Narrow Results

We derive a Bouchard–Eynard type topological recursion for the total descendant potential of $A_N$-singularity. Our argument relies on a certain twisted representation of a Heisenberg Vertex Operator Algebra (VOA) constructed via the periods of $A_N$-singularity. In particular, our approach allows us to prove that the topological recursion for the total descendant potential is equivalent to a certain generating set of ${𝒲}$-algebra constraints.

We first review the spontaneous Lorentz symmetry breaking in the presence of massless gauge fields and infraparticles. This result was obtained long time ago in the context of rigorous quantum field theory (QFT) by Fröhlich, Morchio and Strocchi [Ann. Phys.119, 241 (1979); Phys. Lett. B89, 61 (1979)] and reformulated by Balachandran and Vaidya (arXiv:1302.3406) using the notion of superselection sectors and direction-dependent test functions at spatial infinity for gauge transformations. Inspired by these developments and under the assumption that the spectrum of the electric charge is quantized (in units of a fundamental charge e), we construct a family of vertex operators which create winding number k, electrically charged Abelian vortices from the vacuum (zero winding number sector) and/or shift the winding number by k units. Vortices created by this vertex operator may be viewed both as a source and as a probe for inducing and detecting the breaking of spontaneous Lorentz symmetry.

We find that for rotating vortices, the vertex operator at level k shifts the angular momentum of the vortex by , where is the electric charge of the quantum state of the vortex and q is the charge of the vortex scalar field under the U(1) gauge field. We also show that, for charged-particle-vortex composites, angular momentum eigenvalues shift by being the electric charge of the charged-particle-vortex composite. This leads to the result that for half-odd integral and for odd k, our vertex operators flip the statistics of charged-particle-vortex composites from bosons to fermions and vice versa. For fractional values of , application of vertex operator on charged-particle-vortex composite leads in general to composites with anyonic statistics.

The Adler–Shiota–van Moerbeke formula is employed to derive the W-constraints for the p-reduced BKP hierarchy constrained by the string equation. We also provide the Grassmannian description of the string equation in terms of the spectral parameter.

We calculate the interaction 3-vertex of two massless spin-3 particles with a graviton using vertex operators for spin-3 fields in open string theory, constructed in our previous work. The massless spin-3 fields are shown to interact with the graviton through the linearized Weyl tensor, reproducing the result by Boulanger, Leclercq and Sundell. This is consistent with the general structure of the non-Abelian 2-s-s couplings, implying that the minimal number of space–time derivatives in the interaction vertices of two spin-s and one spin-2 particle is equal to 2s-2.

The purpose of this paper is to present a pedagogical review of T-duality in string theory. The evolution of the closed string is envisaged on the worldsheet in the presence of its massless excitations. The duality symmetry is studied when some of the spacial coordinates are compactified on d-dimensional torus, T^d. The known results are reviewed to elucidate that equations of motion for the compact coordinates are O(d, d) covariant, d being the number of compact directions. Next, the vertex operators of excited massive levels are considered in a simple compactification scheme. It is shown that the vertex operators for each massive level can be cast in a T-duality invariant form in such a case. Subsequently, the duality properties of superstring is investigated in the NSR formulation for the massless backgrounds such as graviton and antisymmetric tensor. The worldsheet superfield formulation is found to be very suitable for our purpose. The Hassan–Sen compactification is adopted and it is shown that the worldsheet equations of motion for compact superfields are O(d, d) covariant when the backgrounds are independent of superfields along compact directions. The vertex operators for excited levels are presented in the NS–NS sector and it is shown that they can be cast in T-duality invariant form for the case of Hassan–Sen compactification scheme. An illustrative example is presented to realize our proposal.

We use vertex operator algebras and intertwining operators to study certain substructures of standard -modules, allowing us to conceptually obtain the classical Rogers–Ramanujan recursion. As a consequence we recover Feigin–Stoyanovsky's character formulas for the principal subspaces of the level 1 standard -modules.

A construction of the Virasoro algebra in terms of free massless two-dimensional boson fields is studied. The ansatz for the Virasoro field contains the most general unitary scaling dimension 2 expression built from vertex operators. The ansatz leads in a natural way to a concept of a quasi root systems. This is a new notion generalizing the notion of a root system in the theory of Lie algebras. We introduce a definition of a quasi root systems and provide an extensive list of examples. Explicit solutions of the ansatz are presented for a range of quasi root systems.

In this paper, vertex representations of the 2-toroidal Lie superalgebras of type $D(m,n)$ are constructed using both bosonic fields and vertex operators based on their loop algebraic presentation.

A novel approach — based upon vertex operator representation — is devised to study the AKNS hierarchy. It is shown that this method reveals the remarkable properties of the AKNS hierarchy in relatively simple, rather natural and particularly effective ways. In addition, the connection of this vertex operator based approach with Lie-algebraic integrability schemes is analyzed and its relationship with τ-function representations is briefly discussed.

We summarize the recent development of realizing arbitrary Jack symmetric functions by a sequence of vertex operators associated with rectangular Young tableaux.