242,464 research outputs found
Curve-counting invariants for crepant resolutions
We construct curve counting invariants for a Calabi-Yau threefold
equipped with a dominant birational morphism . Our invariants
generalize the stable pair invariants of Pandharipande and Thomas which occur
for the case when is the identity. Our main result is a PT/DT-type
formula relating the partition function of our invariants to the
Donaldson-Thomas partition function in the case when is a crepant
resolution of , the coarse space of a Calabi-Yau orbifold
satisfying the hard Lefschetz condition. In this case, our partition function
is equal to the Pandharipande-Thomas partition function of the orbifold
. Our methods include defining a new notion of stability for
sheaves which depends on the morphism . Our notion generalizes slope
stability which is recovered in the case where is the identity on .
Our proof is a generalization of Bridgeland's proof of the PT/DT correspondence
via the Hall algebra and Joyce's integration map.Comment: In this version, Jim Bryan has been added as an author and the
required boundedness result for our stability condition has been added. arXiv
admin note: text overlap with arXiv:1002.4374 by other author
Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures
We prove two identities of Hall-Littlewood polynomials, which appeared
recently in a paper by two of the authors. We also conjecture, and in some
cases prove, new identities which relate infinite sums of symmetric polynomials
and partition functions associated with symmetry classes of alternating sign
matrices. These identities generalize those already found in our earlier paper,
via the introduction of additional parameters. The left hand side of each of
our identities is a simple refinement of a relevant Cauchy or Littlewood
identity. The right hand side of each identity is (one of the two factors
present in) the partition function of the six-vertex model on a relevant
domain.Comment: 34 pages, 14 figure
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