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 TMF, 2015, Volume 184, Number 1, Pages 3–40 (Mi tmf8856)

Matching branches of a nonperturbative conformal block at its singularity divisor

H. Itoyamaab, A. D. Mironovcde, A. Yu. Morozovde

a Osaka City University Advanced Mathematical Institute (OCAMI), Osaka, Japan
b Department of Mathematics and Physics, Osaka City University, Osaka, Japan
c Lebedev Physical Institute, Moscow, Russia
d Institute for Experimental and Theoretical Physics, Moscow, Russia
e National Research Nuclear University MEPhI, Moscow, Russia

Abstract: A conformal block is a function of many variables, usually represented as a formal series with coefficients that are certain matrix elements in the chiral {(}i.e., Virasoro{\rm)} algebra. A nonperturbative conformal block is a multivalued function defined globally over the space of dimensions and has many branches and, perhaps, additional free parameters not seen at the perturbative level. We discuss additional complications of the nonperturbative description that arise because all the best-studied examples of conformal blocks are at the singularity locus in the moduli space {\rm(}at divisors of the coefficients or, simply, at zeros of the Kac determinant{\rm).} A typical example is the Ashkin–Teller point, where at least two naive nonperturbative expressions are provided by the elliptic Dotsenko–Fateev integral and by the celebrated Zamolodchikov formula in terms of theta constants, and they differ. The situation is somewhat similar at the Ising and other minimal model points.

Keywords: two-dimensional conformal theory, conformal block

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation ÍØ-1500.2014.2 Russian Foundation for Basic Research 13-02-0045713-02-0047813-02-91371-ST14-01-92691-Ind12-02-92108-ßô_a National Council for Scientific and Technological Development (CNPq) Ministry of Education, Culture, Sports, Science and Technology, Japan 23540316 Japan Society for the Promotion of Science FY2010-2011

DOI: https://doi.org/10.4213/tmf8856

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English version:
Theoretical and Mathematical Physics, 2015, 184:1, 891–923

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Citation: H. Itoyama, A. D. Mironov, A. Yu. Morozov, “Matching branches of a nonperturbative conformal block at its singularity divisor”, TMF, 184:1 (2015), 3–40; Theoret. and Math. Phys., 184:1 (2015), 891–923

Citation in format AMSBIB
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This publication is cited in the following articles:
1. A. Morozov, Y. Zenkevich, “Decomposing Nekrasov decomposition”, J. High Energy Phys., 2016, no. 2, 098
2. M. Beccaria, A. Fachechi, G. Macorini, L. Martina, “Exact partition functions for deformed $\mathcal{N}=2$ theories with ${\mathcal{N}}_f=4$ flavours”, J. High Energy Phys., 2016, no. 12, 029
3. N. Nemkov, “On new exact conformal blocks and Nekrasov functions”, J. High Energy Phys., 2016, no. 12
4. A. Mironov, A. Morozov, “On determinant representation and integrability of Nekrasov functions”, Phys. Lett. B, 773 (2017), 34–46
5. H. Itoyama, A. Mironov, A. Morozov, “Rainbow tensor model with enhanced symmetry and extreme melonic dominance”, Phys. Lett. B, 771 (2017), 180–188
6. H. Itoyama, A. Mironov, A. Morozov, “Cut and join operator ring in tensor models”, Nucl. Phys. B, 932 (2018), 52–118
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