Virasoro blocks and quasimodular forms

Published for SISSA by Springer Received: August 6, 2020 Accepted: September 30, 2020 Published: November 4, 2020 Diptarka Das,a Shouvik Dattab and Madhusudhan Ramanc a Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India b Department of Theoretical Physics, CERN, 1 Esplanade des Particules, Geneva 23 CH-1211, Switzerland c Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Navy Nagar, Colaba, Mumbai 400 005, India E-mail: , , Abstract: We analyse Virasoro blocks in the regime of heavy intermediate exchange (hp → ∞). For the 1-point block on the torus and the 4-point block on the sphere, we show that each order in the large-hp expansion can be written in closed form as polynomials in the Eisenstein series. The appearance of this structure is explained using the fusion kernel and, more markedly, by invoking the modular anomaly equations via the 2d/4d correspondence. The existence of these constraints allows us to develop a faster algorithm to recursively construct the blocks in this regime. We then apply our results to find corrections to averaged heavy-heavy-light OPE coefficients. Keywords: Conformal and W Symmetry, Conformal Field Theory, Field Theories in Lower Dimensions ArXiv ePrint: 2007.10998 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP11(2020)010 JHEP11(2020)010 Virasoro blocks and quasimodular forms Contents 1 2 Virasoro blocks in the heavy exchange regime 2.1 Torus 1-point block 2.2 Sphere 4-point block 2.3 Further comments 4 5 7 8 3 Modular features and the 2d/4d correspondence 3.1 Parameter maps 3.2 From modular anomalies to the diffusion equation 3.3 An algorithm for faster computations of Virasoro blocks 13 14 15 17 4 Application: heavy-heavy-light OPE coefficients 18 5 Conclusions 22 A Eisenstein series 23 B Higher orders in the 1/hp expansion 24 1 Introduction Correlation functions in conformal field theories are built out of fundamental objects known as conformal blocks. These objects are fixed perturbatively by conformal symmetries and play a key role in various lines of research, including the AdS/CFT correspondence and the conformal bootstrap. In two spacetime dimensions, owing to an infinite-dimensional Virasoro symmetry, the conformal block has a rich structure. A closed form for the blocks, however, still eludes us except in a handful of special cases. For example, at large-c (with conformal dimensions also scaling with c) the Virasoro blocks exponentiate [1, 2], and in this case a closed form for the blocks can be found, by using the monodromy method or the oscillator formalism [3].1 The monodromy method has also been used to determine blocks for heavy-light external dimensions [4], as well as to determine the blocks for asymptotically heavy intermediate dimensions [5]. In this paper, we study Virasoro conformal blocks in the regime of heavy intermediate exchange, i.e. we study the blocks as an expansion in inverse powers of the conformal dimension of the exchanged operator. This study is similar in spirit to other asymptotic analyses 1 For contrast, when conformal dimensions are held fixed, the leading answer for blocks on the sphere in the large-c limit is given by the global block, for which the closed form is known in terms of hypergeometric functions. –1– JHEP11(2020)010 1 Introduction in physics, where various simplifications and interesting features arise when quantities of interest are expanded in a large parameter, for example the rank of a gauge group. Our work draws on a number of themes, each of which we now discuss. Modularity. It is well-known that conformal correlators enjoy modular properties. In the context of torus 1- and 2-point functions, this property has been used to find asymptotic formulae for OPE coefficients, pioneered by [10], and later adapted to various other cases [11–14]. Additionally, since crossing symmetry of the full 4-point sphere correlator can be expressed as a modular property, asymptotic constraints can also be obtained from bootstrapping the high “temperature” result [15]. Conformal correlators are built out of conformal blocks weighted by OPE coefficients. On either side of the crossing equation, blocks in dual channels (or dual tori) appear. A remarkable fact about two-dimensional CFTs is the existence of integral kernels which relate S-dual blocks [16, 17]. These have been used recently to bootstrap the CFT data [18– 20]. It seems unlikely, however, that the Virasoro blocks themselves (on the torus, or on the sphere in the elliptic representation) will have any definite modular properties. 2 If such a property exists in general, even partially, one might hope that a closed form expression for these blocks is possible. In this work we demonstrate that • when the Virasoro blocks are expanded in a specific linear combination of the intermediate conformal dimension and the central charge, the coefficients of the expansion can be resummed into quasimodular forms of PSL(2, Z). • Further, these coefficients are constrained to satisfy a “modular anomaly equation” that one can use to recursively determine higher orders in this expansion, with minimal input from Zamolodchikov recursion. 2 See, however, [21] for a recent development. –2– JHEP11(2020)010 Zamolodchikov recursion. An efficient means of computing the 4-point Virasoro block on the plane is via the recursion relations discovered by Zamolodchikov [1, 6]. This recursion is based on the structure of poles and residues of the block arising due to the presence of degenerate representations. The block can then be written as a sum over appropriately weighted poles of either the exchanged conformal dimension hp , or the central charge c. In either case, the terms in the sum can be recursively evaluated, allowing for a perturbative determination of the block as an expansion in the cross-ratio z, or in the elliptic nome q associated to the pillow coordinates. Similar strategies have been used to derive recursive representations of Virasoro blocks on the torus [7, 8] and on higher genus Riemann surfaces [9]. In both cases, however, the full non-perturbative (in either z or in q) answer for the block is beyond reach. It would therefore be of interest to find further constraints satisfied by conformal blocks that, when combined with Zamolodchikov recursion, can be used to determine the block non-perturbatively (at least in principle). This brings us to the additional asset of modularity, to which we now turn. • Finally, from this expansion one can read off the coefficients of the large-hp expansion straightforwardly. Gauge theories, the 2d/4d correspondence, and a synthesis. Much effort has been directed towards understanding instanton effects in N = 2 supersymmetric gauge theories. Notably, techniques have been developed to localise path integrals onto instanton moduli spaces, and further onto sets of isolated poi (...truncated)