Torus fixed points of moduli spaces of stable bundles of rank three

Weist, Thorsten

doi:10.1016/j.jpaa.2010.12.020

Cited by 18 publications

(27 citation statements)

References 17 publications

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“…the other one is the spectral flow isomorphism of the N = (0, 4) superconformal algebra, which we want to recall for r M5-branes here, building on refs. [53,60], see also [57]. Proposition 2.9 of ref.…”

Section: The Decomposition Of the Elliptic Genusmentioning

confidence: 88%

Wall-Crossing Holomorphic Anomaly and Mock Modularity of Multiple M5-Branes

Alim

Haghighat

Hecht³

et al. 2015

Commun. Math. Phys.

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Using wall-crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes wrapping a rigid divisor inside a Calabi-Yau manifold. The anomaly originates from restoring modularity of an indefinite theta-function capturing the wall-crossing of BPS invariants associated to D4-D2-D0 brane systems. We show the compatibility of this equation with anomaly equations previously observed in the context of N = 4 topological Yang-Mills theory on È 2 and E-strings obtained from wrapping M5-branes on a del Pezzo surface. The non-holomorphic part is related to the contribution originating from bound-states of singly wrapped M5-branes on the divisor. We show in examples that the information provided by the anomaly is enough to compute the BPS degeneracies for certain charges. We further speculate on a natural extension of the anomaly to higher D4-brane charge.shown in ref. [18]. 3 A similar story showed up in N = 4 topological U (2) SYM theory on È 2 [5], where it was shown that different sectors of the partition function need a nonholomorphic completion which was found earlier in ref. [22] in order to restore S-dualtiy invariance. An anomaly equation describing this non-holomorphicity was expected [5] in the cases where b + 2 (P ) = 1. In these cases holomorphic deformations of the canonical bundle are absent. The non-holomorphic contributions were associated with reducible connections U (n) → U (m) × U (n − m) [5,4]. In ref.[4] this anomaly was furthermore related to an anomaly appearing in the context of E-strings [23]. These strings arise from an M5-brane wrapping a del Pezzo surface B 9 , also called 1 2 K3. The anomaly in this context was related to the fact that n of these strings can form bound-states of m and (n − m) strings. Furthermore, the anomaly could also be related to the one appearing in topological string theory.The anomaly thus follows from the formation of bound-states. Although the holomorphic expansion would not know about the contribution from bound-states, the restoration of duality symmetry forces one to take these contributions into account. The nonholomorphicity can be understood physically as the result of a regularization procedure. The path integral produces objects like theta-functions associated to indefinite quadratic forms which need to be regularized to avoid divergences. This regularization breaks the modular symmetry, restoring the symmetry gives non-holomorphic objects. The general mathematical framework to describe these non-holomorphic completions is the theory of mock modular forms developed by Zwegers in ref. [24]. 4 A mock modular form h(τ ) of weight k is a holomorphic function which becomes modular after the addition of a function g * (τ ), at the cost of losing its holomorphicity. Here, g * (τ ) is constructed from a modular form g(τ ) of weight 2 − k, which is referred to as shadow.Another manifestation of the background dependence of the holomorphic expansions of the topological theories are wall-cross...

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Section: The Decomposition Of the Elliptic Genusmentioning

confidence: 88%

Wall-Crossing Holomorphic Anomaly and Mock Modularity of Multiple M5-Branes

Alim

Haghighat

Hecht³

et al. 2015

Commun. Math. Phys.

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“…Expressions for f 3,1 (τ ) were also determined using localization with respect to the toric symmetry of P 2 by Weist[77] and Kool[78].…”

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confidence: 99%

Vafa–Witten Theory and Iterated Integrals of Modular Forms

Manschot

2019

Commun. Math. Phys.

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Vafa-Witten (VW) theory is a topologically twisted version of N = 4 supersymmetric Yang-Mills theory. S-duality suggests that the partition function of VW theory with gauge group SU (N ) transforms as a modular form under duality transformations. Interestingly, Vafa and Witten demonstrated the presence of a modular anomaly, when the theory has gauge group SU (2) and is considered on the complex projective plane P 2 . This modular anomaly could be expressed as an integral of a modular form, and also be traded for a holomorphic anomaly. We demonstrate that the modular anomaly for gauge group SU (3) involves an iterated integral of modular forms. Moreover, the modular anomaly for SU (3) can be traded for a holomorphic anomaly, which is shown to factor into a product of the partition functions for lower rank gauge groups. The SU (3) partition function is mathematically an example of a mock modular form of depth two.

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“…The theory of (mixed) mock modular forms has developed within the past two decades following the seminal work of Zwegers [30]. The next obvious generalization is to U(3) Vafa-Witten theory on P 2 for which the relevant partition functions are where the leading Fourier coefficients of h 3,µ are given by [13,15,16,17,24] h 3,0 (τ ) = 1 9 −q+3q 2 +17q 3 +41q 4 +78q 5 +120q 6 +193q 7 +240q 8 +359q 9 +414q 10 +O q 11 , (1.4) 26 3 . (1.5)…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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

Bringmann

Nazaroglu

2019

Trans. Amer. Math. Soc.

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Topologically twisted N = 4 super Yang-Mills theory has a partition function that counts Euler numbers of instanton moduli spaces. On the manifold P 2 and with gauge group U(3) this partition function has a holomorphic anomaly which makes it a mock modular form of depth two. We employ the Circle Method to find a Rademacher expansion for the Fourier coefficients of this partition function. This is the first example of the use of Circle Method for a mock modular form of a higher depth. 1 We use the notation fN,µ(τ ) for the generating function of Vafa-Witten invariants and define the related function hN,µ(τ ) through fN,µ(τ ) := h N,µ (τ ) η(τ ) 3N , where η(τ ) is Dedekind's eta function. This notation is consistent with that of [7] but differs from that of [17], where hN,µ(τ ) = f N,µ (τ ) η(τ ) 3N is used to denote the generating function of Vafa-Witten invariants. arXiv:1803.09270v2 [math.NT]

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Torus fixed points of moduli spaces of stable bundles of rank three

Cited by 18 publications

References 17 publications

Wall-Crossing Holomorphic Anomaly and Mock Modularity of Multiple M5-Branes

Wall-Crossing Holomorphic Anomaly and Mock Modularity of Multiple M5-Branes

Vafa–Witten Theory and Iterated Integrals of Modular Forms

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

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