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Semi-classical BMS$_3$ blocks and flat holography

Eliot Hijano

TL;DR

This work constructs closed-form semi-classical BMS$_3$ blocks for four-point functions in BMS$_3$ field theories by extending the geometric monodromy method and introducing BMS$_3$ multiplets, which resolve the nontrivial monodromy structure absent in irreducible representations. It then matches these field-theory blocks to holographic computations in flat space, using geodesic networks in flat space cosmologies (FSC) with backreacted heavy operators, and shows that heavy microstates encode thermal-like correlators for light probes, supporting an ETH-like interpretation in flat holography. The results unify a field-theoretic monodromy approach with a bulk extrapolate dictionary for FSC geometries and provide a concrete realization of flat-space holography, including spinful extensions. The discussion outlines future directions, including moving beyond the probe limit and exploring the role of BMS$_3$ multiplets in more general contexts.

Abstract

We present the construction of BMS$_3$ blocks in a two-dimensional field theory and compare the results with holographic computations involving probe particles propagating in flat space cosmologies. On the field theory side, we generalize the monodromy method used in the context of AdS/CFT to theories with BMS symmetry. On the bulk side we consider geodesic Feynman diagrams, recently introduced in [arXiv:1712.07131], evaluated in locally flat geometries generated by backreaction of heavy BMS primary operators. We comment on the implications of these results for the eigenstate thermalization hypothesis in flat holography.

Semi-classical BMS$_3$ blocks and flat holography

TL;DR

This work constructs closed-form semi-classical BMS blocks for four-point functions in BMS field theories by extending the geometric monodromy method and introducing BMS multiplets, which resolve the nontrivial monodromy structure absent in irreducible representations. It then matches these field-theory blocks to holographic computations in flat space, using geodesic networks in flat space cosmologies (FSC) with backreacted heavy operators, and shows that heavy microstates encode thermal-like correlators for light probes, supporting an ETH-like interpretation in flat holography. The results unify a field-theoretic monodromy approach with a bulk extrapolate dictionary for FSC geometries and provide a concrete realization of flat-space holography, including spinful extensions. The discussion outlines future directions, including moving beyond the probe limit and exploring the role of BMS multiplets in more general contexts.

Abstract

We present the construction of BMS blocks in a two-dimensional field theory and compare the results with holographic computations involving probe particles propagating in flat space cosmologies. On the field theory side, we generalize the monodromy method used in the context of AdS/CFT to theories with BMS symmetry. On the bulk side we consider geodesic Feynman diagrams, recently introduced in [arXiv:1712.07131], evaluated in locally flat geometries generated by backreaction of heavy BMS primary operators. We comment on the implications of these results for the eigenstate thermalization hypothesis in flat holography.

Paper Structure

This paper contains 26 sections, 198 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summary of results
  3. BMS$_3$ field theory
  4. The BMS$_3$ algebra
  5. Finite BMS$_3$ transformations
  6. BMS$_3$ currents
  7. Ward identities
  8. Geometric monodromy method
  9. Three-point function monodromy
  10. Four-point function monodromy
  11. The monodromy of ${\cal L}$ - Three-point function
  12. The monodromy of $\mathcal{L}$ - Four-point function
  13. Monodromy method from BMS$_3$ null vectors
  14. CFT$_2$ null vectors
  15. BMS$_3$ null vectors
  16. ...and 11 more sections

Figures (3)

  • Figure 1: BMS block expansion of a four-point function. The heavy operators are labeled as "H", while the light ones are labeled by "L". The choice of channel corresponds to the two heavy operators interchanging a BMS highest weight representation with the light operators. The exchanged representation is labeled by $\alpha$.
  • Figure 2: a) Minkowski Space coordinates. The gray planes correspond to the Cauchy horizon. The planes have been spelled out in equation \ref{['eq:FSCMINK']}. They intersect at the special geodesic $\gamma_{\cal H}$ placed at $t=x=0$. The dotted surface corresponds to the origin in ADM coordinates. The green line represents a null geodesic. The dashed part is time-like separated from the Cauchy Horizon. b) ADM coordinates. The gray cylinder is the Cauchy Horizon at $r=R_c$. The dotted blue line is the origin $r=0$. The green line is a null geodesic that intersects the null boundary at $u_i,\phi_i$.
  • Figure 3: a) Holographic BMS$_3$ block in the Minkowski coordinates \ref{['eq:FSCMINK']}. The gray planes correspond to the horizon of the FSC. The singularity at $r=0$ in ADM coordinates corresponds to the surface $x^2-t^2=R_c^2/M$. The purple dot represents the vertex point $x_i$, whose location after extremizing the on-shell action is given explicitly in \ref{['eq:SOLXI']}. The common vertex point lies past the singularity. b) FSC in ADM coordinates. The gray cylinder represents the horizon at $r=R_c$. The origin at $r=0$ is represented by a dashed blue line. The green null lines fall from the null boundary at the points $u_i,\phi_i$.