391 research outputs found
Formulae of one-partition and two-partition Hodge integrals
Based on the duality between open-string theory on noncompact Calabi-Yau
threefolds and Chern-Simons theory on three manifolds, M Marino and C Vafa
conjectured a formula of one-partition Hodge integrals in term of invariants of
the unknot (hep-th/0108064). Many Hodge integral identities, including the
lambda_g conjecture and the ELSV formula, can be obtained by taking limits of
the Marino-Vafa formula.
Motivated by the Marino-Vafa formula and formula of Gromov-Witten invariants
of local toric Calabi-Yau threefolds predicted by physicists, J Zhou
conjectured a formula of two-partition Hodge integrals in terms of invariants
of the Hopf link (math.AG/0310282) and used it to justify physicists'
predictions (math.AG/0310283).
In this expository article, we describe proofs and applications of these two
formulae of Hodge integrals based on joint works of K Liu, J Zhou and the
author (math.AG/0306257, math.AG/0306434, math.AG/0308015, math.AG/0310272).
This is an expansion of the author's talk of the same title at the BIRS
workshop: "The Interaction of Finite Type and Gromov-Witten Invariants",
November 15--20, 2003.Comment: This is the version published by Geometry & Topology Monographs on 22
April 200
Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc
In this paper, we present foundational material towards the development of a
rigorous enumerative theory of stable maps with Lagrangian boundary conditions,
ie stable maps from bordered Riemann surfaces to a symplectic manifold, such
that the boundary maps to a Lagrangian submanifold. Our main application is to
a situation where our proposed theory leads to a well-defined algebro-geometric
computation very similar to well-known localization techniques in Gromov-Witten
theory. In particular, our computation of the invariants for multiple covers of
a generic disc bounding a special Lagrangian submanifold in a Calabi-Yau
threefold agrees completely with the original predictions of Ooguri and Vafa
based on string duality. Our proposed invariants depend more generally on a
discrete parameter which came to light in the work of Aganagic, Klemm, and Vafa
which was also based on duality, and our more general calculations agree with
theirs up to sign.Comment: This is the version published by Geometry & Topology Monographs on 22
April 200
The Yang-Mills equations over Klein surfaces
Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed
topological type admit a presentation as Lagrangian quotients, and can be
embedded into the symplectic quotient corresponding to the moduli variety of
semi-stable holomorphic vector bundles of fixed rank and degree on a smooth
complex projective curve. From the algebraic point of view, these Lagrangian
quotients are connected sets of real points inside a complex moduli variety
endowed with a real structure; when the rank and the degree are coprime, they
are in fact the connected components of the fixed-point set of the real
structure. This presentation as a quotient enables us to generalize the methods
of Atiyah and Bott to a setting with involutions, and compute the mod 2
Poincare polynomials of these moduli spaces in the coprime case. We also
compute the mod 2 Poincare series of moduli stacks of all real and quaternionic
vector bundles of a fixed topological type. As an application of our
computations, we give new examples of maximal real algebraic varieties.Comment: Final version, 72 pages; formulae in the quaternionic, n>0 case
corrected; proof of Theorem 1.3 revised; references adde
Yang-Mills Connections on Nonorientable Surfaces
In "The Yang-Mills equations over Riemann surfaces", Atiyah and Bott studied
Yang-Mills functional over a Riemann surface from the point of view of Morse
theory. We generalize their study to all closed, compact, connected, possibly
nonorientable surfaces. We introduce the notion of "super central extension" of
the fundamental group of a surface. It is the central extension when the
surface is orientable. We establish a precise correspondence between Yang-Mills
connections and representations of super central extension. Knowing this exact
correspondence, we work mainly at the level of representation varieties which
are finite dimensional instead of the level of strata which are infinite
dimensional.Comment: 45 pages, 1 figur
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