Boundary Liouville Field Theory I. Boundary State and Boundary Two-point Function

TL;DR

This work develops boundary Liouville field theory on the disk, providing exact boundary data by exploiting degenerate operator techniques. It derives closed-form expressions for the bulk one-point function $U(\alpha|\mu_B)$ and the boundary two-point function $d(\beta|\mu_1,\mu_2)$, along with the necessary structure constants to bootstrap multipoint functions, all while revealing a duality under $b\to1/b$ and a key relation between bulk and boundary cosmological constants through a parameter $s$ defined by $\cosh(\pi b s)^2=(\mu_B^2/\mu)\sin(\pi b^2)$. The results are consistent with reflection and minisuperspace limits and have direct relevance to the boundary sine-Gordon model as a reflection amplitude, offering a foundation for further explorations of degenerate boundary operators and exact cross-matrices. These findings deepen the nonperturbative understanding of boundary conformal field theories and their connections to 2D gravity and lattice models.

Abstract

Liouville conformal field theory is considered with conformal boundary. There is a family of conformal boundary conditions parameterized by the boundary cosmological constant, so that observables depend on the dimensional ratios of boundary and bulk cosmological constants. The disk geometry is considered. We present an explicit expression for the expectation value of a bulk operator inside the disk and for the two-point function of boundary operators. We comment also on the properties of the degenrate boundary operators. Possible applications and further developments are discussed. In particular, we present exact expectation values of the boundary operators in the boundary sin-Gordon model.

Figures (2)

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• Figure 2: Contours of integration $C_{i}$ in (\ref{['cint']}) together with the corresponding values of the boundary cosmological constants $\mu_{i}$.