I. Yu. Tipunin

Logarithmic extensions of minimal models: characters and modular transformations.

Abstract 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 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.

Nonsemisimple Fusion Algebras and the Verlinde Formula

Abstract:We find a nonsemisimple fusion algebra associated with each (1, p) Virasoro model. We present a nonsemisimple generalization of the Verlinde formula which allows us to derive from modular transformations of characters.

Kazhdan-Lusztig-dual quantum group for logarithimic extensions of Virasoro minimal models.

We derive and study a quantum group gp,q that is Kazhdan-Lusztig dual to the W-algebra Wp,q of the logarithmic (p,q) conformal field theory model. The algebra Wp,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 property that the quotient Wp,q∕R is the vertex-operator algebra of the (p,q) Virasoro minimal model. The number (2pq) of irreducible gp,q representations is the same as the number of irreducible Wp,q representations on which R acts nontrivially. We find the center of gp,q and show that the modular group representation on it is equivalent to the modular group representation on the Wp,q characters and “pseudocharacters.” The factorization of the gp,q ribbon element leads to a factorization of the modular group representation on the center. We also find the gp,q Grothendieck ring, which is presumably the “logarithmic” fusion of the (p,q) model.

Logarithmic conformal field theories via logarithmic deformations

We construct logarithmic conformal field theories starting from an ordinary conformal field theory -- with a chiral algebra C and the corresponding space of states V -- via a two-step construction: ...

Equivalence between chain categories of representations of affine sl(2) and N=2 superconformal algebras

Highest-weight-type representation theories of the affine sl(2) and N=2 superconformal algebras are shown to be equivalent modulo the respective spectral flows.

Radford, Drinfeld and Cardy boundary states in the (1, p) logarithmic conformal field models

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 Verlinde formula that gives these structure constants. In the context of theories with boundaries, we identify the space of vacuum torus amplitudes with the space of Ishibashi states. Then, we propose (3p - 1) boundary states satisfying the Cardy condition.

Resolutions and characters of irreducible representations of the N=2 superconformal algebra

Abstract We evaluate characters of irreducible representations of the N = 2 supersymmetric extension of the Virasoro algebra. We do so by deriving the BGG resolution of the admissible N = 2 representations and also a new “3, 5, 7, …” resolution in terms of twisted massive Verma modules. We analyse how the characters behave under the automorphisms of the algebra, whose most significant part is the spectral flow transformations. The possibility to express the characters in terms of theta functions is determined by their behaviour under the spectral flow. We also derive the identity expressing every sl (2) character as a linear combination of spectral-flow transformed N = 2 characters; this identity involves a finite number of N = 2 characters in the case of unitary representations. Conversely, we find an integral representation for the admissible N = 2 characters as contour integrals of admissible sl (2) characters.

Higher-Level Appell Functions, Modular Transformations, and Characters

We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The level-ℓ Appell functions satisfy open quasiperiodicity relations with additive theta-function terms emerging in translating by the “period.” Generalizing the well-known interpretation of theta functions as sections of line bundles, the function enters the construction of a section of a rank-(ℓ+1) bundle . We evaluate modular transformations of the functions and construct the action of an SL(2,ℤ) subgroup that leaves the section of constructed from invariant.Modular transformation properties of are applied to the affine Lie superalgebra at a rational level k>−1 and to the N=2 super-Virasoro algebra, to derive modular transformations of “admissible” characters, which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. This gives an example where constructing a modular group action involves extensions among representations in a nonrational conformal model.

Lusztig limit of quantum sl(2) at root of unity and fusion of (1, p) Virasoro logarithmic minimal models

Abstract 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 p th 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.

Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models

Abstract 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 s l ( 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 finitely-generated 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.