Pub Date : 2022-03-22DOI: 10.1142/S0129055X23500095
H. Sati, U. Schreiber
We demonstrate that twisted equivariant differential K-theory of transverse complex curves accommodates exotic charges of the form expected of codimension=2 defect branes, such as of D7-branes in IIB/F-theory on A-type orbifold singularities, but also of their dual 3-brane defects of class-S theories on M5-branes. These branes have been argued, within F-theory and the AGT correspondence, to carry special SL(2)-monodromy charges not seen for other branes, but none of these had previously been identified in the expected brane charge quantization law given by K-theory. Here we observe that it is the subtle (and previously somewhat neglected) twisting of equivariant K-theory by flat complex line bundles appearing inside orbi-singularities ("inner local systems") that makes the secondary Chern character on a punctured plane inside an A-type singularity evaluate to the twisted holomorphic de Rham cohomology which Feigin, Schechtman&Varchenko showed realizes sl(2,C)-conformal blocks, here in degree 1 -- in fact it gives the direct sum of these over all admissible fractional levels. The remaining higher-degree conformal blocks appear similarly if we assume our previously discussed"Hypothesis H"about brane charge quantization in M-theory. Since conformal blocks -- and hence these twisted equivariant secondary Chern characters -- solve the Knizhnik-Zamolodchikov equation and thus constitute representations of the braid group of motions of defect branes inside their transverse space, this provides a concrete first-principles realization of anyon statistics of -- and hence of topological quantum computation on -- defect branes in string/M-theory.
{"title":"Anyonic Defect Branes and Conformal Blocks in Twisted Equivariant Differential (TED) K-theory","authors":"H. Sati, U. Schreiber","doi":"10.1142/S0129055X23500095","DOIUrl":"https://doi.org/10.1142/S0129055X23500095","url":null,"abstract":"We demonstrate that twisted equivariant differential K-theory of transverse complex curves accommodates exotic charges of the form expected of codimension=2 defect branes, such as of D7-branes in IIB/F-theory on A-type orbifold singularities, but also of their dual 3-brane defects of class-S theories on M5-branes. These branes have been argued, within F-theory and the AGT correspondence, to carry special SL(2)-monodromy charges not seen for other branes, but none of these had previously been identified in the expected brane charge quantization law given by K-theory. Here we observe that it is the subtle (and previously somewhat neglected) twisting of equivariant K-theory by flat complex line bundles appearing inside orbi-singularities (\"inner local systems\") that makes the secondary Chern character on a punctured plane inside an A-type singularity evaluate to the twisted holomorphic de Rham cohomology which Feigin, Schechtman&Varchenko showed realizes sl(2,C)-conformal blocks, here in degree 1 -- in fact it gives the direct sum of these over all admissible fractional levels. The remaining higher-degree conformal blocks appear similarly if we assume our previously discussed\"Hypothesis H\"about brane charge quantization in M-theory. Since conformal blocks -- and hence these twisted equivariant secondary Chern characters -- solve the Knizhnik-Zamolodchikov equation and thus constitute representations of the braid group of motions of defect branes inside their transverse space, this provides a concrete first-principles realization of anyon statistics of -- and hence of topological quantum computation on -- defect branes in string/M-theory.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48445396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-06DOI: 10.1142/S0129055X22500428
K. Grabowska, J. Grabowski, M. Ku's, G. Marmo
We consider some natural (functorial) lifts of geometric objects associated with statistical manifolds (metric tensor, dual connections, skewness tensor, etc.) to higher tangent bundles. It turns out that the lifted objects form again a statistical manifold structure, this time on the higher tangent bundles, with the only difference that the metric tensor is pseudo-Riemannian. What is more, natural lifts of potentials (called also divergence or contrast functions) turn out to be again potentials, this time for the lifted statistical structures. We propose an analogous procedure for lifting statistical structures on Lie algebroids and lifting contrast functions which are defined on Lie groupoids. In particular, we study in detail Lie groupoid structures of higher tangent bundles of Lie groupoids. Our geometric constructions of lifts are illustrated by explicit examples, including some important statistical models and potential functions on Lie groupoids. MSC
{"title":"Lifting statistical structures","authors":"K. Grabowska, J. Grabowski, M. Ku's, G. Marmo","doi":"10.1142/S0129055X22500428","DOIUrl":"https://doi.org/10.1142/S0129055X22500428","url":null,"abstract":"We consider some natural (functorial) lifts of geometric objects associated with statistical manifolds (metric tensor, dual connections, skewness tensor, etc.) to higher tangent bundles. It turns out that the lifted objects form again a statistical manifold structure, this time on the higher tangent bundles, with the only difference that the metric tensor is pseudo-Riemannian. What is more, natural lifts of potentials (called also divergence or contrast functions) turn out to be again potentials, this time for the lifted statistical structures. We propose an analogous procedure for lifting statistical structures on Lie algebroids and lifting contrast functions which are defined on Lie groupoids. In particular, we study in detail Lie groupoid structures of higher tangent bundles of Lie groupoids. Our geometric constructions of lifts are illustrated by explicit examples, including some important statistical models and potential functions on Lie groupoids. MSC","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42509202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-20DOI: 10.1142/S0129055X22500192
F. Han, V. Mathai
This paper is a step towards realizing T-duality and Hori formulae for loop spaces. Here, we prove T-duality and Hori formulae for winding [Formula: see text]-loop spaces, which are infinite dimensional subspaces of loop spaces.
{"title":"T-duality, vertical holonomy line bundles and loop Hori formulae","authors":"F. Han, V. Mathai","doi":"10.1142/S0129055X22500192","DOIUrl":"https://doi.org/10.1142/S0129055X22500192","url":null,"abstract":"This paper is a step towards realizing T-duality and Hori formulae for loop spaces. Here, we prove T-duality and Hori formulae for winding [Formula: see text]-loop spaces, which are infinite dimensional subspaces of loop spaces.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44460870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-17DOI: 10.1142/s0129055x2250012x
Manseob Lee, Jumi Oh, Junmi Park
Expansiveness has been used to study dynamic systems and has been developed for various forms of expansiveness. In this paper, we introduce the concept of kinematic [Formula: see text]-expansiveness for flows on a [Formula: see text] compact connected manifold [Formula: see text], which is an extension of [Formula: see text]-expansive homeomorphisms. We prove that if a vector field [Formula: see text] on [Formula: see text] is [Formula: see text] robustly kinematic [Formula: see text]-expansive, then it is quasi-Anosov. Furthermore, we consider the divergence-free vector fields and Hamiltonian systems with the kinematic [Formula: see text]-expansive property; then, we study their robustness.
{"title":"Kinematic N-expansive continuous dynamical systems","authors":"Manseob Lee, Jumi Oh, Junmi Park","doi":"10.1142/s0129055x2250012x","DOIUrl":"https://doi.org/10.1142/s0129055x2250012x","url":null,"abstract":"Expansiveness has been used to study dynamic systems and has been developed for various forms of expansiveness. In this paper, we introduce the concept of kinematic [Formula: see text]-expansiveness for flows on a [Formula: see text] compact connected manifold [Formula: see text], which is an extension of [Formula: see text]-expansive homeomorphisms. We prove that if a vector field [Formula: see text] on [Formula: see text] is [Formula: see text] robustly kinematic [Formula: see text]-expansive, then it is quasi-Anosov. Furthermore, we consider the divergence-free vector fields and Hamiltonian systems with the kinematic [Formula: see text]-expansive property; then, we study their robustness.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41360257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-10DOI: 10.1142/s0129055x22500283
Jean-François Bougron, A. Joye, C. Pillet
We study a class of dynamical semigroups (L)n∈N that emerge, by a Feynman–Kac type formalism, from a random quantum dynamical system (Lωn ◦ · · · ◦Lω1 (ρω0 ))n∈N driven by a Markov chain (ωn)n∈N. We show that the almost sure large time behavior of the system can be extracted from the large n asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator L. As a physical application, we consider the case where the Lω’s are the reduced dynamical maps describing the repeated interactions of a system S with thermal probes Eω. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.
{"title":"Markovian Repeated Interaction Quantum Systems","authors":"Jean-François Bougron, A. Joye, C. Pillet","doi":"10.1142/s0129055x22500283","DOIUrl":"https://doi.org/10.1142/s0129055x22500283","url":null,"abstract":"We study a class of dynamical semigroups (L)n∈N that emerge, by a Feynman–Kac type formalism, from a random quantum dynamical system (Lωn ◦ · · · ◦Lω1 (ρω0 ))n∈N driven by a Markov chain (ωn)n∈N. We show that the almost sure large time behavior of the system can be extracted from the large n asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator L. As a physical application, we consider the case where the Lω’s are the reduced dynamical maps describing the repeated interactions of a system S with thermal probes Eω. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48725668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-01DOI: 10.1142/S0129055X23500058
B. Bhat, Repana Devendra, N. Mallick, K. Sumesh
In this paper, we study the [Formula: see text]-convex set of unital entanglement breaking (EB-)maps on matrix algebras. General properties and an abstract characterization of [Formula: see text]-extreme points are discussed. By establishing a Radon–Nikodym-type theorem for a class of EB-maps we give a complete description of the [Formula: see text]-extreme points. It is shown that a unital EB-map [Formula: see text] is [Formula: see text]-extreme if and only if it has Choi-rank equal to [Formula: see text]. Finally, as a direct consequence of the Holevo form of EB-maps, we derive a non-commutative analog of the Krein–Milman theorem for [Formula: see text]-convexity of the set of unital EB-maps.
{"title":"C∗-extreme points of entanglement breaking maps","authors":"B. Bhat, Repana Devendra, N. Mallick, K. Sumesh","doi":"10.1142/S0129055X23500058","DOIUrl":"https://doi.org/10.1142/S0129055X23500058","url":null,"abstract":"In this paper, we study the [Formula: see text]-convex set of unital entanglement breaking (EB-)maps on matrix algebras. General properties and an abstract characterization of [Formula: see text]-extreme points are discussed. By establishing a Radon–Nikodym-type theorem for a class of EB-maps we give a complete description of the [Formula: see text]-extreme points. It is shown that a unital EB-map [Formula: see text] is [Formula: see text]-extreme if and only if it has Choi-rank equal to [Formula: see text]. Finally, as a direct consequence of the Holevo form of EB-maps, we derive a non-commutative analog of the Krein–Milman theorem for [Formula: see text]-convexity of the set of unital EB-maps.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49124701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-22DOI: 10.1142/s0129055x2250009x
X. Geng, Jiao Wei
The Itoh–Narita–Bogoyavlensky lattice hierarchy associated with a discrete [Formula: see text] matrix spectral problem is derived by using Lenard recursion equations. Resorting to the characteristic polynomial of Lax matrix for the lattice hierarchy, we introduce a three-sheeted Riemann surface [Formula: see text] of arithmetic genus [Formula: see text] and construct the corresponding Baker–Akhiezer function and meromorphic function on it. On the basis of the theory of Riemann surface, the continuous flow and discrete flow related to the lattice hierarchy are straightened with the help of the Abel map. Quasi-periodic solutions of the lattice hierarchy in terms of the Riemann theta function are constructed by using the asymptotic properties and the algebro-geometric characters of the meromorphic function and Riemann surface.
{"title":"Three-sheeted Riemann surface and solutions of the Itoh–Narita–Bogoyavlensky lattice hierarchy","authors":"X. Geng, Jiao Wei","doi":"10.1142/s0129055x2250009x","DOIUrl":"https://doi.org/10.1142/s0129055x2250009x","url":null,"abstract":"The Itoh–Narita–Bogoyavlensky lattice hierarchy associated with a discrete [Formula: see text] matrix spectral problem is derived by using Lenard recursion equations. Resorting to the characteristic polynomial of Lax matrix for the lattice hierarchy, we introduce a three-sheeted Riemann surface [Formula: see text] of arithmetic genus [Formula: see text] and construct the corresponding Baker–Akhiezer function and meromorphic function on it. On the basis of the theory of Riemann surface, the continuous flow and discrete flow related to the lattice hierarchy are straightened with the help of the Abel map. Quasi-periodic solutions of the lattice hierarchy in terms of the Riemann theta function are constructed by using the asymptotic properties and the algebro-geometric characters of the meromorphic function and Riemann surface.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48584002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-21DOI: 10.1142/s0129055x22300035
K. Wernli
We give a detailed introduction to the classical Chern–Simons gauge theory, including the mathematical preliminaries. We then explain the perturbative quantization of gauge theories via the Batalin–Vilkovisky (BV) formalism. We then define the perturbative Chern–Simons partition function at any (possibly non-acylic) reference flat connection using the BV formalism, using a Riemannian metric for gauge fixing. We show that it exhibits an anomaly known as the “framing anomaly” when the Riemannian metric is changed, that is, it fails to be gauge invariant. We explain how one can deal with this anomaly to obtain a topological invariant of framed manifolds.
{"title":"Notes on Chern–Simons perturbation theory","authors":"K. Wernli","doi":"10.1142/s0129055x22300035","DOIUrl":"https://doi.org/10.1142/s0129055x22300035","url":null,"abstract":"We give a detailed introduction to the classical Chern–Simons gauge theory, including the mathematical preliminaries. We then explain the perturbative quantization of gauge theories via the Batalin–Vilkovisky (BV) formalism. We then define the perturbative Chern–Simons partition function at any (possibly non-acylic) reference flat connection using the BV formalism, using a Riemannian metric for gauge fixing. We show that it exhibits an anomaly known as the “framing anomaly” when the Riemannian metric is changed, that is, it fails to be gauge invariant. We explain how one can deal with this anomaly to obtain a topological invariant of framed manifolds.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46342159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-17DOI: 10.1142/S0129055X23500083
A. Anastopoulos, M. Benini
The homotopy theory of representations of nets of algebras over a (small) category with values in a closed symmetric monoidal model category is developed. We illustrate how each morphism of nets of algebras determines a change-of-net Quillen adjunction between the model categories of net representations, which is furthermore a Quillen equivalence when the morphism is a weak equivalence. These techniques are applied in the context of homotopy algebraic quantum field theory with values in cochain complexes. In particular, an explicit construction is presented that produces constant net representations for Maxwell $p$-forms on a fixed oriented and time-oriented globally hyperbolic Lorentzian manifold.
{"title":"Homotopy theory of net representations","authors":"A. Anastopoulos, M. Benini","doi":"10.1142/S0129055X23500083","DOIUrl":"https://doi.org/10.1142/S0129055X23500083","url":null,"abstract":"The homotopy theory of representations of nets of algebras over a (small) category with values in a closed symmetric monoidal model category is developed. We illustrate how each morphism of nets of algebras determines a change-of-net Quillen adjunction between the model categories of net representations, which is furthermore a Quillen equivalence when the morphism is a weak equivalence. These techniques are applied in the context of homotopy algebraic quantum field theory with values in cochain complexes. In particular, an explicit construction is presented that produces constant net representations for Maxwell $p$-forms on a fixed oriented and time-oriented globally hyperbolic Lorentzian manifold.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42429569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-06DOI: 10.1142/S0129055X22500258
L. Vinet, A. Zhedanov
The operator that intertwines between the $mathbb{Z}_2$ - Dunkl operator and the derivative is shown to have a realization in terms of the oscillator operators in one dimension. This observation rests on the fact that the Dunkl intertwining operator maps the Hermite polynomials on the generalized Hermite polynomials.
{"title":"Free boson realization of the Dunkl intertwining operator in one dimension","authors":"L. Vinet, A. Zhedanov","doi":"10.1142/S0129055X22500258","DOIUrl":"https://doi.org/10.1142/S0129055X22500258","url":null,"abstract":"The operator that intertwines between the $mathbb{Z}_2$ - Dunkl operator and the derivative is shown to have a realization in terms of the oscillator operators in one dimension. This observation rests on the fact that the Dunkl intertwining operator maps the Hermite polynomials on the generalized Hermite polynomials.","PeriodicalId":54483,"journal":{"name":"Reviews in Mathematical Physics","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2022-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45595765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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