d Σ τ Using the same argument with the above, we conclude that it coincides with the Poincaré Maass series for Σ τ, 6 Σ τ = π γ Γ \Γ 0 [χγ] ϕ cτ + d, γτ, 3 8

17 Superconformal algebras and mock theta functions 57 where the multiplier system χγ is given in Correspondingly the Fourier coefficients of Σ τ can be computed from the Poincaré Maass series, and we obtain 7 A n = π 8n c= c c I π 8n c d mod c c,d= e 3πisd,c+πi d c n Asymptotic behavior of coefficients is given as 8 A n n π 8n I π 8n Numerical checks Now we present some results of numerical calculations and their comparison with exact results in order to confirm the convergence of the series 6 and asymptotic formulas We have plotted the exact values of Fourier coefficients A n together with the values of the asymptotic formula in figure We have also presented the exact values of A n and compared with the predictions given by modified Bessel function 8 We find very good agreements In the tables we present more detailed results We have numerically computed the exact asymptotic series of A n by truncating the infinite Figure : Exact values of A n 39 and values of the asymptotic formula are given by black dots and black line, respectively We have also plotted exact absolute values A n 3 together with their asymptotic values 8 in grey

18 58 Tohru Eguchi and Kazuhiro Hikami sum c= by a finite number of terms For comparison we also list the exact values of A n : n Exact Leading; Sum of 5 terms 0 terms For the case of A n, we also present their exact values and values given by the truncation of the asymptotic expansion 7: n Exact Leading; 8 Sum of 5 terms 0 terms One sees that numerical results support the convergence of the series 6 the validity of our asymptotic formulae and 7 For its direct verification, we have numerically computed 9 to obtain i Iτ τ P 8τ =385q 7096q q q q q 6678q q q q q q 89 + Here we have truncated the sum over c after the first 800 terms Though the convergence is slower than before, this agrees with 0 5 Chern Simons theory A key in our analysis is to complete the massless superconformal character μz; τ by adding the non-holomorphic partner Rτ 9 so that the sum has a nice modular property Here we would like to point out that Rτ

19 Superconformal algebras and mock theta functions 59 has its own meaning as a topological quantum invariant related to a simple singularity The Witten invariant of 3-manifold M [5] is defined by the Chern Simons path integral 5 Z k M = e πik CSA DA, where the coupling constant k Z denotes the level, and A is a G-gauge connection on the trivial bundle over M The Chern Simons action CSA with gauge group G is CSA = 8π M Tr A da + 3 A A A See [] for mathematically rigorous definition of this quantum invariant WRT invariant in terms of the quantum invariants of links to be surgered When M is the Poincaré homology sphere, it was shown that the Witten invariant Z k M with gauge group SU can be regarded as a limiting value of the Eichler integral of a weight-3/ modular form [3] This correspondence has been checked for other Seifert manifolds [6, 7] Let M be the Seifert manifold M,, [35], which is a spherical neighborhood around the isolated singularity of type-d : M = { x y + y 3 + z =0 } S 5, where S 5 denotes a sufficiently small 5-sphere around the origin It was shown [7] that the Witten invariant Z k M with SU gauge group is given by see also [5, 8] 5 Z k M = i [ k + e πi k+ e πi k+ R ] k + Here Rτ is the period integral of the third power of the Dedekind η-function, and we have a limiting value in τ k+ It was shown that the topological invariants, such as the Chern Simons invariant, the Reidemeister torsion and the Ohtsuki invariant, of M are given by the asymptotic expansion of Z k M ink

20 550 Tohru Eguchi and Kazuhiro Hikami It should also be noted that the non-holomorphic partner of the massless character Rτ is related to the knot invariant for the torus link T, [] 53 T, N = Ne πi N R N Here L N denotes a specific value of the N-colored Jones polynomial for link L at q =e πi/n : this quantity receives renewed interests from the viewpoint of the volume conjecture raised by Kashaev [9], and Murakami and Murakami [37] 6 Concluding remarks As an application of our previous result [] that the coefficients of massive characters of the elliptic genera are the holomorphic part of harmonic Maass form, we have obtained their Rademacher-type expansion by computing the Fourier coefficients of the Poincaré Maass series We note that in the elliptic genus the right-moving sector is fixed to Ramond ground state and thus the non-bps states in the left-moving sector are actually the overall half-bps states BPS non-bps in the right left moving sector It is known that asymptotic increase of the number of half- BPS states is related to the entropy of supersymmetric systems In fact, the multiplicity factor A n behaves like an exponential 3 and we may identity 6 S =π n as the entropy of K3 surface Our methods are applicable to higher level superconformal algebras and higher-dimensional hyperkähler manifolds In general, hyperkähler manifolds with complex k-dimensions we find an entropy [3] k 6 S =π k + n If we consider the case of symmetric product of K3 surfaces K3 [k], the above entropy reproduces the black hole entropy of string theory compactified on K3 surface at large k Acknowledgments One of the authors KH would like to thank M Kaneko and in particular K Ono for his useful communications on the issue of convergence of Poincaré

21 Superconformal algebras and mock theta functions 55 series This work is supported in part by Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan Appendix A Jacobi theta functions The Jacobi theta functions are defined by θ z; τ = n Z q n+ e πin+ z+ = θ z; τ, θ 0 z; τ = n Z q n+ e πin+ z = θ z; τ, θ 00 z; τ = n Z q n e πinz = θ 3 z; τ, θ 0 z; τ = n Z q n e πinz+ = θ z; τ, where we have also shown the relation to the conventional notations References [] G E Andrews, On the theorems of Watson and Dragonette for Ramanujan s mock theta functions, Amer J Math , 5 90 [], The theory of partitions, Addison-Wesley, London, 976 [3], Mock theta functions, in Theta Functions Bowdoin 987, eds L Ehrenpreis and R C Gunning, vol 9 part of Proc Symp Pure Math, American Mathematical Society, Providence, 989, [] K Bringmann and K Ono, The fq mock theta function conjecture and partition ranks, Invent Math , 3 66 [5], Coefficients of harmonic Maass forms, 008, preprint [6] J H Bruinier, Borcherds products on O,l and Chern classes of Heegner divisors, Lecture Notes in Mathematics, 780, Springer, Berlin, 00 [7] J H Bruinier and J Funke, On two geometric theta lifts, Duke Math J 5 00, 5 90 [8] H Cohen and J Oesterlé, Dimensions des espaces de formes modulaires, in Modular Functions of One Variable VI, eds J-P Serre and

22 55 Tohru Eguchi and Kazuhiro Hikami D Zagier, Lecture Notes in Mathematics, 67, Springer, Berlin, 977, [9] R Dijkgraaf, J Maldacena, G Moore and E Verlinde, A black hole Farey tail, 007, hep-th/ v3 [0] L Dragonette, Some asymptotic formulae for the mock theta series of Ramanujan, Trans Amer Math Soc 7 95, [] F J Dyson, Some guesses in the theory of partitions, Eureka Cambridge 8 9, 0 5 [] T Eguchi and K Hikami, Superconformal algebras and mock theta functions, J Phys A: Math Theor, , 3 pages [3], preprint in preparation [] T Eguchi, H Ooguri, A Taormina and S-K Yang, Superconformal algebras and string compactification on manifolds with SUn holonomy, Nucl Phys B , 93 [5] T Eguchi, Y Sugawara and A Taormina, Liouville field, modular forms and elliptic genera, JHEP 007, 9 007, pages [6], Modular forms and elliptic genera for ALE spaces, 008, arxiv: [7] T Eguchi and A Taormina, Unitary representations of the N = superconformal algebra, Phys Lett B , 75 8 [8], Character formulas for the N = superconformal algebra, Phys Lett B , 35 3 [9], On the unitary representations of N =and N =superconformal algebras, Phys Lett B 0 988, 5 3 [0] M Eichler and D Zagier, The theory of Jacobi forms, Progress in Mathematics, 55, Birkhäuser, Boston, 985 [] B Gordon and R J McIntosh, A survey of classical mock theta functions, 009, preprint [] G H Hardy and S Ramanujan, Asymptotic formulæ in combinatory analysis, Proc London Math Soc 98, 75 5 [3] D A Hejhal, The Selberg trace formula for PSL, R Vol, Lecture Notes in Mathematics, 00, Springer, Berlin, 983

23 Superconformal algebras and mock theta functions 553 [] K Hikami, Quantum invariant for torus link and modular forms, Commun Math Phys 6 00, 03 6 [5], Mock false theta functions as quantum invariants, Regular Chaotic Dyn 0 005, [6], On the quantum invariant for the Brieskorn homology spheres, Int J Math 6 005, [7], On the quantum invariant for the spherical Seifert manifold, Commun Math Phys , [8], Transformation formula of the nd order mock theta function, Lett Math Phys , [9] R M Kashaev, The hyperbolic volume of knots from quantum dilogarithm, Lett Math Phys , [30] T Kawai, Y Yamada and S-K Yang, Elliptic genera and N = superconformal field theory, Nucl Phys B 99, 9 [3] R Lawrence and D Zagier, Modular forms and quantum invariants of 3-manifolds, Asian J Math 3 999, [3] A Malmendier and K Ono, SO3-Donaldson invariants of CP and mock theta functions, 008, arxiv:0808 [33] A Maloney and E Witten, Quantum gravity partition function in three dimensions, 007, arxiv:07055 [3] J Manschot and G W Moore, A modern Farey tail, 007, arxiv: [35] J Milnor, On the 3-dimensional Brieskorn manifolds Mp, q, r, in Knots, Groups, and 3-Manifolds, ed L P Neuwirth, Princeton University Press, Princeton, 975, 75 5, papers Dedicated to the Memory of R H Fox [36] G Moore, Strings and arithmetic, in Frontiers in Number Theory, Physics and Geometry II, eds P Cartier, B Julia and P Moussa, Springer, Berlin, 006, [37] H Murakami and J Murakami, The colored Jones polynomials and the simplicial volume of a knot, Acta Math 86 00, 85 0

24 55 Tohru Eguchi and Kazuhiro Hikami [38] K Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-series, CBMS Regional Conference Series in Mathematics, 0, American Mathematical Society, Providence, 00 [39], Unearthing the visions of a master: harmonic Maass forms and number theory, Current Developments in Mathematics 008, International Press, Boston, in press [0] D N Page, A physical picture of the K3 gravitational instanton, Phys Lett B , [] H Rademacher, Topics in analytic number theory, Die Grundlehren der mathematischen Wissenschaften, 69, Springer, New York, 973 [] N Yu Reshetikhin and V G Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent Math 03 99, [3] E T Whittaker and G N Watson, A course of modern analysis, th edition, Cambridge University Press, Cambridge, 97 [] E Witten, Elliptic genera and quantum field theory, Commun Math Phys , [5], Quantum field theory and the Jones polynomial, Commun Math Phys 989, [6] D Zagier, Ramanujan s mock theta functions and their applications [d après Zwegers and Bringmann Ono], Séminaire Bourbaki [7] S P Zwegers, Mock theta functions, PhD thesis, Universiteit Utrecht, 00 Yukawa Institute for Theoretical Physics Kyoto University Kyoto Japan address: eguchi@yukawakyoto-uacjp Department of Mathematics Naruto University of Education Tokushima Japan address: KHikami@gmailcom Received April, 009