106 research outputs found
Tau function and the Prym class
We use the formalism of the Bergman tau functions to study the geometry of
moduli spaces of holomorphic quadratic differentials on complex algebraic
curves. We introduce two natural tau functions and interpret them as
holomorphic sections of certain line bundles on the moduli space. Analyzing the
asymptotic behavior of these tau functions near the boundary of the moduli
space we get two non-trivial relation in the rational Picard group of the
moduli space of quadratic differential.Comment: 20 page
Invertible Cohomological Field Theories and Weil-Petersson volumes
We show that the generating function for the higher Weil-Petersson volumes of
the moduli spaces of stable curves with marked points can be obtained from
Witten's free energy by a change of variables given by Schur polynomials. Since
this generating function has a natural extension to the moduli space of
invertible Cohomological Field Theories, this suggests the existence of a
``very large phase space'', correlation functions on which include Hodge
integrals studied by C. Faber and R. Pandharipande. From this formula we derive
an asymptotical expression for the Weil-Petersson volume as conjectured by C.
Itzykson. We also discuss a topological interpretation of the genus expansion
formula of Itzykson-Zuber, as well as a related bialgebra acting upon quantum
cohomology as a complex version of the classical path groupoid.Comment: 16 pages, AMSTex. The article is considerably enlarged. The
derivation of asymptotical formulas for Weil-Petersson volumes is added and
the topological meaning of some Itzykson-Zuber formulas is discussed. Several
misprints are correcte
Isomonodromic tau function on the space of admissible covers
The isomonodromic tau function of the Fuchsian differential equations
associated to Frobenius structures on Hurwitz spaces can be viewed as a section
of a line bundle on the space of admissible covers. We study the asymptotic
behavior of the tau function near the boundary of this space and compute its
divisor. This yields an explicit formula for the pullback of the Hodge class to
the space of admissible covers in terms of the classes of compactification
divisors.Comment: a few misprints corrected, journal reference adde
Clustering of spectra and fractals of regular graphs
We exhibit a characteristic structure of the class of all regular graphs of
degree d that stems from the spectra of their adjacency matrices. The structure
has a fractal threadlike appearance. Points with coordinates given by the mean
and variance of the exponentials of graph eigenvalues cluster around a line
segment that we call a filar. Zooming-in reveals that this cluster splits into
smaller segments (filars) labeled by the number of triangles in graphs. Further
zooming-in shows that the smaller filars split into subfilars labelled by the
number of quadrangles in graphs, etc. We call this fractal structure,
discovered in a numerical experiment, a multifilar structure. We also provide a
mathematical explanation of this phenomenon based on the Ihara-Selberg trace
formula, and compute the coordinates and slopes of all filars in terms of
Bessel functions of the first kind.Comment: 10 pages, 5 figure
Enumeration of chord diagrams on many intervals and their non-orientable analogs
Two types of connected chord diagrams with chord endpoints lying in a
collection of ordered and oriented real segments are considered here: the real
segments may contain additional bivalent vertices in one model but not in the
other. In the former case, we record in a generating function the number of
fatgraph boundary cycles containing a fixed number of bivalent vertices while
in the latter, we instead record the number of boundary cycles of each fixed
length. Second order, non-linear, algebraic partial differential equations are
derived which are satisfied by these generating functions in each case giving
efficient enumerative schemes. Moreover, these generating functions provide
multi-parameter families of solutions to the KP hierarchy. For each model,
there is furthermore a non-orientable analog, and each such model likewise has
its own associated differential equation. The enumerative problems we solve are
interpreted in terms of certain polygon gluings. As specific applications, we
discuss models of several interacting RNA molecules. We also study a matrix
integral which computes numbers of chord diagrams in both orientable and
non-orientable cases in the model with bivalent vertices, and the large-N limit
is computed using techniques of free probability.Comment: 23 pages, 7 figures; revised and extended versio
Tau functions, Prym-Tyurin classes and loci of degenerate differentials
We study the rational Picard group of the projectivized moduli space of holomorphic -differentials on complex genus stable curves. We define natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of -differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the universal curve
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