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The 2-Dimensional Dual of $φ^4$ in AdS$_3$

Weichen Xiao, Ivo Sachs

Abstract

We study the correlation functions of a conformally coupled $φ^4$-interacting theory in AdS$_3$ and its dual CFT$_2$. The one-loop diagram is not expressible in terms of known transcendental functions, but is shown to be expressible as an infinite sum of previously well-studied tree-level diagrams, and we compute this sum using several number-theoretic conjectures. This enables us to extract recursively, the analytic expressions of anomalous dimensions of all dual double-trace operators. In the $s$-channel various consistency checks were performed against established bootstrap method, while our results in the $t$- and $u$-channel are not available in any previous literature to our knowledge.

The 2-Dimensional Dual of $φ^4$ in AdS$_3$

Abstract

We study the correlation functions of a conformally coupled -interacting theory in AdS and its dual CFT. The one-loop diagram is not expressible in terms of known transcendental functions, but is shown to be expressible as an infinite sum of previously well-studied tree-level diagrams, and we compute this sum using several number-theoretic conjectures. This enables us to extract recursively, the analytic expressions of anomalous dimensions of all dual double-trace operators. In the -channel various consistency checks were performed against established bootstrap method, while our results in the - and -channel are not available in any previous literature to our knowledge.
Paper Structure (19 sections, 123 equations, 6 figures, 5 tables)

This paper contains 19 sections, 123 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cross Diagram
  4. Loop diagrams
  5. Fish diagram and tricks with conformal transformation
  6. The $s$- channel as a cross check
  7. Conformal block expansion in the $t$- and $u$-channel
  8. Spectral function does not work easily for t and u channel
  9. Conformal Block Expansion
  10. The $s$ channel as a warm-up
  11. $t$- and $u$- channel anomalous dimensions
  12. Conclusion
  13. Loop calculation
  14. Spectral Function
  15. Disconnected Diagrams
  16. ...and 4 more sections

Figures (6)

  • Figure 1: The cross diagram $\mathcal{I}_4$
  • Figure 2: The fish diagram $\mathcal{J}_4$
  • Figure 3: The full one-loop diagram $\mathcal{K}_4$
  • Figure 4: The intermediate double trace operator dual to a bubble diagram in the bulk
  • Figure 5: Spectral representation of the cross diagram
  • ...and 1 more figures

Theorems & Definitions (2)

  • Conjecture 5.1
  • Conjecture 5.2