104 research outputs found
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
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
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
Open Gromov-Witten Invariants of Toric Calabi-Yau 3-Folds
We present a proof of the mirror conjecture of Aganagic-Vafa
[arXiv:hep-th/0012041] and Aganagic-Klemm-Vafa [arXiv:hep-th/0105045] on disk
enumeration in toric Calabi-Yau 3-folds for all smooth semi-projective toric
Calabi-Yau 3-folds. We consider both inner and outer branes, at arbitrary
framing. In particular, we recover previous results on the conjecture for (i)
an inner brane at zero framing in the total space of the canonical line bundle
of the projective plane (Graber-Zaslow [arXiv:hep-th/0109075]), (ii) an outer
brane at arbitrary framing in the resolved conifold (Zhou [arXiv:1001.0447]),
and (iii) an outer brane at zero framing in the total space of the canonical
line bundle of the projective plane (Brini [arXiv:1102.0281, Section 5.3]).Comment: 39 pages, 11 figure
Exact beta function from the holographic loop equation of large-N QCD_4
We construct and study a previously defined quantum holographic effective
action whose critical equation implies the holographic loop equation of large-N
QCD_4 for planar self-avoiding loops in a certain regularization scheme. We
extract from the effective action the exact beta function in the given scheme.
For the Wilsonean coupling constant the beta function is exacly one loop and
the first coefficient agrees with its value in perturbation theory. For the
canonical coupling constant the exact beta function has a NSVZ form and the
first two coefficients agree with their value in perturbation theory.Comment: 42 pages, latex. The exponent of the Vandermonde determinant in the
quantum effective action has been changed, because it has been employed a
holomorphic rather than a hermitean resolution of identity in the functional
integral. Beta function unchanged. New explanations and references added,
typos correcte
Knaster's problem for -symmetric subsets of the sphere
We prove a Knaster-type result for orbits of the group in
, calculating the Euler class obstruction. Among the consequences
are: a result about inscribing skew crosspolytopes in hypersurfaces in , and a result about equipartition of a measures in
by -symmetric convex fans
Poisson-de Rham homology of hypertoric varieties and nilpotent cones
We prove a conjecture of Etingof and the second author for hypertoric
varieties, that the Poisson-de Rham homology of a unimodular hypertoric cone is
isomorphic to the de Rham cohomology of its hypertoric resolution. More
generally, we prove that this conjecture holds for an arbitrary conical variety
admitting a symplectic resolution if and only if it holds in degree zero for
all normal slices to symplectic leaves.
The Poisson-de Rham homology of a Poisson cone inherits a second grading. In
the hypertoric case, we compute the resulting 2-variable Poisson-de
Rham-Poincare polynomial, and prove that it is equal to a specialization of an
enrichment of the Tutte polynomial of a matroid that was introduced by Denham.
We also compute this polynomial for S3-varieties of type A in terms of Kostka
polynomials, modulo a previous conjecture of the first author, and we give a
conjectural answer for nilpotent cones in arbitrary type, which we prove in
rank less than or equal to 2.Comment: 25 page
Gopakumar-Vafa invariants via vanishing cycles
In this paper, we propose an ansatz for defining Gopakumar-Vafa invariants of
Calabi-Yau threefolds, using perverse sheaves of vanishing cycles. Our proposal
is a modification of a recent approach of Kiem-Li, which is itself based on
earlier ideas of Hosono-Saito-Takahashi. We conjecture that these invariants
are equivalent to other curve-counting theories such as Gromov-Witten theory
and Pandharipande-Thomas theory. Our main theorem is that, for local surfaces,
our invariants agree with PT invariants for irreducible one-cycles. We also
give a counter-example to the Kiem-Li conjectures, where our invariants match
the predicted answer. Finally, we give examples where our invariant matches the
expected answer in cases where the cycle is non-reduced, non-planar, or
non-primitive.Comment: 63 pages, many improvements of the exposition following referee
comments, final version to appear in Inventione
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