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arXiv:2101.08343 · Physical review. D/Physical review. D.

ZN symmetries, anomalies, and the modular bootstrap

Ying-Hsuan Lin, Shu-Heng Shao

Published Jun 1, 2021• 48 citations
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Abstract

We explore constraints on $(1+1)d$ unitary conformal field theory with an internal ${\mathbb{Z}}_{N}$ global symmetry, by bounding the lightest symmetry-preserving scalar primary operator using the modular bootstrap. Among the other constraints we have found, we prove the existence of a ${\mathbb{Z}}_{N}$-symmetric relevant/marginal operator if $N\ensuremath{-}1\ensuremath{\le}c\ensuremath{\le}9\ensuremath{-}N$ for $N\ensuremath{\le}4$, with the end points saturated by various Wess-Zumino-Witten models that can be embedded into $({\mathfrak{e}}_{8}{)}_{1}$. Its existence implies that robust gapless fixed points are not possible in this range of $c$ if only a ${\mathbb{Z}}_{N}$ symmetry is imposed microscopically. We also obtain stronger, more refined bounds that depend on the `t Hooft anomaly of the ${\mathbb{Z}}_{N}$ symmetry.

DOI
10.1103/physrevd.103.125001

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We constrain the spectrum of two-dimensional unitary, compact conformal field theories with central charge c > 1 using modular bootstrap. Upper bounds on the gap in the dimension of primary operators of any spin, as well as in the dimension of scalar primaries, are computed numerically as functions of the central charge using semi-definite programming. Our bounds refine those of Hellerman and Friedan-Keller, and are in some cases saturated by known CFTs. In particular, we show that unitary CFTs with c < 8 must admit relevant deformations, and that a nontrivial bound on the gap of scalar primaries exists for c < 25. We also study bounds on the dimension gap in the presence of twist gaps, bounds on the degeneracy of operators, and demonstrate how "extremal spectra" which maximize the degeneracy at the gap can be determined numerically.
Two-dimensional Conformal Field Theories (CFTs) allow us to describe quantum field theories at critical points and in general systems without a characteristic scale. It is possible to implement such theories on topologically non-trivial surfaces. If we formulate a two-dimensional CFT on a toroidal surface, a new symmetry emerges known as modular invariance. This symmetry corresponds to the set of transformations that act without changing the toroidal surface, consequently, the partition function of the theory must remain invariant under these transformations. This invariance is used as a constraint by the modular bootstrap method to obtain information about the spectrum without using the Lagrangian and operator content of the theory. By imposing that the theory is consistent with modular symmetry, it is possible to determine the limits of the spectrum beyond which modular invariance can no longer be satisfied, thus constraining the space of admissible two-dimensional conformal field theories. To study CFTs that present discrete global symmetries, it is essential to introduce Topological Defect Lines (TDLs). These are one-dimensional, topologically invariant operators that implement the action of the symmetry group on the Hilbert space. The presence of TDLs modifies the topology of the system and introduces new partition functions associated with different symmetry sectors. Under modular transformations, the TDLs mix among themselves, and consequently, the corresponding partition functions also mix, thus promoting the modular constraint to a vector-valued condition. This formalism makes it possible to implement discrete symmetries also considering the possible presence of anomalies. The objective of this thesis is therefore to understand how discrete symmetries and anomalies affect the spectrum of a CFT. Based on the modular bootstrap framework developed by Lin and Shao for the Z2 symmetry, the analysis was first analytically extended to the Abelian case Z3. This was useful in extending the discussion to a more complex case, namely the non-Abelian symmetry S3. This extension constitutes the original theoretical contribution of the thesis, providing the explicit formulation of modular bootstrap equations for CFTs with non-Abelian discrete symmetry. The analytical modular constraints were reformulated numerically, bridging the modular bootstrap framework with semidefinite programming in SDPB. This analysis paves the way for a numerical study based on convex optimization techniques to explore the constraints arising from modular bootstrap equations.