Moonshine, superconformal symmetry, and quantum error correction

Harvey, Jeffrey A.; Moore, Gregory W.

doi:10.1007/jhep05(2020)146

Cited by 24 publications

(29 citation statements)

References 73 publications

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“…codes can be used to define non-chiral vertex operator algebras, a subject we leave for future investigation. Here we only briefly comment on the recent work [76], which establishes a relation between the Hexacode, understood as the quantum stabilizer code, and a particular SCFT. The SCFT in question, the GTVW theory [77], has chiral vertex operators of dimension 3/2 parametrized by vectors k ∈ R 6 with all components being half-integer, k i = ±1/2.…”

Section: Jhep03(2021)160mentioning

confidence: 99%

“…(k 6 + 1/2) , such that any linear combination in the Hilbert space is mapped to a linear combination of vertex operators. Harvey and Moore show that the code subspace ψ C , defined via an analog of (4.44) ( [76] uses a code equivalent to Hexacode (2.66), and therefore the analog of (4.44) includes imaginary coefficients), is mapped to the special vertex operator, the N = 1 supercurrent. They conjecture that other N = 1 SCFTs are related to other stabilizer codes.…”

Section: Jhep03(2021)160mentioning

confidence: 99%

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Quantum stabilizer codes, lattices, and CFTs

Dymarsky

Shapere

2021

J. High Energ. Phys.

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There is a rich connection between classical error-correcting codes, Euclidean lattices, and chiral conformal field theories. Here we show that quantum error-correcting codes, those of the stabilizer type, are related to Lorentzian lattices and non-chiral CFTs. More specifically, real self-dual stabilizer codes can be associated with even self-dual Lorentzian lattices, and thus define Narain CFTs. We dub the resulting theories code CFTs and study their properties. T-duality transformations of a code CFT, at the level of the underlying code, reduce to code equivalences. By means of such equivalences, any stabilizer code can be reduced to a graph code. We can therefore represent code CFTs by graphs. We study code CFTs with small central charge c = n ≤ 12, and find many interesting examples. Among them is a non-chiral E8 theory, which is based on the root lattice of E8 understood as an even self-dual Lorentzian lattice. By analyzing all graphs with n ≤ 8 nodes we find many pairs and triples of physically distinct isospectral theories. We also construct numerous modular invariant functions satisfying all the basic properties expected of the CFT partition function, yet which are not partition functions of any known CFTs. We consider the ensemble average over all code theories, calculate the corresponding partition function, and discuss its possible holographic interpretation. The paper is written in a self-contained manner, and includes an extensive pedagogical introduction and many explicit examples.

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Section: Jhep03(2021)160mentioning

confidence: 99%

Section: Jhep03(2021)160mentioning

confidence: 99%

Quantum stabilizer codes, lattices, and CFTs

Dymarsky

Shapere

2021

J. High Energ. Phys.

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“…This SPT phase arises, for example, as the infra-red limit of two Majorana fermions with masses of opposite sign and, in the condensed matter literature, it is better known as the topological phase of the Kitaev chain [5,8] . Recent applications of this topological field theory can found [9][10][11][12][13][14][15][16][17][18][19][20]. However, on a Riemann surface with boundary, the Arf topological field theory is not well defined: it suffers the same mod 2 anomaly that we saw above.…”

Section: The Mod 2 Anomalymentioning

confidence: 96%

“…The closed string partition function is now easily computed by implementing the shift (19) in our previous result (17). The contribution from the e i(θ −θ +π)•λ term gives an overall phase which we ignore.…”

Section: Adding Fugacitiesmentioning

confidence: 99%

Boundary RG flows for fermions and the mod 2 anomaly

Smith

Tong

2021

SciPost Phys.

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Boundary conditions for Majorana fermions in d=1+1d=1+1 dimensions fall into one of two SPT phases, associated to a mod 2 anomaly. Here we consider boundary conditions for 2N2N Majorana fermions that preserve a U(1)^NU(1)N symmetry. In general, the left-moving and right-moving fermions carry different charges under this symmetry, and implementation of the boundary condition requires new degrees of freedom, which manifest themselves in a boundary central charge gg. We follow the boundary RG flow induced by turning on relevant boundary operators. We identify the infra-red boundary state. In many cases, the boundary state flips SPT class, resulting in an emergent Majorana mode needed to cancel the anomaly. We show that the ratio of UV and IR boundary central charges is given by g^2_{IR} / g^2_{UV} = \mathrm{dim} \, \mathcal{O}gIR2/gUV2=dim𝒪, the dimension of the perturbing boundary operator. Any relevant operator necessarily has \mathrm{dim} \, \mathcal{O} < 1dim𝒪<1, ensuring that the central charge decreases in accord with the gg-theorem.

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“…Using the AdS/CFT dictionary, one can obtain useful informations about the bulk theory by studying the boundary CFT. Solvability of the 3d gravity and the AdS/CFT correspondence, makes this theory more powerful to reveal some fundamental aspects of the quantum gravity [16]- [18].…”

Section: Jhep09(2020)083mentioning

confidence: 99%

Three dimensional pure gravity and generalized Hecke operators

Ashrafi

2020

J. High Energ. Phys.

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In this paper, we study mathematical functions of relevance to pure gravity in AdS3. Modular covariance places stringent constraints on the space of such functions; modular invariance places even stronger constraints on how they may be combined into physically viable candidate partition functions. We explicitly detail the list of holomorphic and anti-holomorphic functions that serve as candidates for chiral and anti-chiral partition functions and note that modular covariance is only consistent with such functions when the left (resp. right) central charge is an integer multiple of 8, c ∈ 8ℕ. We then find related constraints on the symmetry group of the corresponding topological, Chern-Simons, theory in the bulk of AdS. The symmetry group of the theory can be one of two choices: either SO(2; 1) × SO(2; 1) or its three-fold diagonal cover. We introduce the generalized Hecke operators which map the modular covariant functions to the modular covariant functions. With these mathematical results, we obtain conjectural partition functions for extremal CFT2s, and the corresponding microcanonical entropies, when the chiral central charges are multiples of eight. Finally, we compute subleading corrections to the Beckenstein-Hawking entropy in the bulk gravitational theory with these conjectural partition functions.

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Moonshine, superconformal symmetry, and quantum error correction

Cited by 24 publications

References 73 publications

Quantum stabilizer codes, lattices, and CFTs

Quantum stabilizer codes, lattices, and CFTs

Boundary RG flows for fermions and the mod 2 anomaly

Three dimensional pure gravity and generalized Hecke operators

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