An exact formula for $\mathbf {U (3)}$ Vafa-Witten invariants on $\mathbb {P}^2$

Bringmann, Kathrin; Nazaroglu, Caner

doi:10.1090/tran/7714

Cited by 8 publications

(11 citation statements)

References 36 publications

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“…The structure of the completion suggests that, for a divisor decomposable into a sum of n effective divisors, the holomorphic generating function h p,µ is a mixed vector valued mock modular form of higher depth, equal to n − 1. Such objects have recently appeared in various mathematical and physical contexts [39,64,65,66,31] and correspond to holomorphic functions whose completion is constructed from period (or Eichler) integrals of mock modular forms of smaller depth (with depth 0 mock modular forms being synonymous with ordinary modular forms). For instance, in the case of standard mock modular forms of depth one, the completion is given by an Eichler integral of a modular function [27].…”

Section: Discussionmentioning

confidence: 99%

Black Holes and Higher Depth Mock Modular Forms

Alexandrov

Pioline

2019

Commun. Math. Phys.

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By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory compactified on the same space. Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor D, at the large volume attractor point. For D irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on D and is therefore known to be modular. Instead, when D is the sum of n irreducible divisors D i , we show that the generating function acquires a modular anomaly. We characterize this anomaly for arbitrary n by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions. As a result, the generating function turns out to be a (mixed) mock modular form of depth n − 1. BPS indices and wall-crossingIn this section, we review some aspects of BPS indices in theories with N = 2 supersymmetry, including the tree flow formula relating the moduli-dependent index Ω(γ, z a ) to the attractor index Ω ⋆ (γ). We then apply this formalism in the context of Calabi-Yau string vacua, and express the generalized DT invariants Ω(γ, z a ) in terms of their counterparts evaluated at the large volume attractor point (1.1), known as MSW invariants. Wall-crossing and attractor flowsThe BPS index Ω(γ, z a ) counts (with sign) micro-states of BPS black holes with total electromagnetic charge γ = (p Λ , q Λ ), for a given value z a of the moduli at spatial infinity. While Ω(γ, z a ) is a locally constant function over the moduli space, it can jump across real codimension one loci where certain bound states, represented by multi-centered black hole solutions of N = 2 supergravity, become unstable. The positions of these loci, known as walls of marginal stability, are determined by the central charge Z γ (z a ), a complex-valued linear function of γ whose modulus gives the mass of a BPS state of charge γ, while the phase determines the supersymmetry subalgebra preserved by the state. Since a bound state can only decay when its mass becomes equal to the sum of masses of its constituents, it is apparent that the walls correspond to hypersurfaces where the phases of two central charges, say Z γ L (z a ) and Z γ R (z a ), become aligned. The bound states which may become unstable are then those whose constituents have charges in the positive cone spanned by γ L and γ R . We shall assume that the charges γ L , γ R have non-zero Dirac-Schwinger-Zwanziger (DSZ) pairing γ L , γ R = 0, since otherwise marginal bound states may form, whose stability is hard to control.The general relation between the values of Ω(γ, z a ) on the two sides of a wall has been found in the mathematics literature by and Joyce-Song [42,43], and justified physically in a series of works [6,7,8,9]. However, in this work we require a somewhat differe...

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Section: Discussionmentioning

confidence: 99%

Black Holes and Higher Depth Mock Modular Forms

Alexandrov

Pioline

2019

Commun. Math. Phys.

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“…Having characterized the precise modular properties of generating functions of BPS indices, one can in principle determine them from the knowledge of their polar coefficients, which could be computed by generalizing ideas in [82,83]. Corrections to the Bekenstein-Hawking area law could in principle be computed by applying the circle method for the completed partition functions [54,55]. It would be very interesting to understand the physical origin of the coefficient R n in the non-holomorphic completion, which presumably arises from a spectral asymmetry in the continuum of the superconformal field theory describing wrapped five-branes, or alternatively from boundaries of the moduli space of anti-selfdual configurations in the gauge theory description.…”

Section: Discussionmentioning

confidence: 99%

“…The knowledge of the detailed modular properties could in principle be leveraged to determine h VW,ref N,µ from the knowledge of its polar terms, e.g. using a Rademacher sum [54,55].…”

Section: Introductionmentioning

confidence: 99%

S-duality and refined BPS indices

Alexandrov,

Manschot,

Pioline

2019

Preprint

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Whenever available, refined BPS indices provide considerably more information on the spectrum of BPS states than their unrefined version. Extending earlier work on the modularity of generalized Donaldson-Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi-Yau threefold Y, we construct the modular completion of generating functions of refined BPS indices supported on a divisor class. Although for compact Y the refined indices are not protected, switching on the refinement considerably simplifies the construction of the modular completion. Furthermore, it leads to a non-commutative analogue of the TBA equations, which suggests a quantization of the moduli space consistent with S-duality. In contrast, for a local CY threefold given by the canonical bundle over a complex surface S, refined DT invariants are well-defined, and equal to Vafa-Witten invariants of S. Our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank. In cases where all reducible components of the divisor class are collinear (which occurs e.g. when b 2 (Y) = 1, or in the local case), we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order. In the local case, it agrees with an earlier proposal by Minahan et al for unrefined invariants, and extends it to the refined level using the afore-mentioned noncommutative structure. Finally, we show that these general predictions reproduce known results for U(2) and U(3) Vafa-Witten theory on P 2 , and make them explicit for U(4). Contents 1. Introduction 2. DT invariants, MSW invariants and modular completions 3. Refined construction 3.1 Refined BPS indices 3.2 Refined modular completions 3.3 Refined instanton generating potential 3.4 Star product and TBA-like equations 4. Local Calabi-Yau and Vafa-Witten invariants 4.1 Vafa-Witten invariants 4.2 Refined Vafa-Witten invariants 4.3 Local limit of elliptically fibered CY threefolds 5. Holomorphic anomaly for refined Vafa-Witten invariants 5.1 Refined holomorphic anomaly for compact CY threefold 5.2 Collinear charges 5.3 Partition function: unrefined case 5.4 Refined partition function 6. Conclusions and future directions A. Theta series and modularity A.1 Theta series and refinement A.2 Generalized error functions A.3 Matrix of parameters B. Relevant functions C. Refined instanton generating potential C.1 Proof of Eq.(C.1) D. Proof of the truncation theorem E. Geometric data for Hirzebruch and del Pezzo surfaces F. Modular completion of Vafa-Witten invariants on P 2 F.1 Rank 2 F.2 Rank 3 F.3 Rank 4

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“…In fact, such results are intimately related to the finite-dimensionality of the associated vector spaces of modular objects and this property forms the basis for many of the remarkable applications of modularity to different fields of mathematics. The Circle Method has already been applied to a case involving rank one false theta functions in [11] and to one involving depth two mock modular forms in [10]. It would be interesting to extend these results to the class of functions studied in this paper and explore the implications to the different fields considered here.…”

Section: Theorem 13mentioning

confidence: 95%

Integral representations of rank two false theta functions and their modularity properties

et al. 2021

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False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta functions, following the example of higher depth mock modular forms. In particular, we prove that under quite general conditions, a rank two false theta function is determined in terms of iterated, holomorphic, Eichler-type integrals. This provides a new method for examining their modular properties and we apply it in a variety of situations where rank two false theta functions arise. We first consider generic parafermion characters of vertex algebras of type $$A_2$$ A 2 and $$B_2$$ B 2 . This requires a fairly non-trivial analysis of Fourier coefficients of meromorphic Jacobi forms of negative index, which is of independent interest. Then we discuss modularity of rank two false theta functions coming from superconformal Schur indices. Lastly, we analyze $${\hat{Z}}$$ Z ^ -invariants of Gukov, Pei, Putrov, and Vafa for certain plumbing $$\mathtt{H}$$ H -graphs. Along the way, our method clarifies previous results on depth two quantum modularity.

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An exact formula for $\mathbf {U (3)}$ Vafa-Witten invariants on $\mathbb {P}^2$

Cited by 8 publications

References 36 publications

Black Holes and Higher Depth Mock Modular Forms

Black Holes and Higher Depth Mock Modular Forms

S-duality and refined BPS indices

Integral representations of rank two false theta functions and their modularity properties

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