The u-plane integral, mock modularity and enumerative geometry

Aspman, Johannes; Furrer, Elias; Korpas, Georgios; Ong, Zhi-Cong; Tan, Mei Chee

doi:10.1007/s11005-022-01520-7

Cited by 4 publications

(3 citation statements)

References 48 publications

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“…This construction assigns a finite index subgroup of Γ ⊂ PSL(2, Z) to an elliptic surface, with the singular fibers being mapped to cusps and elliptic points of Γ. An important use of modularity is the evaluation of the so-called u-plane integral [28][29][30][31][32][33], which relies on the map from the Coulomb branch to a subspace of the upper half-plane, called the fundamental domain of Γ. This map was more recently studied to great depths in [34][35][36] for 4d SQCD theories, even for non-modular configurations.…”

Section: Jhep05(2022)163 1 Introductionmentioning

confidence: 99%

Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers

Magureanu

2022

J. High Energ. Phys.

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We study the Coulomb branches of four-dimensional supersymmetric quantum field theories with $$ \mathcal{N} $$ N = 2 supersymmetry, including the KK theories obtained from the circle compactification of the 5d $$ \mathcal{N} $$ N = 1 En Seiberg theories, with particular focus on the relation between their Seiberg-Witten geometries and rational elliptic surfaces. More attention is given to the modular surfaces, which we completely classify using the classification of subgroups of the modular group PSL(2, ℤ), deriving closed-form expressions for the modular functions for all congruence and some of the non-congruence subgroups. Moreover, in such cases, we give a prescription for determining the BPS quivers from the fundamental domains of the monodromy groups and study how changes of these domains can be interpreted as quiver mutations. This prescription can be also generalized to theories whose Coulomb branches contain ‘undeformable’ singularities, leading to known quivers of such theories.

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Section: Jhep05(2022)163 1 Introductionmentioning

confidence: 99%

Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers

Magureanu

2022

J. High Energ. Phys.

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“…The recent progress in computing u-plane integrals has been enabled by mapping the u-plane to a modular fundamental domain, on which the u-plane integrand can be related to mock modular forms and thus be efficiently evaluated [24,26,[43][44][45][46][47][48][49][50][51]. It has been known since the 1990s that the u-planes for N = 2 SQCD with N f = 0, 2, 3 massless hypermultiplets are modular and correspond to fundamental domains for congruence subgroups of PSL(2, Z) [80].…”

Section: Fundamental Domainsmentioning

confidence: 99%

“…Recently, progress has been made on evaluating these u-plane integrals using a change of variables from a to the running coupling τ . As a result, the integration domain becomes a fundamental domain F ⊆ H in the upper half-plane H for the running coupling [24,26,[43][44][45][46][47][48][49][50][51]. The integral then takes the form Φ = F dτ ∧ dτ ν(τ ) Ψ(τ, τ ), (1.2) where the measure factor ν(τ ) further contains the Jacobian for the change of variables from a to τ .…”

Section: Introductionmentioning

confidence: 99%

Topological twists of massive SQCD, Part I

Aspman¹,

Furrer²,

Manschot³

2022

Preprint

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We consider topological twists of four-dimensional N = 2 supersymmetric QCD with gauge group SU(2) and N f ≤ 3 fundamental hypermultiplets. The twists are labelled by a choice of background fluxes for the flavour group, which provides an infinite family of topological partition functions. In this Part I, we demonstrate that in the presence of such fluxes the theories can be formulated for arbitrary gauge bundles on a compact four-manifold. Moreover, we consider arbitrary masses for the hypermultiplets, which introduce new intricacies for the evaluation of the low-energy path integral on the Coulomb branch. We develop techniques for the evaluation of these path integrals. In the forthcoming Part II, we will deal with the explicit evaluation.

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Partition functions and fibering operators on the Coulomb branch of 5d SCFTs

Closset

Magureanu

2023

J. High Energ. Phys.

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We study 5d $$ \mathcal{N} $$ N = 1 supersymmetric field theories on closed five-manifolds $$ \mathcal{M} $$ M 5 which are principal circle bundles over simply-connected Kähler four-manifolds, $$ \mathcal{M} $$ M 4, equipped with the Donaldson-Witten twist. We propose a new approach to compute the supersymmetric partition function on $$ \mathcal{M} $$ M 5 through the insertion of a fibering operator, which introduces a non-trivial fibration over $$ \mathcal{M} $$ M 4, in the 4d topologically twisted field theory. We determine the so-called Coulomb branch partition function on any such $$ \mathcal{M} $$ M 5, which is conjectured to be the holomorphic ‘integrand’ of the full partition function. We precisely match the low-energy effective field theory approach to explicit one-loop computations, and we discuss the effect of non-perturbative 5d BPS particles in this context. When $$ \mathcal{M} $$ M 4 is toric, we also reconstruct our Coulomb branch partition function by appropriately gluing Nekrasov partition functions. As a by-product of our analysis, we provide strong new evidence for the validity of the Lockhart-Vafa formula for the five-sphere partition function.

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The u-plane integral, mock modularity and enumerative geometry

Cited by 4 publications

References 48 publications

Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers

Seiberg-Witten geometry, modular rational elliptic surfaces and BPS quivers

Topological twists of massive SQCD, Part I

Partition functions and fibering operators on the Coulomb branch of 5d SCFTs

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