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59
Crystal Melting and Toric CalabiYau Manifolds
"... We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric CalabiYau threefold. The threedimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low ene ..."
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Cited by 30 (7 self)
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We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric CalabiYau threefold. The threedimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. This generalizes the earlier results on the BPS state counting to an arbitrary noncompact toric CalabiYau manifold. We point out that a proper understanding of the relation between the topological In type IIA superstring theory, supersymmetric bound states of D branes wrapping holomorphic cycles on a CalabiYau manifold give rise to BPS particles in four dimensions. In the past few years, remarkable connections have been found between the counting of such bound states and the topological string theory:
Computation of open GromovWitten invariants for toric CalabiYau 3folds by topological recursion, a proof of the BKMP conjecture
, 2013
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The 3fold vertex via stable pairs
"... Abstract. The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3folds. We evaluate the equivariant vertex for stable pairs on toric 3folds in terms of weighted box counting. In the toric CalabiYau case, the result simplifies to a new form of pure box coun ..."
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Cited by 20 (5 self)
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Abstract. The theory of stable pairs in the derived category yields an enumerative geometry of curves in 3folds. We evaluate the equivariant vertex for stable pairs on toric 3folds in terms of weighted box counting. In the toric CalabiYau case, the result simplifies to a new form of pure box counting. The conjectural equivalence with the DT vertex predicts remarkable identities. The equivariant vertex governs primary insertions in the theory of stable pairs for toric varieties. We consider also the descendent vertex and conjecture the complete rationality of the descendent
GromovWitten/Pairs correspondence for the quintic 3fold
, 2012
"... We use the GromovWitten/Pairs descendent correspondence for toric 3folds and degeneration arguments to establish the GW/P correspondence for several compact CalabiYau 3folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of ..."
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Cited by 18 (6 self)
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We use the GromovWitten/Pairs descendent correspondence for toric 3folds and degeneration arguments to establish the GW/P correspondence for several compact CalabiYau 3folds (including all CY complete intersections in products of projective spaces). A crucial aspect of the proof is the study of the GW/P correspondence for descendents in relative geometries. Projective bundles over surfaces relative to a section play a special role. The GW/P correspondence for CalabiYau complete intersections provides a structure result for the GromovWitten invariants in a fixed curve class. After change of variables, the GromovWitten series is a rational function in the variable −q = e iu invariant under q ↔ q −1.
Localization and gluing of orbifold amplitudes: the GromovWitten orbifold vertex
, 2012
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Wall crossing and Mtheory
, 2009
"... We study BPS bound states of D0 and D2 branes on a single D6 brane wrapping a CalabiYau 3fold X. When X has no compact 4cyles, the BPS bound states are organized into a free field Fock space, whose generators correspond to BPS states of spinning M2 branes in Mtheory compactified down to 5 dimens ..."
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Cited by 12 (5 self)
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We study BPS bound states of D0 and D2 branes on a single D6 brane wrapping a CalabiYau 3fold X. When X has no compact 4cyles, the BPS bound states are organized into a free field Fock space, whose generators correspond to BPS states of spinning M2 branes in Mtheory compactified down to 5 dimensions by a CalabiYau 3fold X. The generating function of the Dbrane bound states is expressed as a reduction of the square of the topological string
A matrix model for plane partitions and (T)ASEP
, 2009
"... We construct a matrix model equivalent (exactly, not asymptotically), to the random plane partition model, with almost arbitrary boundary conditions. Equivalently, it is also a random matrix model for a (T)ASEP with arbitrary boundary conditions. Using the known solution of matrix models, this metho ..."
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Cited by 12 (2 self)
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We construct a matrix model equivalent (exactly, not asymptotically), to the random plane partition model, with almost arbitrary boundary conditions. Equivalently, it is also a random matrix model for a (T)ASEP with arbitrary boundary conditions. Using the known solution of matrix models, this method allows to find the large size asymptotic expansion of plane partitions, to ALL orders. It also allows to describe several universal regimes. On the algebraic geometry point of view, this gives the GromovWitten invariants of C³ with branes, i.e. the topological vertex, in terms of the symplectic invariants of the mirror’s spectral curve.