85 research outputs found

### Integral equations PS-3 and moduli of pants

More than a hundred years ago H.Poincare and V.A.Steklov considered a problem
for the Laplace equation with spectral parameter in the boundary conditions.
Today similar problems for two adjacent domains with the spectral parameter in
the conditions on the common boundary of the domains arises in a variety of
situations: in justification and optimization of domain decomposition method,
simple 2D models of oil extraction, (thermo)conductivity of composite
materials. Singular 1D integral Poincare-Steklov equation with spectral
parameter naturally emerges after reducing this 2D problem to the common
boundary of the domains. We present a constructive representation for the
eigenvalues and eigenfunctions of this integral equation in terms of moduli of
explicitly constructed pants, one of the simplest Riemann surfaces with
boundary. Essentially the solution of integral equation is reduced to the
solution of three transcendent equations with three unknown numbers, moduli of
pants. The discreet spectrum of the equation is related to certain surgery
procedure ('grafting') invented by B.Maskit (1969), D.Hejhal (1975) and
D.Sullivan- W.Thurston (1983).Comment: 27 pages, 13 figure

### Determinantal Characterization of Canonical Curves and Combinatorial Theta Identities

We characterize genus g canonical curves by the vanishing of combinatorial
products of g+1 determinants of Brill-Noether matrices. This also implies the
characterization of canonical curves in terms of (g-2)(g-3)/2 theta identities.
A remarkable mechanism, based on a basis of H^0(K_C) expressed in terms of
Szego kernels, reduces such identities to a simple rank condition for matrices
whose entries are logarithmic derivatives of theta functions. Such a basis,
together with the Fay trisecant identity, also leads to the solution of the
question of expressing the determinant of Brill-Noether matrices in terms of
theta functions, without using the problematic Klein-Fay section sigma.Comment: 35 pages. New results, presentation improved, clarifications added.
Accepted for publication in Math. An

### Vertex--IRF correspondence and factorized L-operators for an elliptic R-operator

As for an elliptic $R$-operator which satisfies the Yang--Baxter equation,
the incoming and outgoing intertwining vectors are constructed, and the
vertex--IRF correspondence for the elliptic $R$-operator is obtained. The
vertex--IRF correspondence implies that the Boltzmann weights of the IRF model
satisfy the star--triangle relation. By means of these intertwining vectors,
the factorized L-operators for the elliptic $R$-operator are also constructed.
The vertex--IRF correspondence and the factorized L-operators for Belavin's
$R$-matrix are reproduced from those of the elliptic $R$-operator.Comment: 25 pages, amslatex, no figure

### A Class of Topological Actions

We review definitions of generalized parallel transports in terms of
Cheeger-Simons differential characters. Integration formulae are given in terms
of Deligne-Beilinson cohomology classes. These representations of parallel
transport can be extended to situations involving distributions as is
appropriate in the context of quantized fields.Comment: 41 pages, no figure

### Gunning-Narasimhan's theorem with a growth condition

Given a compact Riemann surface X and a point x_0 in X, we construct a
holomorphic function without critical points on the punctured Riemann surface R
= X - x_0 which is of finite order at the point x_0. This complements the
result of Gunning and Narasimhan from 1967 who constructed a noncritical
holomorphic function on every open Riemann surface, but without imposing any
growth condition. On the other hand, if the genus of X is at least one, then we
show that every algebraic function on R admits a critical point. Our proof also
shows that every cohomology class in H^1(X;C) is represented as a de Rham class
by a nowhere vanishing holomorphic one-form of finite order on the punctured
surface X-x_0.Comment: J. Geom. Anal., in pres

### An interpolation theorem for proper holomorphic embeddings

Given a Stein manifold X of dimension n>1, a discrete sequence a_j in X, and
a discrete sequence b_j in C^m where m > [3n/2], there exists a proper
holomorphic embedding of X into C^m which sends a_j to b_j for every j=1,2,....
This is the interpolation version of the embedding theorem due to Eliashberg,
Gromov and Schurmann. The dimension m cannot be lowered in general due to an
example of Forster

### New Jacobi-Like Identities for Z_k Parafermion Characters

We state and prove various new identities involving the Z_K parafermion
characters (or level-K string functions) for the cases K=4, K=8, and K=16.
These identities fall into three classes: identities in the first class are
generalizations of the famous Jacobi theta-function identity (which is the K=2
special case), identities in another class relate the level K>2 characters to
the Dedekind eta-function, and identities in a third class relate the K>2
characters to the Jacobi theta-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.Comment: 72 pages (or 78/2 = 39 pages in reduced format

### Genus Two Partition and Correlation Functions for Fermionic Vertex Operator Superalgebras I

We define the partition and $n$-point correlation functions for a vertex
operator superalgebra on a genus two Riemann surface formed by sewing two tori
together. For the free fermion vertex operator superalgebra we obtain a closed
formula for the genus two continuous orbifold partition function in terms of an
infinite dimensional determinant with entries arising from torus Szeg\"o
kernels. We prove that the partition function is holomorphic in the sewing
parameters on a given suitable domain and describe its modular properties.
Using the bosonized formalism, a new genus two Jacobi product identity is
described for the Riemann theta series. We compute and discuss the modular
properties of the generating function for all $n$-point functions in terms of a
genus two Szeg\"o kernel determinant. We also show that the Virasoro vector one
point function satisfies a genus two Ward identity.Comment: A number of typos have been corrected, 39 pages. To appear in Commun.
Math. Phy

### Espaces de Berkovich sur Z : \'etude locale

We investigate the local properties of Berkovich spaces over Z. Using
Weierstrass theorems, we prove that the local rings of those spaces are
noetherian, regular in the case of affine spaces and excellent. We also show
that the structure sheaf is coherent. Our methods work over other base rings
(valued fields, discrete valuation rings, rings of integers of number fields,
etc.) and provide a unified treatment of complex and p-adic spaces.Comment: v3: Corrected a few mistakes. Corrected the proof of the Weierstrass
division theorem 7.3 in the case where the base field is imperfect and
trivially value

### Stein structures and holomorphic mappings

We prove that every continuous map from a Stein manifold X to a complex
manifold Y can be made holomorphic by a homotopic deformation of both the map
and the Stein structure on X. In the absence of topological obstructions the
holomorphic map may be chosen to have pointwise maximal rank. The analogous
result holds for any compact Hausdorff family of maps, but it fails in general
for a noncompact family. Our main results are actually proved for smooth almost
complex source manifolds (X,J) with the correct handlebody structure. The paper
contains another proof of Eliashberg's (Int J Math 1:29--46, 1990) homotopy
characterization of Stein manifolds and a slightly different explanation of the
construction of exotic Stein surfaces due to Gompf (Ann Math 148 (2):619--693,
1998; J Symplectic Geom 3:565--587, 2005). (See also the related preprint
math/0509419).Comment: The original publication is available at http://www.springerlink.co

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