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551
Homological Algebra of Mirror Symmetry
 in Proceedings of the International Congress of Mathematicians
, 1994
"... Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
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Cited by 529 (3 self)
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Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual CalabiYau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on CalabiYau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated
Perturbative gauge theory as a string theory in twistor space
 Commun. Math. Phys
"... Perturbative scattering amplitudes in YangMills theory have many unexpected properties, such as holomorphy of the maximally helicity violating amplitudes. To interpret these results, we Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed ..."
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Cited by 388 (0 self)
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Perturbative scattering amplitudes in YangMills theory have many unexpected properties, such as holomorphy of the maximally helicity violating amplitudes. To interpret these results, we Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves. This in turn is apparently a consequence of an equivalence between the perturbative expansion of N = 4 super YangMills theory and the Dinstanton expansion of a certain string theory, namely the topological B model whose target space is the CalabiYau supermanifold CP 34. December
On the gauge theory/geometry correspondence
 Adv. Theor. Math. Phys
, 1999
"... The ’t Hooft expansion of SU(N) ChernSimons theory on S3 is proposed to be exactly dual to the topological closed string theory on the S2 blow up of the conifold geometry. The Bfield on the S2 has magnitude Ngs = λ, the ’t Hooft coupling. We are able to make a number of checks, such as finding exa ..."
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Cited by 280 (37 self)
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The ’t Hooft expansion of SU(N) ChernSimons theory on S3 is proposed to be exactly dual to the topological closed string theory on the S2 blow up of the conifold geometry. The Bfield on the S2 has magnitude Ngs = λ, the ’t Hooft coupling. We are able to make a number of checks, such as finding exact agreement at the level of the partition function computed on both sides for arbitrary λ and to all orders in 1/N. Moreover, it seems possible to derive this correspondence from a linear sigma model description of the conifold. We propose a picture whereby a perturbative Dbrane description, in terms of holes in the closed string worldsheet, arises automatically from the coexistence of two phases in the underlying U(1) gauge theory. This approach holds promise for a derivation of the AdS/CFT correspondence.
Orientifolds and Mirror symmetry
, 2003
"... We study parity symmetries and crosscap states in classes of N = 2 supersymmetric quantum field theories in 1+1 dimensions, including nonlinear sigma models, gauged WZW models, LandauGinzburg models, and linear sigma models. The parity anomaly and its cancellation play important roles in many of t ..."
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Cited by 277 (14 self)
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We study parity symmetries and crosscap states in classes of N = 2 supersymmetric quantum field theories in 1+1 dimensions, including nonlinear sigma models, gauged WZW models, LandauGinzburg models, and linear sigma models. The parity anomaly and its cancellation play important roles in many of them. The case of the N = 2 minimal model are studied in RCFT, and LG models. We also identify mirror pairs of orientifolds, extending the correspondence between symplectic geometry and algebraic geometry by including unorientable worldsheets. Through the analysis in various models and comparison in the overlapping regimes, we obtain a global picture of orientifolds and Dbranes.
Superstrings and topological strings at large
 N”, J. Math. Phys
"... We embed the large N ChernSimons/topological string duality in ordinary superstrings. This corresponds to a large N duality between generalized gauge systems with N = 1 supersymmetry in 4 dimensions and superstrings propagating on noncompact CalabiYau manifolds with certain fluxes turned on. We a ..."
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Cited by 258 (27 self)
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We embed the large N ChernSimons/topological string duality in ordinary superstrings. This corresponds to a large N duality between generalized gauge systems with N = 1 supersymmetry in 4 dimensions and superstrings propagating on noncompact CalabiYau manifolds with certain fluxes turned on. We also show that in a particular limit of the N = 1 gauge theory system, certain superpotential terms in the N = 1 system (including deformations if spacetime is noncommutative) are captured to all orders in 1/N by the amplitudes of noncritical bosonic strings propagating on a circle with selfdual radius. We also consider Dbrane/antiDbrane system wrapped over vanishing cycles of compact CalabiYau manifolds and argue that at large N they induce a shift in the background to a topologically distinct CalabiYau, which we identify as the ground state system of the Brane/antiBrane system. August
Feynman diagrams and lowdimensional topology
, 2006
"... We shall describe a program here relating Feynman diagrams, topology of manifolds, homotopical algebra, noncommutative geometry and several kinds of “topological physics”. The text below consists of 3 parts. The first two parts (topological sigma model and ChernSimons theory) are formally independ ..."
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Cited by 237 (3 self)
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We shall describe a program here relating Feynman diagrams, topology of manifolds, homotopical algebra, noncommutative geometry and several kinds of “topological physics”. The text below consists of 3 parts. The first two parts (topological sigma model and ChernSimons theory) are formally independent and could be read separately. The third part describes the common algebraic background of both theories.
N=4 topological strings
 Nucl. Phys. B
, 1995
"... We show how to make a topological string theory starting from an N = 4 superconformal theory. The critical dimension for this theory is ĉ = 2 (c = 6). It is shown that superstrings (in both the RNS and GS formulations) and critical N = 2 strings are special cases of this topological theory. Applicat ..."
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Cited by 226 (23 self)
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We show how to make a topological string theory starting from an N = 4 superconformal theory. The critical dimension for this theory is ĉ = 2 (c = 6). It is shown that superstrings (in both the RNS and GS formulations) and critical N = 2 strings are special cases of this topological theory. Applications for this new topological theory include: 1) Proving the vanishing to all orders of all scattering amplitudes for the selfdual N = 2 string with flat background, with the exception of the threepoint function and the closedstring partition function; 2) Showing that the topological partition function of the N = 2 string on the K3 background may be interpreted as computing the superpotential in harmonic superspace generated upon compactification of type II superstrings from 10 to 6 dimensions; and 3) Providing a new prescription for calculating superstring amplitudes which appears to be free of totalderivative ambiguities. July
Mirror symmetry, D–branes and counting holomorphic discs
, 2000
"... We consider a class of special Lagrangian subspaces of CalabiYau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of holomorphic disc instantons ending on the Lagrangian subman ..."
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Cited by 206 (22 self)
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We consider a class of special Lagrangian subspaces of CalabiYau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of holomorphic disc instantons ending on the Lagrangian submanifold to the classical AbelJacobi map on the mirror. We recover some results already anticipated as well as obtain some highly nontrivial new predictions.
MHV vertices and tree amplitudes in gauge theory
 JHEP 0409
"... As an alternative to the usual Feynman graphs, tree amplitudes in YangMills theory can be constructed from tree graphs in which the vertices are tree level MHV scattering amplitudes, continued off shell in a particular fashion. The formalism leads to new and relatively simple formulas for many ampl ..."
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Cited by 177 (8 self)
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As an alternative to the usual Feynman graphs, tree amplitudes in YangMills theory can be constructed from tree graphs in which the vertices are tree level MHV scattering amplitudes, continued off shell in a particular fashion. The formalism leads to new and relatively simple formulas for many amplitudes, and can be heuristically derived from twistor space. March