Open-closed String Theory Research Articles

ON NONTRIVIAL SOLVABILITY OF ONE CLASS OF NONLINEAR INTEGRAL EQUATIONS WITH CONSERVATIVE KERNEL ON THE POSITIVE SEMI-AXIS

The work is devoted to a special class of nonlinear integral equations on the positive semi-axis with conservative kernel that corresponds to a nonlinear operator, for which the property of complete continuity in the space of bounded functions fails. In different special cases this class of equations has applications in particular branches of mathematical physics. In particular, this kind of equations can be met in the radiative transfer theory, kinetic theory of gases, kinetic theory of plasma and in the $p$-adic open-closed string theory. Using a combination of special iterations with the monotonic operator theory methods, that work in defined conical segments it is possible to prove a constructive existence theorem of nonnegative nontrivial bounded solution that has finite limit at infinity. The asymptotics of the constructed solution will also be studied. It is also given an example of nonlinear equation, for which the uniqueness of the solution in the space of bounded functions fails. At the end of the paper will consider some classes of equations both applied and pure theoretical character.

Infinite-dimensional topological field theories from Hurwitz numbers

Classical Hurwitz numbers of a fixed degree together with Hurwitz numbers of seamed surfaces give rise to a Klein topological field theory (see [A. Alexeevski and S. Natanzon, The algebra of bipartite graphs and Hurwitz numbers of seamed surfaces, Izv. Math. 72(4) (2008) 627–646]). We extend this construction to Hurwitz numbers of all degrees simultaneously. The corresponding infinite-dimensional Cardy–Frobenius algebra is computed in terms of Young diagrams and bipartite graphs. This algebra turns out to be isomorphic to the algebra of differential operators introduced in [A. Mironov, A. Morozov and S. Natanzon, Cardy–Frobenius extension of algebra of cut-and-join operators, J. Geom. Phys. 73 (2012) 243–251, arXiv:1210.6955; A Hurwitz theory avatar of open-closed string, Eur. Phys. J. C 73(2) (2013) 1–10, arXiv:1208.5057], which serves a model for open-closed string theory. We prove that the operators corresponding to Young diagrams and bipartite graphs give rise to relations between Hurwitz numbers.

On some exact solutions in p-adic open-closed string theory

The paper is concerned with the construction of exact solutions for nonlinear pseudodifferential equations which describe tachyon dynamics of open-closed p-adic strings. Existence of continuous solutions and their properties are discussed.

Open-Closed Moduli Spaces and Related Algebraic Structures

We set up a Batalin–Vilkovisky Quantum Master Equation (QME) for open-closed string theory and show that the corresponding moduli spaces give rise to a solution, a generating function for their fundamental chains. The equation encodes the topological structure of the compactification of the moduli space of bordered Riemann surfaces. The moduli spaces of bordered J-holomorphic curves are expected to satisfy the same equation, and from this viewpoint, our paper treats the case of the target space equal to a point. We also introduce the notion of a symmetric Open-Closed Topological Conformal Field Theory (OC TCFT) and study the L ∞ and A ∞ algebraic structures associated to it.

Toward an open-closed string theoretical description of a rolling tachyon

We consider how the time-dependent decay process of an unstable D-brane should be described in the full (quantum) open-closed string theory. It is argued that the system, starting from the unstable D-brane configuration, will evolve in time into the time-independent open string tachyon vacuum configuration which we assume to be finite, with the total energy conserved. As a concrete realization of this idea, we construct a toy model describing the open and closed string tachyons which admits such a time-dependent solution. The structure of our model has some resemblance to that of open-closed string field theory.

Tachyon condensation in open-closed p-adic string theory

We study a simple model of p-adic closed and open strings. It sheds some light on the dynamics of tachyon condensation for both types of strings. We calculate the effect of static and decaying D-brane configurations on the closed string background. For closed string tachyons we find lumps analogous to D-branes. By studying their fluctuation spectrum and the D-branes they admit, we argue that closed string lumps should be interpreted as spacetimes of lower dimensionality described by some noncritical p-adic string theory.

Oriented Open-Closed String Theory Revisited

String theory on D-brane backgrounds is open-closed string theory. Given the relevance of this fact, we give details and elaborate upon our earlier construction of oriented open-closed string field theory. In order to incorporate explicitly closed strings, the classical sector of this theory is open strings with a homotopy associativeA∞algebraic structure. We build a suitable Batalin–Vilkovisky algebra on moduli spaces of bordered Riemann surfaces, the construction of which involves a few subtleties arising from the open string punctures and cyclicity conditions. All vertices coupling open and closed strings through disks are described explicitly. Subalgebras of the algebra of surfaces with boundaries are used to discuss symmetries of classical open string theory induced by the closed string sector, and to write classical open string field theory on general closed string backgrounds. We give a preliminary analysis of the ghost-dilaton theorem.

On integrable c < 1 open-closed string theory

The integrable structure of open-closed string theories in the ( p, q) conformal minimal model backgrounds is presented. The relation between the τ-function of the closed string theory and that of the open-closed string theory is uncovered. The resulting description of the open-closed string theory is shown to fit very naturally into the framework of the sl(q, C) KdV hierarchies. In particular, the twisted bosons which underlie and organise the structure of the closed string theory play a similar role here and may be employed to derive loop equations and correlation function recursion relations for the open-closed strings in a simple way.

FLOWS FOR RECTANGULAR MATRIX MODELS

Several new results on the multicritical behavior of rectangular matrix models are presented. We calculate the free energy in the saddle point approximation, and show that at the triple-scaling point, the result is the same as that derived from the recursion formulas. In the triple-scaling limit, we obtain the string equation and a flow equation for arbitrary multicritical points. Parametric solutions are also examined for the limit of almost-square matrix models. This limit is shown to provide an explicit matrix model realization of the scaling equations proposed to describe open-closed string theory.

UNITARY MATRIX MODELS AND 2D QUANTUM GRAVITY

The KdV and modified KdV integrable hierarchies are shown to be different descriptions of the same 2D gravitational system — open-closed string theory. Non-perturbative solutions of the multicritical unitary matrix models map to non-singular solutions of the 'renormalization group' equation for the string susceptibility, [Formula: see text]. We also demonstrate that the large-N solutions of unitary matrix integrals in external fields, studied by Gross and Newman, equal the non-singular pure closed-string solutions of [Formula: see text].

Schwinger-Dyson equations of matrix models for open and closed strings

We study the Schwinger-Dyson equations of a matrix model for an open-closed string theory. The free energy with source terms for closed string scaling operators satisfies the same Virasoro conditions as those of the pure closed string and is obtained from that of the pure closed string by giving appropriate nonvanishing background values to all of the sources.

STRING THEORY EFFECTIVE ACTION: STRING LOOP CORRECTIONS

We discuss the general ideology of the computation of string loop corrections to the effective action for the massless modes of the string. Both the S-matrix and the sigma-model approaches are presented. It is emphasized that the effective action is a more general and better defined object than the S-matrix. In particular, it is finite in spite of modular infinities that may be present in loop amplitudes computed near a “wrong” vacuum. The case of the disc topology in the open-closed string theory is treated in some detail. Some issues concerning the soft dilaton vertex operators related to the infinities of the string amplitudes are disucssed.