Chern-Simons Invariants from Ensemble Averages

Abstract

We discuss ensemble averages of two-dimensional conformal field theories associated with an arbitrary indefinite lattice with integral quadratic form Q. We provide evidence that the holographic dual after the ensemble average is the three-dimensional Abelian Chern-Simons theory with kinetic term determined by Q. The resulting partition function can be written as a modular form, expressed as a sum over the partition functions of Chern-Simons theories on lens spaces. For odd lattices, the dual bulk theory is a spin Chern-Simons theory, and we identify several novel phenomena in this case. We also discuss the holographic duality prior to averaging in terms of Maxwell-Chern-Simons theories.

Non-rational Narain CFTs from codes over $F_4$

“…As a result we could construct all codes of small length n and found many interesting examples. In particular, we found that the conjecturally optimal Narain theory with c = 6, based on the rescaled Coxeter-Todd lattice, also known as K 12 , understood as a Narain lattice [1] is associated with the Hexacode, the unique [6,6,4] self-dual code over F 4 . Furthermore, the conjecturally optimal Narain theory with c = 7 is associated with the "septacode" introduced in section 6.2, the unique [7,7,4] N-type code.…”

“…For c = 6, there is an obvious guess for the code associated with it. The corresponding Narain lattice is the Coxeter-Todd lattice understood as a Lorentizan lattice; the latter is known to be related to the hexacode -the unique self-dual [n, k, d] = [6,3,4] code over F 4 . For c = 7 the code we find is less well-known, and we dub it the "septacode."…”

“…Both in the context of holographic correspondence and the modular bootstrap program, Narain theories play the Goldilocks role of theories rich enough and simple enough to be studied. Holographically, the emphasis is on the bulk description of the ensemble of Narain theories [1][2][3][4][5][6][7][8]. As a parallel development, the spectral gap of Narain theories has being studied numerically using the modular bootstrap approach [1,9,10].…”

A note on ensemble holography for rational tori

“…In this section we discuss the tentative bulk dual interpretation of the ensemble average over A k × A k rational torus CFTs encountered in the previous sections. In analogy with Narain duality [11][12][13][14][15][16][17][18][19][20] and earlier examples of rational ensemble holography [24][25][26], the bulk dual would be an exotic gravity theory described by a Chern-Simons action supplemented by a prescription to sum over certain topologies in the path integral. In the case of interest, we have two compact U (1) Chern-Simons theories at levels k and −k, and we will review some of the standard dictionary with rational torus CFT on the boundary [22,23].…”

“…However, the prescription specifying which topologies are to be included is far from clear. Various extensions of this example have been studied since then [13][14][15][16][17][18][19][20].…”