### Citations

779 |
Conformal Field Theory
- Francesco, Mathieu, et al.
- 1997
(Show Context)
Citation Context ...sults Jacobi forms were first systematically studied by Eichler and Zagier [14] and enjoy a wide variety of applications in the theory of modular forms, combinatorics [37, 38], conformal field theory =-=[12, 24]-=-, black hole physics [10, 13], Hilbert schemes of points [18], Donaldson invariants [19], and many other topics. This paper focuses on a class of negative index Jacobi forms with a single order pole i... |

387 |
The Theory of Jacobi Forms
- Eichler, Zagier
- 1985
(Show Context)
Citation Context ...rface S and appear also as counts of BogomolnyPrasad-Sommerfield (BPS) states in physics. 1. Introduction and Statement of Results Jacobi forms were first systematically studied by Eichler and Zagier =-=[14]-=- and enjoy a wide variety of applications in the theory of modular forms, combinatorics [37, 38], conformal field theory [12, 24], black hole physics [10, 13], Hilbert schemes of points [18], Donaldso... |

229 | Geometric engineering of quantum field theories,” Nucl. Phys. B497
- Katz, Klemm, et al.
- 1997
(Show Context)
Citation Context ...d. 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 =-=[23, 29]-=-. If S is a K3 surface, the sheaves correspond to (small) black holes in N = 4 supergravity also known as Dabholkar-Harvey states [8, 9]. We specialize to the moduli space of sheaves with rank r = 1 a... |

179 |
Asymptotic formulae in combinatory analysis
- HARDY, RAMANUJAN
- 1918
(Show Context)
Citation Context ...and in particular bm,k(n) = 0 for m > n. We next compare the asymptotic behavior of the coefficients b0,k(n) with those of pk(n). It is well-known that the asymptotic growth of the latter is given by =-=[21, 30]-=-: pk(n) = 2 ( k 3 ) 1+k 4 (8n)− 3+k 4 eπ √ 2kn 3 +O ( n− 5+k 4 eπ √ 2kn 3 ) . Thus we find for the ratio b0,k(n)/pk(n): b0,k(n) pk(n) = π3 16 ( k 3n ) 3 2 +O ( n−2 ) . 4 KATHRIN BRINGMANN 1 AND JAN MA... |

153 |
Representations of associative algebras and coherent sheaves
- Bondal
- 1990
(Show Context)
Citation Context ...xponentially larger than the number of remaining SL(2) multiplets given by βdimC(M)+1(M). We note that sheaves on toric surfaces relevant for this article also allow a description in terms of quivers =-=[6, 32]-=-. The Hilbert scheme of n ≫ 1 points corresponds to increasing dimensions of the spaces associated the nodes, with the number of arrows kept fixed. Thus we observe that the asymptotic behavior of the ... |

147 |
The Betti numbers of the Hilbert schemes of points on a smooth projective surface
- Gottsche
- 1990
(Show Context)
Citation Context ...nd Zagier [14] and enjoy a wide variety of applications in the theory of modular forms, combinatorics [37, 38], conformal field theory [12, 24], black hole physics [10, 13], Hilbert schemes of points =-=[18]-=-, Donaldson invariants [19], 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 o... |

113 | Extremal transitions and five–dimensional supersymmetric field theories” Nucl. Phys. B483
- Morrison, Seiberg
- 1997
(Show Context)
Citation Context ...d. 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 =-=[23, 29]-=-. If S is a K3 surface, the sheaves correspond to (small) black holes in N = 4 supergravity also known as Dabholkar-Harvey states [8, 9]. We specialize to the moduli space of sheaves with rank r = 1 a... |

93 | Dyson’s crank of a partition
- Andrews, Garvan
- 1988
(Show Context)
Citation Context ... are functions mimicking modular behavior on (subsets of) Q [15]. It is well-known that the inverse theta function θ1(w; τ) −1 can be written as a sum over its poles. See for more details for example =-=[2, 17]-=-. Proposition 2.1. We have for |ζq|, |ζ−1q| < 1: q k 24 ( ζ 1 2 − ζ− 12 ) hk(w; τ) = 1 (q)k∞ ∑ m∈Z ϑm(q)ζ m with (q)∞ := ∏∞ n=1(1− qn). Proof: The inverse theta function θ1(w; τ) −1 is expressed as a ... |

92 | Elliptic genera of symmetric products and second quantized strings
- Dijkgraaf, Moore, et al.
- 1997
(Show Context)
Citation Context ... systematically studied by Eichler and Zagier [14] and enjoy a wide variety of applications in the theory of modular forms, combinatorics [37, 38], conformal field theory [12, 24], black hole physics =-=[10, 13]-=-, Hilbert schemes of points [18], Donaldson invariants [19], 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. Th... |

63 | New combinatorial interpretations of Ramanujan’s partition congruences mod 5
- Garvan
- 1988
(Show Context)
Citation Context ...he number of partitions of n into k colors. These are enumerated by η(τ)−k:∑ n≥0 pk(n) q n− k 24 = 1 η(τ)k . For k = 1, equation (1.1) corresponds to the generating function of the crank of partition =-=[17]-=-, and for k = 2 to the birank [20]. 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−1 and with maxima... |

56 |
Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b
- Göttsche, Zagier
- 1998
(Show Context)
Citation Context ...wide variety of applications in the theory of modular forms, combinatorics [37, 38], conformal field theory [12, 24], black hole physics [10, 13], Hilbert schemes of points [18], Donaldson invariants =-=[19]-=-, 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 co... |

56 |
On two theorems of combinatory analysis and some allied identities
- Rogers
- 1917
(Show Context)
Citation Context ...property ϑm (q) = ϑ−m (q) of ϑm (q), continues to hold when ϑm (q) is defined with |m| replaced by m. The functions ϑm (q) are examples of false theta functions, which were first introduced by Rogers =-=[31]-=- and have attracted a lot of interest recently. Using the Rogers and Fine identity one can relate ϑm to so-called quantum modular forms which are functions mimicking modular behavior on (subsets of) Q... |

52 |
The Betti numbers of the moduli space of stable sheaves of rank 2 on a ruled surface
- Yoshioka
- 1995
(Show Context)
Citation Context ... 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 rank sheaves on surfaces =-=[26, 27, 35]-=-, and partition functions of black holes and quantum geometry [5, 16, 22] will lead to to important novel insights. The paper is organized as follows: In Section 2 we rewrite the functions of interest... |

42 |
Nonrenormalization of the superstring tension
- Dabholkar, Harvey
- 1989
(Show Context)
Citation Context ...nal supersymmetric gauge theory through geometric engineering [23, 29]. If S is a K3 surface, the sheaves correspond to (small) black holes in N = 4 supergravity also known as Dabholkar-Harvey states =-=[8, 9]-=-. We specialize to the moduli space of sheaves with rank r = 1 and 1st and 2nd Chern classes c1 ∈ H2(S,Z) and c2 ∈ H4(S,Z). These moduli spaces are isomorphic to the Hilbert scheme of c2 points on S (... |

41 | Asymptotic level density in heterotic string theory and rotating black holes
- Russo, Susskind
- 1995
(Show Context)
Citation Context ...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, 33, 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(2)spin representations correspond to representations o... |

37 | Arithmetic of Quantum Entropy Function,” JHEP 0908
- Sen
- 2009
(Show Context)
Citation Context ... labeled by their angular momentum or highest weight J . The subset of BPS states with vanishing angular momentum (J = 0), also known as “pure Higgs states” [4], have recently attracted much interest =-=[4, 25, 28, 34]-=-. The states with J = 0 are in some sense more fundamental. In particular in gravity, these states are candidates for microstates of single center black holes, and as such are the states relevant for ... |

36 | Modular Functions and Dirichlet - Apostol - 1990 |

33 |
Elliptic genera and n=2 superconformal field theory
- Kawai, Yamada, et al.
(Show Context)
Citation Context ...sults Jacobi forms were first systematically studied by Eichler and Zagier [14] and enjoy a wide variety of applications in the theory of modular forms, combinatorics [37, 38], conformal field theory =-=[12, 24]-=-, black hole physics [10, 13], Hilbert schemes of points [18], Donaldson invariants [19], and many other topics. This paper focuses on a class of negative index Jacobi forms with a single order pole i... |

28 |
Quantum modular forms
- Zagier
(Show Context)
Citation Context ... 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 =-=[15, 40]-=-. The appearance of these functions in the above mentioned topics calls for an explicit knowledge of their coefficients and in particular of their asymptotic growth. In this paper we provide such asym... |

19 |
Phase transitions in M theory and
- Witten
- 1996
(Show Context)
Citation Context ...es of sheaves supported on a Calabi-Yau manifold. From this perspective the SU(2)spin representations correspond to representations of the Lefshetz SL(2) action on the cohomology of the moduli spaceM =-=[36]-=-. The states with J = 0 correspond to the part of the middle cohomology which is invariant under the Lefshetz action. In the present work, we consider moduli spaces of semi-stable sheaves supported on... |

17 |
Helices and vector bundles: Seminaire Rudakov
- Rudakov
- 1990
(Show Context)
Citation Context ...xponentially larger than the number of remaining SL(2) multiplets given by βdimC(M)+1(M). We note that sheaves on toric surfaces relevant for this article also allow a description in terms of quivers =-=[6, 32]-=-. The Hilbert scheme of n ≫ 1 points corresponds to increasing dimensions of the spaces associated the nodes, with the number of arrows kept fixed. Thus we observe that the asymptotic behavior of the ... |

16 |
Precision counting of small black holes,” JHEP 0510
- Dabholkar, Denef, et al.
- 2005
(Show Context)
Citation Context ...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, 33, 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(2)spin representations correspond to representations o... |

16 |
The M5-brane elliptic genus
- Gaiotto, Strominger, et al.
(Show Context)
Citation Context ... we belief that application of the techniques in the present paper to partition functions for higher rank sheaves on surfaces [26, 27, 35], and partition functions of black holes and quantum geometry =-=[5, 16, 22]-=- will lead to to important novel insights. The paper is organized as follows: In Section 2 we rewrite the functions of interest in terms of false theta functions and determine their Taylor expansion. ... |

14 |
On the Fourier coefficients of certain modular forms of positive dimension
- Rademacher, Zuckerman
- 1938
(Show Context)
Citation Context ...and in particular bm,k(n) = 0 for m > n. We next compare the asymptotic behavior of the coefficients b0,k(n) with those of pk(n). It is well-known that the asymptotic growth of the latter is given by =-=[21, 30]-=-: pk(n) = 2 ( k 3 ) 1+k 4 (8n)− 3+k 4 eπ √ 2kn 3 +O ( n− 5+k 4 eπ √ 2kn 3 ) . Thus we find for the ratio b0,k(n)/pk(n): b0,k(n) pk(n) = π3 16 ( k 3n ) 3 2 +O ( n−2 ) . 4 KATHRIN BRINGMANN 1 AND JAN MA... |

12 | BPS invariants of semi-stable sheaves on rational surfaces
- Manschot
(Show Context)
Citation Context ... 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 rank sheaves on surfaces =-=[26, 27, 35]-=-, and partition functions of black holes and quantum geometry [5, 16, 22] will lead to to important novel insights. The paper is organized as follows: In Section 2 we rewrite the functions of interest... |

9 |
den Bleeken
- Bena, Berkooz, et al.
(Show Context)
Citation Context ...pin 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 known as “pure Higgs states” =-=[4]-=-, have recently attracted much interest [4, 25, 28, 34]. The states with J = 0 are in some sense more fundamental. In particular in gravity, these states are candidates for microstates of single cente... |

9 |
Split states, entropy enigmas
- Denef, Moore
(Show Context)
Citation Context ... a factor n− 3 2 . Moreover, the number of SU(2)spin multiplets increases with J for small J . It is interesting to compare this with other known asymptotics of pure Higgs states. In particular, Ref. =-=[4, 11]-=- 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. [4] demonstrated that the number of pure Higgs states f... |

8 | Black holes and large order quantum geometry, arXiv:0704.2440 [hep-th
- Huang, Klemm, et al.
(Show Context)
Citation Context ... we belief that application of the techniques in the present paper to partition functions for higher rank sheaves on surfaces [26, 27, 35], and partition functions of black holes and quantum geometry =-=[5, 16, 22]-=- will lead to to important novel insights. The paper is organized as follows: In Section 2 we rewrite the functions of interest in terms of false theta functions and determine their Taylor expansion. ... |

7 |
Congruences in ordered pairs of partitions
- Hammond, Lewis
(Show Context)
Citation Context ...k colors. These are enumerated by η(τ)−k:∑ n≥0 pk(n) q n− k 24 = 1 η(τ)k . For k = 1, equation (1.1) corresponds to the generating function of the crank of partition [17], and for k = 2 to the birank =-=[20]-=-. 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−1 and with maximal degree n. Therefore, we can expr... |

7 |
The Mellin transform and other useful analytic techniques, Appendix to E. Zeidler, Quantum Field Theory I: Basics in Mathematics and Physics. A Bridge Between Mathematicians and Physicists SpringerVerlag, Berlin-Heidelberg-New York (2006), 305–323
- Zagier
(Show Context)
Citation Context ...m. From this the statement of the proposition easily follows. The function ϑm is not modular but may be nicely approximated by its Taylor expansion. For this we use the following general lemma (see =-=[39]-=- for the case of real functions). Lemma 2.2. Let f : C→ C be a C∞ function. Furthermore, we require that f(x) and all its derivatives are of rapid decay for Re(x) → ∞. Then for t → 0 with Re(t) > 0 7a... |

4 |
BPS invariants of N=4 gauge theory on a surface
- Manschot
(Show Context)
Citation Context ... 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 rank sheaves on surfaces =-=[26, 27, 35]-=-, and partition functions of black holes and quantum geometry [5, 16, 22] will lead to to important novel insights. The paper is organized as follows: In Section 2 we rewrite the functions of interest... |

3 |
A Farey Tail for Attractor
- Boer, Cheng, et al.
(Show Context)
Citation Context ... we belief that application of the techniques in the present paper to partition functions for higher rank sheaves on surfaces [26, 27, 35], and partition functions of black holes and quantum geometry =-=[5, 16, 22]-=- will lead to to important novel insights. The paper is organized as follows: In Section 2 we rewrite the functions of interest in terms of false theta functions and determine their Taylor expansion. ... |

1 |
q-series and quantum modular forms, submitted for publication
- Folsom, Ono, et al.
(Show Context)
Citation Context ... 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 =-=[15, 40]-=-. The appearance of these functions in the above mentioned topics calls for an explicit knowledge of their coefficients and in particular of their asymptotic growth. In this paper we provide such asym... |