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2021 | Buch

Kapitel lesen Erstes Kapitel lesen

Representation Theory, Mathematical Physics, and Integrable Systems

In Honor of Nicolai Reshetikhin

herausgegeben von: Prof. Anton Alekseev, Edward Frenkel, Marc Rosso, Ben Webster, Prof. Milen Yakimov

Verlag: Springer International Publishing

Buchreihe : Progress in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Over the course of his distinguished career, Nicolai Reshetikhin has made a number of groundbreaking contributions in several fields, including representation theory, integrable systems, and topology. The chapters in this volume – compiled on the occasion of his 60th birthday – are written by distinguished mathematicians and physicists and pay tribute to his many significant and lasting achievements.
Covering the latest developments at the interface of noncommutative algebra, differential and algebraic geometry, and perspectives arising from physics, this volume explores topics such as the development of new and powerful knot invariants, new perspectives on enumerative geometry and string theory, and the introduction of cluster algebra and categorification techniques into a broad range of areas. Chapters will also cover novel applications of representation theory to random matrix theory, exactly solvable models in statistical mechanics, and integrable hierarchies. The recent progress in the mathematical and physicals aspects of deformation quantization and tensor categories is also addressed.
Representation Theory, Mathematical Physics, and Integrable Systems will be of interest to a wide audience of mathematicians interested in these areas and the connections between them, ranging from graduate students to junior, mid-career, and senior researchers.

Inhaltsverzeichnis

Frontmatter

Examples of Finite-Dimensional Pointed Hopf Algebras in Positive Characteristic

Abstract

We present new examples of finite-dimensional Nichols algebras over fields of positive characteristic. The corresponding braided vector spaces are not of diagonal type, admit a realization as Yetter-Drinfeld modules over finite abelian groups, and are analogous to braidings over fields of characteristic zero whose Nichols algebras have finite Gelfand-Kirillov dimension.

We obtain new examples of finite-dimensional pointed Hopf algebras by bosonization with group algebras of suitable finite abelian groups.

Nicolás Andruskiewitsch, Iván Angiono, István Heckenberger

Poisson Vertex Algebra Cohomology and Differential Harrison Cohomology

Abstract

We construct a canonical map from the Poisson vertex algebra cohomology complex to the differential Harrison cohomology complex, which restricts to an isomorphism on the top degree. This is an important step in the computation of Poisson vertex algebra and vertex algebra cohomologies.

Bojko Bakalov, Alberto De Sole, Victor G. Kac, Veronica Vignoli

Theta Invariants of Lens Spaces via the BV-BFV Formalism

Abstract

The goal of this paper is to investigate the Theta invariant—an invariant of framed 3-manifolds associated with the lowest order contribution to the Chern-Simons partition function—in the context of the quantum BV-BFV formalism. Namely, we compute the state on the solid torus to low degree in ħ and apply the gluing procedure to compute the Theta invariant of lens spaces. We use a distributional propagator which does not extend to a compactified configuration space, so to compute loop diagrams we have to define a regularization of the product of the distributional propagators, which is done in an ad hoc fashion. Also, a polarization has to be chosen for the quantization process. Our results agree with results in the literature for one type of polarization, but for another type of polarization there are extra terms.

Alberto S. Cattaneo, Pavel Mnev, Konstantin Wernli

Generalized Demazure Modules and Prime Representations in Type D n

Abstract

The goal of this paper is to understand the graded limit of a family of irreducible prime representations of the quantum affine algebra associated with a simply laced simple Lie algebra \(\mathfrak {g}\). This family was introduced by Hernandez and Leclerc (Duke Math J 154:265–341, 2010; Monoidal categorifications of cluster algebras of type A and D, Symmetries, Integrable Systems and Representations, Springer Proceedings in Mathematics & Statistics, vol. 40, pp. 175–193, 2013) in the context of monoidal categorification of cluster algebras. The graded limit of a member of this family is an indecomposable graded module for the current algebra \(\mathfrak {g}[t]\); or equivalently a module for the maximal standard parabolic subalgebra in the affine Lie algebra \(\widehat {\mathfrak {g}}\). In the case when \(\mathfrak {g}\) is of type A _n the problem was studied by Brito et al. (J Inst Math Jussieu, 31 pp., 2015), where it was shown that the graded limit is isomorphic to a level two Demazure module. In this paper we study the case when \(\mathfrak {g}\) is of type D _n. We show that in certain cases the limit is a generalized Demazure module, i.e., it is a submodule of a tensor product of level one Demazure modules. We give a presentation of these modules and compute their graded character (and hence also the character of the prime representations) in terms of Demazure modules of level two.

Vyjayanthi Chari, Justin Davis, Ryan Moruzzi Jr.

Cylindric Rhombic Tableaux and the Two-Species ASEP on a Ring

Abstract

The asymmetric simple exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice of n sites. It was introduced around 1970 (Macdonald et al., Biopolymers, 6, 1968; Spitzer, Adv Math, 5:246–290, 1970), and since then has been extensively studied by researchers in statistical mechanics, probability, and combinatorics. Recently the ASEP on a lattice with open boundaries has been linked to Koornwinder polynomials (Corteel and Williams, to appear in Selecta Math, 2015; Cantini, Ann Henri Poincaré, 18(4):1121–1151, 2017), and the ASEP on a ring has been linked to Macdonald polynomials (Cantini et al., J Phys A, 48(38):384001, 25, 2015). In this article we study the two-species asymmetric simple exclusion process (ASEP) on a ring, in which two kinds of particles (“heavy” and “light”), as well as “holes,” can hop both clockwise and counterclockwise (at rates 1 or t depending on the particle types) on a ring of n sites. We introduce some new tableaux on a cylinder called cylindric rhombic tableaux (CRT) and use them to give a formula for the stationary distribution of the two-species ASEP—each probability is expressed as a sum over all CRT of a fixed type. When λ is a partition in {0, 1, 2}ⁿ, we then give a formula for the nonsymmetric Macdonald polynomial E _λ and the symmetric Macdonald polynomial P _λ by refining our tableaux formulas for the stationary distribution.

Sylvie Corteel, Olya Mandelshtam, Lauren Williams

Macdonald Operators and Quantum Q-Systems for Classical Types

Abstract

We propose solutions of the quantum Q-systems of types B _N, C _N, D _N in terms of q-difference operators, generalizing our previous construction for the Q-system of type A. The difference operators are interpreted as q-Whittaker limits of discrete time evolutions of Macdonald-van Diejen type operators. We conjecture that these new operators act as raising and lowering operators for q-Whittaker functions, which are special cases of graded characters of fusion products of KR-modules.

Philippe Di Francesco, Rinat Kedem

The Meromorphic R-Matrix of the Yangian

Abstract

Let \(\mathfrak {g}\) be a complex semisimple Lie algebra and \(Y_{\hbar }(\mathfrak {g})\) its Yangian. Drinfeld proved that the universal R-matrix \(\mathcal {R}(s)\) of \(Y_{\hbar }(\mathfrak {g})\) gives rise to rational solutions of the \(\operatorname {QYBE}\) on irreducible, finite-dimensional representations of \(Y_{\hbar }(\mathfrak {g})\). This result was recently extended by Maulik–Okounkov to symmetric Kac–Moody algebras and representations arising from geometry. We show that rationality ceases to hold on arbitrary finite-dimensional representations, if one requires such solutions to be natural and compatible with tensor products. Equivalently, the tensor category of finite-dimensional representations of \(Y_{\hbar }(\mathfrak {g})\) does not admit rational commutativity constraints. We construct instead two meromorphic commutativity constraints, which are related by a unitarity condition. Each possesses an asymptotic expansion in s which has the same formal properties as \(\mathcal {R}(s)\), and therefore coincides with it by uniqueness. In particular, we give a constructive proof of the existence of \(\mathcal {R}(s)\). Our construction relies on the Gauss decomposition \(\mathcal {R}^+(s)\cdot \mathcal {R}^0(s)\cdot \mathcal {R}^-(s)\) of \(\mathcal {R}(s)\). The divergent abelian term \(\mathcal {R}^0\) was resummed on finite-dimensional representations by the first two authors in Gautam and Toledano Laredo (Publ Math Inst Hautes Études Sci 125:267–337, 2017). In the present paper, we construct \(\mathcal {R}^{\pm }(s)\), prove that they are rational on finite-dimensional representations, and that they intertwine the standard coproduct of \(Y_{\hbar }(\mathfrak {g})\) and the deformed Drinfeld coproduct introduced in loc. cit.

Sachin Gautam, Valerio Toledano Laredo, Curtis Wendlandt

On Spectral Cover Equations in Simpson Integrable Systems

Abstract

Following Simpson we consider the integrable system structure on the moduli spaces of Higgs bundles on a compact Kähler manifold X. We propose a description of the corresponding spectral cover of X as the fiberwise projective dual to a hypersurface in the projectivization \(\mathbb {P}(\mathcal {T}_{X} \oplus \mathcal {O}_X)\) of the tangent bundle \(\mathcal {T}_X\) to X. The defining equation of the hypersurface dual to the Simpson spectral cover is explicitly constructed in terms of the Higgs fields.

Anton A. Gerasimov, Samson L. Shatashvili

Peter-Weyl Bases, Preferred Deformations, and Schur-Weyl Duality

Abstract

We discuss the deformed function algebra \(\mathcal {O}_{\hbar }(G)\) of a simply connected reductive algebraic group G over \({\mathbb {C}}\) using a basis consisting of matrix elements of finite dimensional representations. This leads to a preferred deformation, meaning one where the structure constants of comultiplication are unchanged. The structure constants of multiplication are controlled by quantum 3j symbols. We then discuss connections earlier work on preferred deformations that involved Schur-Weyl duality.

Anthony Giaquinto, Alex Gilman, Peter Tingley

Quantum Periodicity and Kirillov–Reshetikhin Modules

Abstract

We give a proof of the periodicity of quantum T-systems of type A _n × A _ℓ with certain spiral boundary conditions. Our proof is based on the categorification of the T-system in terms of the representation theory of quantum affine algebras, more precisely on relations between classes of Kirillov–Reshetikhin modules and of evaluation modules.

David Hernandez

A Note on the E-Polynomials of a Stratification of the Hilbert Scheme of Points

Abstract

The stratification associated with the number of generators of the ideals of the punctual Hilbert scheme of points on the affine plane has been studied since the 1970s. In this paper, we present an elegant formula for the E-polynomials of these strata.

Yi-Ning Hsiao, Andras Szenes

Galois Action on VOA Gauge Anomalies

Abstract

Assuming regularity of the fixed subalgebra, any action of a finite group G on a holomorphic VOA V determines a gauge anomaly \(\alpha \in \operatorname {\mathrm {H}}^3(G; \boldsymbol {\mu })\), where \(\boldsymbol {\mu } \subset \mathbb C^\times \) is the group of roots of unity. We show that under Galois conjugation V ↦^γ V , the gauge anomaly transforms as α↦γ ²(α). This provides an a priori upper bound of 24 on the order of anomalies of actions preserving a \(\mathbb Q\)-structure, for example the Monster group \(\mathbb M\) acting on its Moonshine VOA V ^♮. We speculate that each field \(\mathbb K\) should have a “vertex Brauer group” isomorphic to \( \operatorname {\mathrm {H}}^3( \operatorname {\mathrm {Gal}}(\bar {\mathbb K}/\mathbb K); \boldsymbol {\mu }^{\otimes 2})\). In order to motivate our constructions and speculations, we warm up with a discussion of the ordinary Brauer group, emphasizing the analogy between VOA gauging and quantum Hamiltonian reduction.

Theo Johnson-Freyd

Heisenberg-Picture Quantum Field Theory

Abstract

What we should mean by “Heisenberg-picture quantum field theory”? Atiyah–Segal-type axioms do a good job of capturing the “Schrödinger picture”: these axioms define a “d-dimensional quantum field theory” to be a symmetric monoidal functor from an (∞, d)-category of “spacetimes” to an (∞, d)-category which at the second-from-top level consists of vector spaces, so at the top level consists of numbers. This paper argues that the appropriate parallel notion “Heisenberg picture” should also be defined in terms of symmetric monoidal functors from the category of spacetimes, but the target should be an (∞, d)-category that in top dimension consists of pointed vector spaces instead of numbers; the second-from-top level can be taken to consist of associative algebras or of pointed categories. The paper ends by outlining two sources of such Heisenberg-picture field theories: factorization algebras and skein theory.

Theo Johnson-Freyd

Irreducibility of the Wysiwyg Representations of Thompson’s Groups

Abstract

We prove irreducibility and mutual inequivalence for certain unitary representations of R. Thompson’s groups F and T.

Vaughan F. R. Jones

Invariants of Long Knots

Abstract

By using the notion of a rigid R-matrix in a monoidal category and the Reshetikhin–Turaev functor on the category of tangles, we review the definition of the associated invariant of long knots. In the framework of the monoidal categories of relations and spans over sets, by introducing racks associated with pointed groups, we illustrate the construction and the importance of consideration of long knots. Else, by using the restricted dual of algebras and Drinfeld’s quantum double construction, we show that to any Hopf algebra H with invertible antipode, one can associate a universal long knot invariant Z _H(K) taking its values in the convolution algebra ((D(H))^o)^∗ of the restricted dual Hopf algebra (D(H))^o of the quantum double D(H) of H. This extends the known constructions of universal invariants previously considered mostly either in the case of finite-dimensional Hopf algebras or by using some topological completions.

Rinat Kashaev

Rigged Configurations and Unimodality

Abstract

For a given partition λ and a dominant sequence of rectangular shape partitions {R _a}, we give sufficient conditions that imply that the corresponding parabolic Kostka polynomial K _λ,{R}(q) is symmetric and unimodal. Examples found to satisfy these conditions include:

principal specialization of the internal product of Schur functions \(s_{\alpha }*s_{\beta }(\frac {q-q^N}{1-q})\), in particular, q-Gaussian polynomials [N λ] _q (the case α = λ, β = (|λ|);
generalized q-Narayana numbers/polynomials N _ℓ((λ;{R});q) associated with a partition λ and a sequence of rectangular shape partitions {R} = (R ₁, …, R _n);
symmetry and unimodality of the statistics charge generating function \({\mathcal {K}}_{\lambda ,1^{|\lambda |}}^{[d]}(q)\) defined on the set of standard Young tableaux of a given shape λ and a fixed number of descents d;
Schröder–Narayana SchN _k(n.d;q), Kirkman–Caley KC _d(n;q), and Motzkin sum MS _d(n;q) polynomials;
a (q, t)-deformation of the Euler polynomials A _n(t); and
reduced decomposition polynomials RD ^[ℓ](n;q). As a corollary of our general result, we prove symmetry and unimodality of
classical and rectangular q-Narayana numbers and
a certain q-deformation of the odd Eulerian numbers A(2n + 1, k). We also prove the strict log-concavity of q-binomial coefficients, and we introduce and initiate the study of double Liskova polynomials \(L_{\alpha ,\beta }^{\gamma }(q,t)\), which are a natural generalization of the Kostka–Macdonald polynomials K _α,β(q, t).

Anatol N. Kirillov

Turning Point Processes in Plane Partitions with Periodic Weights of Arbitrary Period

Abstract

I study random plane partitions with respect to volume measures with periodic weights of arbitrarily high period. I show that near the vertical boundary the system develops up to as many turning points as the period of the weights and that these turning points are separated by vertical facets which can have arbitrary rational slope. In the lozenge tiling formulation of the model, the facets consist of only two types of lozenges arranged in arbitrary periodic deterministic patterns. We compute the correlation functions near turning points and show that the point processes at the turning points can be described as several GUE-corners processes which are non-trivially correlated.

The weights we study introduce a first-order phase transition in the system. We compute the limiting correlation functions near this phase transition and obtain a process which is translation invariant in the vertical direction but not the horizontal.

Sevak Mkrtchyan

The Skein Category of the Annulus

Abstract

We construct the skein category \(\mathcal {S}\) of the annulus and show that it is equivalent to the affine Temperley-Lieb category of Graham and Lehrer. It leads to a skein theoretic description of the extended affine Temperley-Lieb algebras. We construct an endofunctor of \(\mathcal {S}\) that corresponds, on the level of tangle diagrams, to the insertion of an arc connecting the inner and outer boundary of the annulus. We use it to define and construct towers of extended affine Temperley-Lieb algebra modules. It allows us to construct a tower of modules acting on spaces of link patterns on the punctured disc which play an important role in the study of loop models. In case of trivial Dehn twist we show that the direct sum of the representation spaces of the link pattern tower defines a graded algebra that may be regarded as a relative version of the Roger-Yang skein algebra of arcs and links on the punctured disc. We also describe the link pattern tower in terms of fused extended affine Temperley-Lieb algebra modules.

K. Al Qasimi, J. V. Stokman

Tensor Product of the Fock Representation with Its Dual and the Deligne Category

Abstract

We describe \(\mathfrak {sl}(\infty )\)-module structure of the tensor product of the Fock representation and its shifted dual using action of \(\mathfrak {sl}(\infty )\) on the abelian envelope of the Deligne’s category GL(t).

Vera Serganova

Exact Density Matrix for Quantum Group Invariant Sector of XXZ Model

Abstract

Using the fermionic basis, we obtain the expectation values of all \(U_q(\mathfrak {sl}_2)\)-invariant local operators on 8 sites for the anisotropic six-vertex model on a cylinder with generic Matsubara data. In the case when the \(U_q(\mathfrak {sl}_2)\)-symmetry is not broken, this computation is equivalent to finding the entire density matrix up to 8 sites. As application, we compute the entanglement entropy without and with temperature and compare the results with CFT predictions.

F. Smirnov

Loops in Surfaces and Star-Fillings

Abstract

We discuss a new approach to computing the standard algebraic operations on homotopy classes of loops in a surface: the homological intersection number, Goldman’s Lie bracket, and the author’s Lie cobracket. Our approach uses fillings of the surface by certain graphs.

Vladimir Turaev

Titel: Representation Theory, Mathematical Physics, and Integrable Systems
herausgegeben von: Prof. Anton Alekseev
Edward Frenkel
Marc Rosso
Ben Webster
Prof. Milen Yakimov
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-78148-4
Print ISBN: 978-3-030-78147-7
DOI: https://doi.org/10.1007/978-3-030-78148-4

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Inhaltsverzeichnis

Frontmatter

Examples of Finite-Dimensional Pointed Hopf Algebras in Positive Characteristic

Poisson Vertex Algebra Cohomology and Differential Harrison Cohomology

Theta Invariants of Lens Spaces via the BV-BFV Formalism

Generalized Demazure Modules and Prime Representations in Type D n

Cylindric Rhombic Tableaux and the Two-Species ASEP on a Ring

Macdonald Operators and Quantum Q-Systems for Classical Types

The Meromorphic R-Matrix of the Yangian

On Spectral Cover Equations in Simpson Integrable Systems

Peter-Weyl Bases, Preferred Deformations, and Schur-Weyl Duality

Quantum Periodicity and Kirillov–Reshetikhin Modules

A Note on the E-Polynomials of a Stratification of the Hilbert Scheme of Points

Galois Action on VOA Gauge Anomalies

Heisenberg-Picture Quantum Field Theory

Irreducibility of the Wysiwyg Representations of Thompson’s Groups

Invariants of Long Knots

Rigged Configurations and Unimodality

Turning Point Processes in Plane Partitions with Periodic Weights of Arbitrary Period

The Skein Category of the Annulus

Tensor Product of the Fock Representation with Its Dual and the Deligne Category

Exact Density Matrix for Quantum Group Invariant Sector of XXZ Model

Loops in Surfaces and Star-Fillings

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