Confluent conformal blocks of the second kind
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Springer
Received: April 23, 2020 Accepted: June 3, 2020 Published: June 22, 2020
Jonatan Lenells and Julien Roussillon Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden
E-mail: [email protected], [email protected] Abstract: We construct confluent conformal blocks of the second kind of the Virasoro algebra. We also construct the Stokes transformations which map such blocks in one Stokes sector to another. In the BPZ limit, we verify explicitly that the constructed blocks and the associated Stokes transformations reduce to solutions of the confluent BPZ equation and its Stokes matrices, respectively. Both the confluent conformal blocks and the Stokes transformations are constructed by taking suitable confluent limits of the crossing transformations of the four-point Virasoro conformal blocks. Keywords: Conformal Field Theory, Supersymmetric Gauge Theory ArXiv ePrint: 2004.09278
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)133
JHEP06(2020)133
Confluent conformal blocks of the second kind
Contents 1 4
2 Confluence of the hypergeometric BPZ equation 2.1 Hypergeometric BPZ equation 2.2 Confluent BPZ equation 2.2.1 Degenerate confluent conformal blocks of the first kind 2.2.2 Degenerate confluent conformal blocks of the second kind 2.2.3 Stokes matrices 2.2.4 Connection matrices 2.3 Confluence of the solutions
4 5 6 6 6 7 8 9
3 Four-point Virasoro conformal blocks 3.1 Crossing transformations
14 15
4 Confluent conformal blocks of the first kind
16
5 Main results 5.1 Assumptions 5.2 Confluent conformal blocks of the second kind 5.3 First main result 5.4 Second main result 5.5 Remarks
17 17 17 18 19 20
6 Proofs 6.1 Proof 6.2 Proof 6.2.1 6.2.2
21 21 25 25 26
7 The 7.1 7.2 7.3 7.4
of theorem 1 of theorem 2 Derivation of the Stokes transformations for n = 2j − 1 Derivation of the Stokes transformations for n = 2j
BPZ limit BPZ limit of Cn BPZ limit of Dn (t) BPZ limit of the Stokes transformations Summary
27 27 32 33 37
8 Conclusions and perspectives
37
A Two special functions
38
–i–
JHEP06(2020)133
1 Introduction 1.1 Organization of the paper
1
Introduction
F 0 (e2iπ z) = M0 F 0 (z),
F 1 (1 + e2iπ z) = M1 F 1 (1 + z),
F ∞ (e−2iπ z −1 ) = M∞ F ∞ (z −1 ), (1.1)
where the three monodromy matrices Mp , p = 0, 1, ∞, are diagonal. The solutions F±p (z) can be expressed in terms of hypergeometric functions. The bases F p , p = 0, 1, ∞, are known as the s-channel, t-channel, and u-channel degenerate conformal blocks, respectively, see e.g. [22]. In a similar way, the general (i.e. nondegenerate) s-channel, t-channel, and u-channel conformal blocks form infinite-dimensional bases for the space of four-point conformal blocks. The purpose of this paper is to describe what happens to these bases as the regular singular point at 1 tends to infinity and merges with the regular singular point at ∞ to form an irregular singularity. In this limit, the hypergeometric equation degenerates into the confluent hypergeometric equatio
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