Liouville correlation functions from four-dimensional gauge theories
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 produce an \(S\)-duality invariant object, which coincides with the Liouville correlation function on the corresponding punctured Riemann surface. One sees that the product of the Liouville three-point functions neatly recombines into the modulus squared of the one-loop contribution to Nekrasov's partition function.
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