A. M. Semikhatov

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.

Parent Field Theory and Unfolding in BRST First-Quantized Terms

For free-field theories associated with BRST first-quantized gauge systems, we identify generalized auxiliary fields and pure gauge variables already at the first-quantized level as the fields associated with algebraically contractible pairs for the BRST operator. Locality of the field theory is taken into account by separating the space–time degrees of freedom from the internal ones. A standard extension of the first-quantized system, originally developed to study quantization on curved manifolds, is used here for the construction of a first-order parent field theory that has a remarkable property: by elimination of generalized auxiliary fields, it can be reduced both to the field theory corresponding to the original system and to its unfolded formulation. As an application, we consider the free higher-spin gauge theories of Fronsdal.

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.

Triplectic quantization: A geometrically covariant description of the Sp(2)-symmetric Lagrangian formalism

A geometric description is given for the Sp(2) covariant version of the field-antifield quantization of general constrained systems in the Lagrangian formalism. We develop differential geometry on manifolds in which a basic set of coordinates (“fields”) have two superpartners (“antifields”). The quantization on such a triplectic manifold requires introducing several specific differential-geometric objects, whose properties we study. These objects are then used to impose a set of generalized master equations that ensure gauge-independence of the path integral. The theory thus quantized is shown to extend to a level-1 theory formulated on a manifold that includes antifields to the Lagrange multipliers. We also observe intriguing relations between triplectic and ordinary symplectic geometry.

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.

The sℓ̂(2)⊕sℓ̂(2)/sℓ̂(2) coset theory as a Hamiltonian reduction of D̂(2|1;α)

We show that the coset ^sl(2)+^sl(2)/^sl(2) is a quantum Hamiltonian reduction of the exceptional affine Lie superalgebra ^D(2|1;\alpha) and that the corresponding W algebra is the commutant of the U_{q}D(2|1;\alpha) quantum group.