Liouville correlation functions from four-dimensional gauge theories
type: Item
instance of: scholarly article
title: Liouville correlation functions from four-dimensional gauge theories
zbMATH Open document ID: 1185.81111
author: Luis F. Alday
author: Davide Gaiotto
author: Yuji Tachikawa
published in: Letters in Mathematical Physics
publication date: 2010-03-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/0906.3219
review text: From the text: The authors conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus \(g\) and \(n\) punctures as the Nekrasov partition function of a certain class of \(\mathcal{N}=2\) SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus \(0, 1\). There is a huge physical and mathematical literature on \(\mathcal{N} =2\) field theories. It is natural to wonder if this class of SCFTs may provide a connection with the large literature on objects defined through the sewing of Riemann surfaces, in particular, the theory of two-dimensional conformal field theories. This paper is devoted to test a specific realization of this general idea: the identification of the Nekrasov partition function [\textit{N. A. Nekrasov}, Adv. Theor. Math. Phys. 7, 831--864 (2003; Zbl 1056.81068)], and with \textit{A. Okounkov}, Prog. Math. 244, 525--596 (2006; Zbl 1233.14029)] of these \(\mathcal{N}=2\) SCFTs and the Liouville theory correlation functions on the corresponding Riemann surfaces. The crucial idea is that for each sewing of the Riemann surface one is given two natural objects: Nekrasov's instanton computation in the corresponding Lagrangian description of the theory and the ``Liouville conformal block'' defined by a sum over Virasoro descendants of a primary field in each of the sewing channels. With a judicious identification of the parameters on the two sides, the authors demonstrate by explicit examples that the two objects coincide at genus \(g=0,1\) for various \(n\). They also conjecture the general map at higher genus and number of punctures. Taking inspiration from \textit{V. Pestun}'s computation [Commun. Math. Phys. 313, No. 1, 71--129 (2012; Zbl 1257.81056)] of the \(S^4\) partition function of \(\mathcal{N}=2\) SCFTs they assemble together the squared modulus of Nekrasov's instanton partition function together with tree level and one-loop contributions to …
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 81T40
Mathematics Subject Classification ID: 32G81
Mathematics Subject Classification ID: 81T13
Mathematics Subject Classification ID: 81T60
zbMATH DE Number: 5681462
zbMATH Keywords: conformal field theory
zbMATH Keywords: gauge theory
zbMATH Keywords: AGT conjecture
MaRDI profile type: Publication
OpenAlex ID: W2107702031
Wikidata QID: Q56057582
arXiv ID: 0906.3219
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DOI: 10.1007/S11005-010-0369-5
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description: scientific article