# 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:
- 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 = {2017},

month = {9}

}

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Works referenced in this record:

##
Quantum gravity partition functions in three dimensions

journal, February 2010

- Maloney, Alexander; Witten, Edward
- Journal of High Energy Physics, Vol. 2010, Issue 2

##
Thermalization of the spectral function in strongly coupled two dimensional conformal field theories

journal, April 2013

- Balasubramanian, V.; Bernamonti, A.; Craps, B.
- Journal of High Energy Physics, Vol. 2013, Issue 4

##
Universal spectrum of 2d conformal field theory in the large c limit

journal, September 2014

- Hartman, Thomas; Keller, Christoph A.; Stoica, Bogdan
- Journal of High Energy Physics, Vol. 2014, Issue 9

##
Localization & exact holography

journal, April 2013

- Dabholkar, Atish; Gomes, João; Murthy, Sameer
- Journal of High Energy Physics, Vol. 2013, Issue 4

##
Universality of long-distance AdS physics from the CFT bootstrap

journal, August 2014

- Fitzpatrick, A. Liam; Kaplan, Jared; Walters, Matthew T.
- Journal of High Energy Physics, Vol. 2014, Issue 8

##
Conformal symmetry in two-dimensional space: Recursion representation of conformal block

journal, October 1987

- Zamolodchikov, Al. B.
- Theoretical and Mathematical Physics, Vol. 73, Issue 1

##
Causality constraints in conformal field theory

journal, May 2016

- Hartman, Thomas; Jain, Sachin; Kundu, Sandipan
- Journal of High Energy Physics, Vol. 2016, Issue 5

##
Creating a (Quantum?) Constraint, in Pre Planckian Space-Time Early Universe via the Einstein Cosmological Constant in a One to One and Onto Comparison between Two Action Integrals

journal, January 2017

- Beckwith, Andrew Walcott
- Journal of High Energy Physics, Gravitation and Cosmology, Vol. 03, Issue 02

##
Conformal bootstrap, universality and gravitational scattering

journal, December 2015

- Jackson, Steven; McGough, Lauren; Verlinde, Herman
- Nuclear Physics B, Vol. 901

##
A natural language for AdS/CFT correlators

journal, November 2011

- Fitzpatrick, A. Liam; Kaplan, Jared; Penedones, Joao
- Journal of High Energy Physics, Vol. 2011, Issue 11

##
Holography and Riemann surfaces

journal, January 2000

- Krasnov, Kirill
- Advances in Theoretical and Mathematical Physics, Vol. 4, Issue 4

##
On information loss in AdS3/CFT2

journal, May 2016

- Fitzpatrick, A. Liam; Kaplan, Jared; Li, Daliang
- Journal of High Energy Physics, Vol. 2016, Issue 5

##
Conformal field theory, two-dimensional quantum gravity and quantization of Teichmüller space

journal, June 1990

- Verlinde, Herman
- Nuclear Physics B, Vol. 337, Issue 3

##
Black hole collapse in the 1/c expansion

journal, July 2016

- Anous, Tarek; Hartman, Thomas; Rovai, Antonin
- Journal of High Energy Physics, Vol. 2016, Issue 7

##
Entwinement and the emergence of spacetime

journal, January 2015

- Balasubramanian, Vijay; Chowdhury, Borun D.; Czech, Bartłlomiej
- Journal of High Energy Physics, Vol. 2015, Issue 1

##
The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant

journal, December 1995

- Coussaert, Oliver; Henneaux, Marc; Driel, Peter van
- Classical and Quantum Gravity, Vol. 12, Issue 12

##
Thermalization of Green functions and quasinormal modes

journal, July 2015

- David, Justin R.; Khetrapal, Surbhi
- Journal of High Energy Physics, Vol. 2015, Issue 7

##
Entwinement in discretely gauged theories

journal, December 2016

- Balasubramanian, V.; Bernamonti, A.; Craps, B.
- Journal of High Energy Physics, Vol. 2016, Issue 12

##
Black holes from CFT: universality of correlators at large c

journal, August 2017

- Kraus, Per; Sivaramakrishnan, Allic; Snively, River
- Journal of High Energy Physics, Vol. 2017, Issue 8

##
A conformal block Farey tail

journal, June 2017

- Maloney, Alexander; Maxfield, Henry; Ng, Gim Seng
- Journal of High Energy Physics, Vol. 2017, Issue 6

##
Witten diagrams revisited: the AdS geometry of conformal blocks

journal, January 2016

- Hijano, Eliot; Kraus, Per; Perlmutter, Eric
- Journal of High Energy Physics, Vol. 2016, Issue 1

##
On the late-time behavior of Virasoro blocks and a classification of semiclassical saddles

journal, April 2017

- Fitzpatrick, A. Liam; Kaplan, Jared
- Journal of High Energy Physics, Vol. 2017, Issue 4

##
Partition functions of three-dimensional pure gravity

journal, January 2008

- Yin, Xi
- Communications in Number Theory and Physics, Vol. 2, Issue 2

##
Infinite conformal symmetry in two-dimensional quantum field theory

journal, July 1984

- Belavin, A. A.; Polyakov, A. M.; Zamolodchikov, A. B.
- Nuclear Physics B, Vol. 241, Issue 2

##
Clebsch–Gordan and Racah–Wigner Coefficients for a Continuous Series of Representations of ?q (??(2, ℝ))

journal, December 2001

- Ponsot, B.; Teschner, J.
- Communications in Mathematical Physics, Vol. 224, Issue 3

##
Holography from conformal field theory

journal, October 2009

- Heemskerk, Idse; Penedones, Joao; Polchinski, Joseph
- Journal of High Energy Physics, Vol. 2009, Issue 10

##
Non-equilibrium scalar two point functions in AdS/CFT

journal, April 2015

- Keränen, Ville; Kleinert, Philipp
- Journal of High Energy Physics, Vol. 2015, Issue 4

Works referencing / citing this record:

##
Covariant holographic entanglement negativity for disjoint intervals in $$AdS_3/CFT_2$$ A d S 3 / C F T 2

journal, June 2019

- Malvimat, Vinay; Mondal, Sayid; Paul, Boudhayan
- The European Physical Journal C, Vol. 79, Issue 6