Coupling of Two Conformal Field Theories and Nakajima–Yoshioka Blow-Up Equations

Bershtein, Mikhail; Feigin, Boris; Litvinov, A

doi:10.1007/s11005-015-0802-x

Cited by 18 publications

(30 citation statements)

References 30 publications

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“…The contribution of each sector to the correlation number is given by a copy 2 ) under change of patch is related to the intersection of the corresponding divisors. Investigations in similar directions for Hirzebruch surfaces have been pioneered in [64].…”

Section: Discussionmentioning

confidence: 99%

Exact results for N $$ \mathcal{N} $$ = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

Bershtein

Bonelli

Ronzani

et al. 2016

J. High Energ. Phys.

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We provide a contour integral formula for the exact partition function of N = 2 supersymmetric U(N ) gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for U(2) N = 2 * theory on P 2 for all instanton numbers. In the zero mass case, corresponding to the N = 4 supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a longstanding conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new.

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Section: Discussionmentioning

confidence: 99%

Exact results for N $$ \mathcal{N} $$ = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

Bershtein

Bonelli

Ronzani

et al. 2016

J. High Energ. Phys.

Self Cite

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“…This will play a relevant role in the subsequent discussion. It was observed in [28] that (5.1) suggests a link to Liouville gravity. In the following we will show that indeed three-point number and conformal blocks of this CFT arise as building blocks of the supersymmetric partition function of N = 2 U(2) gauge theory on S 2 × S 2 .…”

Section: Jhep07(2015)054mentioning

confidence: 97%

N = 2 $$ \mathcal{N}=2 $$ supersymmetric gauge theories on S2 × S2 and Liouville Gravity

Bawane

Bonelli

Ronzani

et al. 2015

J. High Energ. Phys.

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We consider N = 2 supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a U(1) isometry. This is used to explicitly compute the supersymmetric path integral on S 2 × S 2 via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity.

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“…The functions Z(a; 1 , 2 |Λ) are known to satisfy Nakajima-Yoshioka blowup relations [NY03, (6.13)] (see also [BFL13,(5.3)] for CFT interpretation). We write them in terms of the full partition functions as in [NY03L,(5.2)]…”

Section: Power Series Representation For the Tau Functionmentioning

confidence: 99%

“…The naming for "short" and "long" c = −2 tau functions is inspired by the length of shifts of σ in the sums (3.8) and (3.10) correspondingly. There are also differential Nakajima-Yoshioka blowup relations on 4d pure gauge N = 2 SU (2) Nekrasov partition functions [NY03], [NY03L], [BFL13]. Relations [NY03L,(5.2)] ([NY03, (6.14)]) have the form n∈Z Z(a + 2 1 n; 1 , 2 − 1 |Λe − 1 2 1 α )Z(a + 2 2 n;…”

Section: Power Series Representation For the Tau Functionmentioning

confidence: 99%

Painlevé equations from Nakajima–Yoshioka blowup relations

Bershtein

Shchechkin

2019

Lett Math Phys

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Gamayun, Iorgov and Lisovyy in 2012 proposed that tau function of the Painlevé equation equals to the series of c = 1 Virasoro conformal blocks. We study similar series of c = −2 conformal blocks and relate it to Painlevé theory. The arguments are based on Nakajima-Yoshioka blowup relations on Nekrasov partition functions.We also study series of q-deformed c = −2 conformal blocks and relate it to q-Painlevé equation. As an application we prove formula for the tau function of q-Painlevé A (1) 7 equation.

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Coupling of Two Conformal Field Theories and Nakajima–Yoshioka Blow-Up Equations

Abstract: We study the conformal vertex algebras which naturally arise in relation to the Nakajima-Yoshioka blow-up equations.2010 Mathematics Subject Classification. 17B68, 81R10.

Cited by 18 publications

References 30 publications

Exact results for N $$ \mathcal{N} $$ = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

Exact results for N $$ \mathcal{N} $$ = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

N = 2 $$ \mathcal{N}=2 $$ supersymmetric gauge theories on S2 × S2 and Liouville Gravity

Painlevé equations from Nakajima–Yoshioka blowup relations

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