Instanton counting with a surface operator and the chain-saw quiver

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 .

“…Our approach should be compared with the results of [67]. Furthermore, we remark that the above constitutes a useful set-up to study the AGT correspondence [49,63,64].…”

Section: Jhep01(2014)038mentioning

confidence: 99%

The stringy instanton partition function

Bonelli

Sciarappa

Tanzini

et al. 2014

“…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.

“…In the limit that β → 0, it is well-known [74] that 2 , a) can be written as 27) where ω is a symplectic form on U(M G,m ), invariant under the U (1) 2 × T action, and…”

Section: The Spectrum Of Spacetime Bps States On the Lhs Of (512)mentioning

confidence: 99%

“…"Ramified" generalizations of the AGT correspondence for pure SU (N ) to include surface operators were also proposed and checked to some extent in [26,27], although the 1 A written account of these lectures can also be found in [14].…”

Section: Introduction Summary and Acknowledgementsmentioning

confidence: 99%

See 1 more Smart Citation

M-theoretic derivations of 4d-2d dualities: from a geometric Langlands duality for surfaces, to the AGT correspondence, to integrable systems

Tan

2013

In Part I, we extend our analysis in [arXiv:0807.1107], and show that a mathematically conjectured geometric Langlands duality for complex surfaces in [1], and its generalizations -which relate some cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine (sub) G-algebras, where G is any compact Lie group -can be derived, purely physically, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.In Part II, to the setup in Part I, we introduce Omega-deformation via fluxbranes and add half-BPS boundary defects via M9-branes, and show that the celebrated AGT correspondence in [2,3], and its generalizations -which essentially relate, among other things, some equivariant cohomology of the moduli space of certain ("ramified") G-instantons to the integrable representations of the Langlands dual of certain affine W-algebras -can likewise be derived from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent.In Part III, we consider various limits of our setup in Part II, and connect our story to chiral fermions and integrable systems. Among other things, we derive the NekrasovOkounkov conjecture in [4] -which relates the topological string limit of the dual Nekrasov partition function for pure G to the integrable representations of the Langlands dual of an affine G-algebra -and also demonstrate that the Nekrasov-Shatashvili limit of the "fullyramified" Nekrasov instanton partition function for pure G is a simultaneous eigenfunction of the quantum Toda Hamiltonians associated with the Langlands dual of an affine G-algebra.Via the case with matter, we also make contact with Hitchin systems and the "ramified" geometric Langlands correspondence for curves.