Azat M. Gainutdinov

Azat M. Gainutdinov
French National Centre for Scientific Research | CNRS · Laboratoire de Mathématiques et Physique Théorique, Tours

PhD

About

64
Publications
4,564
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1,507
Citations
Citations since 2017
36 Research Items
814 Citations
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
2017201820192020202120222023020406080100120
Introduction
I currently work at the Laboratoire de Mathématiques et Physique Théorique, Tours, French National Centre for Scientific Research. I do research in Mathematical Physics, Algebra, Representation theory and Category theory. My current project is 'Modified trace.'
Additional affiliations
October 2016 - present
French National Centre for Scientific Research
Position
  • Researcher CR2
October 2015 - October 2016
Deutsches Elektronen-Synchrotron
Position
  • PostDoc Position
October 2013 - October 2015
University of Hamburg
Position
  • PostDoc Position

Publications

Publications (64)
Article
Full-text available
Using results of Shimizu on internal characters we prove a useful non-semisimple variant of the categorical Verlinde formula for factorisable finite tensor categories. Conjecturally, examples of such categories are given by the representations RepV of a vertex operator algebra V subject to certain finiteness conditions. Combining this with results...
Article
Full-text available
The periodic sℓ(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the continuum limit by a logarithmic conformal field theory (LCFT) at central charge c = 0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace CP 1|1 = U(2|1)/(U(...
Article
In the context of Conformal Field Theory (CFT), many results can be obtained from the representation theory of the Virasoro algebra. While the interest in Logarithmic CFTs has been growing recently, the Virasoro representations corresponding to these quantum field theories remain dauntingly complicated, thus hindering our understanding of various c...
Article
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We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed spin-chain and its continuum limit-the symplectic fermions theory-and rely on two technical companion papers, Gainutdinov et al. (Nucl Phys B 871:245-288, 2013) and Gainutdinov et al. (Nucl...
Article
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The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2p th root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the ribbon element determines the decomposition of π i...
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We derive by the traditional algebraic Bethe ansatz method the Bethe equations for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [J. Phys. A 37 (2004), 433-440, arXiv:hep-th/0304092]. The technical difficulties due to the breaking of $\mathsf{U}(1)$ symmetry and the absence of a reference state are...
Preprint
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We consider the link and three-manifold invariants in [DGGPR], which are defined in terms of certain non-semisimple finite ribbon categories $\mathcal{C}$ together with a choice of tensor ideal and modified trace. If the ideal is all of $\mathcal{C}$, these invariants agree with those defined by Lyubashenko in the 90's. We show that in that case th...
Article
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This paper is the first in a series where we attempt to define defects in critical lattice models that give rise to conformal field theory topological defects in the continuum limit. We focus mostly on models based on the Temperley–Lieb algebra, with future applications to restricted solid-on-solid (also called anyonic chains) models, as well as no...
Preprint
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We study universal solutions to reflection equations with a spectral parameter, so-called K-operators, within a general framework of universal K-matrices - an extended version of the approach introduced by Appel-Vlaar. Here, the input data is a quasi-triangular Hopf algebra $H$, its comodule algebra $B$ and a pair of consistent twists. In our setti...
Preprint
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We derive by the traditional Algebraic Bethe Ansatz method the Bethe equations for the general open XXZ spin chain with non-diagonal boundary terms under the Nepomechie constraint [arXiv:hep-th/0304092]. The technical difficulties due to the breaking of $U(1)$ symmetry and the absence of a reference state are overcome by an algebraic construction w...
Article
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A bstract We introduce new $$ {U}_{\mathfrak{q}}{\mathfrak{sl}}_2 $$ U q sl 2 -invariant boundary conditions for the open XXZ spin chain. For generic values of $$ \mathfrak{q} $$ q we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $$ \mathfrak{q}={e}^{\frac{i\pi}{p}} $$ q = e...
Preprint
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We introduce new $U_q\mathfrak{sl}_2$-invariant boundary conditions for the open XXZ spin chain. For generic values of $q$ we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $q=e^{\frac{i\pi}{p}}$ a $2p$-th root of unity $ - $ to its $p$-dimensional analogue. Both cases are pa...
Article
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We use modified traces to renormalize Lyubashenko’s closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2 + 1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions of Lyubashe...
Article
We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice H having a Uq(sln) quantum group symmetry. We focus here on the n=3 case of cubic webs, based on the Kuperberg A2 spider, and illustrate its properties by comparisons with the well-known dilute l...
Preprint
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Davydov--Yetter (DY) cohomology classifies infinitesimal deformations of the monoidal structure of tensor functors and tensor categories. We consider such deformations of finite tensor categories and exact tensor functors between them. In arXiv:1910.06094, DY cohomology with coefficients was introduced and related to the comonad cohomology for a ce...
Article
In [arXiv:1912.02063], we constructed 3-dimensional Topological Quantum Field Theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modula...
Preprint
Full-text available
We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice H having a U_q(sl_n) quantum group symmetry. We focus here on the n=3 case of cubic webs, based on the Kuperberg A_2 spider, and illustrate its properties by comparisons with the well-known dilut...
Article
Full-text available
A modified trace for a finite k-linear pivotal category is a family of linear forms on endomorphism spaces of projective objects which has cyclicity and so-called partial trace properties. We show that a non-degenerate modified trace defines a compatible with duality Calabi-Yau structure on the subcategory of projective objects. The modified trace...
Article
This is the first in a series of papers devoted to generalisations of statistical loop models. We define a lattice model of U q ( s l n ) webs on the honeycomb lattice, for n ⩾ 2. It is a statistical model of closed, cubic graphs with certain non-local Boltzmann weights that can be computed from spider relations. For n = 2, the model has no branchi...
Article
For C a finite tensor category we consider four versions of the central monad, A1,…,A4 on C. Two of them are Hopf monads, and for C pivotal, so are the remaining two. In that case all Ai are isomorphic as Hopf monads. We define a monadic cointegral for Ai to be an Ai-module morphism 1→Ai(D), where D is the distinguished invertible object of C. We r...
Preprint
Full-text available
This is the first in a series of papers devoted to generalisations of statistical loop models. We define a lattice model of $U_q(\mathfrak{sl}_n)$ webs on the honeycomb lattice, for $n \ge 2$. It is a statistical model of closed, cubic graphs with certain non-local Boltzmann weights that can be computed from spider relations. For $n=2$, the model h...
Preprint
Full-text available
In [arXiv:1912.02063], we constructed 3-dimensional Topological Quantum Field Theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modula...
Preprint
Full-text available
For $\mathcal{C}$ a finite tensor category we consider four versions of the central monad, $A_1, \dots, A_4$ on $\mathcal{C}$. Two of them are Hopf monads, and for $\mathcal{C}$ pivotal, so are the remaining two. In that case all $A_i$ are isomorphic as Hopf monads. We define a monadic cointegral for $A_i$ to be an $A_i$-module morphism $\mathbf{1}...
Preprint
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We provide a lattice regularization of all topological defects in minimal models CFTs using RSOS and anyonic spin chains. For defects of type $(1,s)$, we connect our result with the "topological symmetry" initially identified in Fibonacci anyons [Phys. Rev. Lett. 98, 160409 (2007)], and the center of the affine Temperley-Lieb algebra discussed in [...
Preprint
Full-text available
We use modified traces to renormalize Lyubashenko's closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2+1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions of Lyubashenk...
Article
Let H be a finite-dimensional unimodular pivotal quasi-Hopf algebra over a field k, and let H-mod be the pivotal tensor category of finite-dimensional H-modules. We give a bijection between left (resp. right) modified traces on the tensor ideal H-pmod of projective modules and left (resp. right) cointegrals for H. The non-zero left/right modified t...
Preprint
Full-text available
Davydov-Yetter cohomology classifies infinitesimal deformations of tensor categories and of tensor functors. Our first result is that Davydov-Yetter cohomology for finite tensor categories is equivalent to the cohomology of a comonad arising from the central Hopf monad. This has several applications: First, we obtain a short and conceptual proof of...
Preprint
Full-text available
Let H be a finite-dimensional unimodular pivotal quasi-Hopf algebra over a field k, and let H-mod be the pivotal tensor category of finite-dimensional H-modules. We give a bijection between left (resp. right) modified traces on the tensor ideal H-pmod of projective modules and left (resp. right) cointegrals for H. The non-zero left/right modified t...
Preprint
Full-text available
Chern-Simons Theories with gauge super-groups appear naturally in string theory and they possess interesting applications in mathematics, e.g. for the construction of knot and link invariants. This paper is the first in a series where we propose a new quantisation scheme for such super-group Chern-Simons theories on 3-manifolds of the form $\Sigma...
Preprint
Full-text available
This paper is the first in a series where we attempt to define defects in critical lattice models that give rise to conformal field theory topological defects in the continuum limit. We focus mostly on models based on the Temperley-Lieb algebra, with future applications to restricted solid-on-solid (also called anyonic chains) models, as well as no...
Preprint
Full-text available
This paper is the first in a series where we attempt to define defects in critical lattice models that give rise to conformal field theory topological defects in the continuum limit. We focus mostly on models based on the Temperley-Lieb algebra, with future applications to restricted solid-on-solid (also called anyonic chains) models, as well as no...
Preprint
Full-text available
We construct a large family of ribbon quasi-Hopf algebras related to small quantum groups, with a factorizable R-matrix. Our main purpose is to obtain non-semisimple modular tensor categories for quantum groups at even roots of unity, where typically the initial representation category is not even braided. The technique we use is a modularization o...
Preprint
Full-text available
Let H be a finite-dimensional pivotal and unimodular Hopf algebra over a field k. It was shown in [BBGa] that the projective tensor ideal in H-mod admits a unique non-degenerate modified trace, a natural generalisation of the categorical trace. This paper provides an extension of this result to a much more general setting. We first extend the notio...
Article
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A bstract The equivalent of fusion in boundary conformal field theory (CFT) can be realized quite simply in the context of lattice models by essentially glueing two open spin chains. This has led to many developments, in particular in the context of chiral logarithmic CFT. We consider in this paper a possible generalization of the idea to the case...
Article
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We give a new factorisable ribbon quasi-Hopf algebra U, whose underlying algebra is that of the restricted quantum group for sl(2) at a 2p'th root of unity. The representation category of U is conjecturally ribbon-equivalent to that of the triplet vertex operator algebra W(p). We obtain U via a simple current extension from the unrolled restricted...
Article
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We introduce a family of factorisable ribbon quasi-Hopf algebras Q(N) for N a positive integer: as an algebra, Q(N) is the semidirect product of CZ2 with the direct sum of a Graßmann and a Clifford algebra in 2N generators. We show that RepQ(N) is ribbon equivalent to the symplectic fermion category SF(N) that was computed in [54] from conformal bl...
Article
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The relationship between bulk and boundary properties is one of the founding features of (Rational) Conformal Field Theory. Our goal in this paper is to explore the possibility of having an equivalent relationship in the context of lattice models. We focus on models based on the Temperley-Lieb algebra, and use the concept of braid translation, whic...
Article
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For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show: (1) ${\mathcal{C}}$ always contains a simple projective object; (2) if ${\mathcal{C}}$ is in addition ribbon, the internal characters of projective modules span a submodule for the projective $\text{SL}(2,\mathbb{...
Article
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We study the universal Hopf algebra L of Majid and Lyubashenko in the case that the underlying ribbon category is the category of representations of a finite dimensional ribbon quasi-Hopf algebra A. We show that L=A* with coadjoint action and compute the Hopf algebra structure morphisms of L in terms of the defining data of A. We give explicitly th...
Article
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The concept of cyclic tridiagonal pairs is introduced, and explicit examples are given. For a fairly general class of cyclic tridiagonal pairs with cyclicity N, we associate a pair of `divided polynomials'. The properties of this pair generalize the ones of tridiagonal pairs of Racah type. The algebra generated by the pair of divided polynomials is...
Article
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Finite Temperley-Lieb (TL) algebras are diagram-algebra quotients of (the group algebra of) the famous Artin's braid group $B_N$, while the affine TL algebras arise as diagram algebras from a generalized version of the braid group. We study asymptotic `$N\to\infty$' representation theory of these quotients (parametrized by $q\in\mathbb{C}^{\times}$...
Article
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For generic values of q, all the eigenvectors of the transfer matrix of the U_q sl(2)-invariant open spin-1/2 XXZ chain with finite length N can be constructed using the algebraic Bethe ansatz (ABA) formalism of Sklyanin. However, when q is a root of unity (q=exp(i pi/p) with integer p>1), the Bethe equations acquire continuous solutions, and the t...
Article
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We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain of finite length N. For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the Temperley-Lieb algebra; and a formula for the degeneracies of the tr...
Article
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We consider the (finite-dimensional) small quantum group $\bar{U}_q sl(2)$ at $q=i$. We show that $\bar{U}_i sl(2)$ does not allow for an R-matrix, even though $U \otimes V \cong V \otimes U$ holds for all finite-dimensional representations $U,V$ of $\bar{U}_i sl(2)$. We then give an explicit coassociator $\Phi$ and an R-matrix $R$ such that $\bar{...
Article
The periodic sℓ(2|1) alternating spin chain encodes (some of) the properties of hulls of percolation clusters, and is described in the continuum limit by a logarithmic conformal field theory (LCFT) at central charge c = 0. This theory corresponds to the strong coupling regime of a sigma model on the complex projective superspace CP$^{1|1}$ = U(2|1)...
Article
Full-text available
Conformal field theory (CFT) has proven to be one of the richest and deepest subjects of modern theoretical and mathematical physics research, especially as regards statistical mechanics and string theory. It has also stimulated an enormous amount of activity in mathematics, shaping and building bridges between seemingly disparate fields through th...
Article
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Logarithmic Conformal Field Theories (LCFT) play a key role, for instance, in the description of critical geometrical problems (percolation, self avoiding walks, etc.), or of critical points in several classes of disordered systems (transition between plateaus in the integer and spin quantum Hall effects). Much progress in their understanding has b...
Article
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This paper is part of an effort to gain further understanding of 2D Logarithmic Conformal Field Theories (LCFTs) by exploring their lattice regularizations. While all work so far has dealt with the Virasoro algebra (or the product of left and right Virasoro), the best known (although maybe not the most relevant physically) LCFTs in the continuum ar...
Article
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Nontrivial critical models in 2D with a central charge c=0 are described by logarithmic conformal field theories (LCFTs), and exhibit, in particular, mixing of the stress-energy tensor with a "logarithmic" partner under a conformal transformation. This mixing is quantified by a parameter (usually denoted b), introduced in Gurarie [Nucl. Phys. B546,...
Article
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The interest in Logarithmic Conformal Field Theories (LCFTs) has been growing over the last few years thanks to recent developments coming from various approaches. A particularly fruitful point of view consists in considering lattice models as regularizations for such quantum field theories. The indecomposability then encountered in the representat...
Article
This paper is the first in a series devoted to the study of logarithmic conformal field theories (LCFT) in the bulk. Building on earlier work in the boundary case, our general strategy consists in analyzing the algebraic properties of lattice regularizations (quantum spin chains) of these theories. In the boundary case, a crucial step was the ident...
Article
This paper is second in a series devoted to the study of periodic super-spin chains. In our first paper at 2011, we have studied the symmetry algebra of the periodic gl(1|1) spin chain. In technical terms, this spin chain is built out of the alternating product of the gl(1|1) fundamental representation and its dual. The local energy densities - the...
Article
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The subject of our study is the Kazhdan-Lusztig (KL) equivalence in the context of a one-parameter family of logarithmic CFTs based on Virasoro symmetry with the (1,p) central charge. All finite-dimensional indecomposable modules of the KL-dual quantum group - the "full" Lusztig quantum sl(2) at the root of unity - are explicitly described. These a...
Article
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We propose a generalized Verlinde formula associated with (1, p) logarithmic models of two-dimensional conformal field theories, which have applications in statistical physics problems such as the sand-pile model and phase transitions in polymers. This formula gives the integer structure constants in the whole (3p-1)-dimensional space of vacuum tor...
Article
We introduce a Kazhdan--Lusztig-dual quantum group for (1,p) Virasoro logarithmic minimal models as the Lusztig limit of the quantum sl(2) at pth root of unity and show that this limit is a Hopf algebra. We calculate tensor products of irreducible and projective representations of the quantum group and show that these tensor products coincide with...
Article
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We introduce (p-1) pseudocharacters in the space of (1,p) model vacuum torus amplitudes to complete the distinguished basis in the 2p-dimensional fusion algebra to a basis in the whole (3p-1)-dimensional space of torus amplitudes, and the structure constants in this basis are (not necessarily non-negative) integer numbers. We obtain a generalized V...
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We derive and study a quantum group g(p,q) that is Kazhdan--Lusztig-dual to the W-algebra W(p,q) of the logarithmic (p,q) conformal field theory model. The algebra W(p,q) is generated by two currents $W^+(z)$ and $W^-(z)$ of dimension (2p-1)(2q-1), and the energy--momentum tensor T(z). The two currents generate a vertex-operator ideal $R$ with the...
Article
We study logarithmic conformal field models that extend the (p,q) Virasoro minimal models. For coprime positive integers p and q, the model is defined as the kernel of the two minimal-model screening operators. We identify the field content, construct the W-algebra Wp,q that is the model symmetry (the maximal local algebra in the kernel), describe...
Article
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To study the representation category of the triplet W-algebra W(p) that is the symmetry of the (1,p) logarithmic conformal field theory model, we propose the equivalent category C(p) of finite-dimensional representations of the restricted quantum group $U_q SL2$ at $q=e^{\frac{i\pi}{p}}$. We fully describe the category C(p) by classifying all indec...

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