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The Mellin Formalism for Boundary CFT$_d$

Leonardo Rastelli, Xinan Zhou

TL;DR

<3-5 sentence high-level summary> The paper extends the Mellin-space approach to conformal field theories with boundaries and interfaces by formulating a Mellin representation tailored to ICFTs, using a holographic dual with an $AdS_d$ brane in $AdS_{d+1}$. It demonstrates that ICFT conformal blocks are captured by geodesic Witten diagrams, provides general expressions and explicit evaluations for contact diagrams, and develops bulk and interface exchange Witten diagrams via both truncation and spectral representations. The results establish a coherent framework where contact diagrams are Mellin-constant and exchange diagrams possess a controlled pole structure, enabling potential applications to ICFT bootstrap and holographic computations across dimensions.

Abstract

We extend the Mellin representation of conformal field theory (CFT) to allow for conformal boundaries and interfaces. We consider the simplest holographic setup dual to an interface CFT - a brane filling an $AdS_{d}$ subspace of $AdS_{d+1}$ - and perform a systematic study of Witten diagrams in this setup. As a byproduct of our analysis, we show that geodesic Witten diagrams in this geometry reproduce interface CFT$_d$ conformal blocks, generalizing the analogous statement for CFTs with no defects.

The Mellin Formalism for Boundary CFT$_d$

TL;DR

<3-5 sentence high-level summary> The paper extends the Mellin-space approach to conformal field theories with boundaries and interfaces by formulating a Mellin representation tailored to ICFTs, using a holographic dual with an brane in . It demonstrates that ICFT conformal blocks are captured by geodesic Witten diagrams, provides general expressions and explicit evaluations for contact diagrams, and develops bulk and interface exchange Witten diagrams via both truncation and spectral representations. The results establish a coherent framework where contact diagrams are Mellin-constant and exchange diagrams possess a controlled pole structure, enabling potential applications to ICFT bootstrap and holographic computations across dimensions.

Abstract

We extend the Mellin representation of conformal field theory (CFT) to allow for conformal boundaries and interfaces. We consider the simplest holographic setup dual to an interface CFT - a brane filling an subspace of - and perform a systematic study of Witten diagrams in this setup. As a byproduct of our analysis, we show that geodesic Witten diagrams in this geometry reproduce interface CFT conformal blocks, generalizing the analogous statement for CFTs with no defects.

Paper Structure

This paper contains 20 sections, 167 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Mellin formalism
  3. Conformal covariance in embedding space
  4. Mellin formalism: review of the bulk case
  5. Mellin formalism for Interface CFTs
  6. Conformal blocks as geodesic Witten diagrams
  7. Conformal blocks in ICFT
  8. Witten diagrams and geodesic Witten diagrams
  9. Proof by conformal Casimir equation
  10. Proof by explicit evaluation
  11. Contact Witten diagrams
  12. Contact diagrams without cross ratio
  13. Contact diagrams with one cross ratio
  14. General cases
  15. Exchange Witten diagrams
  16. ...and 5 more sections

Figures (7)

  • Figure 1: Three types of Witten diagrams
  • Figure 2: The bulk-channel geodesic Witten diagram.
  • Figure 3: The interface-channel geodesic Witten diagram.
  • Figure 4: A bulk exchange Witten diagram is replaced by a sum of contact Witten diagrams when $\Delta_1+\Delta_2-\Delta$ is a positive even integer.
  • Figure 5: Using the split representation of the bulk-to-bulk propagator the bulk exchange Witten diagram is reduced to the product of a three-point contact Witten diagram and an one-point contact Witten diagram.
  • ...and 2 more figures