Three dimensional pure gravity and generalized Hecke operators
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Springer
Received: May 5, Revised: July 29, Accepted: August 13, Published: September 11,
2020 2020 2020 2020
M. Ashrafi Department of Physics, University of California Davis, One Shields Avenue, Davis, California 95616, U.S.A.
E-mail: [email protected] Abstract: In this paper, we study mathematical functions of relevance to pure gravity in AdS3. Modular covariance places stringent constraints on the space of such functions; modular invariance places even stronger constraints on how they may be combined into physically viable candidate partition functions. We explicitly detail the list of holomorphic and anti-holomorphic functions that serve as candidates for chiral and anti-chiral partition functions and note that modular covariance is only consistent with such functions when the left (resp. right) central charge is an integer multiple of 8, c ∈ 8N. We then find related constraints on the symmetry group of the corresponding topological, Chern-Simons, theory in the bulk of AdS. The symmetry group of the theory can be one of two choices: either SO(2; 1) × SO(2; 1) or its three-fold diagonal cover. We introduce the generalized Hecke operators which map the modular covariant functions to the modular covariant functions. With these mathematical results, we obtain conjectural partition functions for extremal CFT2s, and the corresponding microcanonical entropies, when the chiral central charges are multiples of eight. Finally, we compute subleading corrections to the Beckenstein-Hawking entropy in the bulk gravitational theory with these conjectural partition functions. Keywords: AdS-CFT Correspondence, Black Holes, Conformal Field Theory ArXiv ePrint: 2004.04424
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)083
JHEP09(2020)083
Three dimensional pure gravity and generalized Hecke operators
Contents 1
2 Holomorphically factorizable partition function 2.1 Partition function 2.2 Modularity of partition function 2.3 The basis for the modular covariant functions
5 5 5 6
3 The Hecke operators 3.1 The order n transformations
7 9
4 Three dimensional gravity
12
5 Summary
16
1
Intoduction
In three dimensional classical Einstein gravity, every two solutions are locally isometric and there is no propagating degrees of freedom. In 3d gravity with negative cosmological constant (AdS gravity), the existence of the BTZ black hole [1, 2] makes the theory more interesting to consider this theory as a toy model to understand the higher dimensional gravity [3]–[10]. Three dimensional AdS gravity has asymptotic Virasoro symmetry. During the quantization of this theory, Brown-Henneaux showed that the theory has left and right moving Virasoro algebras which are part of the structure of the conformal field theory [14]. The 3l corresponding central charge is c = 2g (where l is the length of the AdS space). This shows the existence of the boundary conformal field theory. This duality is an example of the AdS/CFT correspondence, which is a correspondence between a bulk gravity, a
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