Generalized string equations for double Hurwitz numbers

The generating function of double Hurwitz numbers is known to become a tau function of the Toda hierarchy. The associated Lax and Orlov–Schulman operators turn out to satisfy a set of generalized string equations. These generalized string equations resemble those of c=1$c=1$ string theory except that the Orlov–Schulman operators are contained therein in an exponentiated form. These equations are derived from a set of intertwining relations for fermion bilinears in a two-dimensional free fermion system. The intertwiner is constructed from a fermionic counterpart of the cut-and-join operator. A classical limit of these generalized string equations is also obtained. The so-called Lambert curve emerges in a specialization of its solution. This seems to be another way of deriving the spectral curve of the random matrix approach to Hurwitz numbers.     

The homogeneous realization of the basic representation of A inf1 sup(1) and the toda lattice

It is well-known that the principal realization of the basic module L(₀) over A _inf1 ^sup(1) gives rise to the KdV hierarchy of partial differential equations. Here we use the homogeneous realization of the same module to construct a hierarchy of differential-difference equations, the first member of which turns out to be the equation for the Toda lattice.     

The kirkwood-salsburg equations for random continuum percolation

We develop two different hierarchies of Kirkwood-Salsburg equations for the connectedness functions of random continuum percolation. These equations are derived by writing the Kirkwood-Salsburg equations for the distribution functions of thes-state continuum Potts model (CPM), taking thes1 limit, and forming appropriate linear combinations. The first hierarchy is satisfied by a subset of the connectedness functions used in previous studies. It gives rigorous, low-order bounds for the mean number of clusters n _c and the two-point connectedness function. The second hierarchy is a closed set of equations satisfied by the generalized blocking functions, each of which is defined by the probability that a given set of connections between particles is absent. These auxiliary functions are shown to be a natural basis for calculating the properties of continuum percolation models. They are the objects naturally occurring in integral equations for percolation theory. Also, the standard connectedness functions can be written as linear combinations of them. Using our second Kirkwood-Salsburg hierarchy, we show the existence of an infinite sequence of rigorous, upper and lower bounds for all the quantities describing random percolation, including the mean cluster size and mean number of clusters. These equations also provide a rigorous lower bound for the radius of convergence of the virial series for the mean number of clusters. Most of the results obtained here can be readily extended to percolation models on lattices, and to models with positive (repulsive) pair potentials.     

Representation theoretical meaning of the initial value problem for the Toda lattice hierarchy: I

The Toda lattice hierarchy is shown to have the Bruhat decomposition of the A group as its parameter space instead of the Grassmann manifold for the KP hierarchy. Takasaki's work on the initial value problem for the Toda lattice hierarchy is reinterpreted from this point of view.     

Integrability of a family of quantum field theories related to sigma models

A method is introduced for constructing lattice discretizations of large classes of integrable quantum field theories. The method proceeds in two steps: The quantum algebraic structure underlying the integrability of the model is determined from the algebra of the interaction terms in the light-cone representation. The representation theory of the relevant quantum algebra is then used to construct the basic ingredients of the quantum inverse scattering method, the lattice Lax matrices and R-matrices. This method is illustrated with four examples: The sinh-Gordon model, the affine sl(3) Toda model, a model called the fermionic sl(2|1) Toda theory, and the N=2 supersymmetric sine-Gordon model. These models are all related to sigma models in various ways. The N=2 supersymmetric sine-Gordon model, in particular, describes the Pohlmeyer reduction of string theory on AdS₂×S², and is dual to a supersymmetric non-linear sigma model with a sausage-shaped target space.     

Decoupling of holes in thetJ model atJ=2t fromspl(2, 1) symmetry

The binding energy of twop=0 holes in thetJ model on a square lattice is shown to vanish atJ=2t due to thespl(2, 1) symmetry at this point. In any back-ground of antiferromagnetic type, with finite total spin in an infinite volume, the binding energy atx=0 hole doping vanishes at least as (J–2t)² asJ2t. These exact results may serve as tests of more comprehensive calculations of interactions of holes in thetJ model.     

ClassicalA n -W-geometry

By analyzing theextrinsic geometry of two dimensional surfaces chirally embedded inC P ⁿ (theC P ⁿ W-surface [1]), we give exact treatments in various aspects of the classical W-geometry in the conformal gauge: First, the basis of tangent and normal vectors are defined at regular points of the surface, such that their infinitesimal displacements are given by connections which coincide with the vector potentials of the (conformal)A _n -Toda Lax pair. Since the latter is known to be intrinsically related with the W symmetries, this gives the geometrical meaning of theA _n W-Algebra. Second, W-surfaces are put in one-to-one correspondence with solutions of the conformally-reduced WZNW model, which is such that the Toda fields give the Cartan part in the Gauss decomposition of its solutions. Third, the additional variables of the Toda hierarchy are used as coordinates ofC P ⁿ . This allows us to show that W-transformations may be extended as particular diffeomorphisms of this target-space. Higher-dimensional generalizations of the WZNW equations are derived and related with the Zakharov-Shabat equations of the Toda hierarchy. Fourth, singular points are studied from a global viewpoint, using our earlier observation [1] that W-surfaces may be regarded as instantons. The global indices of the W-geometry, which are written in terms of the Toda fields, are shown to be the instanton numbers for associated mappings of W-surfaces into the Grassmannians. The relation with the singularities of W-surface is derived by combining the Toda equations with the Gauss-Bonnet theorem.     

Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes

We develop techniques to compute higher loop string amplitudes for twistedN=2 theories with=3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particular realization of theN=2 theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira-Spencer theory, which may be viewed as the closed string analog of the Chern-Simons theory. Using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in Calabi-Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingN=2 theory. Relations withc=1 strings are also pointed out.This article was processed by the author using the Springer-Verlag TEX CoMaPhy macro package 1991.     

High-density percolation: Exact solution on a Bethe lattice

A new percolation problem is posed where the sites on a lattice are randomly occupied but where only those occupied sites with at least a given numberm of occupied neighbors are included in the clusters. This problem, which has applications in magnetic and other systems, is solved exactly on a Bethe lattice. The classical percolation critical exponents=gg=1 are found. The percolation thresholds vary between the ordinary percolation thresholdp _c (m=1)=l/(z – 1) andp _c(m=z) =[l/(z – 1)]¹/(z–1). The cluster size distribution asymptotically decays exponentially withn, for largen, p p _c .Supported in part by National Science Foundation grant DMR78-10813.     

Extended Discrete KP Hierarchy and its Reductions from a Geometric Viewpoint

We interpret the recently suggested extended discrete KP (Toda lattice) hierarchy from a geometrical point of view. We show that the latter corresponds to the union of invariant submanifolds S ₀ ⁿ of the system which is a chain of infinitely many copies of Darboux–KP hierarchy, while the intersections yields a number of reduction s to l-field lattices.     

On a point of order

A method of using algebraic curves to obtain estimates of critical points accurate enough to identify them as simple algebraic numbers (if that is what they are) is discussed and illustrated with an application to the (q-state Potts model on the triangular lattice for cases of pure two-site interactions and pure three-site interactions. In the latter case the critical point is conjectured to be . In a similar conjecture for the critical percolation probability on thedirected square lattice,q _c ^1/2 (q _c +3)=2(q _c +p _c =1).     

Study of the three-dimensional Ising model on film geometry with the cluster Monte Carlo method

Topological properties of clusters are used to extract critical parameters. This method is tested for the bulk properties ofd=2 percolation and thed=2, 3 Ising model. For the latter we obtain an accurate value of the critical temperatureJ/k _B T _c=0.221617(18). In the case of thed=3 Ising model with film geometry the critical value of the surface coupling at the special transitions is determined as J_1c/J=1.5004(20) together with the critical exponents ₁ ^m =0.237(5) and=0.461(15).     

Pressure-volume relationship,quasiparticle density of states and hybridization gap of mixed valence EuO

Thes-f model, extended by a hybridization term, is used to investigate the pressure-dependence of the quasiparticle density of states of paramagnetic EuO (T=300K). We find a semiconductor-metal transition of first order atp _c =301 kbar, which manifests itself by a striking discontinuity in the pressure-volume relationship, in almost exact agreement with the experiment [5]. The transition is due to the valence-change Eu²⁺Eu³⁺ of the Europium ion.-Special attention is devoted to the competing influence of thes-f exchange interaction and thes-f hybridization between extendeds-band states and localizedf-levels on characteristic details of the phase transition. Exchange and hybridization are therefore taken into account exactly, while the electron hopping is treated in an approximate many body theory.-As a typical feature we find a small hybridization gap in the quasiparticle density of states. Forp<p _c the chemical potential lies always within this gap, so that the material is a narrow gap semiconductor. Forp<p _c the gap does survive, but then is below the gap allowing metallic conductivity.     

The Relativistic non-abelian Chern-Simons Equations

We study ther xr system of nonlinear elliptic equations ,a=1,2,...,r,x∈R ², where λ τ 0 is a constant parameter,K = (K_ab) is the Cartan matrix of a semi-simple Lie algebra, and β_p is the Dirac measure concentrated atp R ². This system of equations arises in the relativistic non-Abelian Chern-Simons theory and may be viewed as a nonintegrable deformation of the integrable Toda system. We establish the existence of a class of solutions known as topological multivortices. The crucial step in our method is the use of the decomposition theorem of Cholesky for positive definite matrices so that a variational principle can be formulated. Research supported in part by the National Science Foundation under grant DMS-9596041     

p-Adic string compactified on a torus

U(1) ^xD model with the Villain action on ag-loop generalizationF _g of the Bruhat-Tits tree for thep-adic linear groupGL(2, _p ) is considered. All correlation functions and the statistical sum are calculated. We compute also the averages of these correlation functions forN vertices attached to the boundary ofF _g. When the compactification radius tends to infinity the averages provide theg-loopN-point amplitudes of the uncompactifiedp-adic string theory, in particular forg=0 the Freund-Olson amplitudes.     

Macdonald polynomials from Sklyanin algebras: A conceptual basis for thep-adics-quantum group connection

We establish a previously conjectured connection betweenp-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which interpolate between the zonal spherical functions of related real andp-adic symmetric spaces. The elliptic quantum algebras underlie theZ _n -Baxter models. We show that in then limit, the Jost function for the scattering offirst level excitations in the 1+1 dimensional field theory model associated to theZ _n -Baxter model coincides with the Harish-Chandra-likec-function constructed from the Macdonald polynomials associated to the root systemA ₁. The partition function of theZ ₂-Baxter model itself is also expressed in terms of this Macdonald-Harish-Chandrac-function, albeit in a less simple way. We relate the two parametersq andt of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular thep-adic regimes in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of q-deforming Euler products.Work supported in part by the NSF: PHY-9000386     

Exponent inequalities for the bulk conductivity of a hierarchical model

The bulk conductivity ^*(p) of the bond lattice in ^d is considered, where the bonds have conductivity 1 with probabilityp or 0 with probability 1-p Various representations of the derivatives of ^*(p) are developed. These representations are used to analyze the behavior of ^*(p) for =0 near the percolation thresholdp _c , when the conducting backbone is assumed to have a hierarchical node-link-blob (NLB) structure. This model has loops on arbitrarily many length scales and contains both singly and multiply connected bonds. Exact asymptotics of for the NLB model are proven under some technical assumptions. The proof employs a novel technique whereby for the NLB model with =0 andp nearp _c is computed using perturbation theory for ^*(p) (for two- and three-component resistor lattices) aroundp=1 with a sequence of s converging to 1 as one goes deeper in the hierarchy. These asymptotics establish convexity of ^*(p) (for the NLB model) nearp _c , and that its critical exponentt obeys the inequalities 1t2 ford=2,3, while 2t3 ford4. The upper boundt=2 ind=3, which is realizable in the NLB class, virtually coincides with two very recent numerical estimates obtained from simulation and series expansion for the original model.Supported in part by NSF Grant DMS-8801673 and AFOSR Grant AFOSR-90-0203     

New Jacobi-like identities for Z K parafermion characters

We state and prove various new identities involving theZ _K parafermion characters (or level-K string functions)c _n ^l for the casesK=4,K=8, andK=16. These identities fall into three classes: identities in the first class are generalizations of the famous Jacobi -function identity (which is theK=2 special case), identities in another class relate the levelK>2 characters to the Dedekind -function, and identities in a third class relate theK>2 characters to the Jacobi -functions. These identities play a crucial role in the interpretation of fractional superstring spectra by indicating spacetime supersymmetry and aiding in the identification of the spacetime spin and statistics of fractional superstring states.     

Fluctuations and lack of self-averaging in the kinetics of domain growth

The fluctuations occurring when an initially disordered system is quenched at timet=0 to a state, where in equilibrium it is ordered, are studied with a scaling theory. Both the mean-sizel(t) ^d of thed-dimensional ordered domains and their fluctuations in size are found to increase with the same power of the time; their relative size fluctuations are independent of the total volumeL ^d of the system. This lack of self-averaging is tested for both the Ising model and the ⁴ model on the square lattice. Both models exhibit the same lawl(t)=(Rt) ^x withx=1/2, although the ⁴ model has soft walls. However, spurious results withx1/2 are obtained if bad pseudorandom numbers are used, and if the numbern of independent runs is too small (n itself should be of the order of 10³). We also predict a critical singularity of the rateR(1–T/T _c) ^v(z–1/x),v being the correlation length exponent,z the dynamic exponent.Also quenches to the critical temperatureT _c itself are considered, and a related lack of self-averaging in equilibrium computer simulations is pointed out for quantities sampled from thermodynamic fluctuation relations.