Abstract
Modular invariance imposes rigid constraints on the partition functions of two-dimensional conformal field theories (CFTs). Many fundamental results follow strictly from modular invariance and unitarity, giving rise to the numerical modular bootstrap program. Here we report on a way to relate a particular family of quantum error correcting codes to a family of "code CFTs," which forms a subset of the space of Narain CFTs. This correspondence reduces modular invariance of the 2D CFT partition function to a few simple algebraic relations obeyed by a multivariate polynomial characterizing the corresponding code. Using this correspondence, we construct many explicit examples of physically distinct isospectral theories, as well as many examples of nonholomorphic functions, which satisfy all the basic properties of a 2D CFT partition function, yet are not associated with any known CFT.
Highlights
Two-dimensional conformal field theories (CFTs) enjoy an exceptionally wide range of applications, from condensed matter physics to string theory and quantum gravity
We report on a way to relate a particular family of quantum error correcting codes to a family of “code CFTs,” which forms a subset of the space of Narain CFTs
This correspondence reduces modular invariance of the 2D CFT partition function to a few simple algebraic relations obeyed by a multivariate polynomial characterizing the corresponding code
Summary
Modular invariance imposes rigid constraints on the partition functions of two-dimensional conformal field theories (CFTs). It has been observed numerous times, including in the context of the conformal bootstrap in d > 2, that a robust solution of the bootstrap constraints, e.g., a “kink” in the exclusion plot, reflects the presence of an actual theory This picture is consistent with another observation that, with the exception of a limited family of examples related to chiral models [18] and a class of candidate partition functions for rational CFTs with two characters [19], all currently known nonchiral candidate partition functions—nonholomorphic modular-invariant functions Zðτ; τÞ, which can be expanded in (Virasoro) characters with non-negative integral coefficients and with leading coefficient equal to one (reflecting the requirement of a unique CFT vacuum)—are partition functions of actual 2D theories. These “fake” polynomials provide thousands of examples of modular-invariant Zðτ; τÞ, which are sums of Uð1Þn × Uð1Þn characters, Zðτ; τÞ
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