Invertible Topological Field Theories

Schommer-Pries, Christopher

doi:10.48550/arxiv.1712.08029

Cited by 11 publications

(10 citation statements)

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“…Locality and unitarity are believed to be necessary conditions for relativistic quantum systems. It is not clear whether they are also sufficient or not, especially because the locality in the above sense might be weaker than physically expected, and it may be weaker than the fully extended version of locality [28,29] (see also [30] for invertible field theory) used in [3]. However, locality and unitarity in the above sense may be sufficient for invertible field theories.…”

Section: The Principles Of Locality and Unitarity In Invertible Field...mentioning

confidence: 97%

On the Cobordism Classification of Symmetry Protected Topological Phases

Yonekura

2019

Commun. Math. Phys.

102

120

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In the framework of Atiyah's axioms of topological quantum field theory with unitarity, we give a direct proof of the fact that symmetry protected topological (SPT) phases without Hall effects are classified by cobordism invariants. We first show that the partition functions of those theories are cobordism invariants after a tuning of the Euler term. Conversely, for a given cobordism invariant, we construct a unitary topological field theory whose partition function is given by the cobordism invariant, assuming that a certain bordism group is finitely generated. Two theories having the same cobordism invariant partition functions are isomorphic.

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Section: The Principles Of Locality and Unitarity In Invertible Field...mentioning

confidence: 97%

On the Cobordism Classification of Symmetry Protected Topological Phases

Yonekura

2019

Commun. Math. Phys.

102

120

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“…Therefore up to the issue of defining the relevant notions, we can indirectly study symmetry-protected topological phases by studying invertible (roughly) topological field theories. 6 This turns out to be useful, since topological field theories can be defined rigorously and the homotopy type of invertible field theories is very well-understood [18]. Three nontrivial problems addressed in [8] are 1. to define the appropriate 'target spectrum' for the field theory; 2. to define when an invertible topological field theory is unitary; 3. given an internal fermionic symmetry group G, they define Euclidean ddimensional spacetime structure groups H d (G) which they use to equip their bordisms with the appropriate structure (such as spin).…”

Section: Low-energy Effective Theories Of Invertible Lattice Modelsmentioning

confidence: 99%

Interacting SPT phases are not Morita invariant

Stehouwer

2021

Preprint

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Class D topological superconductors have been described as invertible topological phases protected by charge Q and particle-hole symmetry C. A competing description is that class D has no internal symmetries except for the fermion parity group Z F 2 = {1, (−1) F }. In the weakly interacting setting, it can be argued that 'particle-hole symmetry cancels charge' in a suitable sense. Namely, the classification results are independent of which of the two internal symmetry groups are taken because of a Morita equivalence. However, we argue that for strongly interacting particles, the group of symmetry-protected topological phases in the two cases are nonisomorphic in dimension 2 + 1. This shows that in contrast to the free case, interacting phases are not Morita invariant. To accomplish this, we use the approach to interacting phases using invertible field theories and bordism. We give explicit expressions of invertible field theories which have the two different groups Z F 2 and U (1)Q ⋊ Z C 2 as internal symmetries and give spacetime manifolds on which their partition functions disagree. Techniques from algebraic topology are used to compute the relevant bordism groups, most importantly the James spectral sequence. The result is that there are both a new Z2-and a new Z-invariant for U (1)Q ⋊ Z F 2 that are not present for Z F 2 .

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“…The higher categorical incarnations of these extended cobordism categories are of course also the main objects of study in Lurie's (sketch of a) solution to the cobordism hypothesis [Lur09c], and the results of Bökstedt-Madsen have been reproven in the language of higher categories by Schommer-Pries in [SP17]. Now, denote by PS the category of pre-spectra, that is the lax limit of the diagram…”

Section: 42mentioning

confidence: 99%

Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity

Calmès¹,

Dotto²,

Harpaz³

et al. 2020

Preprint

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We define Grothendieck-Witt spectra in the setting of Poincaré ∞-categories, show that they fit into an extension with a K-and an L-theoretic part and deduce localisation sequences for Verdier quotients. As special cases we obtain generalisations of Karoubi's fundamental and periodicity theorems for rings in which 2 need not be invertible.A novel feature of our approach is the systematic use of ideas from cobordism theory by interpreting the hermitian Q-construction as an algebraic cobordism category. We also use this to give a new description of the LA-spectra of Weiss and Williams. CONTENTS IntroductionRecollection 1 Poincaré-Verdier sequences and additive functors 1.1 Poincaré-Verdier sequences 1.2 Split Poincaré-Verdier sequences and Poincaré recollements 1.3 Poincaré-Karoubi sequences 1.4 Examples of Poincaré-Verdier sequences 1.5 Additive and localising functors 2 The hermitian Q-construction and algebraic cobordism categories 2.1 The hermitian Q-construction 2.2 The cobordism category of a Poincaré ∞-category 2.3 Algebraic surgery 2.4 The additivity theorem 2.5 Fibrations between cobordism categories 2.6 Additivity in K-Theory 3 Structure theory for additive functors 3.1 Cobordisms of Poincaré functors 3.2 Isotropic decompositions of Poincaré ∞-categories 3.3 The group-completion of an additive functor 3.4 The spectrification of an additive functor 3.5 Bordism invariant functors 3.6 The bordification of an additive functor 3.7 The genuine hyperbolisation of an additive functor 4 Grothendieck-Witt theory 4.1 The Grothendieck-Witt space 4.2 The Grothendieck-Witt spectrum 4.3 The Bott-Genauer sequence and Karoubi's fundamental theorem 4.4 L-theory and the fundamental fibre square 4.5 The real algebraic K-theory spectrum and Karoubi periodicity 4.6 LA-theory after Weiss and Williams Date: September 16, 2020. 144 B.2 Schlichting's Grothendieck-Witt-spectrum of a ring with 2 invertible 146 References 150which we term the fundamental fibre square. Now, in [HM] Hesselholt and Madsen promoted the Grothendieck-Witt spectrum GW s cl ( , ) into the genuine fixed points of what they termed the real algebraic K-theory KR s cl ( , ), a genuine C 2 -spectrum. We similarly produce a functor KR ∶ Cat p ∞ ⟶ Sp gC 2 using the language of spectral Mackey functors, with the property that the isotropy separation square of KR(C, Ϙ) is precisely the fundamental fibre square above, so that in particular KR(C, Ϙ) gC 2 ≃ GW(C, Ϙ) and KR(C, Ϙ) C 2 ≃ L(C, Ϙ); here (−) gC 2 and (−) C 2 ∶ S gC 2 → S denote the genuine and geometric fixed points functors, respectively. Combined with the comparison results of [HS20] this affirms the conjecture of Hesselholt and Madsen, that the geometric fixed points of the real algebraic K-theory spectrum of a discrete ring are a version of Ranicki's L-theory.As the ultimate expression of periodicity, we then enhance our extension of Karoubi's periodicity to the following statement in the language of genuine homotopy theory: Theorem D. The boundary map of the metabolic Poincaré-Verdier sequence provides a...

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Invertible Topological Field Theories

Cited by 11 publications

References 0 publications

On the Cobordism Classification of Symmetry Protected Topological Phases

On the Cobordism Classification of Symmetry Protected Topological Phases

Interacting SPT phases are not Morita invariant

Hermitian K-theory for stable $\infty$-categories II: Cobordism categories and additivity

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