Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Semigroup of annuli in Liouville CFT

Guillaume Baverez, Colin Guillarmou, Antti Kupiainen, Rémi Rhodes

Abstract

In conformal field theory, the semigroup of annuli with boundary parametrization plays a special role, in that it generates the whole algebra of local conformal symmetries, the so-called Virasoro algebra. The subgroup of elements $\mathbb{A}_f=\mathbb{D}\setminus f(\mathbb{D}^\circ)$ for contracting biholomorphisms $f:\mathbb{D}\to f(\mathbb{D})\subset \mathbb{D}^\circ$ with $f(0)=0$ is called the holomorphic semigroup of annuli. In this article, we construct a differentiable representation of the holomorphic semigroup into the space of bounded operators on the Hilbert space $\mathcal{H}$ of Liouville Conformal Field Theory and show it generates under differentiation the positive Virasoro elements ${\bf L}_n,\tilde{\bf L}_n$ for $n\geq 0$. We also construct a projective representation of the semigroup of annuli in the space of bounded operators on $\mathcal{H}$ in terms of Segal amplitudes and show that all Virasoro elements ${\bf L}_n,\tilde{\bf L}_n$ for $n\in \mathbb{Z}$ are generated by differentiation of these annuli amplitudes. Finally, we use this to show that the Segal amplitudes for Liouville theory are differentiable with respetc to their boundary parametrizations, and the differential is computed in terms of Virasoro generators. This paper will serve, in a forthcoming work, as a fundamental tool in the construction of the conformal blocks as globally defined holomorphic sections of a holomorphic line bundle on Teichmuller space and satisfying the Ward identities.

Semigroup of annuli in Liouville CFT

Abstract

In conformal field theory, the semigroup of annuli with boundary parametrization plays a special role, in that it generates the whole algebra of local conformal symmetries, the so-called Virasoro algebra. The subgroup of elements for contracting biholomorphisms with is called the holomorphic semigroup of annuli. In this article, we construct a differentiable representation of the holomorphic semigroup into the space of bounded operators on the Hilbert space of Liouville Conformal Field Theory and show it generates under differentiation the positive Virasoro elements for . We also construct a projective representation of the semigroup of annuli in the space of bounded operators on in terms of Segal amplitudes and show that all Virasoro elements for are generated by differentiation of these annuli amplitudes. Finally, we use this to show that the Segal amplitudes for Liouville theory are differentiable with respetc to their boundary parametrizations, and the differential is computed in terms of Virasoro generators. This paper will serve, in a forthcoming work, as a fundamental tool in the construction of the conformal blocks as globally defined holomorphic sections of a holomorphic line bundle on Teichmuller space and satisfying the Ward identities.
Paper Structure (36 sections, 36 theorems, 290 equations, 4 figures)

This paper contains 36 sections, 36 theorems, 290 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Liouville Conformal field theory and its conformal blocks
  3. Background
  4. Notations
  5. Riemann surfaces
  6. Regularized determinants
  7. Green functions
  8. Gaussian Free Fields
  9. $g$-Regularizations
  10. Construction of Liouville Conformal Field Theory
  11. Gaussian multiplicative chaos
  12. Liouville CFT path integral
  13. Vertex operators
  14. Amplitudes
  15. Riemann surfaces with marked points
  16. ...and 21 more sections

Key Result

Theorem 1.1

The map ${\bf T}: f\mapsto {\bf T}_f$ is a representation of the semigroup $\mathcal{S}$ in the sense that for $f_1,f_2\in \mathcal{S}$ Moreover for $\varepsilon>0$, the map $f\mapsto {\bf T}_f$ is differentiable as map where $\mathcal{D}(\mathcal{Q})\subset \mathcal{H}$ is the domain of the quadratic form $\mathcal{Q}(F)=\langle {\bf H}F,F\rangle_{\mathcal{H}}$ associated to the Hamiltonian ${\b

Figures (4)

  • Figure 1: A genus $2$ surface cut into two pairs of pants $\Sigma_1,\Sigma_2$ along $3$ curves $\mathcal{C}=(\mathcal{C}_1,\mathcal{C}_2,\mathcal{C}_3)$.
  • Figure 2: Varying boundaries by gluing annuli
  • Figure 3: In/out boundaries on an annulus $\mathbb{A}$
  • Figure 4: Grey area: the annulus $\mathbb{A}_f$. White area: the image $f(\mathbb{D})=\mathbb{D}_f$.

Theorems & Definitions (73)

  • Theorem 1.1: Representation of Segal semigroup of annuli
  • Theorem 1.2: Differential of Segal amplitudes
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 63 more