Workshop on Quantum Cohomology of Stacks
Feb. 12 - 16, 2007
Abstracts
Linda Chen
Enumerative geometry and Gromov-Witten theory of
stacks
We give an approach to an enumerative geometry problem using
Gromov-Witten theory of stacks. In particular, we compute the
number of
rational plane curves with prescribed tangencies to a smooth
cubic. We
show that certain invariants are enumerative, and obtain a formula for
these numbers. This is joint work with Chuck Cadman.
Weimin Chen
Pseudoholomorphic curves in the context of
orbifolds
One area of applications of Gromov's pseudoholomorphic
curve theory is in the study of symplectic topology itself, which,
for a number of reasons, was particularly powerful in the case of
dimension 4. In this talk we will describe an extension of the theory
to a context where spaces are allowed to have quotient singularities.
Such singular spaces naturally occured, in particular, in problems
involving actions of a compact Lie group. One basic conviction is that
pseudoholomorphic curves in such a singular space encode certain global
information about the singularities of the space, which is useful in
various applications.
Alessandro Chiodo
Mumford formula generalised to rth roots: crepant and r-spin
applications
I will present the generalisation [math/0607324] of Mumford's formula
describing the Chern character of the Hodge bundle. The generalised
formula describes the Chern character of the direct image of universal rth
roots. It has two natural applications: to moduli of r-spin curves and to
quantum cohomology of orbifolds.
Tom Coates
Mirror Symmetry and The Crepant Resolution Conjecture
I will explain how to use mirror symmetry to determine how
genus-zero Gromov--Witten invariants of toric orbifolds are related to
those of their crepant resolutions. A key ingredient is the genus-zero
version of Givental's quantization formalism.
This talk is about joint work with Alessio Corti, Hiroshi Iritani, and
Hsian-Hua Tseng. It introduces material which will be used in the talks
by Corti, Iritani, Ruan, and Tseng.
Alessio Corti
Quantum orbifold cohomology of weak Fano toric DM
stacks
I outline a recipe for computing the small quantum orbifold cohomology
of a weak Fano toric DM stack X. This is completely explicit if X is
Fano and has terminal singularities. In general, the recipe requires the
inversion of a "mirror map". This is joint work with T Coates, H Iritani
and H-H Tseng.
Rebecca Goldin
Chen-Ruan Cohomology for global quotients by abelian Lie groups
Let X be an orbifold obtained as a global quotient of a manifold by a
finite group, which is not necessarily abelian. Fantechi and Gottsche
devised an approach to calculate its orbifold (or Chen-Ruan)
cohomology, using a larger ring with a G-action and calculating its
invariants.
In recent work, Tara Holm, Allen Knutson and I developed a way to
generalize these techniques in certain cases. Let M be an almost
complex manifold with an action by an abelian Lie group T. We define
the inertial cohomology of the pair (M,T). In the case that T acts
with finite stabilizers, this cohomology coincides with the
Fantechi-Gottsche description of Chen-Ruan cohomology. However, in the
case T does not act locally freely, this is a new ring worth studying
in its own right. In particular, when M is a symplectic manifold and T
acts in a Hamiltonian fashion, the inertial cohomology /surjects/ onto
the Chen-Ruan cohomology of the orbifold M//T, the symplectic quotient
of M by T. When M is the cotangent bundle of $C^n$, the inertial
cohomology of (M,T) surjects onto the hyperkahler quotient M////T,
also called a hypertoric variety (this latter result is joint with
Megumi Harada). In both the symplectic and the hyperkahler case, these
surjections lead to new ways of describing Chen-Ruan cohomology for
the orbifolds obtained as quotients.
In this talk, we will present the construction of the inertial
cohomology of (M,T) including its ring structure, show how it is
related to the work of Fantechi and Gottsche, and (briefly) discuss
these surjectivity theorems.
Megumi Harada
The K-theory of symplectic quotients
Let G be a compact connected Lie group, and $(M,\omega)$ a Hamiltonian
G-space with proper moment map $\mu$. A classical theorem of Kirwan
states that there is a surjective ring map $\kappa$ from the
G-equivariant rational cohomology of M surjects onto the ordinary
rational cohomology ring of the symplectic quotient M//G. The Kirwan
surjectivity theorem, in addition to computations of the kernel of
$\kappa$, give powerful methods for explicit computations of the
cohomology rings of symplectic quotients.
We present integral K-theoretic analogues of this theory which
therefore gives methods for computing the integral K-theory of
symplectic quotients. More specifically: (1) we prove a K-theoretic
Kirwan surjectivity theorem; (2) give a relationship between the
kernel of the Kirwan map $\kappa_G$ for a nonabelian Lie group and the
kernel of the Kirwan map $\kappa_T$ for its maximal torus (thus allowing
us to reduce computations to the abelian case; and (3) in the abelian
case, give methods for explicit computations of the kernel of
$\kappa_T$. Our results are K-theoretic analogues of the
rational-cohomological theory developed by Kirwan, Martin,
Tolman-Weitsman, and others.
Tara Holm
Symplectic techniques for computing the cohomology of orbifolds
We present techniques for computing the various cohomology rings
associated to an orbifold X that arises as a symplectic quotient
M//G. Building on the themes developed in Goldin's talk, we first
define an alternative version of the stringy product on the inertial
cohomology of a Hamiltonian T-space. This new product avoids mention
of an obstruction bundle, is clearly associative, and is
combinatorial in nature. We use this and surjectivity results to
give a combinatorial description of the Chen-Ruan cohomology of an
abelian symplectic reduction. We conclude with several examples, and
a brief discussion of coefficient rings. This talk is based on joint
projects with Goldin and Knutson; Sjamaar; and Tolman.
Hiroshi Iritani
Wall-crossing in the quantum cohomology of toric orbifolds
My talk represents joint work with Tom Coates, Alessio Corti and
Hsian-Hua Tseng. We study the change in quantum cohomology under a
wall-crossing of Kaehler classes. Quantum cohomology is considered to be
a family of rings over $H^{1,1}(X)$, the complexified Kaehler moduli
space. Moreover, this family of rings comes from Hodge theory
called "quantum D-module". In the toric case, using an explicit mirror
family, we can construct a global "Kaehler moduli" and a global quantum
D-module over it. This global moduli space has several cusps,
each of which corresponds to a toric orbifold. The different orbifolds
are related by wall-crossings. A neighbourhood of each cusp is the
Kaehler moduli of the corresponding toric orbifold. The Fourier
expansion of the global D-module connection at each cusp gives the genus
zero Gromov-Witten invariants of the corresponding toric orbifold.
This (modulo a mirror conjecture) establishes a quantum cohomology
version of the McKay correspondence (Ruan's conjecture) in case of toric
orbifolds. One interesting observation here is that whereas the family
of rings is globally well-defined, the Frobenius (or flat) structure
associated with the quantum cohomology varies from cusp to cusp.
Dagan Karp
Large N duality and orbifold quotients of the resolved conifold
Large N duality is a conjectural correspondence relating SU(N)
Chern-Simons theory of real 3-manifolds to Gromov-Witten theory of
complex 3-folds. In this talk I'll discuss CS/GW duality for certain
lens spaces, whose duals are (resolutions of) cyclic quotients of the
resolved conifold. Removing the parentheses in the previous sentence
necessitates orbifold Gromov-Witten theory. The (resolved) lens space
duals are toric Calabi-Yau threefolds; hence the method on the GW side
consists of the topological vertex. This confirms a conjecture of
Aganagic-Klemm-Marino-Vafa posed in 2002, and is joint work with Dave
Auckly and Sergiy Koshkin.
Ralph Kaufmann
The global orbifold approach to stringy
geometry
As has been demonstrated in many constructions such as
orbifold cohomology by Fantechi and Goettsche or in orbifold K-theory
by T. Jarvis, T. Kimura and myself, there are certain richer
structures for an orbifold which admits a representation as a
global quotient. We discuss old and new stringy phenomena from this
global orbifold perspective.
Takashi Kimura
Stringy Algebraic Structures and Orbifolds
Associated to a smooth, projective variety with the action of a
finite group
G is its stringy cohomology ring, a G-Frobenius algebra introduced by
Fantechi-Goettsche whose G-coinvariants yield the Chen-Ruan orbifold
cohomology of the quotient orbifold. We contrast the different
definitions of this structure and explain the role of the moduli space of pointed
admissible G-covers. We will also discuss some questions and generalizations
of these constructions.
Etienne Mann
Orbifold quantum cohomology of weighted projective spaces
In 2001, S. Barannikov showed that the Frobenius manifold coming
from the quantum cohomology of the complex projective
space of dimension n is isomorphic to the
Frobenius manifold associated to the Laurent polynomial
$x_{1}+...+x_{n}+(x_{1}...x_{n})^{-1}$
We propose to explain how we can extend this correspondence to
weighted projective space and a certain Laurent polynomial.
Fabio Perroni
Chen-Ruan cohomology of ADE singularities
We study Ruan's "cohomological crepant resolution conjecture" for orbifolds
with transversal ADE singularities. In the $A_n$-case we compute both the
Chen-Ruan cohomology ring $H^*_{\rm CR}([Y])$ and the quantum corrected
cohomology ring $H^*(Z)(q_1,...,q_n)$. The former is achieved in general, the
later up to some additional, technical assumptions. We construct an explicit
isomorphism between $H^*_{\rm CR}([Y])$ and $H^*(Z)(-1)$ in the $A_1$-case,
verifying Ruan's conjecture. In the $A_n$-case, the family
$H^*(Z)(q_1,...,q_n)$ is not defined for $q_1=...=q_n=-1$ as the
conjecture would require. We propose a slightly modified conjecture in
the $A_n$-case which we prove in the $A_2$-case, and for $P(1,3,4,4)$,
by constructing explicit isomorphisms.
Michael Rose
Arithmetic mirror symmetry and l-adic Chen-Ruan
cohomology
The Weil conjectures provide a technique to translate certain mirror
theorems into the context of arithmetic algebraic geometry. I will demonstrate
this strategy and give a survey of results in this direction.
Yongbin Ruan
Quantum birational geometry
In the early day of quantum cohomology era, there was an interesting
question called "naturality of quantum cohomology". Namely,
what are the "morphism"s between symplectic/projective manifolds such
that they induces a homomorphism on quantum cohomology. It was quickly
realized that this is a very deep question and related to birational
geometry. In the mid-90's, a set of conjectures were made to pin down
these relationships. Very little progress was made on these conjectures
over the last ten years, partially due to the fact that there are still
some pieces of information missing from the general
formulation of the conjecture. With the advance in orbifold theory, I
will explain that the B-field and the symplectic transformation are
precisely the missing ingredients.
Hsian-Hua Tseng
Twisted Orbifold Gromov-Witten Invariants in
Genus
Zero
In this talk we will discuss an explicit procedure relating genus-zero
twisted orbifold Gromov-Witten invariants to usual orbifold
Gromov-Witten invariants. We will also discuss its application to the
calculation of genus zero orbifold Gromov-Witten invariants of $C^3/Z_3$.
This is joint work with Tom Coates, Alessio Corti, and Hiroshi
Iritani.
Angelo Vistoli
Essential dimension and algebraic stacks
I will report on joint work with Patrick Brosnan and Zinovy
Reichstein. We extend the notion of "essential dimension" that has
been studied so far for algebraic group, to algebraic stacks. The
problem is the following: given a geometric object X over a field K
(e.g., an algebraic variety), what is the least transcendence degree
of a field of definition of X over the prime field? In other words,
how many independent parameters do we need to define X? We have
complete results for smooth, or stable, curves. Furthermore the
stack-theoretic machinery that we develop can also be applied to the
case of case of algebraic groups, showing for example that the
essential dimension of $Spin_n$ grows exponentially with n.