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We consider 1D quantum scattering problem for a Hamiltonian with symmetries. We show that the proper treatment of symmetries in the spirit of homological algebra leads to new objects, generalizing the well known T- and K-matrices. Homological treatment implies that old objects and new ones are be combined in a differential. This differential arises from homotopy transfer of induced interaction and symmetries on solutions of free equations of motion. Therefore, old and new objects satisfy remarkable quadratic equations. We construct an explicit example in SUSY QM on $S^1$ demonstrating nontriviality of the above relation.

We study the field theory localizing to holomorphic maps from a complex manifold of complex dimension 2 to a toric target (a generalization of A model). Fields are realized as maps to $(\mathbb{C}^*)^N$ where one includes special observables supported on (1,1)-dimensional submanifolds to produce maps to the toric compactification. We study the mirror of this model. It turns out to be a free theory interacting with $N_\mathrm{comp}$ topological strings of type A. Here $N_\mathrm{comp}$ is the number of compactifying divisors of the toric target. Before the mirror transformation these strings are vortex (actually, holomortex) strings.

We demonstrate the effect of interference, similar to the Hong-Ou-Mandel effect, for two single-photon pulses by writing them sequentially inside a quantum memory cell in a tripod configuration., Comment: 4 pages, 1 figure

We study two-dimensional non-abelian BF theory in Lorenz gauge and prove that it is a topological conformal field theory. This opens the possibility to compute topological string amplitudes (Gromov-Witten invariants). We found that the theory is exactly solvable in the sense that all correlators are given by finite-dimensional convergent integrals. Surprisingly, this theory turns out to be logarithmic in the sense that there are correlators given by polylogarithms and powers of logarithms. Furthermore, we found fields with "logarithmic conformal dimension" (elements of a Jordan cell for $L_0$). We also found certain vertex operators with anomalous dimensions that depend on the non-abelian coupling constant. The shift of dimension of composite fields may be understood as arising from the dependence of subtracted singular terms on local coordinates. This generalizes the well-known explanation of anomalous dimensions of vertex operators in the free scalar field theory., Comment: Ver. 2: Section 6.3 ("A remark on logarithmic phenomena") expanded. An expository tweak: switched to the index-free notations throughout the text

In this paper we discuss the possibility of building a linear cluster state based on modes with a certain orbital angular momentum (OAM). We show that in the system under consideration a field with a rich mode structure is generated in the cavity. We also analyze the conditions under which an infinite system of Heisenberg-Langevin equations describing the dynamics of intracavity fields can be shortened and solved analytically. To analyze the genuine number of quantum degrees of freedom, we use the supermodes technique. This approach allows us to build the most entangled cluster state., Comment: 11 pages, 7 figures, submitted to Phys. Rev. A

We study the two-dimensional topological abelian BF theory in the Lorenz gauge and, surprisingly, we find that the gauged-fixed theory is a free type B twisted N=(2,2) superconformal theory with odd linear target space, with the ghost field c being the pullback of the linear holomorphic coordinate on the target. The BRST operator of the gauge-fixed theory equals the total Q of type B twisted theory. This unexpected identification of two different theories opens a way for nontrivial deformations of both of these theories., Comment: Ver. 2: a discussion of the relation of descent to AKSZ construction added. Ver. 3: section 2.5.2 (a toy example of Gromov-Witten invariant) corrected; section 2.5.1 expanded with comments on the BV generating function for the action of a vector field

In this article, we consider the possibility of manipulation of quantum signals, ensured by the use of the tripod-type atomic memory cell. We show that depending on a configuration of driving fields at the writing and reading, such a cell allows both to store and transform the signal. It is possible to provide the operation of the memory cell in a Mach-Zehnder interferometer mode passing two successive pulses at the input. We proposed a procedure of partial signal read out that provides entanglement between the retrieved light and the atomic ensemble. Thus, we have shown that tripod atomic cell is a promising candidate to implement of quantum logical operations, including two-qubit ones, that can be performed on the basis of only one cell., Comment: 11 pages, 1 figure

We have considered theoretically the feasibility of the broadband quantum memory based on the resonant tripod-type atomic configuration. In this case, the writing of a signal field is carried out simultaneously into two channels, and characterized by an excitation of two spin waves of the atomic ensemble. With simultaneous read out from both channels quantum properties of the original signal are mapped on the retrieval pulse no worse than in the case of memory based on Lambda-type atomic configuration. At the same time new possibilities are opened up for manipulation of quantum states associated with sequential reading out (and/or sequential writing) of signal pulses. For example, the pulse in squeezed state is converted into two partially entangled pulses with partially squeezed quadratures. Alternatively, two independent signal pulses with orthogonal squeezed quadratures can be converted into two entangled pulses., Comment: 14 pages, 4 figures

We consider first-order bosonic string theory, perturbed by the primary operator, corresponding to deformation of the target-space complex structure. We compute the effective action in this theory and find that its consistency with the world-sheet conformal invariance requires necessarily the Kodaira-Spencer equations to be satisfied by target-space Beltrami differentials. We discuss the symmetries of the theory and its reformulation in terms of the vielbein background fields., Comment: 13 pages, 2 figures

We consider the first order formalism in string theory, providing a new off-shell description of the nontrivial backgrounds around an "infinite metric". The OPE of the vertex operators, corresponding to the background fields in some "twistor representation", and conditions of conformal invariance results in the quadratic equation for the background fields, which appears to be equivalent to the Einstein equations with a Kalb-Ramond B-field and a dilaton. Using a new representation for the Einstein equations with B-field and dilaton we find a new class of solutions including the plane waves for metric (graviton) and the B-field. We discuss the properties of these background equations and main features of the BRST operator in this approach., Comment: LaTeX2e, 18 pages, Phys. Lett. B, in press, corrected typos

We present new evidence for the conjecture that BPS correlation functions in the N=2 supersymmetric gauge theories are described by an auxiliary two dimensional conformal field theory. We study deformations of the N=2 supersymmetric gauge theory by all gauge-invariant chiral operators. We calculate the partition function of the N=2 theory on R^4 with appropriately twisted boundary conditions. For the U(1) theory with instantons (either noncommutative, or D-instantons, depending on the construction) the partition function has a representation in terms of the theory of free fermions on a sphere, and coincides with the tau-function of the Toda lattice hierarchy. Using this result we prove to all orders in string loop expansion that the effective prepotential (for U(1) with all chiral couplings included) is given by the free energy of the topological string on CP^1. Gravitational descendants play an important role in the gauge fields/string correspondence. The dual string is identified with the little string bound to the fivebrane wrapped on the two-sphere. We also discuss the theory with fundamental matter hypermultiplets., Comment: 39 pages, TeX; substantial revision of the exposition, misprints corrected, refs added