Chern-Simons and twisted supersymmetry in various dimensions

Bethe/gauge correspondence identifies supersymmetric vacua of massive gauge theories invariant under the two dimensional N = 2 Poincare supersymmetry with the stationary states of some quantum integrable system. The supersymmetric theory can be twisted in a number of ways, producing a topological field theory. For these theories we compute the handle gluing operator H. We also discuss the Gaudin conjecture on the norm of Bethe states and its connection to H.

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“…[18,19,21,25]. Of course the physical derivation skips various subtleties, which are by now thoroughly addressed in [26,27].…”

Section: Topological Metric and The Handle Gluing Operatormentioning

confidence: 99%

Bethe/gauge correspondence on curved spaces

Nekrasov

Shatashvili

2015

J. High Energ. Phys.

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200

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“…The resulting four dimensional theory (for gauge group SU (2)) is related to the theory of the so-called E-strings [37][38]. The instanton contributions to the correlation functions of the chiral operators in this theory are related to the elliptic genera of the instanton moduli space [39] and could be summed up, giving rise to the Seiberg-Witten curves for these theories. However, in this paper we shall neither discuss elliptic, nor trigonometric limits, even though they lead to interesting integrable systems [40].…”

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confidence: 99%

Small Instantons, Little Strings and Free Fermions

Losev

Marshakov

Nekrasov

2005

From Fields to Strings: Circumnavigating Theoretical Physics

204

369

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We present new evidence for the conjecture that BPS correlation functions in the N = 2 supersymmetric gauge theories are described by an auxiliary two dimensional conformal field theory. We study deformations of the N = 2 supersymmetric gauge theory by all gauge-invariant chiral operators. We calculate the partition function of the N = 2 theory on R 4 with appropriately twisted boundary conditions. For the U (1) theory with instantons (either noncommutative, or D-instantons, depending on the construction) the partition function has a representation in terms of the theory of free fermions on a sphere, and coincides with the tau-function of the Toda lattice hierarchy. Using this result we prove to all orders in string loop expansion that the effective prepotential (for U (1) with all chiral couplings included) is given by the free energy of the topological string on CP 1 .Gravitational descendants play an important rôle in the gauge fields/string correspondence. The dual string is identified with the little string bound to the fivebrane wrapped on the two-sphere. We also discuss the theory with fundamental matter hypermultiplets.February 2003 † On leave of absence from: ITEP, Moscow, 117259, Russia INTRODUCTIONThe Holy Grail of the theoretical physics is the nonperturbative theory which includes quantum gravity, sometimes called M-theory [1]. The current wisdom says there is no fundamental coupling constant. Whatever (string) perturbation theory is used depends on the particular solution one expands about. The expansion parameter is one of the geometric characteristics of the background. It is obviously interesting to look for simplified string and field theoretic models, which have string loop expansion, and where the string coupling constant has a geometric interpretation. String expansion in gauge theoryLarge N gauge theories are the most popular, and the most elusive models with string representation. In the gauge/string duality [2][3] one matches the connected correlation functions of the gauge theory observables with the partition function of the string theory in the bulk. The closed string dual has 1 N 2 as a string coupling constant. Advances in the studies of the type II string compactifications on Calabi-Yau manifolds led to another class of models, which in the low-energy limit reduce to N = 2 supersymmetric gauge theories, with a novel type of string loop expansion. Namely, certain couplings F g in the low-energy effective action are given by the genus g partition function of the topologically twisted string on Calabi-Yau. The gauge group of the N = 2 theory does not have to be U (N ) with large N . It is determined by the geometry of Calabi-Yau manifold [4][5] [6].The rôle of effective string coupling is played by the vev of the graviphoton field strength [7], which is usually assumed to be constant [8]. Generalized Scherk-Schwarz constructionIn this paper we shall explain that there exists another, natural from the gauge theory point of view, way to flesh out these couplings. The idea is to put the t...

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“…Moreover, independent of that development, it has been shown [2,3,4] how the notion of topological quantum field theories [5] or, more specifically, cohomological gauge theories being related to supersymmetric gauge theories by twisting, can be extended to dimensions greater than four. Such higher-dimensional (untwisted) cohomological theories are obtained when Euclidean supersymmetric gauge theories are considered on manifolds with reduced holonomy group H ⊂ SO(D).…”

Section: Introductionmentioning

confidence: 99%

“…Then, in fact, the G 2 -invariant theory can be obtained by ordinary dimensional reduction. Moreover, that theory has a nice interpretation: It may be regarded as the seven-dimensional analogue of the N T = 2, D = 3 super-BF theory [9], just as the Spin (7)-invariant theory may be regarded as the eightdimensional analogue of the N T = 1, D = 4 Donaldson-Witten theory [3]. Namely, replacing in the G 2 -invariant theory the octonionic through the quaternionic structure constants and considering all the fields as three-dimensional ones one gets exactly the N T = 2, D = 3 super-BF theory (without matter).…”

Section: Introductionmentioning

confidence: 99%

G₂-INVARIANT 7D EUCLIDEAN SUPER YANG–MILLS THEORY AS 7-DIMENSIONAL ANALOGUE OF 3D SUPER-BF THEORY

Mülsch¹,

Geyer²

2004

Int. J. Geom. Methods Mod. Phys.

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A formulation of N T = 1, D = 8 Euclidean super Yang-Mills theory with generalized selfduality and reduced Spin(7)-invariance is given which avoids the peculiar extra constraints of Ref. [7]. Its reduction to seven dimensions leads to the G 2 -invariant N T = 2, D = 7 super Yang-Mills theory which may be regarded as a higher-dimensional analogue of the N T = 2, D = 3 super-BF theory. When reducing further that G 2 -invariant theory to three dimensions one gets the N T = 2 super-BF theory coupled a spinorial hypermultiplet.

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Chern-Simons and twisted supersymmetry in various dimensions

Cited by 77 publications

References 42 publications

Bethe/gauge correspondence on curved spaces

Bethe/gauge correspondence on curved spaces

Small Instantons, Little Strings and Free Fermions

G₂-INVARIANT 7D EUCLIDEAN SUPER YANG–MILLS THEORY AS 7-DIMENSIONAL ANALOGUE OF 3D SUPER-BF THEORY

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Chern-Simons and twisted supersymmetry in various dimensions

Cited by 77 publications

References 42 publications

Bethe/gauge correspondence on curved spaces

Bethe/gauge correspondence on curved spaces

Small Instantons, Little Strings and Free Fermions

G2-INVARIANT 7D EUCLIDEAN SUPER YANG–MILLS THEORY AS 7-DIMENSIONAL ANALOGUE OF 3D SUPER-BF THEORY

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Product

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G₂-INVARIANT 7D EUCLIDEAN SUPER YANG–MILLS THEORY AS 7-DIMENSIONAL ANALOGUE OF 3D SUPER-BF THEORY