Cohomological localization of $$ \mathcal{N} $$ = 2 gauge theories with matter

Festuccia, Guido; Gorantis, Anastasios; Pittelli, Antonio; Polydorou, Konstantina; Ruggeri, Lorenzo

doi:10.1007/jhep09(2020)133

Cited by 7 publications

(14 citation statements)

References 65 publications

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“…choose f + at SD fixed points and f − at ASD ones (or vice versa). For the hypermultiplet, the complex at the fixed points roughly equates to the one of a chiral/anti-chiral Dirac operator (cf [11]). Hence, upon applying appropriate transition maps (similar to (2.5)), the two symbols, once extended to large enough open subsets, should be related in terms of (3.11) on the intersection.…”

Section: Remark 45 (Non-trivial Gauge Backgrounds)mentioning

confidence: 99%

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Index of the transversally elliptic complex in Pestunization

Mauch

Ruggeri

2022

J. Phys. A: Math. Theor.

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In this note we present a formula for the equivariant index of the cohomological complex obtained from localization of $\mathcal{N}=2$ SYM on simply-connected compact four-manifolds with a $T^2$-action. Knowledge of said index is essential to compute the perturbative part of the partition function for the theory. In the topologically twisted case, the complex is elliptic and its index can be computed in a standard way using the Atiyah-Bott localization formula. Recently, a framework for more general types of twisting, so-called cohomological twisting, was introduced for which the complex turns out to be only transversally elliptic. While the index of such a complex has been computed for some cases where the manifold can be lifted to a Sasakian $S^1$-fibration in five dimensions, a general four-dimensional treatment was still lacking. We provide a formal, purely four-dimensional treatment of the cohomological complex, showing that the Laplacian part can be globally split off while the remaining part can be trivialized uniquely in the group-direction. This ultimately produces a simple formula for the index applicable for any compact simply-connected four-manifold. Finally, the index formula is applied to examples on $S^4$, $\mathbb{CP}^2$ and $\mathbb{F}^1$. For the latter, we use the result to compute the perturbative partition function.

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Section: Remark 45 (Non-trivial Gauge Backgrounds)mentioning

confidence: 99%

“…More precisely, in the following we use the vector field that enters in the definition of P + ω which does not necessarily have to be v. However, both choices yield isomorphic subbundles of Ω 2[10] 11. Use a retraction r : U → Y which induces r * : K H (TY) → K H (TU).…”

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confidence: 99%

Index of the transversally elliptic complex in Pestunization

Mauch

Ruggeri

2022

J. Phys. A: Math. Theor.

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“…In this section we review the general aspects of N = 2 supersymmetric theories on curved manifolds following [10,12].…”

Section: N = Supersymmetric Theories On Curved Spacementioning

confidence: 99%

“…The supersymmetric backgrounds are then obtained by requiring that the fermionic fields ψ i µ , χ i and their supersymmetry variations vanish. These conditions are analyzed in detail in [10,12] and explicit expressions for various background fields are obtained 2 . The supersymmetric backgrounds are obtained by requiring that • The metric g admits a smooth real Killing vector field v with (at-most) isolated fixed points.…”

Section: Lagrangians and Background Fieldsmentioning

confidence: 99%

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Flavor deformations and supersymmetry enhancement in $4d$ $\mathcal{N}=2$ theories

Naseer

Thull

2022

SciPost Phys.

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We study \mathcal{N}=2𝒩=2 theories on four-dimensional manifolds that admit a Killing vector vv with isolated fixed points. It is possible to deform these theories by coupling position-dependent background fields to the flavor current multiplet. The partition function of the deformed theory only depends on the value of the background scalar fields at the fixed points of vv. For a single adjoint hypermultiplet, the partition function becomes independent of the supergravity as well as the flavor background if the scalars attain special values at the fixed points. For these special values, supersymmetry at the fixed points enhances from the Donaldson-Witten twist to the Marcus twist or the Vafa-Witten twist of \mathcal{N}=4𝒩=4 SYM. Our results explain the recently observed squashing independence of \mathcal{N}=2^*𝒩=2* theory on the squashed sphere and provide a new squashing independent point. Interpreted through the AGT-correspondence, this implies the bb-independence of torus one-point functions of certain operators in Liouville/Toda CFT. The position-dependent deformations imply relations between correlators of partially integrated operators in \mathcal{N}=2𝒩=2 SCFT with flavor symmetries.

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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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Cohomological localization of $$ \mathcal{N} $$ = 2 gauge theories with matter

Cited by 7 publications

References 65 publications

Index of the transversally elliptic complex in Pestunization

Index of the transversally elliptic complex in Pestunization

Flavor deformations and supersymmetry enhancement in $4d$ $\mathcal{N}=2$ theories

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

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