Field Theory of Open and Closed Strings with Discrete Target Space

We study a $U(N)$-invariant vector+matrix chain with the color structure of a lattice gauge theory with quarks and interpret it as a theory of open andclosed strings with target space $\Z$. The string field theory is constructed as a quasiclassical expansion for the Wilson loops and lines in this model. In a particular parametrization this is a theory of two scalar massless fields defined in the half-space $\{x\in \Z , \tau >0\}$. The extra dimension $\tau$ is related to the longitudinal mode of the strings. The topology-changing string interactions are described by a local potential. The closed string interaction is nonzero only at boundary $\tau =0$ while the open string interaction falls exponentially with $\tau$.Comment: 15 pages, harvmac. no figures; some typos corrected and a reference adde

Loop Gas Model for Open Strings

The open string with one-dimensional target space is formulated in terms of an SOS, or loop gas, model on a random surface. We solve an integral equation for the loop amplitude with Dirichlet and Neumann boundary conditions imposed on different pieces of its boundary. The result is used to calculate the mean values of order and disorder operators, to construct the string propagator and find its spectrum of excitations. The latter is not sensible neither to the string tension \L nor to the mass $\mu$ of the ``quarks'' at the ends of the string. As in the case of closed strings, the SOS formulation allows to construct a Feynman diagram technique for the string interaction amplitudes

Gauge Invariant Matrix Model for the \^A-\^D-\^E Closed Strings

The models of triangulated random surfaces embedded in (extended) Dynkin diagrams are formulated as a gauge-invariant matrix model of Weingarten type. The double scaling limit of this model is described by a collective field theory with nonpolynomial interaction. The propagator in this field theory is essentially two-loop correlator in the corresponding string theory.Comment: 9 pages, SPhT/92-09

Rational Theories of 2D Gravity from the Two-Matrix Model

The correspondence claimed by M. Douglas, between the multicritical regimes of the two-matrix model and 2D gravity coupled to (p,q) rational matter field, is worked out explicitly. We found the minimal (p,q) multicritical potentials U(X) and V(Y) which are polynomials of degree p and q, correspondingly. The loop averages W(X) and \tilde W(Y) are shown to satisfy the Heisenberg relations {W,X} =1 and {\tilde W,Y}=1 and essentially coincide with the canonical momenta P and Q. The operators X and Y create the two kinds of boundaries in the (p,q) model related by the duality (p,q) - (q,p). Finally, we present a closed expression for the two two-loop correlators and interpret its scaling limit.Comment: 24 pages, preprint CERN-TH.6834/9

Some examples of rigid representations

Consider the Deligne-Simpson problem: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset GL(n,{\bf C})$ (resp. $c_j\subset gl(n,{\bf C})$) so that there exist irreducible $(p+1)$-tuples of matrices $M_j\in C_j$ (resp. $A_j\in c_j$) satisfying the equality $M_1... M_{p+1}=I$ (resp. $A_1+... +A_{p+1}=0$)}. The matrices $M_j$ and $A_j$ are interpreted as monodromy operators and as matrices-residua of fuchsian systems on Riemann's sphere. We give new examples of existence of such $(p+1)$-tuples of matrices $M_j$ (resp. $A_j$) which are {\em rigid}, i.e. unique up to conjugacy once the classes $C_j$ (resp. $c_j$) are fixed. For rigid representations the sum of the dimensions of the classes $C_j$ (resp. $c_j$) equals $2n^2-2$

Spatial Weighting Matrix Selection in Spatial Lag Econometric Model

This paper investigates the choice of spatial weighting matrix in a spatial lag model framework. In the empirical literature the choice of spatial weighting matrix has been characterized by a great deal of arbitrariness. The number of possible spatial weighting matrices is large, which until recently was considered to prevent investigation into the appropriateness of the empirical choices. Recently Kostov (2010) proposed a new approach that transforms the problem into an equivalent variable selection problem. This article expands the latter transformation approach into a two-step selection procedure. The proposed approach aims at reducing the arbitrariness in the selection of spatial weighting matrix in spatial econometrics. This allows for a wide range of variable selection methods to be applied to the high dimensional problem of selection of spatial weighting matrix. The suggested approach consists of a screening step that reduces the number of candidate spatial weighting matrices followed by an estimation step selecting the final model. An empirical application of the proposed methodology is presented. In the latter a range of different combinations of screening and estimation methods are employed and found to produce similar results. The proposed methodology is shown to be able to approximate and provide indications to what the ‘true’ spatial weighting matrix could be even when it is not amongst the considered alternatives. The similarity in results obtained using different methods suggests that their relative computational costs could be primary reasons for their choice. Some further extensions and applications are also discussed

Boundary Ground Ring in 2D String Theory

The 2D quantum gravity on a disc, or the non-critical theory of open strings, is known to exhibit an integrable structure, the boundary ground ring, which determines completely the boundary correlation functions. Inspired by the recent progress in boundary Liouville theory, we extend the ground ring relations to the case of non-vanishing boundary Liouville interaction known also as FZZT brane in the context of the 2D string theory. The ring relations yield an over-determined set of functional recurrence equations for the boundary correlation functions. The ring action closes on an infinite array of equally spaced FZZT branes for which we propose a matrix model realization. In this matrix model the boundary ground ring is generated by a pair of complex matrix fields.Comment: sect. 5 extended, appendix adde

Choosing the Right Spatial Weighting Matrix in a Quantile Regression Model

This paper proposes computationally tractable methods for selecting the appropriate spatial weighting matrix in the context of a spatial quantile regression model. This selection is a notoriously difficult problem even in linear spatial models and is even more difficult in a quantile regression setup. The proposal is illustrated by an empirical example and manages to produce tractable models. One important feature of the proposed methodology is that by allowing different degrees and forms of spatial dependence across quantiles it further relaxes the usual quantile restriction attributable to the linear quantile regression. In this way we can obtain a more robust, with regard to potential functional misspecification, model, but nevertheless preserve the parametric rate of convergence and the established inferential apparatus associated with the linear quantile regression approach