Symmetries of Heterotic String Theory

We study the symmetries of the two dimensional Heterotic string theory by following the approach of Kinnersley et al for the study of stationary-axially symmetric Einstein-Maxwell equations. We identify the finite dimensional groups $G'$ and $H'$ for the Einstein-Maxwell equations. We also give the constructions for the infinite number of conserved currents and the affine $\hat{o}(8, 24)$ symmetry algebra in this formulation. The generalized Ehlers and Harrison transformations are identified and a parallel between the infinite dimensional symmetry algebra for the heterotic string case with $\hat{sl}(3, R)$ that arise in the case of Einstein-Maxwell equations is pointed out.Comment: 26 pages, Few comments added, version to appear in Nuclear Physics

Integrals of Motion in the Two Killing Vector Reduction of General Relativity

We apply the inverse scattering method to the midi-superspace models that are characterized by a two-parameter Abelian group of motions with two spacelike Killing vectors. We present a formulation that simplifies the construction of the soliton solutions of Belinski\v i and Zakharov. Furthermore, it enables us to obtain the zero curvature formulation for these models. Using this, and imposing periodic boundary conditions corresponding to the Gowdy models when the spatial topology is a three torus $T ^3$, we show that the equation of motion for the monodromy matrix is an evolution equation of the Heisenberg type. Consequently, the eigenvalues of the monodromy matrix are the generating functionals for the integrals of motion. Furthermore, we utilise a suitable formulation of the transition matrix to obtain explicit expressions for the integrals of motion. This involves recursion relations which arise in solving an equation of Riccati type. In the case when the two Killing vectors are hypersurface orthogonal the integrals of motion have a particularly simple form.Comment: 20 pages, plain TeX, SU-GP-93/7-8, UM-P-93/7

Cops and Robbers is EXPTIME-complete

We investigate the computational complexity of deciding whether k cops can capture a robber on a graph G. In 1995, Goldstein and Reingold conjectured that the problem is EXPTIME-complete when both G and k are part of the input; we prove this conjecture.Comment: v2: updated figures and slightly clarified some minor point

Ehlers Transformations and String Effective Action

We explicitly obtain the generalization of the Ehlers transformation for stationary axisymmetric Einstein equations to string theory. This is accomplished by finding the twist potential corresponding to the moduli fields in the effective two dimensional theory. Twist potential and symmetric moduli are shown to transform under an $O(d,d)$ which is a manifest symmetry of the equations of motion. The non-trivial action of this $O(d,d)$ is given by the Ehlers transformation and belongs to the set $O(d) \times O(d)\over O(d)$.Comment: 13 pages, minor corrections, version to appear in Physics Letters B35

Gravitational Solitons and Monodromy Transform Approach to Solution of Integrable Reductions of Einstein Equations

In this paper the well known Belinskii and Zakharov soliton generating transformations of the solution space of vacuum Einstein equations with two-dimensional Abelian groups of isometries are considered in the context of the so called "monodromy transform approach", which provides some general base for the study of various integrable space - time symmetry reductions of Einstein equations. Similarly to the scattering data used in the known spectral transform, in this approach the monodromy data for solution of associated linear system characterize completely any solution of the reduced Einstein equations, and many physical and geometrical properties of the solutions can be expressed directly in terms of the analytical structure on the spectral plane of the corresponding monodromy data functions. The Belinskii and Zakharov vacuum soliton generating transformations can be expressed in explicit form (without specification of the background solution) as simple (linear-fractional) transformations of the corresponding monodromy data functions with coefficients, polynomial in spectral parameter. This allows to determine many physical parameters of the generating soliton solutions without (or before) calculation of all components of the solutions. The similar characterization for electrovacuum soliton generating transformations is also presented.Comment: 8 pages, 1 figure, LaTeX2e; based on a talk given at the International Conference 'Solitons, Collapses and Turbulence: Achievements, Developments and Perspectives', (Landau Institute for Theoretical Physics, Chernogolovka, Moscow region, Russia, August 3 -- 10, 1999); as submitted to Physica

To catch a falling robber

We consider a Cops-and-Robber game played on the subsets of an $n$-set. The robber starts at the full set; the cops start at the empty set. On each turn, the robber moves down one level by discarding an element, and each cop moves up one level by gaining an element. The question is how many cops are needed to ensure catching the robber when the robber reaches the middle level. Aaron Hill posed the problem and provided a lower bound of $2^{n/2}$ for even $n$ and $\binom{n}{\lceil n/2 \rceil}2^{-\lfloor n/2 \rfloor}$ for odd $n$. We prove an upper bound (for all $n$) that is within a factor of $O(\ln n)$ times this lower bound.Comment: Minor revision

DUALITY SYMMETRY GROUP OF TWO DIMENSIONAL HETEROTIC STRING THEORY

The equations of motion of the massless sector of the two dimensional string theory, obtained by compactifying the heterotic string theory on an eight dimensional torus, is known to have an affine o(8,24) symmetry algebra generating an O(8,24) loop group. In this paper we study how various known discrete S- and T- duality symmetries of the theory are embedded in this loop group. This allows us to identify the generators of the discrete duality symmetry group of the two dimensional string theory.Comment: LaTeX, 30 page

Matrix Ernst Potentials and Orthogonal Symmetry for Heterotic String in Three Dimensions

A new matrix representation for low-energy limit of heterotic string theory reduced to three dimensions is considered. The pair of matrix Ernst Potentials uniquely connected with the coset matrix is derived. The action of the symmetry group on the Ernst potentials is established.Comment: 10 pages in LaTe