ASYMPTOTIC FORMULAS FOR COEFFICIENTS OF INVERSE THETA FUNCTIONS

Transcription

1 ASYMPTOTIC FORMULAS FOR COEFFICIENTS OF INVERSE THETA FUNCTIONS KATHRIN BRINGMANN AND JAN MANSCHOT Abstract We determine asymptotic formulas for the coefficients of a natural class of negative index and negative weight Jacobi forms These coefficients can be viewed as a refinement of the numbers p n of partitions of n into colors Part of the motivation for this wor is that they are equal to the Betti numbers of the Hilbert scheme of points on an algebraic surface S and appear also as counts of Bogomolny- Prasad-Sommerfield BPS states in physics Introduction and Statement of Results Jacobi forms were first systematically studied by Eichler and Zagier [] and enjoy a wide variety of applications in the theory of modular forms, combinatorics [7, 8], conformal field theory [, ], blac hole physics [0, ], Hilbert schemes of points [8], Donaldson invariants [9], and many other topics This paper focuses on a class of negative index Jacobi forms with a single order pole in the elliptic variable w The analysis of the coefficients of such functions is more complicated then the well-understood class of Jacobi forms which depend holomorphically on w It turns out that these Fourier coefficients in w are not modular but related to quantum modular forms [5, 0] The appearance of these functions in the above mentioned topics calls for an explicit nowledge of their coefficients and in particular of their asymptotic growth In this paper we provide such asymptotic formulas One of the immediate motivations is the counting of BPS states in physics and in particular those with vanishing angular momentum This motivation is explained in more detail after stating the results Statement of results We consider the following class of negative weight / N and index / Jacobi forms h w; τ := i θ w; τητ, 000 Mathematics Subject Classification F7, F50, F0, P8, J60

2 KATHRIN BRINGMANN AND JAN MANSCHOT with q := expπiτ, ζ := expπiw θ w; τ := iζ q 8 q n ζq n ζ q n, ητ := q n q n n We are interested in two expansions of h w; τ The first expansion is in terms of the coefficients a m, n, defined by: q ζ ζ h w; τ =: a m, nζ m q n, n 0 ζq, ζ q < m Z Note that m Z a m,n = p n, where p n denotes the number of partitions of n into colors These are enumerated by ητ : p n q n = ητ n 0 For =, equation corresponds to the generating function of the cran of partition [7], and for = to the biran [0] The second expansion is motivated from physics, and is based on the fact that the coefficients of q are Laurent polynomials, symmetric under ζ ζ and with maximal degree n Therefore, we can express h w; τ as: q ζ ζ h w; τ = b m, n χ m+ ζ q n, with m,n 0 χ m ζ := ζm ζ m ζ ζ, b m,n := a m, n a m+, n Following the approach of Wright [8], we determine all polynomial corrections to the leading exponential of the coefficients a m, n in the large n limit Theorem We have for N : a m, n = π N l= I l d m, ln +l+ π n π 6 +l+ + O n N e π n,

3 where d m, l are defined by equation and I l x is the usual I-Bessel function Here the error term depends on and m Theorem allows us to compare the asymptotic growths of a m, n for different values of m The asymptotic behavior of the Bessel function : directly yields: Corollary We have I l x = a m, n a r, n = π r m 8n 9+ where the error term depends on m,, and r ex πx + O x, +7 e π n + O n e π n From Theorem the asymptotics of b m, n for large n immediately follow: Corollary We have b m, n = m + π 8n 9+ where the error term depends on m and +7 e π n + O n e π n Note that this corollary shows that b m, n increases with m in the limit of large n Beyond the validity of equation, b m, n eventually decreases with increasing m for fixed n, and in particular b m, n = 0 for m > n We next compare the asymptotic behavior of the coefficients b 0, n with those of p n It is well-nown that the asymptotic growth of the latter is given by [, 0]: p n = Thus we find for the ratio b 0, n/p n: + 8n + e π n + O n 5+ e π b 0, n p n = π + O n 6 n n,,

4 KATHRIN BRINGMANN AND JAN MANSCHOT Motivation: Moduli spaces and BPS states BPS states of both gauge theory and gravity have been extensively studied in the past for a variety of reasons These states are representations of the SU spin massive little group in four dimensions labeled by their angular momentum or highest weight J The subset of BPS states with vanishing angular momentum J = 0, also nown as pure Higgs states [], have recently attracted much interest [, 5, 8, ] The states with J = 0 are in some sense more fundamental In particular in gravity, these states are candidates for microstates of single center blac holes, and as such are the states relevant for studies of the Beenstein-Hawing area law of blac hole entropy Within string theory it is possible to obtain exact generating functions of the degeneracies of classes of BPS states The asymptotic growth as function of the angular momentum is for example previously studied in [7,, 9] String theory relates BPS states to the cohomology of moduli spaces of sheaves supported on a Calabi-Yau manifold From this perspective the SU spin representations correspond to representations of the Lefshetz SL action on the cohomology of the moduli space M [6] The states with J = 0 correspond to the part of the middle cohomology which is invariant under the Lefshetz action In the present wor, we consider moduli spaces of semi-stable sheaves supported on a complex algebraic surface S, which can be thought of as being embedded inside a Calabi-Yau manifold If S is one of the rational surfaces, the sheaves can be related to monopole or monopole strings in respectively four and five dimensional supersymmetric gauge theory through geometric engineering [, 9] If S is a K surface, the sheaves correspond to small blac holes in N = supergravity also nown as Dabholar-Harvey states [8, 9] We specialize to the moduli space of sheaves with ran r = and st and nd Chern classes c H S, Z and c H S, Z These moduli spaces are isomorphic to the Hilbert scheme of c points on S viewing c as a number Göttsche has determined the generating function of the Betti numbers of the Hilbert schemes [8] We need to introduce some notation to explain his result Let Mn be the Hilbert scheme of n points Let furthermore P X, ζ := dim C X i=0 β i X ζ i be the Poincaré polynomial of X with β i X the ith Betti number of X We choose the surface S such that β S = β S = 0 Then, we have [8]: n 0 ζ dim C Mn P Mn, ζ q n = q β S+ ζ ζ i θ w; τ ητ β S,

5 which precisely equals the function in with = β S+ The coefficients a m, n are in this context the Betti numbers of the moduli spaces The expansion in terms of b J, n decomposes the cohomology in terms of J + -dimensional SL or SU spin representations For J = 0 and =, our formula confirms nicely the numerical estimates for b 0, n in [9, Appendix C] Note that since this analysis is carried out in the so-called wea coupling or D-brane regime, and the coefficients b J, n are not BPS indices, these coefficients can not be claimed to count blac holes with fixed angular momentum Equation shows that in the context of this paper, the number of pure Higgs states has the same exponential growth as the total number of states, but that the number of pure Higgs states is smaller by a factor n Moreover, the number of SU spin multiplets increases with J for small J It is interesting to compare this with other nown asymptotics of pure Higgs states In particular, Ref [, ] considered this question for quiver moduli spaces in the limit of a large number of arrows between the nodes of a quiver with a potential Ref [] demonstrated that the number of pure Higgs states for these quivers, β dimc MM β dimc M+M, is exponentially larger than the number of remaining SL multiplets given by β dimc M+M We note that sheaves on toric surfaces relevant for this article also allow a description in terms of quivers [6, ] The Hilbert scheme of n points corresponds to increasing dimensions of the spaces associated the nodes, with the number of arrows ept fixed Thus we observe that the asymptotic behavior of the number of pure Higgs states in the two limits, large number of arrows or large dimensions, is rather different It will be interesting to understand better the significance of these different asymptotic behaviors Moreover we belief that application of the techniques in the present paper to partition functions for higher ran sheaves on surfaces [6, 7, 5], and partition functions of blac holes and quantum geometry [5, 6, ] will lead to to important novel insights The paper is organized as follows: In Section we rewrite the functions of interest in terms of false theta functions and determine their Taylor expansion Section uses the Circle Method to prove our main theorem 5 Acnowledgements The research of the authors was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation After this paper was submitted to the arxiv, we learned from Paul de Lange that for special cases the main terms were also obtained by [7] We are grateful to him for the correspondence We also than Roland Maina, Boris Pioline, Rob Rhoades and Miguel Zapata Rolón for useful correspondence

6 6 KATHRIN BRINGMANN AND JAN MANSCHOT Relation to false theta functions We start by writing the generating function of a m, n for fixed m and in terms of the functions ϑ m q defined by: ϑ m q := + q m n q nn+ +n m n 0 Remar We note that the property ϑ m q = ϑ m q of ϑ m q, continues to hold when ϑ m q is defined with m replaced by m The functions ϑ m q are examples of false theta functions, which were first introduced by Rogers [] and have attracted a lot of interest recently Using the Rogers and Fine identity one can relate ϑ m to so-called quantum modular forms which are functions mimicing modular behavior on subsets of Q [5] It is well-nown that the inverse theta function θ w; τ can be written as a sum over its poles See for more details for example [, 7] Proposition We have for ζq, ζ q < : q ζ ζ h w; τ = q with q := n= qn ϑ m qζ m Proof: The inverse theta function θ w; τ is expressed as a sum over its poles by: iq 8 ζ ζ θ w; τ = ζ q n Z m Z n q nn+ ζq n Using geometric series expansion, we may rewrite as + ζ n q nn+ q q +nm ζ m + ζ q n>0 m 0 From this the statement of the proposition easily follows n>0 m 0 n q nn+ +nm ζ m The function ϑ m is not modular but may be nicely approximated by its Taylor expansion For this we use the following general lemma see [9] for the case of real functions Lemma Let f : C C be a C function Furthermore, we require that fx and all its derivatives are of rapid decay for Rex Then for t 0 with Ret > 0

7 7 and a > 0, we have for any N N 0 : f m + at = fxdx t m=0 0 N n=0 Here B n x denotes the nth Bernoulli polynomial To use Lemma we write for fixed N and q = e z q ϑm q =: Lemma then gives Lemma We have for N f n 0 B n+ a n! n + tn + O t N+ N d m, lz l + O z N+ l= ϑ m q = + q m q m+ N l=0 c m lz l + O z N+ with c m l := l l m B l+ l!l + + m B l+ + In particular the first values for d m, l are: d m, =, d m, = , d m, = m Proof: We may write ϑ m q as: ϑ m q = + q m q m+ f n + m + z n 0 f n + m + z with fx := e x Substitution of Lemma gives the desired result Remar We note that the case m = 0 can be easily concluded from [8], where the asymptotics of the coefficients of / ϑ 0 q/q for =, are determined Use of the Circle Method In this section, we prove Theorem following an approach by Wright [8] To prove the theorem, we assume via symmetry that m 0 and set F m, q := n 0 a m, nq n

8 8 KATHRIN BRINGMANN AND JAN MANSCHOT By Proposition, we obtain that F m, q = ϑ q m q By Cauchy s Theorem, we have for n a m, n = πi C F m, q dq, q n+ where C is a circle surrounding 0 counterclocwise We choose e η for the radius of C with η = π and split C into two arcs C = C 6n + C, where C is the arc going counterclocwise from phase η to η and C is its complement in C Consequently, we have a m, n = M + E with M := F m,q dq, πi C q n+ E := F m,q dq πi C q n+ We will show that the main asymptotic contribution comes from M Moreover we parametrize q = e z with Rez = η The integral along C In the integral along C, we approximate F m, by simpler functions Firstly, recall that from the transformation law of the η-function [, Theorem ] we obtain: e z ; e z = Thus we want to approximate q where To be more precise, we split P q := z by π e z + π 6z z π e z P + l <0 l>0 M = M + E e π z e π z, ; e π z p lq l q

9 9 with M := z e z πi C q n+ P e π z ϑ m qdq, π E := πi C q ϑ z mq e z n+ e z ; e z P e π z dq π We first bound E which turns into the error term Firstly we obtain from z e z e z ; e z P e π z = O π To bound ϑ m, we use that on C z = η + Imz η + η Thus, by Lemma, ϑ m q z η, where throughout gx fx has the same meaning as gx = Ofx Using that the length of C is Oη, we may thus bound E n e π n 6 We next investigate M We aim to approximate ϑ m by its Taylor expansion given in Lemma and thus we split M = M + E with M := N z d m, l P πi l= C q n+ e π z z l dq, π E := z N P πi C q n+ e π z e z ϑm q d m, lz l dq π l= We first estimate E and show that it contributes to the error term By Lemma E e nη z +N+ e π Re 6 z dz C Since Rez z Rez = η we may bound nη + π n 6 Re π z

10 0 KATHRIN BRINGMANN AND JAN MANSCHOT Moreover on C z = x + y η + η η As before, the path of integration may be estimated against η Thus We next decompose M = π E η N+ + e π n n N e π n N d m, l l= j 0 p ji l+,j, where for s > 0 I s,j := z s e π 6z π j z +n+z dq πi C These integrals may now be written in terms of the classical I-Bessel functions Lemma We have I s,j = n s π j 6 s+ I s π j n + O n s π e n Proof: Let D be the rectangular counterclocwise path from iη to + iη with endpoints η iη and η + iη Denote by D i, i =,,, the paths from iη to η iη, from η iη to η + iη, and from η + iη to + iη Maing the change of variables q = e z gives that I s,j = πi D z s e π 6z π j z +nz dz We next use the Residue Theorem to turn this integral into an integral over D For this we bound the integrals along D and D We only give the details for D On this path we may bound Re z z η Writing z = η + i u, 0 u < gives z = η u + η η + u Thus the integral along D may be bounded by e π η j η η + u s e nη u du e π η +nη η s e nu du η u s e nu du

11 The second term is an incomplete Gamma function and thus exponentially small Thus may up to an exponentially small error be bounded by η s e π η +nη e nη n s π n e n In the remaining integral we mae the change of variables z = t to get n I s,j = n s t s e t+ π n π jn 6t t dt + O n s π n e πi D We now use the following representation of the I-Bessel function [] I l z = z l 0+ t l exp t + z dt, πi t where the integral is along any path looping from around 0 bac to counterclocwise Substitution into gives the claim Substitution of Lemma in equation yields M = π l N j 0 I l d m, l p j n +l+ π j n π j 6 +l+ + O n π e n The integral along C On C, Imz varies from η to π + η Using a rough bound for the theta function, we find ϑ m q n Using we obtain the bound 5 Now e n n++mrez + n 0 e nrez = Re e π Re e z ; e z 6 z = z η η + Imz 5η Thus 5 may be estimated against exp π This gives that: 0η E e π 0η +nη e π 5 6n e η η

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14 KATHRIN BRINGMANN AND JAN MANSCHOT Mathematical Institute, University of Cologne, Weyertal 86-90, 509 Cologne, Germany address: Camille Jordan Institute, University of Lyon, boulevard du novembre 98, 696 Villeurbanne cedex, France address: