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Isomonodromic Tau-Functions from Liouville Conformal Blocks

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Abstract

The goal of this note is to show that the Riemann–Hilbert problem to find multivalued analytic functions with \({{\rm SL}(2,\mathbb{C})}\)-valued monodromy on Riemann surfaces of genus zero with n punctures can be solved by taking suitable linear combinations of the conformal blocks of Liouville theory at c = 1. This implies a similar representation for the isomonodromic tau-function. In the case n = 4 we thereby get a proof of the relation between tau-functions and conformal blocks discovered in Gamayun et al. (J High Energy Phys, 10:038, 2012). We briefly discuss a possible application of our results to the study of relations between certain \({\mathcal{N}=2}\) supersymmetric gauge theories and conformal field theory.

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Authors and Affiliations

  1. Bogolyubov Institute for Theoretical Physics, Kyiv, 03680, Ukraine

    N. Iorgov

  2. Laboratoire de Mathématiques et Physique Théorique, Université de Tours, 37200, Tours, France

    O. Lisovyy

  3. DESY Theory, Notkestr. 85, 22603, Hamburg, Germany

    J. Teschner

Authors
  1. N. Iorgov
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  2. O. Lisovyy
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  3. J. Teschner
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Correspondence to J. Teschner.

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Communicated by N. Reshetikhin

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Iorgov, N., Lisovyy, O. & Teschner, J. Isomonodromic Tau-Functions from Liouville Conformal Blocks. Commun. Math. Phys. 336, 671–694 (2015). https://doi.org/10.1007/s00220-014-2245-0

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