# From conformal blocks to path integrals in the Vaidya geometry

## Abstract

Correlators in conformal field theory are naturally organized as a sum over conformal blocks. In holographic theories, this sum must reorganize into a path integral over bulk fields and geometries. We explore how these two sums are related in the case of a point particle moving in the background of a 3d collapsing black hole. The conformal block expansion is recast as a sum over paths of the first-quantized particle moving in the bulk geometry. Off-shell worldlines of the particle correspond to subdominant contributions in the Euclidean conformal block expansion, but these same operators must be included in order to correctly reproduce complex saddles in the Lorentzian theory. During thermalization, a complex saddle dominates under certain circumstances; in this case, the CFT correlator is not given by the Virasoro identity block in any channel, but can be recovered by summing heavy operators. This effectively converts the conformal block expansion in CFT from a sum over intermediate states to a sum over channels that mimics the bulk path integral.

- Authors:

- Univ. of British Columbia, Vancouver, BC (Canada). Dept. of Physics and Astronomy
- Cornell Univ., Ithaca, NY (United States). Dept. of Physics
- Univ. of Geneva (Switzerland). Dept. of Theoretical Physics

- Publication Date:

- Research Org.:
- Cornell Univ., Ithaca, NY (United States); Univ. of British Columbia, Vancouver, BC (Canada); Univ. of Geneva (Switzerland)

- Sponsoring Org.:
- USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); Natural Sciences and Engineering Research Council of Canada (NSERC); Simons Foundation (United States); Swiss National Science Foundation (SNSF)

- OSTI Identifier:
- 1425794

- Grant/Contract Number:
- SC0014123; 376206; 200021 162796; NCCR 51NF40-141869

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2017; Journal Issue: 9; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; 79 ASTRONOMY AND ASTROPHYSICS; AdS-CFT correspondence; black holes; conformal field theory

### Citation Formats

```
Anous, Tarek, Hartman, Thomas, Rovai, Antonin, and Sonner, Julian.
```*From conformal blocks to path integrals in the Vaidya geometry*. United States: N. p., 2017.
Web. doi:10.1007/JHEP09(2017)009.

```
Anous, Tarek, Hartman, Thomas, Rovai, Antonin, & Sonner, Julian.
```*From conformal blocks to path integrals in the Vaidya geometry*. United States. doi:10.1007/JHEP09(2017)009.

```
Anous, Tarek, Hartman, Thomas, Rovai, Antonin, and Sonner, Julian. Mon .
"From conformal blocks to path integrals in the Vaidya geometry". United States.
doi:10.1007/JHEP09(2017)009. https://www.osti.gov/servlets/purl/1425794.
```

```
@article{osti_1425794,
```

title = {From conformal blocks to path integrals in the Vaidya geometry},

author = {Anous, Tarek and Hartman, Thomas and Rovai, Antonin and Sonner, Julian},

abstractNote = {Correlators in conformal field theory are naturally organized as a sum over conformal blocks. In holographic theories, this sum must reorganize into a path integral over bulk fields and geometries. We explore how these two sums are related in the case of a point particle moving in the background of a 3d collapsing black hole. The conformal block expansion is recast as a sum over paths of the first-quantized particle moving in the bulk geometry. Off-shell worldlines of the particle correspond to subdominant contributions in the Euclidean conformal block expansion, but these same operators must be included in order to correctly reproduce complex saddles in the Lorentzian theory. During thermalization, a complex saddle dominates under certain circumstances; in this case, the CFT correlator is not given by the Virasoro identity block in any channel, but can be recovered by summing heavy operators. This effectively converts the conformal block expansion in CFT from a sum over intermediate states to a sum over channels that mimics the bulk path integral.},

doi = {10.1007/JHEP09(2017)009},

journal = {Journal of High Energy Physics (Online)},

number = 9,

volume = 2017,

place = {United States},

year = {Mon Sep 04 00:00:00 EDT 2017},

month = {Mon Sep 04 00:00:00 EDT 2017}

}

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