108 research outputs found
Mixed Hodge polynomials of character varieties
We calculate the E-polynomials of certain twisted GL(n,C)-character varieties
M_n of Riemann surfaces by counting points over finite fields using the
character table of the finite group of Lie-type GL(n,F_q) and a theorem proved
in the appendix by N. Katz. We deduce from this calculation several geometric
results, for example, the value of the topological Euler characteristic of the
associated PGL(n,C)-character variety. The calculation also leads to several
conjectures about the cohomology of M_n: an explicit conjecture for its mixed
Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture
relating the pure part to absolutely indecomposable representations of a
certain quiver. We prove these conjectures for n = 2.Comment: with an appendix by Nicholas M. Katz; 57 pages. revised version: New
definition for homogeneous weight in Definition 4.1.6, subsequent arguments
modified. Some other minor changes. To appear in Invent. Mat
Pure O-sequences and matroid h-vectors
We study Stanley's long-standing conjecture that the h-vectors of matroid
simplicial complexes are pure O-sequences. Our method consists of a new and
more abstract approach, which shifts the focus from working on constructing
suitable artinian level monomial ideals, as often done in the past, to the
study of properties of pure O-sequences. We propose a conjecture on pure
O-sequences and settle it in small socle degrees. This allows us to prove
Stanley's conjecture for all matroids of rank 3. At the end of the paper, using
our method, we discuss a first possible approach to Stanley's conjecture in
full generality. Our technical work on pure O-sequences also uses very recent
results of the third author and collaborators.Comment: Contains several changes/updates with respect to the previous
version. In particular, a discussion of a possible approach to the general
case is included at the end. 13 pages. To appear in the Annals of
Combinatoric
Topology of character varieties and representations of quivers
In arXiv:0810.2076 we presented a conjecture generalizing the Cauchy formula
for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials
of the representation varieties of Riemann surfaces with semi-simple conjugacy
classes at the punctures. We proved several results which support this
conjecture. Here we announce new results which are consequences of those of
arXiv:0810.2076
Arithmetic harmonic analysis on character and quiver varieties
We present a conjecture generalizing the Cauchy formula for Macdonald
polynomials. This conjecture encodes the mixed Hodge polynomials of the
character varieties of representations of the fundamental group of a Riemann
surface of genus g to GL_n(C) with fixed generic semi-simple conjugacy classes
at k punctures. Using the character table of GL_n(F_q) we calculate the
E-polynomial of these character varieties and confirm that it is as predicted
by our main conjecture. Then, using the character table of gl_n(F_q), we
calculate the E-polynomial of certain associated comet-shaped quiver varieties,
the additive analogues of our character variety, and find that it is the pure
part of our conjectured mixed Hodge polynomial. Finally, we observe that the
pure part of our conjectured mixed Hodge polynomial also equals certain
multiplicities in the tensor product of irreducible representations of
GL_n(F_q). This implies a curious connection between the representation theory
of GL_n(F_q) and Kac-Moody algebras associated with comet-shaped, typically
wild, quivers.Comment: To appear in Duke Math. Journal + a section with examples is adde
Linear Sigma Models of H and KK Monopoles
We propose a gauged linear sigma model of k H-monopoles. We also consider the
T-dual of this model describing KK-monopoles and clarify the meaning of
"winding coordinate" studied recently in hep-th/0507204.Comment: 13 pages, lanlmac; V3:added argument on the nature of disk instanto
Morse theory of the moment map for representations of quivers
The results of this paper concern the Morse theory of the norm-square of the
moment map on the space of representations of a quiver. We show that the
gradient flow of this function converges, and that the Morse stratification
induced by the gradient flow co-incides with the Harder-Narasimhan
stratification from algebraic geometry. Moreover, the limit of the gradient
flow is isomorphic to the graded object of the
Harder-Narasimhan-Jordan-H\"older filtration associated to the initial
conditions for the flow. With a view towards applications to Nakajima quiver
varieties we construct explicit local co-ordinates around the Morse strata and
(under a technical hypothesis on the stability parameter) describe the negative
normal space to the critical sets. Finally, we observe that the usual Kirwan
surjectivity theorems in rational cohomology and integral K-theory carry over
to this non-compact setting, and that these theorems generalize to certain
equivariant contexts.Comment: 48 pages, small revisions from previous version based on referee's
comments. To appear in Geometriae Dedicat
Tree modules and counting polynomials
We give a formula for counting tree modules for the quiver S_g with g loops
and one vertex in terms of tree modules on its universal cover. This formula,
along with work of Helleloid and Rodriguez-Villegas, is used to show that the
number of d-dimensional tree modules for S_g is polynomial in g with the same
degree and leading coefficient as the counting polynomial A_{S_g}(d, q) for
absolutely indecomposables over F_q, evaluated at q=1.Comment: 11 pages, comments welcomed, v2: improvements in exposition and some
details added to last sectio
On Non-Abelian Symplectic Cutting
We discuss symplectic cutting for Hamiltonian actions of non-Abelian compact
groups. By using a degeneration based on the Vinberg monoid we give, in good
cases, a global quotient description of a surgery construction introduced by
Woodward and Meinrenken, and show it can be interpreted in algebro-geometric
terms. A key ingredient is the `universal cut' of the cotangent bundle of the
group itself, which is identified with a moduli space of framed bundles on
chains of projective lines recently introduced by the authors.Comment: Various edits made, to appear in Transformation Groups. 28 pages, 8
figure
Multigraded Castelnuovo-Mumford Regularity
We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated
by toric geometry, we work with modules over a polynomial ring graded by a
finitely generated abelian group. As in the standard graded case, our
definition of multigraded regularity involves the vanishing of graded
components of local cohomology. We establish the key properties of regularity:
its connection with the minimal generators of a module and its behavior in
exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove
that its multigraded regularity bounds the equations that cut out the
associated subvariety. We also provide a criterion for testing if an ample line
bundle on X gives a projectively normal embedding.Comment: 30 pages, 5 figure
Global Structure of Moduli Space for BPS Walls
We study the global structure of the moduli space of BPS walls in the Higgs
branch of supersymmetric theories with eight supercharges. We examine the
structure in the neighborhood of a special Lagrangian submanifold M, and find
that the dimension of the moduli space can be larger than that naively
suggested by the index theorem, contrary to previous examples of BPS solitons.
We investigate BPS wall solutions in an explicit example of M using Abelian
gauge theory. Its Higgs branch turns out to contain several special Lagrangian
submanifolds including M. We show that the total moduli space of BPS walls is
the union of these submanifolds. We also find interesting dynamics between BPS
walls as a byproduct of the analysis. Namely, mutual repulsion and attraction
between BPS walls sometimes forbid a movement of a wall and lock it in a
certain position; we also find that a pair of walls can transmute to another
pair of walls with different tension after they pass through.Comment: 42 pages, 11 figures; a few comments adde
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