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arXiv:2309.07528 · Journal of High Energy Physics

The linear property of genus-g, n-point, b-boundary, c-crosscap correlation functions in two-dimensional conformal field theory

Xun Liu

Published Sep 18, 2024• 0 citations
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Abstract

A bstract We propose a method to challenge the calculation of genus- g , bulk n -point, b -boundary, c -crosscap correlation functions with x boundary operators $$ {\mathcal{F}}_{g,n,b,c}^x $$ F g , n , b , c x in two-dimensional conformal field theories (CFT 2 ). We show that $$ {\mathcal{F}}_{g,n,b,c}^x $$ F g , n , b , c x are infinite linear combinations of genus- g , bulk ( n + b + c )-point functions F g , ( n + b + c ) , and try to obtain the linear coefficients in this work. We show the existence of a single pole structure in the linear coefficients at degenerate limits. A practical method to obtain the infinite linear coefficients is the free field realizations of Ishibashi states. We review the results in Virasoro minimal models $$ \mathcal{M} $$ M ( p, p ′) and extend it to the N = 1 minimal models $$ \mathcal{SM} $$ SM ( p, p ′).

DOI
10.1007/jhep09(2024)108

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A bstract We propose a method to challenge the calculation of genus- g , bulk n -point, b -boundary, c -crosscap correlation functions with x boundary operators Fg,n,b,cx{\mathcal{F}}_{g,n,b,c}^xFg,n,b,cx​ F g , n , b , c x in two-dimensional conformal field theories (CFT 2 ). We show that Fg,n,b,cx{\mathcal{F}}_{g,n,b,c}^xFg,n,b,cx​ F g , n , b , c x are infinite linear combinations of genus- g , bulk ( n + b + c )-point functions F g , ( n + b + c ) , and try to obtain the linear coefficients in this work. We show the existence of a single pole structure in the linear coefficients at degenerate limits. A practical method to obtain the infinite linear coefficients is the free field realizations of Ishibashi states. We review the results in Virasoro minimal models M\mathcal{M}M M ( p, p ′) and extend it to the N = 1 minimal models SM\mathcal{SM}SM SM ( p, p ′).
The existence of a positive linear functional acting on the space of (differences between) conformal blocks has been shown to rule out regions in the parameter space of conformal field theories (CFTs). We argue that at the boundary of the allowed region the extremal functional contains, in principle, enough information to determine the dimensions and operator product expansion (OPE) coefficients of an infinite number of operators appearing in the correlator under analysis. Based on this idea we develop the extremal functional method (EFM), a numerical procedure for deriving the spectrum and OPE coefficients of CFTs lying on the boundary (of solution space). We test the EFM by using it to rederive the low lying spectrum and OPE coefficients of the two-dimensional Ising model based solely on the dimension of a single scalar quasiprimary--no Virasoro algebra required. Our work serves as a benchmark for applications to more interesting, less known CFTs in the near future.
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This thesis explores the application of numerical bootstrap techniques to two-point functions of bulk insertions in the presence of a boundary in two-dimensional conformal field theories. The first part of this thesis is devoted to presenting a comprehensive introduction to essential concepts in the numerical bootstrap of two-dimensional CFTs. Namely, an extensive introduction to the basics of conformal field theories (CFTs), their properties arising from the algebraic structure of conformal transformations, consitency conditions on the observables in a CFT, as well as the effects of the presence of defects in two-dimensional CFTs, is given. This is followed by a quick recount of the numerical techniques employed in the literature to solve for the aforementioned consistency conditions. These chapters form an overview of the seminal works that set the backdrop of the contributions made in the second part of the thesis. In the contributions of the thesis, we discuss the progress made in the investigations in (Meineri and Radhakrishnan 2025) where two-point functions in BCFTs are bootstrapped by treating them as a part of a larger system of consistency conditions involving the partition function on an annulus, the two-point function in the presence of a boundary, and four-point functions on the infinite plane. This perspective of treating the two-point function as a part of a larger system makes accessible parts of the BCFT data that have so-far remained inaccessible, and we present results on the boundary gap in the two-point function, and improvements in existing numerical bounds on the boundary entropy.