Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model

Even though matrix model partition functions do not exhaust the entire set of tau-functions relevant for string theory, they seem to be elementary building blocks for many others and they seem to properly capture the fundamental symplicial nature of quantum gravity and string theory. We propose to consider matrix model partition functions as new special functions. This means they should be investigated and put into some standard form, with no reference to particular applications. At the same time, the tables and lists of properties should be full enough to avoid discoveries of unexpected peculiarities in new applications. This is a big job, and the present paper is just a step in this direction. Here we restrict our consideration to the finite-size Hermitean 1-matrix model and concentrate mostly on its phase/branch structure arising when the partition function is considered as a D-module. We discuss the role of the CIV-DV prepotential (as generating a possible basis in the linear space of solutions to the Virasoro constraints, but with a lack of understanding of why and how this basis is distinguished) and evaluate first few multiloop correlators, which generalize semicircular distribution to the case of multitrace and non-planar correlators.Comment: 64 pages, LaTe

Analytical Solutions of Open String Field Theory

In this work we review Schnabl's construction of the tachyon vacuum solution to bosonic covariant open string field theory and the results that followed. We survey the state of the art of string field theory research preceding this construction focusing on Sen's conjectures and the results obtained using level truncation methods. The tachyon vacuum solution can be described in various ways. We describe its geometric representation using wedge states, its formal algebraic representation as a pure-gauge solution and its oscillator representation. We also describe the analytical proofs of some of Sen's conjectures for this solution. The tools used in the context of the vacuum solution can be adapted to the construction of other solutions, namely various marginal deformations. We present some of the approaches used in the construction of these solutions. The generalization of these ideas to open superstring field theory is explained in detail. We start from the exposition of the problems one faces in the construction of superstring field theory. We then present the cubic and the non-polynomial versions of superstring field theory and discuss a proposal suggesting their classical equivalence. Finally, the bosonic solutions are generalized to this case. In particular, we focus on the (somewhat surprising) generalization of the tachyon solution to the case of a theory with no tachyons.Comment: Invited review for Physics Reports. v1: 106 p., 8 fig. v2: 108 p., minor changes. v3: 117 p., 9 fig., presentation modified and expanded in several places, typos corrected, ref. added and updated. v4: Published version. 125 p., 10 fig., further modifications of the presentation, ref. added and update

Evaluation of Correlation Functions in Integrable Quantum Field Theories

The aim of this thesis is to explore correlation functions in two dimensional quantum field theories in two distinct ways. In part I a new method for calculating the differential equations parametrising the correlation functions of twist fields associated with the U (1) symmetry of the Dirac model is presented. While developing this method a new family of descendent twist fields are identified and their form factors calculated. This provides a novel way of calculating the vacuum expectation values of the primary twist fields and is shown to be entirely consistent with known results. The method of calculating the correlation functions of twist fields provides a parametrisation of several other correlation functions for various quantum states. Since this method relies on the Ward identities found in a double copy model it is hoped to have wider applications in other free fermion models. Part II concerns the truncated conformal space approach which has been developed to approximate perturbed conformal field theories. In this part the theory underpinning the approach is discussed and a working algorithm is developed for both bulk and boundary perturbed minimal models. The energy levels, mass gaps and one point functions of various models are computed using the truncated conformal space approach and are shown to be in good agreement with previous calculations. A possible method for using this approach to approximate two point functions in perturbed conformal field theories is discussed

Alternative Symmetries in Quantum Field Theory and Gravity

A general, incomplete and partisan overview of various areas of the theoretical investigation is presented. Most of this activity stems from the search for physics beyond quantum field theory and general relativity, a titanic struggle that, in my opinion, empowered the symmetry principles to a dangerous level of speculation. In the works (that are my own) commented upon here the attempt has been to proceed by holding to certain epistemological pillars (usually absent from the too speculative theories) such as, e.g., four or less dimensions, proposals for experimental tests of radical ideas, wide cross-fertilization, etc.. As for the latter, the enterprise is undertaken within a theoretical perspective that pushes till condensed matter and even biology the cross-fertilization between ``branches of physics''.Comment: 42 pages, 1 figure, Habilitation (associate professorship) dissertation at Charles University in Prague, the papers of Section 5 are not included but easy to fin