Abstract
In this paper, we investigate the correlation functions of the conformal field theory (CFT) with the $T\bar{T}$ deformation on torus in terms of perturbative CFT approach, which is the extension of the previous investigations on correlation functions defined on a plane. We systematically obtain the first order correction to the correlation functions of the CFTs with $T\bar{T}$ deformation in both operator formalism and path integral language. As a consistency check, we compute the deformed partition function, namely the zero-point correlation function, up to the first order, which is consistent with results in literature. Moreover, we obtain a new recursion relation for correlation functions with multiple $T$'s and $\bar{T}$'s insertion in generic CFTs on torus. Base on the recursion relations, we study some correlation functions of stress tensors up to the first order under $T\bar{T}$ deformation.
Highlights
A class of exactly solvable deformation of ∂LðλÞ ∂λ 1⁄4 −Z d2zTTðzÞ; ð1Þ where the composite operator TTðzÞ constructed from stress tensor within the theory LðλÞ was first introduced in Ref. [1]
We investigate the correlation functions of the conformal field theory (CFT) with the TTdeformation on a torus in terms of the perturbative CFT approach, which is the extension of the previous investigations on correlation functions defined on a plane
To construct the correlation functions of the CFTs on a torus with a TTdeformation, we apply the Ward identity on a torus and do a proper regularization procedure to figure out the correlation functions with TTdeformation in terms of the perturbative field theory approach
Summary
Z d2zTTðzÞ; ð1Þ where the composite operator TTðzÞ constructed from stress tensor within the theory LðλÞ was first introduced in Ref. [1]. The other motivation to study the correlation functions in the deformed theory on the torus is associated with reading the information about multiple entanglement entropy of the multi-interval, since the multi-interval Renyi entropy can be related to the computation of a partition function (i.e., zero-point function) on a torus or correlation functions of twist operators on a torus [73,74,75,76]. In terms of the perturbative approach, we obtain the correlation functions with TTdeformation systematically by using both operator formalism and path integral language following the analysis in Refs. In Appendixes, we list the notations and some relevant techniques which are very useful in our analysis
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