16 research outputs found
Bimodule structure in the periodic gl(1|1) spin chain
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 nearest neighbor Heisenberg-like
couplings - provide a representation of the Jones Temperley Lieb (JTL) algebra.
The symmetry algebra is then the centralizer of JTL, and turns out to be
smaller than for the open chain, since it is now only a subalgebra of U_q sl(2)
at q=i, dubbed U_q^{odd} sl(2). 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} sl(2) and JTL. 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 gl(1|1) spin chain over JTL for
any even number N of tensorands 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.Comment: latex, 42 pp., 13 figures + 5 figures in color, many comments adde
Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models
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 the fusion of irreducible and
logarithmic modules in the (1,p) Virasoro logarithmic minimal models.Comment: 19 page
Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the gl(1|1) periodic spin chain, Howe duality and the interchiral algebra
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, "Continuum limit
and symmetries of the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013)
245-288] and "Bimodule structure in the periodic gl(1|1) spin chain" [Nucl.
Phys. B 871 (2013) 289-329]. Our main result is that the algebra of local
Hamiltonians, the Jones-Temperley-Lieb algebra JTL_N, goes over in the
continuum limit to a bigger algebra than the product of the left and right
Virasoro algebras. This algebra, S - which we call interchiral, mixes the left
and right moving sectors, and is generated, in the symplectic fermions case, by
the additional field , with
a symmetric form and conformal weights (1,1). We discuss in details
how the Hilbert space of the LCFT decomposes onto representations of this
algebra, and how this decomposition is related with properties of the finite
spin-chain. We show that there is a complete correspondence between algebraic
properties of finite periodic spin chains and the continuum limit. An important
technical aspect of our analysis involves the fundamental new observation that
the action of JTL_N in the spin chain is in fact isomorphic to an
enveloping algebra of a certain Lie algebra, itself a non semi-simple version
of . The semi-simple part of JTL_N is represented by ,
providing a beautiful example of a classical Howe duality, for which we have a
non semi-simple version in the full JTL image represented in the spin-chain. On
the continuum side, simple modules over the interchiral algebra S are
identified with "fundamental" representations of .Comment: 69 pp., 10 figs, v2: the paper has been substantially modified - new
proofs, new refs, new App C with inductive limits construction, et
Logarithmic extensions of minimal models: characters and modular transformations
We study logarithmic conformal field models that extend the (p,q) Virasoro
minimal models. For coprime positive integers and , the model is defined
as the kernel of the two minimal-model screening operators. We identify the
field content, construct the W-algebra W(p,q) that is the model symmetry (the
maximal local algebra in the kernel), describe its irreducible modules, and
find their characters. We then derive the SL(2,Z) representation on the space
of torus amplitudes and study its properties. From the action of the
screenings, we also identify the quantum group that is Kazhdan--Lusztig-dual to
the logarithmic model.Comment: 43pp., AMSLaTeX++. V3: Some explanatory comments added, notational
inaccuracies corrected, references adde
Lattice fusion rules and logarithmic operator product expansions
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 [Nucl. Phys. B 777, 316 (2007)]. Using
quantum group results, we provide rigorous calculations of the fusion of
various TL modules. 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.Comment: 54pp, many comments adde
Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models
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 are exhausted by projective
modules and four series of modules that have a functorial correspondence with
any quotient or a submodule of Feigin-Fuchs modules over the Virasoro algebra.
Our main result includes calculation of tensor products of any pair of the
indecomposable modules. Based on the Kazhdan-Lusztig equivalence between
quantum groups and vertex-operator algebras, fusion rules of Kac modules over
the Virasoro algebra in the (1,p) LCFT models are conjectured.Comment: 40pp. V2: a new introduction, corrected typos, some explanatory
comments added, references adde
The non-semisimple Verlinde formula and pseudo-trace functions
International audienceUsing 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 on pseudo-trace functions by Miyamoto and Arike–Nagatomo, one can make a precise conjecture for a non-semisimple modular Verlinde formula which relates modular properties of pseudo-trace functions for V and the product in the Grothendieck ring of RepV . We test this conjecture in the example of the vertex operator algebra of N pairs of symplectic fermions by explicitly computing the modular S -transformation of the pseudo-trace functions
The symplectic fermion ribbon quasi-Hopf algebra and the -action on its centre
International audienceWe 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 by the third author from conformal blocks of the corresponding logarithmic conformal field theory. The latter category in turn is conjecturally ribbon equivalent to representations of Vev, the even part of the symplectic fermion vertex operator super algebra.Using the formalism developed in our previous paper we compute the projective SL(2,ℤ)-action on the centre of Q(N) as obtained from Lyubashenko's general theory of mapping class group actions for factorisable finite ribbon categories. This allows us to test a conjectural non-semisimple version of the modular Verlinde formula: we verify that the SL(2,ℤ)-action computed from Q(N) agrees projectively with that on pseudo trace functions of V_{ev}
Topological defects in periodic RSOS models and anyonic chains
We provide a lattice regularization of all topological defects in minimal models CFTs using RSOS and anyonic spin chains. For defects of type , 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 [1811.02551]. We show that the topological nature of the defects is exact on the lattice as well. Our defects of type , in contrast, are only topological in the continuum limit. Identifications are obtained by a mix of algebraic and Bethe-ansatz techniques. Most of our discussion is framed in a Hamiltonian (or transfer matrix) formalism, and direct and crossed channel are both discussed in detail. For defects of type , we also show how to implement their fusion, which turns out to reproduce the tensor product of the underlying monoidal category used to build the anyonic chain