Scheme dependence of instanton counting in ALE spaces

Ito, Youichi; Maruyoshi, Kazunobu; Okuda, Takuya

doi:10.1007/jhep05(2013)045

Cited by 24 publications

(35 citation statements)

References 27 publications

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“…Nevertheless conjectural formula (B.5) states that corresponding equiavariant Euler characteristics are the same. In the conformal (or cohomology, or just q 1 , q 2 → 1) limit the corresponding statement was made in [3, Sec 4.1] (see also [17], [23]).…”

Section: (B2)mentioning

confidence: 70%

q-deformed Painlevéτfunction andq-deformed conformal blocks

Bershtein

Shchechkin

2017

J. Phys. A: Math. Theor.

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We propose q-deformation of the Gamayun-Iorgov-Lisovyy formula for Painlevé τ function. Namely we propose the formula for τ function for q-difference Painlevé equation corresponding to A(1)′ 7 surface (and A (1) 1 symmetry) in Sakai classification. In this formula τ function equals the series of q-Virasoro Whittaker conformal blocks (equivalently Nekrasov partition functions for pure SU (2) 5d theory). [arXiv:q-alg/9605002v2].

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Section: (B2)mentioning

confidence: 70%

q-deformed Painlevéτfunction andq-deformed conformal blocks

Bershtein

Shchechkin

2017

J. Phys. A: Math. Theor.

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“…The Nekrasov partition functions in the generic case of U (N ) gauge theory on [C 2 /Z k ] can be similarly defined and regarded as T-equivariant integrals over the moduli spaces M ξ 0 ( v, w ), see e.g. [114,11,5,61,24]; the T-fixed points in this case are vectors of w-coloured Young diagrams Y with k colours. However, in this case the torus fixed point sets…”

Section: N = 2 Gauge Theory On X Kmentioning

confidence: 99%

N=2 gauge theories, instanton moduli spaces and geometric representation theory

Szabo

2016

Journal of Geometry and Physics

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We survey some of the AGT relations between N = 2 gauge theories in four dimensions and geometric representations of symmetry algebras of two-dimensional conformal field theory on the equivariant cohomology of their instanton moduli spaces. We treat the cases of gauge theories on both flat space and ALE spaces in some detail, and with emphasis on the implications arising from embedding them into supersymmetric theories in six dimensions. Along the way we construct new toric noncommutative ALE spaces using the general theory of complex algebraic deformations of toric varieties, and indicate how to generalise the construction of instanton moduli spaces. We also compute the equivariant partition functions of topologically twisted six-dimensional Yang-Mills theory with maximal supersymmetry in a general Ω-background, and use the construction to obtain novel reductions to theories in four dimensions.

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“…Nonetheless the results of the JK residue calculation exactly match those of the Moyal products. Compared with instanton counting which also exhibits wall-crossing when the stability parameters are varied [23,24,25,20,26,27], for 't Hooft operators the physical meaning of the stability parameters (FI parameters) is simpler; they are the positions of the operators.…”

Section: Introductionmentioning

confidence: 99%

Wall-crossing and operator ordering for ’t Hooft operators in $$ \mathcal{N} $$ = 2 gauge theories

Hayashi¹,

Okuda²,

Yoshida³

2019

J. High Energ. Phys.

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We study half-BPS 't Hooft line operators in 4d N = 2 U (N ) gauge theories on S 1 ×R 3 with an Ω-deformation. The recently proposed brane construction of 't Hooft operators shows that non-perturbative contributions to their correlator are identified with the Witten indices of quiver supersymmetric quantum mechanics. For the products of minimal 't Hooft operators, a chamber in the space of Fayet-Iliopoulos parameters in the quantum mechanics corresponds to an ordering of the operators inserted along a line. These considerations lead us to conjecture that the Witten indices can be read off from the Moyal products of the expectation values of the minimal 't Hooft operators, and also that wall-crossing occurs in the quantum mechanics only when the ordering of the operators changes. We confirm the conjectures by explicitly computing the Witten indices for the products of two and three minimal 't Hooft operators in all possible chambers.2 In [18] the authors also observed the correspondence between the ordering and wall-crossing for the product of T and T . We complete their analysis by including the product of two minimal operators of the same type. We further investigate the products of three minimal operators of all types, which exhibit a richer structure.3 A primitive version of the conjectures and some evidence were presented by T.O. in the workshop "Representation Theory, Gauge Theory, and Integrable Systems" held at the Kavli IPMU in February, 2019. 5 The same symbol e i (or e a ) will denote an orthonormal basis of a Euclidean space other than R N . 6 To compare wtih [12,14] note the following. Λ cort in (2.3) is also the coroot lattice of SU (N ). The cocharacter lattice of SU (N ) is the sublattice { i z i = 0} of Λ cochar in (2.2). Defining z U (1) := (1/N ) i z i j e j and z SU (N ) := z − z U (1) for z = i z i e i , we have |v| ≤ |B| ⇔ |v SU (N ) | ≤ |B SU (N ) | for v ∈ B + Λ cort .7 Since the Langlands dual group of SU (N ) is SU (N )/Z N , 't Hooft operators corresponding to B = ±e i do not exist in an SU (N ) gauge theory.8 Section 5 discusses subtleties for N F odd.12 The D3-branes are assumed to be at generic positions in the (x 4 , x 5 )-space. 13 Due to the s-rule a single NS5-brane, with K D1-branes (1 ≤ K ≤ N ) attached and placed on the right of the stack of all the D3-and D7-branes, gives an 't Hooft operator with magnetic charge K i=1 e i , which seems to correspond to the K-th exterior power of the fundamental representation. Similarly an NS5-brane

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Scheme dependence of instanton counting in ALE spaces

Cited by 24 publications

References 27 publications

q-deformed Painlevéτfunction andq-deformed conformal blocks

q-deformed Painlevéτfunction andq-deformed conformal blocks

N=2 gauge theories, instanton moduli spaces and geometric representation theory

Wall-crossing and operator ordering for ’t Hooft operators in $$ \mathcal{N} $$ = 2 gauge theories

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