88 research outputs found
Geometry of Character Varieties of Surface Groups
This article is based on a talk delivered at the RIMS--OCAMI Joint
International Conference on Geometry Related to Integrable Systems in
September, 2007. Its aim is to review a recent progress in the Hitchin
integrable systems and character varieties of the fundamental groups of Riemann
surfaces. A survey on geometric aspects of these character varieties is also
provided as we develop the exposition from a simple case to more elaborate
cases.Comment: 21 page
Volume of representation varieties
We introduce the notion of volume of the representation variety of a finitely
presented discrete group in a compact Lie group using the push-forward measure
associated to a map defined by a presentation of the discrete group. We show
that the volume thus defined is invariant under the Andrews-Curtis moves of the
generators and relators of the discrete group, and moreover, that it is
actually independent of the choice of presentation if the difference of the
number of generators and the number of relators remains the same. We then
calculate the volume of the representation variety of a surface group in an
arbitrary compact Lie group using the classical technique of Frobenius and
Schur on finite groups. Our formulas recover the results of Witten and Liu on
the symplectic volume and the Reidemeister torsion of the moduli space of flat
G-connections on a surface up to a constant factor when the Lie group G is
semisimple.Comment: 27 pages in AMS-LaTeX forma
Commutative Algebras of Ordinary Differential Operators with Matrix Coefficients
A classification of commutative integral domains consisting of ordinary
differential operators with matrix coefficients is established in terms of
morphisms between algebraic curves.Comment: AMS-TeX format, 16 page
Edge contraction on dual ribbon graphs and 2D TQFT
We present a new set of axioms for 2D TQFT formulated on the category of cell
graphs with edge-contraction operations as morphisms. We construct a functor
from this category to the endofunctor category consisting of Frobenius
algebras. Edge-contraction operations correspond to natural transformations of
endofunctors, which are compatible with the Frobenius algebra structure. Given
a Frobenius algebra A, every cell graph determines an element of the symmetric
tensor algebra defined over the dual space A*. We show that the
edge-contraction axioms make this assignment depending only on the topological
type of the cell graph, but not on the graph itself. Thus the functor generates
the TQFT corresponding to A.Comment: accepted in Journal of Algebra (22 pages, 13 figures
An invitation to 2D TQFT and quantization of Hitchin spectral curves
This article consists of two parts. In Part 1, we present a formulation of
two-dimensional topological quantum field theories in terms of a functor from a
category of Ribbon graphs to the endofuntor category of a monoidal category.
The key point is that the category of ribbon graphs produces all Frobenius
objects. Necessary backgrounds from Frobenius algebras, topological quantum
field theories, and cohomological field theories are reviewed. A result on
Frobenius algebra twisted topological recursion is included at the end of Part
1.
In Part 2, we explain a geometric theory of quantum curves. The focus is
placed on the process of quantization as a passage from families of Hitchin
spectral curves to families of opers. To make the presentation simpler, we
unfold the story using SL_2(\mathbb{C})-opers and rank 2 Higgs bundles defined
on a compact Riemann surface of genus greater than . In this case,
quantum curves, opers, and projective structures in all become the same
notion. Background materials on projective coordinate systems, Higgs bundles,
opers, and non-Abelian Hodge correspondence are explained.Comment: 53 pages, 6 figure
Ribbon Graphs, Quadratic Differentials on Riemann Surfaces, and Algebraic Curves Defined over
It is well known that there is a bijective correspondence between metric
ribbon graphs and compact Riemann surfaces with meromorphic Strebel
differentials. In this article, it is proved that Grothendieck's correspondence
between dessins d'enfants and Belyi morphisms is a special case of this
correspondence. For a metric ribbon graph with edge length 1, an algebraic
curve over and a Strebel differential on it is constructed. It is also
shown that the critical trajectories of the measured foliation that is
determined by the Strebel differential recover the original metric ribbon
graph. Conversely, for every Belyi morphism, a unique Strebel differential is
constructed such that the critical leaves of the measured foliation it
determines form a metric ribbon graph of edge length 1, which coincides with
the corresponding dessin d'enfant.Comment: Higher resolution figures available at
http://math.ucdavis.edu/~mulase
Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy
We present in this paper a differential version of Mirzakhani's recursion
relation for the Weil-Petersson volumes of the moduli spaces of bordered
Riemann surfaces. We discover that the differential relation, which is
equivalent to the original integral formula of Mirzakhani, is a Virasoro
constraint condition on a generating function for these volumes. We also show
that the generating function for psi and kappa_1 intersections on the moduli
space of stable algebraic curves is a 1-parameter solution to the KdV
hierarchy. It recovers the Witten-Kontsevich generating function when the
parameter is set to be 0.Comment: 21 pages, 3 figures; v3. new introduction, minor revision
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