17 research outputs found
Diagrams for Symmetric Product Orbifolds
We develop a diagrammatic language for symmetric product orbifolds of
two-dimensional conformal field theories. Correlation functions of twist
operators are written as sums of diagrams: each diagram corresponds to a
branched covering map from a surface where the fields are single-valued to the
base sphere where twist operators are inserted. This diagrammatic language
facilitates the study of the large N limit and makes more transparent the
analogy between symmetric product orbifolds and free non-abelian gauge
theories. We give a general algorithm to calculate the leading large N
contribution to four-point correlators of twist fields.Comment: 43 pages, 19 figures, v2: references adde
Genus expansion for real Wishart matrices
We present an exact formula for moments and cumulants of several real
compound Wishart matrices in terms of an Euler characteristic expansion,
similar to the genus expansion for complex random matrices. We consider their
asymptotic values in the large matrix limit: as in a genus expansion, the terms
which survive in the large matrix limit are those with the greatest Euler
characteristic, that is, either spheres or collections of spheres. This
topological construction motivates an algebraic expression for the moments and
cumulants in terms of the symmetric group. We examine the combinatorial
properties distinguishing the leading order terms. By considering higher
cumulants, we give a central limit-type theorem for the asymptotic distribution
around the expected value
Exact 2-point function in Hermitian matrix model
J. Harer and D. Zagier have found a strikingly simple generating function for
exact (all-genera) 1-point correlators in the Gaussian Hermitian matrix model.
In this paper we generalize their result to 2-point correlators, using Toda
integrability of the model. Remarkably, this exact 2-point correlation function
turns out to be an elementary function - arctangent. Relation to the standard
2-point resolvents is pointed out. Some attempts of generalization to 3-point
and higher functions are described.Comment: 31 pages, 1 figur
BGWM as Second Constituent of Complex Matrix Model
Earlier we explained that partition functions of various matrix models can be
constructed from that of the cubic Kontsevich model, which, therefore, becomes
a basic elementary building block in "M-theory" of matrix models. However, the
less topical complex matrix model appeared to be an exception: its
decomposition involved not only the Kontsevich tau-function but also another
constituent, which we now identify as the Brezin-Gross-Witten (BGW) partition
function. The BGW tau-function can be represented either as a generating
function of all unitary-matrix integrals or as a Kontsevich-Penner model with
potential 1/X (instead of X^3 in the cubic Kontsevich model).Comment: 42 page
Computing pseudotriangulations via branched coverings
We describe an efficient algorithm to compute a pseudotriangulation of a
finite planar family of pairwise disjoint convex bodies presented by its
chirotope. The design of the algorithm relies on a deepening of the theory of
visibility complexes and on the extension of that theory to the setting of
branched coverings. The problem of computing a pseudotriangulation that
contains a given set of bitangent line segments is also examined.Comment: 66 pages, 39 figure