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arXiv:2112.01485

Anomalies of fermionic CFTs via cobordism and bootstrap

Andrea Grigoletto

Published Dec 2, 2021• 0 citations
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Abstract

We study constraints on the space of $d=2$ fermionic CFTs as a function of non-perturbative anomalies exhibited under a fermionic discrete symmetry group $G^f$, focusing our attention also on cases where $G^f$ is non-abelian or presents a non-trivial twist of the $\mathbb{Z}^f_2$ subgroup. For the cases we selected, among our results we find that modular bootstrap consistency bounds predict the presence of relevant/marginal operators only for some groups and anomalies. From this point of view, the appearance in the analysis of several kinks around irrelevant operators with $Δ>2$ means that for fermionic systems with increasingly larger symmetry groups modular bootstrap is able to give less constraining bounds than its bosonic counterpart. Within our analysis we show how the anomaly constraints on fermionic CFTs can be effectively recovered from the structure of the abelian subgroups of $G^f$. Finally, we extend the previous surgery description of bordism invariants that describe $3d$ abelian spin-TQFTs, in order to include the case of theories with $\mathrm{Spin}\text{-}\mathbb{Z}^f_{2^{l+1}}$ structures.

DOI
10.48550/arxiv.2112.01485

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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.
We explore constraints on (1+1)d(1+1)d(1+1)d unitary conformal field theory with an internal ZN{\mathbb{Z}}_{N}ZN​ 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 ZN{\mathbb{Z}}_{N}ZN​-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 (e8)1({\mathfrak{e}}_{8}{)}_{1}(e8​)1​. Its existence implies that robust gapless fixed points are not possible in this range of ccc if only a ZN{\mathbb{Z}}_{N}ZN​ symmetry is imposed microscopically. We also obtain stronger, more refined bounds that depend on the `t Hooft anomaly of the ZN{\mathbb{Z}}_{N}ZN​ symmetry.
Abstract We provide a mathematical proposal for the anomaly indicators of symmetries of (2+1)-D fermionic topological orders, and work out the consequences of our proposal in several nontrivial examples. Our proposal is an invariant of a super modular tensor category with a fermionic group action, which gives a (3+1)-D topological field theory (TFT) that we conjecture to be invertible; the anomaly indicators are partition functions of this TFT on 4-manifolds generating the corresponding twisted spin bordism group. Our construction relies on a bosonization construction due to Gaiotto–Kapustin and Tata–Kobayashi–Bulmash–Barkeshli, together with a “bosonization conjecture” which we explain in detail. In the second half of the paper, we discuss several examples of our invariants relevant to condensed-matter physics. The most important example we consider is Z/4T×Z/2f\mathbb {Z}/4^T\times \mathbb {Z}/2^fZ/4T×Z/2f Z / 4 T × Z / 2 f time-reversal symmetry with symmetry algebra T2=(−1)FC{\mathcal {T}}^2 = (-1)^FCT2=(−1)FC T 2 = ( - 1 ) F C , which many fermionic topological orders enjoy, including the U(1)5\textrm{U}(1)_5U(1)5​ U ( 1 ) 5 spin Chern–Simons theory. Using newly developed tools involving the Smith long exact sequence, we calculate the cobordism group that classifies its anomaly, present the generating manifold, and calculate the partition function on the generating manifold which serves as our anomaly indicator. Our approach allows us to reproduce anomaly indicators known in the literature with simpler proofs, including Z/4Tf\mathbb {Z}/4^{Tf}Z/4Tf Z / 4 Tf time-reversal symmetry with symmetry algebra T2=(−1)F\mathcal T^2 = (-1)^FT2=(−1)F T 2 = ( - 1 ) F , and other symmetry groups in the 10-fold way involving Lie group symmetries.
We incorporate the microscopic assumptions that lead to a certain generalization of the Lieb-Schultz-Mattis (LSM) theorem for one-dimensional spin chains into the conformal bootstrap. Our approach accounts for the ``LSM anomaly'' possessed by these spin chains through a combination of modular bootstrap and correlator bootstrap of symmetry defect operators. Specifically, we obtain universal bounds on the local operator content of (1+1)d(1+1)d(1+1)d conformal field theories (CFTs) that could describe translationally invariant lattice Hamiltonians with a ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N}$ symmetry realized projectively at each site. We assume, for such models, that in the CFT the lattice translation symmetry is realized as an emanant internal ZN{\mathbb{Z}}_{N}ZN​ symmetry. We present bounds on local operators both with and without refinement by their global symmetry representations. Interestingly, we can obtain nontrivial bounds on charged operators when NNN is odd, which turns out to be impossible with modular bootstrap alone. Our bounds exhibit distinctive kinks, some of which are approximately saturated by known theories and others that are unexplained. We discuss additional scenarios with the properties necessary for our bounds to apply, including certain multicritical points between (1+1)d(1+1)d(1+1)d symmetry protected topological phases, where we argue that the anomaly studied in our bootstrap calculations should emerge.
In this dissertation we describe progress towards carving out the space of non-integrable conformal field theories (CFTs) in 1+1 dimensions using conformal bootstrap techniques. We develop and apply numerical and analytical techniques to explore the non-perturbative consequences of conformal symmetry, unitarity and locality on the spectrum and dynamics of 2D CFTs. Throughout we focus on compact irrational CFTs, which have a discrete spectrum consisting of an infinite number of primary operators and exhibit chaotic dynamics. In the first part of the dissertation we report on numerical advances in charting the space of 2D CFTs. We use semidefinite programming to compute bounds on the gap in the spectrum of primary operators as a function of the central charge and on spectral functions that capture the spectral density of primary operators. We then develop recursion relations for arbitrary Virasoro conformal blocks, and explore the constraints of modular invariance of the genus-two partition function, leading to ``critical surfaces'' that bound the heavy structure constants of a CFT. In the second part of the dissertation we develop the analytic bootstrap program for two dimensional conformal field theories, focusing on universal aspects of irrational unitary CFTs and holography. We show that the large-spin sector of irrational CFTs is universally governed by a ``Virasoro mean field theory,'' incorporating exact stress tensor dynamics into the analytic bootstrap and leading to non-perturbative insights about the bound-state dynamics of three-dimensional quantum gravity. We then leverage a generalized notion of modular invariance to derive a universal asymptotic formula for the average value of the structure constants whenever one or more of the operators has large conformal dimension. The large central charge limit makes contact with quantum gravity in three dimensions, where the averaging over heavy states corresponds to coarse-graining over black hole microstates. Finally, we study the Euclidean gravitational path integral of pure AdS3 quantum gravity, and use a modular bootstrap analysis to suggest new saddle point configurations that can be consistently included in the gravitational path integral and whose inclusion renders the dual torus partition function unitary.
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.