Ivan K. Kostov

Strings with discrete target space

We investigate the field theory of strings having as a target space an arbitrary discrete one- dimensional manifold. The existence of the continuum limit is guaranteed if the target space is a Dynkin diagram of a simply laced Lie algebra or its affine extension. In this case the theory can be mapped onto the theory of strings embedded in the infinite discrete line Z Z which is the target space of the SOS model. On the regular lattice this mapping is known as Coulomb gas picture. Introducing a quantum string fieldx(l) depending on the position x and the length l of the closed string, we give a formal definition of the string field theory in terms of a functional integral. The classical string background is found as a solution of the saddle-point equation which is equivalent to the loop equation we have previously considered (1). The continuum limit exists in the vicinity of the singular points of this equation. We show that for given target space there are many ways to achieve the continuum limit; they are related to the multicritical points of the ensemble of surfaces without embedding. Once the classical background is known, the amplitudes involving propagation of strings can be evaluated by perturbative expansion around the saddle point of the functional integral. For example, the partition function of the noninteracting closed string (toroidal world sheet) is the contribution of the gaussian fluctuations of the string field. The vertices in the corresponding Feynman diagram technique are constructed as the loop amplitudes in a random matrix model with suitably chosen potential.

Gauge invariant matrix model for the Â-D̂-Ê closed strings

Abstract 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 the two-loop correlator in the corresponding string theory.

Loop gas model for open strings

Abstract 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 sensitive either to the string tension λ or to the mass μ of the “quarks” at the ends of the string. As in the case of closed strings, the SOS formulation allows us to construct a Feynman-diagram technique for the string interaction amplitudes.

Geometrical Critical Phenomena on a Random Surface of Arbitrary Genus

Abstract The statistical mechanics of self-avoiding walks (SAW) or of the O( n )-loop model on a two-dimensional random surface are shown to be exactly solvable. The partition functions of SAW and surface configurations (possibly in the presence of vacuum loops) are calculated by planar diagram enumeration techniques. Two critical regimes are found: a dense phase where the infinite walks and loops fill the infinite surface, the non-filled part staying finite, and a dilute phase where the infinite surface singularity on the one hand, and walk and loop singularities on the other, merge together. The configuration critical exponents of self-avoiding networks of any fixed topology G, on a surface with arbitrary genus H , are calculated as universal functions of G and H . For self-avoiding walks, the exponents are built from an infinite set of basic conformal dimensions associated with central charges c = −2 (dense phase) and c = 0 (dilute phase). The conformal spectrum Δ L , L ⩾ 1 associated with L -leg star polymers is calculated exactly, for c = −2 and c = 0. This is generalized to the set of L -line “watermelon” exponents Δ L of the O( n ) model on a random surface. The results are in perfect agreement with the conformal theory of Knizhnik, Polyakov and Zamolodchikov describing matter fields coupled to 2D quantum gravity. The infinite series of dimensions Δ L dressed by gravity calculated here, together with the corresponding SAW conformal dimensions Δ L (0) in the plane, known independently from Coulomb-gas techniques, match the KPZ relation Δ − Δ (0) = Δ(1 − Δ) κ , where c = 1 − 6(1 − κ) 2 k . This provides a cross check of Coulomb-gas techniques, the KPZ conformal theory of matter fields with 2D quantum gravity and the universality of random lattices. The divergences of the partition functions of self-avoiding networks on the random surface, possibly in the presence of vacuum loops, are shown to satisfy a factorization theorem over the vertices of the network. This provides a proof, in the presence of a fluctuating metric, of a result conjectured earlier in the standard plane. From this, the value of the string susceptibility γ str ( H , c ) is extracted for a random surface of arbitrary genus H , bearing a field theory of central charge c , or equivalently, embedded in d  c dimensions. Lastly, by enumerating spanning trees on a random lattice, we solve the similar problem of hamiltonian walks on the (fluctuating) Manhattan covering lattice. We also obtain new results for dilute trees on a random surface.

THE ADE FACE MODELS ON A FLUCTUATING PLANAR LATTICE

Abstract The ADE two-dimensional interaction-round-a-face statistical models are formulated on a fluctuating planar lattice. The continuum limit of such systems is described by the minimal conformal theories coupled to quantum gravity. All these models can be reformulated in terms of a gas of self-avoiding noninteresecting loops on a random planar graph. This representation allows us to calculate the partition function and the susceptibilities of the order parameters in the case of lattices with spherical topology. The scaling dimensions of the order parameters are shown to form a linear spectrum.

Free field presentation of the An coset models on the torus

Abstract Partition functions of a family of conformal invariant models obtained through the Goddard-Kent-Olive coset construction A n (1) + A n (1) /A n (1) are expressed in terms of a Gaussian free field with magnetic defects on the torus. It is argued that the same relation holds for the corresponding solvable IRF lattice models (where the role of the free field is played by the underlying vertex model with Z n + 1 symmetry).

TWO-POINT CORRELATOR FOR THE D = 1 CLOSED BOSONIC STRING

Abstract We calculate exactly the two-point correlation function for a random surface discretized via dense planar Feynman graphs when the dimension of the target space is D = 1. The same quantity is computed for the D = 1 bosonic string within the approach of Knizhnik, Polyakov and Zamolodchikov. The agreement is perfect.

Loop amplitudes for nonrational string theories

Abstract We investigate the loop equations for 2D quantum gravity+matter with running central charge D . The matter field is constructed as an SOS model with floating electric charge α 0 . We find the one-loop amplitudes for different boundary conditions and conjecture a compact formula for multiloop correlators. Finally, we discuss the limit D →1 ( α 0 →0) and the origin of the scaling violation at D =1.

Exactly solvable field theory of D=0 closed and open strings

Abstract The field theory of interacting closed and open strings is solved using the equivalence with a random matrix model. The partition function of the theory is given in the scaling limit by an universal function of the renormalized cosmological constant Λ, mass M at the ends of the open string, and couplings G and Γ for the closed- and open-string interactions, correspondingly. Its second derivative in Λ obeys an ordinary differential equation of fourth degree.

Field theory of open and closed strings with discrete target space

Abstract We study a U ( N )-invariantvector+matrix chain with the color structure of a lattice gauge theory with quarks and interpret it as a theory of open and closed 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 ∈ Z , τ > 0}. The extra dimension τ 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 τ =0 while the open string interaction falls exponentially with τ.