Anomalous dimensions from thermal AdS partition functions
- PDF / 524,631 Bytes
- 35 Pages / 595.276 x 841.89 pts (A4) Page_size
- 89 Downloads / 199 Views
Springer
Received: September 4, 2020 Accepted: September 7, 2020 Published: October 23, 2020
Per Kraus,a Stathis Megasa and Allic Sivaramakrishnanb a
Mani L. Bhaumik Institute for Theoretical Physics, University of California, 475 Portola Plaza, Los Angeles, CA 90095, U.S.A. b Department of Physics and Astronomy, University of Kentucky, 505 Rose Street, Lexington, KY 40506, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: We develop an efficient method for computing thermal partition functions of weakly coupled scalar fields in AdS. We consider quartic contact interactions and show how to evaluate the relevant two-loop vacuum diagrams without performing any explicit AdS integration, the key step being the use of K¨all´en-Lehmann type identities. This leads to a simple method for extracting double-trace anomalous dimensions in any spacetime dimension, recovering known first-order results in a streamlined fashion. Keywords: AdS-CFT Correspondence, 1/N Expansion ArXiv ePrint: 2004.08635
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)149
JHEP10(2020)149
Anomalous dimensions from thermal AdS partition functions
Contents 1 Introduction 1.1 General method and summary of results
1 2
2 CFT partition functions
7 9 12
4 Quartic contact interactions: simple examples 4.1 Non-derivative interaction 4.2 Two-derivative interaction
12 13 14
5 Spinning propagators 5.1 Embedding space 5.2 Spinning propagators
16 16 17
6 General results 6.1 Interaction vertices 6.2 Decomposition into spinning propagators 6.3 Highest spin contribution 6.4 Another example
18 18 18 19 21
7 (∇µ φ∇µ φ)2 interaction
23
8 Discussion
26
A Scalar partition function with angular potentials A.1 CFT character A.2 Free partition function in AdS
27 27 27
B Free spin-1 partition function
28
C Anomalous dimensions from four-point Witten diagrams
29
1
Introduction
It is interesting both for its own sake and in connection to the AdSd+1 /CFTd correspondence to ask how the spectrum of a weakly coupled quantum field theory in AdS behaves as a function of its coupling. On the AdS side, this amounts to computing binding energies
–1–
JHEP10(2020)149
3 Computation of free field partition functions in AdSd+1 3.1 Higher spin fields
1.1
General method and summary of results
Thermal AdSd+1 is described by the Euclidean signature line element ds2 =
1 dρ2 + dt2 + sin2 ρdΩ2d−1 , 2 cos ρ
(1.1)
with periodic imaginary time, t∼ =t+β .
(1.2)
Here and elsewhere we are setting the AdS radius to unity. We consider some weakly interacting quantum field theory with coupling constant λ living on this background geometry, and seek to compute the thermal partition sum Z(β) = Tre−βH , 1
(1.3)
Constraints on the AdS3 /CFT2 spectrum implied by modular invariance of the partition function form the basis of the modular bootstrap program [21]. Modular invariance (for d = 2) will play no role for us, since we focus on the low energy spectrum without regard to issues of
Data Loading...
