3,292 research outputs found
Field Theory of Open and Closed Strings with Discrete Target Space
We study a -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 . 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 . The extra dimension
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 while the open string
interaction falls exponentially with .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 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
Feynman rules for string field theories with discrete target space
We derive a minimal set of Feynman rules for the loop amplitudes in unitary
models of closed strings, whose target space is a simply laced (extended)
Dynkin diagram. The string field Feynman graphs are composed of propagators,
vertices (including tadpoles) of all topologies, and leg factors for the
macroscopic loops. A vertex of given topology factorizes into a fusion
coefficient for the matter fields and an intersection number associated with
the corresponding punctured surface. As illustration we obtain explicit
expressions for the genus-one tadpole and the genus-zero four-loop amplitude.Comment: 19 pages, harvmac, 4 uuencoded figures included using epsf. A missing
term added to the expression for the genus-one tadpole and Fig.3 modified
correspondingl
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
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
Boundary Correlators in 2D Quantum Gravity: Liouville versus Discrete Approach
We calculate a class of two-point boundary correlators in 2D quantum gravity
using its microscopic realization as loop gas on a random surface. We find a
perfect agreement with the two-point boundary correlation function in Liouville
theory, obtained by V. Fateev, A. Zamolodchikov and Al. Zamolodchikov. We also
give a geometrical meaning of the functional equation satisfied by this
two-point function.Comment: 21 pages, 5 figures, harvmac, eqs. (2.11) and (5.11) correcte
Exact Solution of the Six-Vertex Model on a Random Lattice
We solve exactly the 6-vertex model on a dynamical random lattice, using its
representation as a large N matrix model. The model describes a gas of dense
nonintersecting oriented loops coupled to the local curvature defects on the
lattice. The model can be mapped to the c=1 string theory, compactified at some
length depending on the vertex coupling. We give explicit expression for the
disk amplitude and evaluate the fractal dimension of its boundary, the average
number of loops and the dimensions of the vortex operators, which vary
continuously with the vertex coupling.Comment: typos corrected and a figure added in Appendix
Some examples of rigid representations
Consider the Deligne-Simpson problem: {\em give necessary and sufficient
conditions for the choice of the conjugacy classes
(resp. ) so that there exist irreducible
-tuples of matrices (resp. ) satisfying the
equality (resp. )}. The matrices
and are interpreted as monodromy operators and as matrices-residua of
fuchsian systems on Riemann's sphere.
We give new examples of existence of such -tuples of matrices
(resp. ) which are {\em rigid}, i.e. unique up to conjugacy once the
classes (resp. ) are fixed. For rigid representations the sum of the
dimensions of the classes (resp. ) equals
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