Abelian BF theory and Turaev-Viro invariant

Mathieu, Philippe; Thuillier, Frank

doi:10.1063/1.4942046

Cited by 14 publications

(46 citation statements)

References 80 publications

(147 reference statements)

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“…This trivial fibre is the (only) one that constitutes Deligne bundle if we consider a theory over 3 . This choice enables establishing a link with Reshetikhin-Turaev abelian invariant (see [1]). Note that usually the normalization of Reshetikhin-Turaev invariant is chosen to be related to 1 × 2 .…”

Section: (1) Chern-simons and Bf Theoriesmentioning

confidence: 99%

“…Then, by Pontrjagin duality, we can associate to it a unique element ∈ ( [1] ) # . Thus, for a fibre over ∑ ∈ 1 ≃ 2 we will consider as origin the element…”

Section: Decomposition Of Deligne Cohomology Classesmentioning

confidence: 99%

“…for a fixed connection and a translation , then, for a translation in ( [1] ) # associated with a cycle :…”

Section: (1) Chern-simons and Bf Theoriesmentioning

confidence: 99%

“…These rules define the addition of a Deligne coboundary to a Deligne cocycle. The quotient set of Deligne cocycles by Deligne coboundaries is the first Deligne cohomology group [1] . [1] is naturally endowed with a structure of Z-modulus.…”

Section: Advances In High Energy Physicsmentioning

confidence: 99%

“…The quotient set of Deligne cocycles by Deligne coboundaries is the first Deligne cohomology group [1] . [1] is naturally endowed with a structure of Z-modulus. It can be described in particular through two exact sequences.…”

Section: Advances In High Energy Physicsmentioning

confidence: 99%

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Exact Computations in Topological Abelian Chern-Simons and BF Theories

Mathieu

2017

Advances in High Energy Physics

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We introduce Deligne cohomology that classifies (1) fibre bundles over 3 manifolds endowed with connections. We show how the structure of Deligne cohomology classes provides a way to perform exact (nonperturbative) computations in (1) Chern-Simons theory (BF theory, resp.) at the level of functional integrals. The partition functions (and observables) of these theories are strongly related to topological invariants well known to the mathematicians.

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Section: (1) Chern-simons and Bf Theoriesmentioning

confidence: 99%

“…Then, by Pontrjagin duality, we can associate to it a unique element ∈ ( [1] ) # . Thus, for a fibre over ∑ ∈ 1 ≃ 2 we will consider as origin the element…”

Section: Decomposition Of Deligne Cohomology Classesmentioning

confidence: 99%

“…for a fixed connection and a translation , then, for a translation in ( [1] ) # associated with a cycle :…”

Section: (1) Chern-simons and Bf Theoriesmentioning

confidence: 99%

Section: Advances In High Energy Physicsmentioning

confidence: 99%

Section: Advances In High Energy Physicsmentioning

confidence: 99%

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Exact Computations in Topological Abelian Chern-Simons and BF Theories

Mathieu

2017

Advances in High Energy Physics

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On 2-form gauge models of topological phases

Delcamp

Tiwari

2019

J. High Energ. Phys.

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We explore various aspects of 2-form topological gauge theories in (3+1)d. These theories can be constructed as sigma models with target space the second classifying space B 2 G of the symmetry group G, and they are classified by cohomology classes of B 2 G. Discrete topological gauge theories can typically be embedded into continuous quantum field theories. In the 2-form case, the continuous theory is shown to be a strict 2-group gauge theory. This embedding is studied by carefully constructing the space of q-form connections using the technology of Deligne-Beilinson cohomology. The same techniques can then be used to study more general models built from Postnikov towers. For finite symmetry groups, 2-form topological theories have a natural lattice interpretation, which we use to construct a lattice Hamiltonian model in (3+1)d that is exactly solvable. This construction relies on the introduction of a cohomology, dubbed 2-form cohomology, of algebraic cocycles that are identified with the simplicial cocycles of B 2 G as provided by the so-called W -construction of Eilenberg-MacLane spaces. We show algebraically and geometrically how a 2-form 4-cocycle reduces to the associator and the braiding isomorphisms of a premodular category of G-graded vector spaces. This is used to show the correspondence between our 2-form gauge model and the Walker-Wang model. arXiv:1901.02249v2 [hep-th] 27 May 2019 7 Correspondence with the Walker-Wang model 54 7.1 Braided monoidal categories 54 7.2 Walker-Wang model for the category of G-graded vector spaces 56 7.3 From the 2-form gauge model to the Walker-Wang model 59 8 Conclusion 61 A Postnikov towers and sigma models 63 B Pontrjagin square 64 C Operators, quantization and invertibility of 2-form topological theories 65 D Topological actions in terms of Deligne-Beilinson cocycles 69 ∼ 1 ∼ SECTION 1 6We define non-trivial topological orders as the ones that have long-range entanglement, non-trivial ground state degeneracy that depends on the topology and fractionalized excitations.7 By premodular category we mean a braided fusion category. A premodular category is then modular if its S-matrix is non-degenerate.∼ 4 ∼ premodular category is a finite abelian group and a quadratic form, the Walker-Wang model provides a Hamiltonian realization of a 2-form gauge theory that describes the topological order mentioned above.Our study pursues two complementary approaches: The first one relies on a formulation of 2-form gauge theories in the continuum. Indeed, it is often possible to embed discrete gauge theories, especially the ones built from abelian groups, into continuous gauge theories. This embedding, if possible, is such that partition function of the discrete gauge theory and the one of the continuous theory are equal. A well-known example of such a procedure is the embedding of a Z n -gauge theory in (d+1)-dimensions into a BF theory with a U(1)-connection 1-form A and a U(1)-dynamical field (d−1)-form B. 8 A special example of this scenario is the embedding of the toric code model,...

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