A bstract There is a rich connection between classical errorcorrecting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum errorcorrecting codes, those of the stabilizer type, are related to Lorentzian lattices and nonchiral CFTs. More specifically, real selfdual stabilizer codes can be associated with even selfdual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. Tduality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge c = n ≤ 12, and find many interesting examples. Among them is a nonchiral E 8 theory, which is based on the root lattice of E 8 understood as an even selfdual Lorentzian lattice. By analyzing all graphs with n ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. Wemore »
Conformal Field Theories with Sporadic Group Symmetry
The monster sporadic group is the automorphism group of a central charge $c=24$ vertex operator algebra (VOA) or meromorphic conformal field theory (CFT). In addition to its $c=24$ stress tensor $T(z)$, this theory contains many other conformal vectors of smaller central charge; for example, it admits $48$ commuting $c=\frac12$ conformal vectors whose sum is $T(z)$. Such decompositions of the stress tensor allow one to construct new CFTs from the monster CFT in a manner analogous to the GoddardKentOlive (GKO) coset method for affine Lie algebras. We use this procedure to produce evidence for the existence of a number of CFTs with sporadic symmetry groups and employ a variety of techniques, including Hecke operators and modular linear differential equations, to compute the characters of these CFTs. Our examples include (extensions of) nine of the sporadic groups appearing as subquotients of the monster, as well as the simple groups ${}^2\tsl{E}_6(2)$ and $\tsl{F}_4(2)$ of Lie type. Many of these examples are naturally associated to McKay's $\widehat{E_8}$ correspondence, and we use the structure of Norton's monstralizer pairs more generally to organize our presentation.
 Award ID(s):
 1818875
 Publication Date:
 NSFPAR ID:
 10299493
 Journal Name:
 Communications in Mathematical Physics
 ISSN:
 00103616
 Sponsoring Org:
 National Science Foundation
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