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arXiv:2312.13337 · Journal of High Energy Physics

Conformal perturbation theory for n-point functions: structure constant deformation

Benjamin A. Burrington, Ida G. Zadeh

Published Jun 13, 2024• 2 citations
arXivHTMLPDF

Abstract

A bstract We consider conformal perturbation theory for n -point functions on the sphere in general 2D CFTs to first order in coupling constant. We regulate perturbation integrals using canonical hard disk excisions of size ϵ around the fixed operator insertions, and identify the full set of counter terms which are sufficient to regulate all such integrated n -point functions. We further explore the integrated 4-point function which computes changes to the structure constants of the theory. Using an sl (2) map, the three fixed locations of operators are mapped to 0, 1, and ∞. We show that approximating the mapped excised regions to leading order in ϵ does not lead to the same perturbative shift to the structure constant as the exact in ϵ region. We explicitly compute the correction back to the exact in ϵ region of integration in terms of the CFT data. We consider the compact boson, and show that one must use the exact in ϵ region to obtain agreement with the exact results for structure constants in this theory.

DOI
10.1007/jhep06(2024)078

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A bstract We consider conformal perturbation theory for n -point functions on the sphere in general 2D CFTs to first order in coupling constant. We regulate perturbation integrals using canonical hard disk excisions of size ϵ around the fixed operator insertions, and identify the full set of counter terms which are sufficient to regulate all such integrated n -point functions. We further explore the integrated 4-point function which computes changes to the structure constants of the theory. Using an sl (2) map, the three fixed locations of operators are mapped to 0, 1, and ∞. We show that approximating the mapped excised regions to leading order in ϵ does not lead to the same perturbative shift to the structure constant as the exact in ϵ region. We explicitly compute the correction back to the exact in ϵ region of integration in terms of the CFT data. We consider the compact boson, and show that one must use the exact in ϵ region to obtain agreement with the exact results for structure constants in this theory.
A bstract We consider a perturbative expansion of the Lanczos coefficients and the Krylov complexity for two-dimensional conformal field theories under integrable deformations. Specifically, we explore the consequences of TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ , JT‾ \textrm{J}\overline{\textrm{T}} JT J T ¯ , and JJ‾ \textrm{J}\overline{\textrm{J}} JJ J J ¯ deformations, focusing on first-order corrections in the deformation parameter. Under TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ deformation, we demonstrate that the Lanczos coefficients b n exhibit unexpected behavior, deviating from linear growth within the valid perturbative regime. Notably, the Krylov exponent characterizing the rate of exponential growth of complexity surpasses that of the undeformed theory for positive value of deformation parameter, suggesting a potential violation of the conjectured operator growth bound within the realm of perturbative analysis. One may attribute this to the existence of logarithmic branch points along with higher order poles in the autocorrelation function compared to the undeformed case. In contrast to this, both JJ‾ \textrm{J}\overline{\textrm{J}} JJ J J ¯ and JT‾ \textrm{J}\overline{\textrm{T}} JT J T ¯ deformations induce no first order correction to either the linear growth of Lanczos coefficients at large- n or the Krylov exponent and hence the results for these two deformations align with those of the undeformed theory.
A bstract The entanglement entropy of spacetime regions A in odd-dimensional conformal field theories (CFTs) contains a universal constant term, (−1)d−12 F(A) {\left(-1\right)}^{\frac{d-1}{2}}\ F(A) (−1)2d−1​ F(A) − 1 d − 1 2 F A . This quantity can be robustly defined by considering the mutual information of pairs of slightly deformed versions of A . In the case of general three-dimensional CFTs, F ( A ) is positive definite and bounded below by the round disk result, F ( A ) ≥ F 0 ≡ F ( ∂A = 𝕊 1 ). Additionally, strong evidence has been provided that for every region A , F ( A ) /F 0 is maximized, within the space of CFT 3 ’s, by the free scalar field result. In this paper we show that while F ( A ) remains a local minimum around F 0 ≡ F ( ∂A = 𝕊 3 ) for small deformations of the spherical entangling surface, it can take values of arbitrarily large magnitude with either sign for more general regions, and hence it is neither upper- nor lower-bounded in general CFT 5 ’s. We argue that an analogous conjecture regarding the extremization of F ( A ) /F 0 for general regions within the space of theories fails in d = 5. We instead analyze the viability of the weaker bound, F ϵ /F 0 ≤ F ϵ /F 0 | free scalar , ∀CFT 5 for general small geometric deformations of the spherical entangling surface. This is equivalent to a general constraint involving the stress-tensor two-point function C T and the Euclidean partition function on the sphere, namely, C T /F 0 ≤ C T /F 0 | free scalar ≈ 0 . 314, which we show to hold for all known CFT 5 ’s. We also comment on possible extensions of this result to higher dimensions.
A bstract For QFTs in AdS the boundary correlation functions remain conformal even if the bulk theory has a scale. This allows one to constrain RG flows with numerical conformal bootstrap methods. We apply this idea to flows between two-dimensional CFTs, focusing on deformations of the tricritical and ordinary Ising model. We provide non-perturbative constraints for the boundary correlation functions of these flows and compare them with conformal perturbation theory in the vicinity of the fixed points. We also reproduce a completely general constraint on the sign of the TT‾T\overline{T }TT deformation in two dimensions.
In this dissertation we describe progress towards carving out the space of non-integrable conformal field theories (CFTs) in 1+1 dimensions using conformal bootstrap techniques. We develop and apply numerical and analytical techniques to explore the non-perturbative consequences of conformal symmetry, unitarity and locality on the spectrum and dynamics of 2D CFTs. Throughout we focus on compact irrational CFTs, which have a discrete spectrum consisting of an infinite number of primary operators and exhibit chaotic dynamics. In the first part of the dissertation we report on numerical advances in charting the space of 2D CFTs. We use semidefinite programming to compute bounds on the gap in the spectrum of primary operators as a function of the central charge and on spectral functions that capture the spectral density of primary operators. We then develop recursion relations for arbitrary Virasoro conformal blocks, and explore the constraints of modular invariance of the genus-two partition function, leading to ``critical surfaces'' that bound the heavy structure constants of a CFT. In the second part of the dissertation we develop the analytic bootstrap program for two dimensional conformal field theories, focusing on universal aspects of irrational unitary CFTs and holography. We show that the large-spin sector of irrational CFTs is universally governed by a ``Virasoro mean field theory,'' incorporating exact stress tensor dynamics into the analytic bootstrap and leading to non-perturbative insights about the bound-state dynamics of three-dimensional quantum gravity. We then leverage a generalized notion of modular invariance to derive a universal asymptotic formula for the average value of the structure constants whenever one or more of the operators has large conformal dimension. The large central charge limit makes contact with quantum gravity in three dimensions, where the averaging over heavy states corresponds to coarse-graining over black hole microstates. Finally, we study the Euclidean gravitational path integral of pure AdS3 quantum gravity, and use a modular bootstrap analysis to suggest new saddle point configurations that can be consistently included in the gravitational path integral and whose inclusion renders the dual torus partition function unitary.
In this paper, we investigate the correlation functions of the conformal field theory (CFT) with the TT‾T\overline{T}TT deformation 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. We systematically obtain the first-order correction to the correlation functions of the CFTs with a TT‾T\overline{T}TT 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 the literature. Moreover, we obtain a new recursion relation for correlation functions with multiple TTT's and T‾\overline{T}T's inserted in generic CFTs on a torus. Based on the recursion relations, we study some correlation functions of stress tensors up to the first order under TT‾T\overline{T}TT deformation.
We consider perturbation defects obtained by perturbing a 2D conformal field theory (CFT) by a relevant operator on a half-plane.If the perturbed bulk theory flows to an infrared fixed point described by another CFT, the defect flows to a conformal defect between the ultraviolet and infrared fixed point CFTs.For short bulk renormalization group flows connecting two fixed points which are close in theory space we find a universal perturbative formula for the boundary entropy of the corresponding conformal perturbation defect.We compare the value of the boundary entropy that our formula gives for the flows between nearby Virasoro minimal models  m with the boundary entropy of the defect constructed by Gaiotto (2012 J. High Energy Phys. JHEP12(2012) 103) and find a match at the first two orders in the m 1 expansion.