Higher central charges and Witt groups

Ng, Siu-Hung; Rowell, Eric C.; Wang, Yilong; Zhang, Qing

doi:10.48550/arxiv.2002.03570

Cited by 3 publications

(7 citation statements)

References 32 publications

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“…The proof of this theorem relies on general aspects of gauging a one-form global symmetry. When the three-manifold M is the lens space L(n, 1), this phase is known as the higher central charge [23,24], which we denote by ξ n . The higher central charge admits a simple expression in terms of the spins θ(a) of the anyons a:…”

Section: Obstructions Beyond Anomaliesmentioning

confidence: 99%

“…These quantities have made a previous appearance in the math literature, where they went under the name of higher central charges [23,24]. By using various techniques from Galois theory, the authors of those references were able to show that the higher central charges with gcd(n, |G|) = 1 are indeed obstructions to having a Lagrangian subgroup.…”

Section: Obstructions From 3-manifold Invariantsmentioning

confidence: 99%

“…But as we will now see, this assumption is incorrect. There are Abelian TQFTs with trivial ξ n for all gcd(n, |G|) = 1, and yet they do not admit any gapped boundary [24].…”

Section: Abelian Tqfts With Vanishing Higher Central Chargesmentioning

confidence: 99%

“…In particular, we will show the following: Theorem 2.5 An Abelian TQFT T with Frobenius-Schur exponent N FS admits a gapped boundary if and only if ξ n (T ) = 1 for all n such that gcd n, N FS gcd(n,N FS ) = 1. 13 In fact these theories and their products generate all the elements in the Witt group of Abelian theories with trivial higher central charges [24].…”

Section: Complete Obstructions To Gapped Boundariesmentioning

confidence: 99%

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Higher central charges and topological boundaries in 2+1-dimensional TQFTs

Kaidi,

Komargodski,

Ohmori

et al. 2021

Preprint

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A 2+1-dimensional topological quantum field theory (TQFT) may or may not admit topological (gapped) boundary conditions. A famous necessary, but not sufficient, condition for the existence of a topological boundary condition is that the chiral central charge c − has to vanish. In this paper, we consider conditions associated with "higher" central charges, which have been introduced recently in the math literature. In terms of these new obstructions, we identify necessary and sufficient conditions for the existence of a topological boundary in the case of bosonic, Abelian TQFTs, providing an alternative to the identification of a Lagrangian subgroup. Our proof relies on general aspects of gauging generalized global symmetries. For non-Abelian TQFTs, we give a geometric way of studying topological boundary conditions, and explain certain necessary conditions given again in terms of the higher central charges. Along the way, we find a curious duality in the partition functions of Abelian TQFTs, which begs for an explanation via the 3d-3d correspondence.

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Section: Obstructions Beyond Anomaliesmentioning

confidence: 99%

Section: Obstructions From 3-manifold Invariantsmentioning

confidence: 99%

“…But as we will now see, this assumption is incorrect. There are Abelian TQFTs with trivial ξ n for all gcd(n, |G|) = 1, and yet they do not admit any gapped boundary [24].…”

Section: Abelian Tqfts With Vanishing Higher Central Chargesmentioning

confidence: 99%

Section: Complete Obstructions To Gapped Boundariesmentioning

confidence: 99%

See 2 more Smart Citations

Higher central charges and topological boundaries in 2+1-dimensional TQFTs

Kaidi,

Komargodski,

Ohmori

et al. 2021

Preprint

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“…See [42] for an example. In a broader context, these obstructions come from the higher central charges [43][44][45]. It remains an interesting question whether it is possible to determine a necessary and sufficient condition for gapped boundary and domain walls to exist just from a ground state wave function.…”

Section: Gapped Domain Wallmentioning

confidence: 99%

Modular commutator in gapped quantum many-body systems

Kim,

Shi,

Kato

et al. 2021

Preprint

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In arXiv:2110.06932, we argued that the chiral central charge -a topologically protected quantity characterizing the edge theory of a gapped (2+1)-dimensional system -can be extracted from the bulk by using an order parameter called the modular commutator. In this paper, we reveal general properties of the modular commutator and strengthen its relationship with the chiral central charge. First, we identify connections between the modular commutator and conditional mutual information, time reversal, and modular flow. Second, we prove, within the framework of the entanglement bootstrap program, that two topologically ordered media connected by a gapped domain wall must have the same modular commutator in their respective bulk. Third, we numerically calculate the value of the modular commutator for a bosonic lattice Laughlin state for finite sizes and extrapolate to the infinite-volume limit. The result of this extrapolation is consistent with the proposed formula up to an error of about 0.7%.

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Extracting higher central charge from a single wave function

Kobayashi¹,

Wang²,

Soejima³

et al. 2023

Preprint

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A (2+1)D topologically ordered phase may or may not have a gappable edge, even if its chiral central charge c− is vanishing. Recently, it is discovered that a quantity regarded as a "higher" version of chiral central charge gives a further obstruction beyond c− to gapping out the edge. In this Letter, we show that the higher central charges can be characterized by the expectation value of the partial rotation operator acting on the wavefunction of the topologically ordered state. This allows us to extract the higher central charge from a single wavefunction, which can be evaluated on a quantum computer. Our characterization of the higher central charge is analytically derived from the modular properties of edge conformal field theory, as well as the numerical results with the ν = 1/2 bosonic Laughlin state and the non-Abelian gapped phase of the Kitaev honeycomb model, which corresponds to U(1)2 and Ising topological order respectively. The letter establishes a numerical method to obtain a set of obstructions to the gappable edge of (2+1)D bosonic topological order beyond c−. We also point out that the expectation values of the partial rotation on a single wavefunction put a constraint on the low-energy spectrum of the bulk-boundary system of (2+1)D bosonic topological order, reminiscent of the Lieb-Schultz-Mattis type theorems.

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Higher central charges and Witt groups

Cited by 3 publications

References 32 publications

Higher central charges and topological boundaries in 2+1-dimensional TQFTs

Higher central charges and topological boundaries in 2+1-dimensional TQFTs

Modular commutator in gapped quantum many-body systems

Extracting higher central charge from a single wave function

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