Contour integrals and the modular S $$ \mathcal{S} $$ -matrix
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Springer
Received: December 27, 2019 Accepted: May 27, 2020 Published: July 8, 2020
Sunil Mukhi, Rahul Poddar and Palash Singh Indian Institute of Science Education and Research, Homi Bhabha Rd, Pashan, Pune 411 008, India
E-mail: [email protected], [email protected], [email protected] Abstract: We investigate a conjecture to describe the characters of large families of RCFT’s in terms of contour integrals of Feigin-Fuchs type. We provide a simple algorithm to determine the modular S-matrix for arbitrary numbers of characters as a sum over paths. Thereafter we focus on the case of 2, 3 and 4 characters, where agreement between the critical exponents of the integrals and the characters implies that the conjecture is true. In these cases, we compute the modular S-matrix explicitly, verify that it agrees with expectations for known theories, and use it to compute degeneracies and multiplicities of primaries. We verify that our algorithm reproduces the correct S-matrix for SU(2)k for all k ≤ 18 which provides additional evidence for the original conjecture. On the way we note that the Verlinde formula provides interesting constraints on the critical exponents of RCFT in this context. Keywords: Conformal and W Symmetry, Conformal Field Theory, Field Theories in Lower Dimensions, Integrable Field Theories ArXiv ePrint: 1912.04298
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)045
JHEP07(2020)045
Contour integrals and the modular S-matrix
Contents 1 Introduction
1
2 Contour-integral representation of characters
3 8 8 14 15
4 Applications: theories with ≤ 4 characters 4.1 2 characters 4.2 3 characters 4.3 4 characters
16 17 20 22
5 Greater than 5 characters
24
6 Conclusions and open problems
25
A Review of the “unfolding” procedure
26
B Relation between ordered and unordered integrals
28
C Degeneracies of quasi-characters
30
1
Introduction
A procedure for the classification of admissible characters for rational conformal field theories in 2 dimensions was proposed in [1]. This starts by writing the most general Modular Linear Differential equation (MLDE) for a given class of theories, labelled by the number n of characters and the Wronskian index `. This MLDE depends on a finite number of parameters. The n independent solutions automatically transform as vector-valued modular functions, but in general they do not have integral coefficients in their expansion in powers of the parameter q = e2πiτ . Thus they cannot be interpreted as counting the degeneracy of states in a physical system. One then varies the parameters in the MLDE until these coefficients become non-negative integers. At this point one has admissible characters and can try to identify the RCFT that they potentially describe. A recent status report on this programme can be found in [2]. One deficiency of the MLDE approach is that, because we solve the equation as a qseries, it is difficult or impossible to actually compute the modular transformation matrix Sij on the char
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