Azat M. Gainutdinov

Lattice fusion rules and logarithmic operator product expansions

Abstract 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 representation theory of the corresponding finite-dimensional associative algebras exactly mimics the Virasoro indecomposable modules expected to arise in the continuum limit. In this paper, we study in detail the so-called Temperley–Lieb (TL) fusion functor introduced in physics by Read and Saleur [N. Read and H. Saleur, Associative-algebraic approach to logarithmic conformal field theories, Nucl. Phys. B 777 (2007) 316]. Using quantum group results, we provide rigorous calculations of the fusion of various TL modules at roots of unity cases. Our results are illustrated by many explicit examples relevant for physics. We discuss how indecomposability arises in the “lattice” fusion and compare the mechanisms involved with similar observations in the corresponding field theory. We also discuss the physical meaning of our lattice fusion rules in terms of indecomposable operator product expansions of quantum fields.

Continuum limit and symmetries of the periodic gℓ(1|1) spin chain

Abstract 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 identification of the space of states as a bimodule over the Temperley–Lieb (TL) algebra and the quantum group U q s l ( 2 ) . The extension of this analysis in the bulk case involves considerable difficulties, since the U q s l ( 2 ) symmetry is partly lost, while the TL algebra is replaced by a much richer version (the Jones–Temperley–Lieb — JTL — algebra). Even the simplest case of the g l ( 1 | 1 ) spin chain — corresponding to the c = − 2 symplectic fermions theory in the continuum limit — presents very rich aspects, which we will discuss in several papers. In this first work, we focus on the symmetries of the spin chain, that is, the centralizer of the JTL algebra in the alternating tensor product of the g l ( 1 | 1 ) fundamental representation and its dual. We prove that this centralizer is only a subalgebra of U q s l ( 2 ) at q = i that we dub U q odd s l ( 2 ) . We then begin the analysis of the continuum limit of the JTL algebra: using general arguments about the regularization of the stress–energy tensor, we identify families of JTL elements going over to the Virasoro generators L n , L ¯ n in the continuum limit. We then discuss the s l ( 2 ) symmetry of the (continuum limit) symplectic fermions theory from the lattice and JTL point of view. The analysis of the spin chain as a bimodule over U q odd s l ( 2 ) and JTL N is discussed in the second paper of this series.

A physical approach to the classification of indecomposable Virasoro representations from the blob algebra

Abstract 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 critical phenomena. We extend in this paper the construction of Read and Saleur (2007) [1] , [2] , and uncover a deep relationship between the Virasoro algebra and a finite-dimensional algebra characterizing the properties of two-dimensional statistical models, the so-called blob algebra (a proper extension of the Temperley–Lieb algebra). This allows us to explore vast classes of Virasoro representations (projective, tilting, generalized staggered modules, etc.), and to conjecture a classification of all possible indecomposable Virasoro modules (with, in particular, L 0 Jordan cells of arbitrary rank) that may appear in a consistent physical Logarithmic CFT where Virasoro is the maximal local chiral algebra. As by-products, we solve and analyze algebraically quantum-group symmetric XXZ spin chains and sl ( 2 | 1 ) supersymmetric spin chains with extra spins at the boundary, together with the “mirror” spin chain introduced by Martin and Woodcock (2003) [3] .

Bimodule structure in the periodic gℓ(1|1) spin chain

Abstract This paper is the second in a series devoted to the study of periodic super-spin chains. In our first paper (Gainutdinov et al., 2013 [3] ), we have studied the symmetry algebra of the periodic g l ( 1 | 1 ) spin chain. In technical terms, this spin chain is built out of the alternating product of the g l ( 1 | 1 ) fundamental representation and its dual. The local energy densities — the nearest neighbour Heisenberg-like couplings — provide a representation of the Jones–Temperley–Lieb (JTL) algebra JTL N . The symmetry algebra is then the centralizer of JTL N , and turns out to be smaller than for the open chain, since it is now only a subalgebra of U q s l ( 2 ) at q = i — dubbed U q odd s l ( 2 ) in Gainutdinov et al. (2013) [3] . A crucial step in our associative algebraic approach to bulk logarithmic conformal field theory (LCFT) is then the analysis of the spin chain as a bimodule over U q odd s l ( 2 ) and JTL N . While our ultimate goal is to use this bimodule to deduce properties of the LCFT in the continuum limit, its derivation is sufficiently involved to be the sole subject of this paper. We describe representation theory of the centralizer and then use it to find a decomposition of the periodic g l ( 1 | 1 ) spin chain over JTL N for any even N and ultimately a corresponding bimodule structure. Applications of our results to the analysis of the bulk LCFT will then be discussed in the third part of this series.

Logarithmic conformal field theory

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 the study of vertex operator algebras, a (partial) axiomatisation of a chiral CFT. One can add to this that the successes of CFT, particularly when applied to statistical lattice models, have also served as an inspiration for mathematicians to develop entirely new fields: Schramm-Loewner evolution and Smirnov’s discrete complex analysis being notable examples. When the energy operator fails to be diagonalisable on the quantum state space, the CFT is said to be logarithmic. Consequently, a logarithmic CFT is one whose quantum space of states is constructed from a collection of representations which includes reducible but indecomposable ones. This qualifier arises because of the consequence that certain correlation functions will possess logarithmic singularities, something that contrasts with the familiar case of power law singularities. While such logarithmic singularities and reducible representations were noted by Rozansky and Saleur in their study of the U(1|1) Wess-Zumino-Witten model in 1992, the link between the non-diagonalisability of the energy operator and logarithmic singularities in correlators is usually ascribed to Gurarie’s 1993 article (his paper also contains the first usage of the term “logarithmic conformal field theory”). The class of CFTs that were under control at this time was quite small. In particular, an enormous amount of work from the statistical mechanics and string theory communities had produced a fairly detailed understanding of the (so-called) rational CFTs. However, physicists from both camps were well aware that applications from many diverse fields required significantly more complicated non-rational theories. Examples include critical percolation, supersymmetric string backgrounds, disordered electronic systems, sandpile models describing avalanche processes, and so on. In each case, the non-rationality and non-unitarity of the CFT suggested that a more general theoretical framework was needed. Driven by the desire to better understand these applications, the mid-nineties saw significant theoretical advances aiming to generalise the constructs of rational CFT to a more general class. In 1994, Nahm introduced an algorithm for computing the fusion product of representations which was significantly generalised two years later by Gaberdiel and Kausch who applied it to explicitly construct (chiral) representations upon which the energy operator acts non-diagonalisably. Their work made it clear that underlying the physically relevant correlation functions are classes of reducible but indecomposable representations that can be investigated mathematically to the benefit of applications. In another direction, Flohr had meanwhile initiated the study of modular properties of the characters of logarithmic CFTs, a topic which had already evoked much mathematical interest in the rational case. Since these seminal theoretical papers appeared, the field has undergone rapid development, both theoretically and with regard to applications. Logarithmic CFTs are now known to describe non-local observables in the scaling limit of critical lattice models, for example percolation and polymers, and are an integral part of our understanding of quantum strings propagating on supermanifolds. They are also believed to arise as duals of three-dimensional chiral gravity models, fill out hidden sectors in non-rational theories with non-compact target spaces, and describe certain transitions in various incarnations of the quantum Hall effect. Other physical

Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity

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 transfer matrix eigenvalues in terms of dimensions of tilting sl(2)_q-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (q=exp(i pi/2)), we give explicit formulas for the number of admissible solutions and degeneracies. We also consider the cases of generic q and the isotropic (q->1) limit. Numerical solutions of the Bethe equations up to N=8 are presented. Our results are consistent with the Bethe ansatz solution being complete.

The periodic sℓ(2|1) alternating spin chain and its continuum limit as a bulk logarithmic conformal field theory at c = 0

A bstractThe 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 CP1|1 = U(2|1)/(U(1) × U(1|1)), and the spectrum of critical exponents can be obtained exactly. In this paper we push the analysis further, and determine the main representation theoretic (logarithmic) features of this continuum limit by extending to the periodic case the approach of [1] [N. Read and H. Saleur, Nucl. Phys. B 777 (2007) 316]. We first focus on determining the representation theory of the finite size spin chain with respect to the algebra of local energy densities provided by a representation of the affine Temperley-Lieb algebra at fugacity one. We then analyze how these algebraic properties carry over to the continuum limit to deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the theory, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.

On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories

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, which is a natural way to close an open spin chain by adding an interaction between the first and last spins using braiding to bring them next to each other. The interaction thus obtained is in general non-local, but has the key feature that it is expressed solely in terms of the algebra for the open spin chain - the ordinary Temperley-Lieb algebra and its blob algebra generalization. This is in contrast with the usual periodic spin chains which involve only local interactions, and are described by the periodic TL algebra. We show that for the Restricted Solid-On-Solid models, which are known to be described by minimal unitary CFTs in the continuum limit, the braid translation in fact does provide the ordinary periodic model starting from the open model with fixed boundary conditions on the two sides of the strip. This statement has a precise mathematical formulation, which is a pull-back map between irreducible modules of, respectively, the blob algebra and the affine TL algebra. We then turn to the same kind of analysis for two models whose continuum limits are Logarithmic CFTs - the alternating gl(1|1) and sl(2|1) spin chains. We find that the result for minimal models does not hold any longer: braid translation of the relevant TL modules does not give rise to the modules known to be present in the periodic chains. In the gl(1|1) case, the content in terms of the irreducibles is the same, as well as the spectrum, but the detailed structure (like logarithmic coupling) is profoundly different. This carries over to the continuum limit.

Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity

Abstract For generic values of q , all the eigenvectors of the transfer matrix of the U q s l ( 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 = e i π / p with integer p ≥ 2 ), the Bethe equations acquire continuous solutions, and the transfer matrix develops Jordan cells. Hence, there appear eigenvectors of two new types: eigenvectors corresponding to continuous solutions (exact complete p -strings), and generalized eigenvectors. We propose general ABA constructions for these two new types of eigenvectors. We present many explicit examples, and we construct complete sets of (generalized) eigenvectors for various values of p and N .

Bimodule structure in the periodic spin chain

Abstract This paper is the second in a series devoted to the study of periodic super-spin chains. In our first paper (Gainutdinov et al., 2013 [3] ), we have studied the symmetry algebra of the periodic g l ( 1 | 1 ) spin chain. In technical terms, this spin chain is built out of the alternating product of the g l ( 1 | 1 ) fundamental representation and its dual. The local energy densities — the nearest neighbour Heisenberg-like couplings — provide a representation of the Jones–Temperley–Lieb (JTL) algebra JTL N . The symmetry algebra is then the centralizer of JTL N , and turns out to be smaller than for the open chain, since it is now only a subalgebra of U q s l ( 2 ) at q = i — dubbed U q odd s l ( 2 ) in Gainutdinov et al. (2013) [3] . A crucial step in our associative algebraic approach to bulk logarithmic conformal field theory (LCFT) is then the analysis of the spin chain as a bimodule over U q odd s l ( 2 ) and JTL N . While our ultimate goal is to use this bimodule to deduce properties of the LCFT in the continuum limit, its derivation is sufficiently involved to be the sole subject of this paper. We describe representation theory of the centralizer and then use it to find a decomposition of the periodic g l ( 1 | 1 ) spin chain over JTL N for any even N and ultimately a corresponding bimodule structure. Applications of our results to the analysis of the bulk LCFT will then be discussed in the third part of this series.