The Liouville side of the vortex

Abstract:We calculate the instanton partition function of the four-dimensional N = 2 ⋆ SU(N ) gauge theory in the presence of a generic surface operator, using equivariant localization. By analyzing the constraints that arise from S-duality, we show that the effective twisted superpotential, which governs the infrared dynamics of the two-dimensional theory on the surface operator, satisfies a modular anomaly equation. Exploiting the localization results, we solve this equation in terms of elliptic and quasi-modular forms which resum all non-perturbative corrections. We also show that our results, derived for monodromy defects in the four-dimensional theory, match the effective twisted superpotential describing the infrared properties of certain two-dimensional sigma models coupled either to pure N = 2 or to N = 2 ⋆ gauge theories.

Section: Jhep07(2017)068mentioning

confidence: 99%

Modular and duality properties of surface operators in N = 2 ⋆ $$ \mathcal{N}={2}^{\star } $$ gauge theories

Ashok

Billó

Dell’Aquila

et al. 2017

“…It now follows from the important identities 12) JHEP02 (2016)155 that Y s jm exists only for j ∈ |s| + Z ≥0 . One can normalize the monopole harmonics so that the following holds, D…”

Section: Jhep02(2016)155mentioning

confidence: 99%

“…It is particularly interesting to study such defects in the classes of supersymmetric theories in 2, 3 or 4 dimensions where various dualities or nontrivial relations among observables are known. For example, the inclusion of surface defects in 4D N = 2 supersymmetric gauge theories has been studied in the computation of instanton partition functions [3][4][5][6][7][8][9][10][11][12], superconformal indices [13][14][15][16][17] or sphere partition functions [18][19][20][21], and the results led to a more detailed understanding of the relation between 4D N = 2 SUSY theories and 2D conformal field theories [22][23][24], topological field theories or topological strings. The loop operators in 4D N = 2 theories were studied from a similar viewpoint in [25][26][27]; see [28] for a review.…”

Section: Introductionmentioning

confidence: 99%

Orbifolds, defects and sphere partition function

Hosomichi

2016

Gauge theories in the presence of codimension two vortex defects are known to be related to the theories on orbifolds. By using this relation we study the localized path integrals of 2D N = (2, 2) SUSY gauge theories with point-like vortex defects. We present a formula for the correlation functions of vortex defects inserted at the north and the south poles of squashed spheres. For Abelian gauge theories the correlators are locally constant as functions of the parameters of the defect, but exhibit discontinuity at some threshold values determined from the R-charges of the matter multiplets. For non-Abelian gauge groups the correlators depend non-trivially on the types of gauge symmetry breaking due to the defects.

“…This was originally computed in [13] and recently discussed in the context of AGT correspondence in [11,[14][15][16].…”

Section: Jhep01(2014)038mentioning

confidence: 99%

The stringy instanton partition function

Bonelli

Sciarappa

Tanzini

et al. 2014

Self Cite

Abstract:We perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A 1 singularity. This is accomplished via supersymmetric localization on the blown-up two-sphere. We show that in the blow-down limit C 2 /Z 2 the partition function reduces to the Nekrasov partition function evaluating the equivariant volume of the instanton moduli space. For finite radius we obtain a tower of world-sheet instanton corrections, that we identify with the equivariant Gromov-Witten invariants of the ADHM moduli space. We show that these corrections can be encoded in a deformation of the Seiberg-Witten prepotential. From the mathematical viewpoint, the D1-D5 system under study displays a twofold nature: the D1-branes viewpoint captures the equivariant quantum cohomology of the ADHM instanton moduli space in the Givental formalism, and the D5-branes viewpoint is related to higher rank equivariant Donaldson-Thomas invariants of P 1 × C 2 .