Semi-classical Virasoro blocks: proof of exponentiation
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Springer
Received: October 23, 2019 Accepted: December 19, 2019 Published: January 20, 2020
Mert Be¸sken,a,b Shouvik Dattaa and Per Krausa a
Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics & Astronomy, University of California Los Angeles, 475 Portola Plaza, Los Angeles, CA 90095, U.S.A. b Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Amsterdam 1098 XH, The Netherlands
E-mail: [email protected], [email protected], [email protected] Abstract: Virasoro conformal blocks are expected to exponentiate in the limit of large central charge c and large operator dimensions hi , with the ratios hi /c held fixed. We prove this by employing the oscillator formulation of the Virasoro algebra and its representations. The techniques developed are then used to provide new derivations of some standard results on conformal blocks. Keywords: Conformal Field Theory, Conformal and W Symmetry, Field Theories in Lower Dimensions ArXiv ePrint: 1910.04169
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP01(2020)109
JHEP01(2020)109
Semi-classical Virasoro blocks: proof of exponentiation
Contents 1
2 Virasoro blocks from the oscillator formalism 2.1 Oscillator formalism 2.2 States and wavefunctions 2.3 Virasoro blocks
2 2 3 4
3 Exponentiation of semi-classical Virasoro blocks
5
4 Examples 4.1 Perturbatively heavy vacuum block 4.2 Heavy-light block 4.3 Blocks with heavy exchange
7 7 9 10
5 Discussion
12
A Consistency of the linear system of equations
13
1
Introduction
Conformal blocks in 2d CFTs are fixed by Virasoro symmetry. However, closed form expressions are known only in some special cases. A general feature of the semi-classical limit, c → ∞,
hi , h → ∞,
hi h , fixed , c c
is that the conformal block is believed to exponentiate, i.e. it takes the form [1] hi h c , ;z . V(hi , h, c; z) ≈ exp − f 6 c c
(1.1)
(1.2)
Here c is the central charge, hi are conformal dimensions of the external operators, h is the conformal dimension of the exchanged primary and z is the cross-ratio. Although there is compelling evidence for (1.2), a first principles derivation of this well-known formula is lacking. The aim of this paper is to close this gap. An intuitively appealing, but somewhat heuristic, argument for exponentiation is provided by Liouville theory. At large c, correlation functions of heavy primary operators may be computed using the saddle point approximation to the Liouville path integral. Assuming that the saddle point picks out a particular Virasoro block, together with the large c behavior of the DOZZ structure constants in this regime [2], the result follows. A strong check
–1–
JHEP01(2020)109
1 Introduction
2
Virasoro blocks from the oscillator formalism
In this section we briefly review the oscillator representation of the Virasoro algebra [11] and its application to the computation of conformal blocks. A detailed discussion of this formalism and its applications, along with its derivation from the linear dil
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