An effective proof of the Cartan formula: The even prime

Medina-Mardones, Anibal M.

doi:10.1016/j.jpaa.2020.106444

Cited by 16 publications

(15 citation statements)

References 11 publications

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“…The parameter k is an integer. In the formulas for O4, ζ = n1 ∪ [(n1 ∪ ω2) ∪2 ω2 + n1 ∪ ω2] is the Cartan coboundary [40], and ω2 ′ = ω2 + s1 ∪ n1. Here P(X) = X ∪ X − X ∪1 dX is the Pontryagin square of the 2-cochain X [41].…”

Section: Data For Invertible Fermion Phasesmentioning

confidence: 99%

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Classification of (2+1)D invertible fermionic topological phases with symmetry

Barkeshli,

Chen,

Hsin

et al. 2021

Preprint

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We provide a classification of invertible topological phases of interacting fermions with symmetry in two spatial dimensions for general fermionic symmetry groups G f and general values of the chiral central charge c−. Here G f is a central extension of a bosonic symmetry group G b by fermion parity, (−1) F , specified by a second cohomology class [ω2] ∈ H 2 (G b , Z2). Our approach proceeds by gauging fermion parity and classifying the resulting G b symmetry-enriched topological orders while keeping track of certain additional data and constraints. We perform this analysis through two perspectives, using G-crossed braided tensor categories and Spin(2c−)1 Chern-Simons theory coupled to a background G gauge field. These results give a way to characterize and classify invertible fermionic topological phases in terms of a concrete set of data and consistency equations, which is more physically transparent and computationally simpler than the more abstract methods using cobordism theory and spectral sequences. Our results also generalize and provide a different approach to the recent classification of fermionic symmetry-protected topological phases by Wang and Gu, which have chiral central charge c− = 0. We show how the 10-fold way classification of topological insulators and superconductors fits into our scheme, along with general non-perturbative constraints due to certain choices of c− and G f . Mathematically, our results also suggest an explicit general parameterization of deformation classes of (2+1)D invertible topological quantum field theories with G f symmetry.

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Section: Data For Invertible Fermion Phasesmentioning

confidence: 99%

“…We can further simplify the expression by introducing the Cartan coboundary ζ, defined as [40]: The c − terms in (C25) can be simplified as…”

Section: (Naturality) For Anymentioning

confidence: 99%

Classification of (2+1)D invertible fermionic topological phases with symmetry

Barkeshli,

Chen,

Hsin

et al. 2021

Preprint

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“…In the paper [19], the second author carried out a program, somewhat similar to the joint program here, for finding an explicit coboundary formula implying the Cartan product formula for Steenrod Squares. In fact, both that work and the work in this paper originated when we were working on the paper [6] and needed a specific coboundary formula for the relation Sq 2 ([a] 2 ) = (Sq 1 [a]) 2 for a cocycle a of degree 2, which is simultaneously a Cartan relation and an Adem relation.…”

Section: 34mentioning

confidence: 99%

“…We have implemented Table Reduction as part of a computer program, along with all the other ingredients needed to make explicit our coboundary formulae for Adem relations. Note that as explained in 5.3.3 and discussed in 4.5.4, for each of the cochains J = J Ψ ( x q × x p ), J Ψ ( x p × x q ), J (23) ( x q × x p ) ∈ N * (EΣ 4 ), 19 It is here that we use the identification of the group operation on triples with D 8 that we discussed in 4.1.3 and 5.1.3.…”

Section: 45mentioning

confidence: 99%

A cochain level proof of Adem relations in the mod 2 Steenrod algebra

Brumfiel

Medina-Mardones

Morgan

2021

J. Homotopy Relat. Struct.

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In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes p. Steenrod’s student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod 2 Adem relations at the cochain level. Specifically, given a mod 2 cocycle, we produce explicit cochain formulae whose coboundaries are the Adem relations among compositions of Steenrod Squares applied to the cocycle, using Steenrod’s original cochain definition of the Square operations.

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“…For cellular chains we have seen explicit cochain level constructions, the cup-(p, i) products, inducing the Steenrod operations, and it is desirable to produce cochains enforcing these relations. For the even prime case, Cartan and Adem coboundaries have been constructed effectively in [Med20c] and [BMM21] respectively. Cartan coboundaries for odd primes can be constructed with the tools already described, but the Adem relation requires additional techniques not yet available.…”

Section: Cartan and Adem Relationsmentioning

confidence: 99%

The diagonal of cellular spaces and effective algebro-homotopical constructions

Medina-Mardones¹

2021

Preprint

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In this survey article we discuss certain homotopy coherent enhancements of the coalgebra structure on cellular chains defined by an approximation to the diagonal. Over the rational numbers, C∞-coalgebra structures control the Q-complete homotopy theory of spaces, and over the integers, E∞coalgebras provide an appropriate setting to model the full homotopy category. Effective constructions of these structures, the focus of this work, carry geometric and combinatorial information which has found applications in various fields including deformation theory, higher category theory, and condensed matter physics.

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An effective proof of the Cartan formula: The even prime

Cited by 16 publications

References 11 publications

Classification of (2+1)D invertible fermionic topological phases with symmetry

Classification of (2+1)D invertible fermionic topological phases with symmetry

A cochain level proof of Adem relations in the mod 2 Steenrod algebra

The diagonal of cellular spaces and effective algebro-homotopical constructions

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