Classical Codes and Chiral CFTs at Higher Genus

Abstract

Higher genus modular invariance of two-dimensional conformal field theories (CFTs) is a largely unexplored area. In this paper, we derive explicit expressions for the higher genus partition functions of a specific class of CFTs: code CFTs, which are constructed using classical error-correcting codes. In this setting, the $\mathrm{Sp}(2g,\mathbb{Z})$ modular transformations of genus $g$ Riemann surfaces can be recast as a simple set of linear maps acting on $2^g$ polynomial variables, which comprise an object called the code enumerator polynomial. The CFT partition function is directly related to the enumerator polynomial, meaning that solutions of the linear constraints from modular invariance immediately give a set of seemingly consistent partition functions at a given genus. We then find that higher genus constraints, plus consistency under degeneration limits of the Riemann surface, greatly reduces the number of possible code CFTs. This work provides a step towards a full understanding of the constraints from higher genus modular invariance on 2d CFTs.

arXiv comment48pp

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Cited by in corpus (10)

• Narain CFTs and error-correcting codes on finite fields
• Fermionic CFTs from classical codes over finite fields
• Narain CFTs and Quantum Codes at Higher Genus
• Holographic description of Narain CFTs and their code-based ensembles
• Fake Z
• TQFT gravity and ensemble holography
• Wormholes and surface defects in rational ensemble holography
• Fermionic CFTs from topological boundaries in abelian Chern-Simons theories
• Optimal Narain CFTs from Codes
• Code CFTs and Topological Matter