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We study the fusion kernel for nondegenerate conformal blocks in the Liouville theory as a solution of difference equations originating from the pentagon identity. We propose an approach for solving these equations based on a “nonperturbative” series expansion that allows calculating the fusion kernel iteratively. We also find exact solutions for the special central charge values c = 1+6(b − b−1)2, b ∈ ℕ. For c = 1, the obtained result reproduces the formula previously obtained from analytic properties of a solution of a Painlev´e equation, but our solution has a significantly simplified form.

We classify deformations of the standard embedding of the Lie superalgebra \( \mathcal{K} \)(2) of contact vector fields on the (1, 2)-dimensional supercircle into the Lie superalgebra SΨD(S1|2) of pseudodifferential operators on the supercircle S1|2. The proposed approach leads to the deformations of the central charge induced on \( \mathcal{K} \)(2) by the canonical central extension of SΨD(S1|2).

We show that the Belavin-Polyakov-Zamolodchikov equation of the minimal model of conformal field theory with the central charge c = 1 for the Virasoro algebra is contained in a system of linear equations that generates the Schlesinger system with 2×2 tmatrices. This generalizes Suleimanov’s result on the Painleve equations. We consider the properties of the solutions, which are expressible in terms of the Riemann theta function.

We present explicit recursion relations for the four-point superconformal block functions that are essentially particular contributions of the given conformal class to the four-point correlation function. The approach is based on the analytic properties of the superconformal blocks as functions of the conformal dimensions and the central charge of the superconformal algebra. We compare the results with the explicit analytic expressions obtained for special parameter values corresponding to the truncated operator product expansion. These recursion relations are an efficient tool for numerically studying the four-point correlation function in superconformal field theory in the framework of the bootstrap approach, similar to that in the case of the purely conformal symmetry.

We consider the partition function of the two-dimensional free massive Majorana fermion coupled to the quantized metric of the spherical topology. By adding an arbitrary conformal “spectator” matter, we gain control over the total matter central charge. This provides an interesting continuously parameterized family of critical points and also allows making a connection with the semiclassical limit. We use the Liouville field theory as the effective description of the quantized gravity. The spherical scaling function is calculated approximately, but (we believe) to a good numerical precision, in almost the whole domain of the spectator parameter. An impressive comparison with the predictions of the exactly solvable matrix model yields a more general model of random-lattice statistics, which is most probably not solvable by the matrix-model technique but reveals a more general pattern of critical behavior. We hope that numerical simulations or series extrapolation will be able to reveal our family of scaling functions.

We compute bulk 3- and 4-point tachyon correlators in the 2d Liouville gravity with non-rational matter central charge c, 14 pages, 2 figures

We study various Sp(2M)-invariant field equations corresponding to rank-r tensor products of the Fock (singleton) representation of Sp(2M). These equations describe localization on “branes” of different dimensions embedded in the generalized space-time \(\mathcal{M}_M\) with matrix (i.e., “central charge”) coordinates. We consider the case of a bilinear tensor product in detail and show that the conserved currents built from bilinears of rank-1 fields in \(\mathcal{M}_M\) satisfy the field equations for rank-2 fields in \(\mathcal{M}_M\). We also show that rank-2 fields in \(\mathcal{M}_M\) are equivalent to rank-1 fields in \(\mathcal{M}_{2M}\).

Generalized Friedman-type cosmological solutions (spatially homogeneous and isotropic) are obtained and classified in a D-dimensional dilaton gravitational model with the central charge deficit A.

We consider a fermionic representation for the integrals of motion of the sine-Gordon equations. We calculate the algebra of noncommuting integrals of motion and discuss the correlation of the integrals of motion to representations of the su(2) and the Virasoro algebras. We also construct the conserved currents.

A twistor-like extension of the N=2 conformal superspace (SSp) with nontrivial central charges is presented. Parameters of new (anti)commutative generators carrying an index of the spin SU (2)-symmetry are used to construct massless harmonics, dramatically reducing the number of essential coordinates. Under this approach, the usual vector coordinates become complex, while the Grassmann coordinates are reduced to a single variable which is scalar under the Lorentz transformation. The choice of the appropriate analytical subspace in the initial SSp is rather difficult, but our closed algebra gives the possibility of covariantly reducing the theory from the initial SSp to an SSp of a smaller Grassmann dimension. A special definition of the coordinates and the flat space for this structure allows one to obtain the Penrose twistor equation and the Ogievetsky harmonics.

A polynomial generalization of supersymmetry in quantum mechanics in one and two dimensions is proposed. Polynomial superalgebras in one dimension are classified. In two dimensions, a detailed analysis is made for supercharges of second order with respect to derivatives and it is shown that in all cases the binomial superalgebra leads to hidden dynamical symmetry generated by the central charge.

We examine the problem of constructing N=2 superconformal algebras out of N=1 non-semi-simple affine Lie algebras. These N=2 superconformal theories share the property that the super Virasoro central charge depends only on the dimension of the Lie algebra. We find, in particular, a construction having a central charge c=9. This provides a possible internal space for string compactification and where mirror symmetry might be explored.

The BRST invariance condition in a highest-weight representation of the topological (≡ twisted N=2) algebra captures the ‘invariant’ content of two-dimensional gravity coupled to matter. The topological algebra allows reductions to either the DDK-dressed matter or the ‘Kontsevich-Miwa’-dressed matter related to Virasoro-constrained KP hierarchy. The standard DDK formulation is recovered by splitting the topological generators into c=−26 reparametrization ghosts + matter + ‘Liouville,’ while a similar splitting involving c=−2 ghosts gives rise to the matter dressed in exactly the way required in order that the theory be equivalent to Virasoro constraints on the KP hierarchy. The two dressings of matter with the ‘Liouville’ differ also by their ‘ghost numbers,’ which is similar to the existence of representatives of BRST cohomologies with different ghost numbers. The topological central charge c≠3 provides a two-fold covering of the allowed regiond≤1 ∪d≥25 of the matter central charge d via d=(c+1)(c+6)/(c−3). The ‘Liouville’ field is identified as the ghost-free part of the topological U(1) current. The construction thus allows one to establish a direct relation (presumably an equivalence) between the Virasoro-constrained KP hierarchies, minimal models, and the BRST invariance condition for highest-weight states of the topological algebra.

In the paper, SO(N)- and U(N)-invariant algebras of extended superconformal symmetry in two dimensions with arbitrary central charge are constructed. Besides the energy-momentum tensor, they contain only currents of dimensions 1 and 3/2.