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FROM EXCEPTIONAL COLLECTIONS TO MOTIVIC DECOMPOSITIONS Via Noncommutative Motives
, 2012
"... Making use of noncommutative motives we relate exceptional collections (and more generally semiorthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X)Q of every smooth and proper DeligneMumford stack X, whose bounded derived category D b (X) of cohere ..."
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Cited by 9 (4 self)
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Making use of noncommutative motives we relate exceptional collections (and more generally semiorthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X)Q of every smooth and proper DeligneMumford stack X, whose bounded derived category D b (X) of coherent schemes admits a full exceptional collection, decomposes into a direct sum of tensor powers of the Lefschetz motive. Examples include projective spaces, quadrics, toric varieties, homogeneous spaces, Fano threefolds, and moduli spaces. On the other hand we prove that if M(X)Q decomposes into a direct direct sum of tensor powers of the Lefschetz motive and moreover D b (X) admits a semiorthogonal decomposition, then the noncommutative motive of each one of the pieces of the semiorthogonal decomposition is a direct sum of ⊗units. As an application we obtain a simplification of Dubrovin’s conjecture.
Profiling the brane drain in a nonsupersymmetric orbifold
 JHEP 0601
, 2006
"... We study Dbranes in a nonsupersymmetric orbifold of type C 2 /Γ, perturbed by a tachyon condensate, using a gauged linear sigma model. The RG flow has both higgs and coulomb branches, and each branch supports different branes. The coulomb branch branes account for the “brane drain ” from the higgs ..."
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Cited by 7 (1 self)
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We study Dbranes in a nonsupersymmetric orbifold of type C 2 /Γ, perturbed by a tachyon condensate, using a gauged linear sigma model. The RG flow has both higgs and coulomb branches, and each branch supports different branes. The coulomb branch branes account for the “brane drain ” from the higgs branch, but their precise relation to fractional branes has hitherto been unknown. Building on the results of hepth/0403016 we construct, in detail, the map between fractional branes and the coulomb/higgs branch branes for two examples in the type 0 theory. This map depends on the phase of the tachyon condensate in a surprising and intricate way. In the mirror LandauGinzburg picture the dependence on the tachyon phase is manifested by discontinuous changes in the shape of the Dbrane. July 20, 2005 1. Introduction and
Stokes Matrix for the Quantum Cohomology of Cubic Surfaces
"... We prove the conjectural relation between the Stokes matrix for the quantum cohomology of X and an exceptional collection generating D b coh(X) when X is a smooth cubic surface. The proof is based on a toric degeneration of a cubic surface, the Givental’s mirror theorem for toric manifolds, and the ..."
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Cited by 4 (0 self)
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We prove the conjectural relation between the Stokes matrix for the quantum cohomology of X and an exceptional collection generating D b coh(X) when X is a smooth cubic surface. The proof is based on a toric degeneration of a cubic surface, the Givental’s mirror theorem for toric manifolds, and the PicardLefschetz theory. 1
Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures. arXiv:1404.6407
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Givental Jfunctions, quantum integrable systems, . . .
, 2014
"... We study 4d N = 2 gauge theories with a codimension two full surface operator, which exhibit a fascinating interplay of supersymmetric gauge theories, equivariant GromovWitten theory and geometric representation theory. For pure YangMills and N = 2 ∗ theory, we describe a full surface operator as ..."
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Cited by 2 (0 self)
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We study 4d N = 2 gauge theories with a codimension two full surface operator, which exhibit a fascinating interplay of supersymmetric gauge theories, equivariant GromovWitten theory and geometric representation theory. For pure YangMills and N = 2 ∗ theory, we describe a full surface operator as the 4d gauge theory coupled to a 2d N = (2, 2) gauge theory. By supersymmetric localizations, we present the exact partition functions of both 4d and 2d theories which satisfy integrable equations. In addition, the form of the structure constants with a semidegenerate field in SL(N,R) WZNW model is predicted from oneloop determinants of 4d gauge theories with a full surface operator via the AGT relation.