Gauge theories on compact toric manifolds

Bonelli, Giulio; Fucito, F.; Morales, José F.; Ronzani, Massimiliano; Sysoeva, Ekaterina; Tanzini, Alessandro

doi:10.1007/s11005-021-01419-9

“…Such non-holomorphic contributions have appeared in similar contexts. In specific cases, the non-holomorphic contributions are derived from different physical points of view, for example the continuum of multi-particle states in R 4 [58], or the quantum field theory on the world volume of the D-branes [44,47,48], or the perspective of D3-instantons in the hypermultiplet moduli space [18].…”

Section: Jhep03(2022)001mentioning

confidence: 99%

“…This implies potentially interesting arithmetic of the BPS indices, while the non-holomorphic contribution is also interesting independently, and a generalization of similar non-holomorphic terms in partition functions of N = 4 Yang-Mills on four-manifolds [40,41]. The origins and explicit expressions of these terms have been understood better recently [30,[42][43][44][45][46][47][48].…”

Section: Introductionmentioning

confidence: 99%

Scaling black holes and modularity

Chattopadhyaya

¹

,

Manschot

²

,

Mondal

³

2022

J. High Energ. Phys.

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Scaling black holes are solutions of supergravity with multiple black hole singularities, which can be adiabatically connected to a single center black hole solution. We develop techniques to determine partition functions for such scaling black holes, if each constituent carries a non-vanishing magnetic charge corresponding to a D4-brane in string theory, or equivalently M5-brane in M-theory. For three constituents, we demonstrate that the partition function is a mock modular form of depth two, and we determine the appropriate non-holomorphic completion using generalized error functions. From the four-dimensional perspective, the modular parameter is the axion-dilaton, and our results show that S-duality leaves this subset of the spectrum invariant. From the five-dimensional perspective, the modular parameter is the complex structure of a torus T2, and the scaling black holes are dual to states in the dimensional reduction of the M5-brane worldvolume theory to T2. As a case study, we specialize the compactification manifold to a K3 fibration, and explicitly evaluate holomorphic parts of partition functions.

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“…Such non-holomorphic contributions have appeared in similar contexts. In specific cases, the non-holomorphic contributions are derived from different physical points of view, for example the continuum of multi-particle states in R 4 [57], or the quantum field theory on the world volume of the D-branes [44,47,48], or the perspective of D3-instantons in the hypermultiplet moduli space [18].…”

Section: Modular Completion Of φ µmentioning

confidence: 99%

“…This implies potentially interesting arithmetic of the BPS indices, while the non-holomorphic contribution is also interesting independently, and a generalization of similar non-holomorphic terms in partition functions of N = 4 Yang-Mills on four-manifolds [40,41]. The origins and explicit expressions of these terms have been understood better recently [30,42,43,44,45,46,47,48].…”

Section: Introductionmentioning

confidence: 99%

Scaling Black Holes and Modularity

Chattopadhyaya,

Manschot,

Mondal

2021

Preprint

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Scaling black holes are solutions of supergravity with multiple black hole singularities, which can be adiabatically connected to a single center black hole solution. We develop techniques to determine partition functions for such scaling black holes, if each constituent carries a non-vanishing magnetic charge corresponding to a D4-brane in string theory, or equivalently M5-brane in M-theory. For three constituents, we demonstrate that the partition function is a mock modular form of depth two, and we determine the appropriate non-holomorphic completion using generalized error functions. From the four-dimensional perspective, the modular parameter is the axion-dilaton, and our results show that S-duality leaves this subset of the spectrum invariant. From the five-dimensional perspective, the modular parameter is the complex structure of a torus T 2 , and the scaling black holes are dual to states in the dimensional reduction of the M5-brane worldvolume theory to T 2 . As a case study, we specialize the compactification manifold to a K3 fibration, and explicitly evaluate holomorphic parts of partition functions.October 13, 2021(2.23)this condition is always satisfied since a + b − c > 0 and the lhs does not have real roots in this case. Moreover if (c − a)(c − b) ≥ 0, the condition is saturated for ε ± c = 1 M c 1 ± (c−a)(c−b) ab and violated for ε − c < ε < ε + c . Both roots are non-negative provided (a, b, c) obeys the triangle inequality a + b ≥ c. Noting that ε + c > 1 M c , we see ε ≥ ε + c makes r 31 negative. Thus we must have ε ≤ ε − c . Using two other triangle inequalities, we have 1 We apologize for the multiple use of a, b and c.

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“…Within the setting of the twisted N = 4 theory and the dual two-dimensional field theory, Dabholkar, Putrov and Witten [30] have given a derivation of the non-holomorphic anomaly. Moreover, Bonelli et al [40] have arrived at the holomorphic anomaly using toric localization in supersymmetric field theory.…”

Section: Introductionmentioning

confidence: 99%

Topological correlators of $SU(2)$, $\mathcal{N}=2^*$ SYM on four-manifolds

Manschot,

Moore

2021

Preprint

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We consider topologically twisted N = 2, SU (2) gauge theory with a massive adjoint hypermultiplet on a smooth, compact four-manifold X. A consistent formulation requires coupling the theory to a Spin c structure, which is necessarily non-trivial if X is non-spin. We derive explicit formulae for the topological correlation functions when b + 2 ≥ 1. We demonstrate that, when the Spin c structure is canonically determined by an almost complex structure and the mass is taken to zero, the path integral reproduces known results for the path integral of the N = 4 gauge theory with Vafa-Witten twist.On the other hand, we reproduce results from Donaldson-Witten theory after taking a suitable infinite mass limit. The topological correlators are functions of the UV coupling constant τ uv and we confirm that they obey the expected S-duality transformation laws. The holomorphic part of the partition function is a generating function for the Euler numbers of the matter (or obstruction) bundle over the instanton moduli space. For b + 2 = 1, we derive a non-holomorphic contribution to the path integral, such that the partition function and correlation functions are mock modular forms rather than modular forms. We comment on the generalization of this work to the large class of N = 2 theories of class S.

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Gauge theories on compact toric manifolds

Cited by 8 publications

References 47 publications

Scaling black holes and modularity

Scaling black holes and modularity

Scaling Black Holes and Modularity

Topological correlators of $SU(2)$, $\mathcal{N}=2^*$ SYM on four-manifolds

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