1,250 research outputs found
Quantum cohomology of the Hilbert scheme of points on A_n-resolutions
We determine the two-point invariants of the equivariant quantum cohomology
of the Hilbert scheme of points of surface resolutions associated to type A_n
singularities. The operators encoding these invariants are expressed in terms
of the action of the affine Lie algebra \hat{gl}(n+1) on its basic
representation. Assuming a certain nondegeneracy conjecture, these operators
determine the full structure of the quantum cohomology ring. A relationship is
proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas
theories of A_n x P^1. We close with a discussion of the monodromy properties
of the associated quantum differential equation and a generalization to
singularities of type D and E.Comment: 37 pages, 2 figures; typos are correcte
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
Calabi-Yau manifolds from pairs of non-compact Calabi-Yau manifolds
Most of Calabi-Yau manifolds that have been considered by physicists are
complete intersection Calabi-Yau manifolds of toric varieties or some quotients
of product types. Purpose of this paper is to introduce a different and rather
new kind of construction method of Calabi-Yau manifolds by pasting two
non-compact Calabi-Yau manifolds. We will also in some details explain a
curious and mysterious similarity with construction of some -manifolds
(also called Joyce manifolds), which are base spaces for M-theory.Comment: 10 pages. Accepted for publication in JHE
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