A Review on Instanton Counting and W-Algebras

Tachikawa, Yuji

doi:10.1007/978-3-319-18769-3_4

Cited by 25 publications

(29 citation statements)

References 171 publications

(250 reference statements)

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“…The Nekrasov subfunctions capture the contribution of the various multiplets to the instanton partition function of the four-dimensional [41], five-dimensional [42] or the six-dimensional gauge theories [47][48][49]. For a review of these see [43,50]. The six-dimensional instanton partition function we can identify the different contribution from the tensor or matter multiplets.…”

Section: B Nekrasov Subfunctionsmentioning

confidence: 99%

Beyond triality: dual quiver gauge theories and little string theories

Bastian

Hohenegger

Iqbal

et al. 2018

J. High Energ. Phys.

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The web of dual gauge theories engineered from a class of toric Calabi-Yau threefolds is explored. In previous work, we have argued for a triality structure by compiling evidence for the fact that every such manifold X N,M (for given (N, M )) engineers three a priori different, weakly coupled quiver gauge theories in five dimensions. The strong coupling regime of the latter is in general described by Little String Theories. Furthermore, we also conjectured that the manifold X N,M is dual to X N ,M if N M = N M and gcd(N, M ) = gcd(N , M ). Combining this result with the triality structure, we currently argue for a large number of dual quiver gauge theories, whose instanton partition functions can be computed explicitly as specific expansions of the topological partition function Z N,M of X N,M . We illustrate this web of dual theories by studying explicit examples in detail. We also undertake first steps in further analysing the extended moduli space of X N,M with the goal of finding other dual gauge theories. the given conditions, the Kähler cones of the Calabi-Yau manifolds X N,M and X N ,M are part of a common extended moduli space. The relation (1.1) was explicitly shown for a large class of examples and passed highly non-trivial consistency checks for generic (N, M ). Moreover, it is expected that the partition functions Z N,M and Z N ,M agree upon taking into account the duality map implicit in (1.1). This fact was explicitly checked for gcd(N, M ) = 1 in [30].Secondly, in [32] the question was analysed what type of (low-energy) gauge theories can be engineered from a given Calabi-Yau manifold X N,M . By studying different series expansions of Z N,M (in terms of different sets of the Kähler parameters of X N,M ) it was found that the Kähler cone of X N,M contains three different regions that engineer five-dimensional quiver gauge theories with gauge groups( 1.2) respectively, where k = gcd(N, M ). It is expected that the UV completions of (at least some of) these gauge theories are LSTs. The relation among the three low-energy theories with gauge groups (1.2) was dubbed triality in [32], reflecting the fact that all three are engineered by the same Calabi-Yau threefold. It is important to realise that the mapping of the various gauge theory parameters is highly non-trivial among the members of the triality: indeed, the coupling constants, mass parameters and other Coulomb branch parameters are mixed in a highly non-trivial fashion when going from one theory to another.Combining the triality discussed in [32] with the observations of [31] that the Calabi-Yau manifold X N,M itself can be dualised to other manifolds in the sense of eq. (1.1) suggests an even more elaborate picture at the level of the low-energy theories: indeed it implies that a (circular) quiver gauge theory with N gauge nodes of type U (M ) (that is engineered by X N,M for arbitrary (N, M )) is dual to a whole web of other quiver gauge theories that have N gauge nodes of type U (M ) such that N M = N M and gcd(N, M ) = gcd(N , M ). In this p...

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Section: B Nekrasov Subfunctionsmentioning

confidence: 99%

Beyond triality: dual quiver gauge theories and little string theories

Bastian

Hohenegger

Iqbal

et al. 2018

J. High Energ. Phys.

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“…is well-defined and non-singular. In this limit only the mass dependent terms of the conformal weights 15 From now on we simplify the notation by omitting the subscript {ξ 1 , . .…”

Section: The Uv Curvementioning

confidence: 99%

“…Various approaches have been pursued: the geometric description of the low-energy effective actionà la Seiberg-Witten (SW) [2,3], the exact computation of instanton corrections by means of localization techniques [4,5], the relations to integrable models [6], the 2d/4d correspondence also known as the AGT correspondence [7,8], the use of β-ensembles and matrix model techniques [9,10]. Moreover, the string embedding of such theories via geometric engineering has led to the possibility of expressing some relevant observables via topological string amplitudes [11,12] 1 , and further insights 1 We refer to the series of recent review articles [13][14][15][16][17] for an extensive discussion of these topics.…”

Section: Introductionmentioning

confidence: 99%

Non-perturbative studies of N=2 conformal quiver gauge theories

et al. 2015

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We study N = 2 super-conformal field theories in four dimensions that correspond to mass-deformed linear quivers with n gauge groups and (bi-)fundamental matter. We describe them using Seiberg-Witten curves obtained from an M-theory construction and via the AGT correspondence. We take particular care in obtaining the detailed relation between the parameters appearing in these descriptions and the physical quantities of the quiver gauge theories. This precise map allows us to efficiently reconstruct the non-perturbative prepotential that encodes the effective IR properties of these theories. We give explicit expressions in the cases n = 1, 2, also in the presence of an Ω-background in the Nekrasov-Shatashvili limit. All our results are successfully checked against those of the direct microscopic evaluation of the prepotentialà la Nekrasov using localization methods.

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“…The functions Ψ(a) can be identified with the instanton partition functions (see [36] for a review and references) that were found to be related to Liouville conformal blocks by AGT. The approach proposed in [25] establishes the relation between the wave-functions Ψ(a) in (1.8) and the Liouville conformal blocks without using the relation to the instanton partition functions observed in [33].…”

Section: Jhep10(2015)143mentioning

confidence: 99%

Line operators in theories of class S $$ \mathcal{S} $$ , quantized moduli space of flat connections, and Toda field theory

Coman

Gabella

Teschner³

2015

J. High Energ. Phys.

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Non-perturbative aspects of N = 2 supersymmetric gauge theories of class S are deeply encoded in the algebra of functions on the moduli space M flat of flat SL(N )-connections on Riemann surfaces. Expectation values of Wilson and 't Hooft line operators are related to holonomies of flat connections, and expectation values of line operators in the low-energy effective theory are related to Fock-Goncharov coordinates on M flat . Via the decomposition of UV line operators into IR line operators, we determine their noncommutative algebra from the quantization of Fock-Goncharov Laurent polynomials, and find that it coincides with the skein algebra studied in the context of Chern-Simons theory. Another realization of the skein algebra is generated by Verlinde network operators in Toda field theory. Comparing the spectra of these two realizations provides non-trivial support for their equivalence. Our results can be viewed as evidence for the generalization of the AGT correspondence to higher-rank class S theories.

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A Review on Instanton Counting and W-Algebras

Cited by 25 publications

References 171 publications

Beyond triality: dual quiver gauge theories and little string theories

Beyond triality: dual quiver gauge theories and little string theories

Non-perturbative studies of N=2 conformal quiver gauge theories

Line operators in theories of class S $$ \mathcal{S} $$ , quantized moduli space of flat connections, and Toda field theory

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