Results 1  10
of
48
Lectures on 2D YangMills Theory, Equivariant Cohomology and Topological Field Theories
, 1996
"... These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying ..."
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Cited by 97 (7 self)
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These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals.
Quantum Cohomology and Virasoro Algebra
"... We propose that the Virasoro algebra controls quantum cohomologies of general Fano manifolds M (c 1 (M) ? 0) and determines their partition functions at all genera. We construct Virasoro operators in the case of complex projective spaces and show that they reproduce the results of KontsevichManin, ..."
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Cited by 40 (3 self)
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We propose that the Virasoro algebra controls quantum cohomologies of general Fano manifolds M (c 1 (M) ? 0) and determines their partition functions at all genera. We construct Virasoro operators in the case of complex projective spaces and show that they reproduce the results of KontsevichManin, Getzler etc. on the genus0,1 instanton numbers. We also construct Virasoro operators for a wider class of Fano varieties. The central charge of the algebra is equal to Ø(M ), the Euler characteristic of the manifold M . As is well known, the quantum cohomology of a (symplectic) manifold M is described by the topological oemodel with M being its target space [1]. The partition function of the topological oemodel is given by the sum over holomorphic maps from a Riemann surface \Sigma g to M . When the degree d of the map is zero (constant map), correlation functions of the oemodel reproduces the classical intersection numbers among homology cycles of M . When the degree is nonzero, h...
Jholomorphic curves, moment maps, and invariants of Hamiltonian group actions
, 1999
"... This paper outlines the construction of invariants of Hamiltonian group actions on symplectic manifolds. The invariants are derived from the solutions of a nonlinear rst order elliptic partial dierential equation involving the CauchyRiemann operator, the curvature, and the moment map (see (17) belo ..."
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Cited by 35 (5 self)
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This paper outlines the construction of invariants of Hamiltonian group actions on symplectic manifolds. The invariants are derived from the solutions of a nonlinear rst order elliptic partial dierential equation involving the CauchyRiemann operator, the curvature, and the moment map (see (17) below). They are related to the Gromov invariants of the reduced spaces. Our motivation arises from the proof of the AtiyahFloer conjecture in [17, 18, 19] which deals with the relation between holomorphic curves ! M S in the moduli space M S of at connections over a Riemann surface S and antiselfdual instantons over the 4manifold S. In [3] Atiyah and Bott interpret the space M S as a symplectic quotient of the space A S of connections on S by the action of the group G S of gauge transformations. A moment's thought shows that the various terms in the antiselfduality equations over S (see equation (64) below) can be interpreted symplectically. Hence they should give rise to meaningful equations in a context where the space A S is replaced by a nite dimensional symplectic manifold M and the gauge group G S by a compact Lie group G with a Hamiltonian action on M . In this paper 2 we show how the resulting equations give rise to invariants of Hamiltonian group actions. The same adiabatic limit argument as in [19] then leads to a correspondence between these invariants and the Gromov{Witten invariants of the quotient M==G (Conjecture 3.6). This correspondence is the subject of the PhD thesis [27] of the second author. In Section 2 we review the relevant background material about Hamiltonian group actions, gauge theory, equivariant cohomology, and holomorphic curves in symplectic quotients. The heart of this paper is Section 3, where we discuss the equations and the...
Local Mirror Symmetry at Higher Genus
"... We discuss local mirror symmetry for highergenus curves. Specifically, we consider the topological string partition function from higher genus curves contained in a Fano surface within a CalabiYau. Our main example is the local P2 case. The KodairaSpencer theory of gravity, tailored to this local ..."
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Cited by 23 (4 self)
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We discuss local mirror symmetry for highergenus curves. Specifically, we consider the topological string partition function from higher genus curves contained in a Fano surface within a CalabiYau. Our main example is the local P2 case. The KodairaSpencer theory of gravity, tailored to this local geometry, can be solved to compute this partition function. Then, using the results of Gopakumar and Vafa [1] and the local mirror map [2], the partition function can be rewritten in terms of expansion coefficients, which are found to be integers. We verify, through localization calculations in the Amodel, many of these GromovWitten predictions. The integrality is a mystery, mathematically speaking. The asymptotic growth (with degree) of the invariants is analyzed. Some suggestions are made towards an enumerative interpretation, following the BPSstate description of Gopakumar and Vafa.
Extended deformation functors
 Int. Math. Res. Not
"... We introduce a precise notion, in terms of some Schlessinger’s type conditions, of extended deformation functors which is compatible with most of recent ideas in the Derived Deformation Theory (DDT) program and with geometric examples. With this notion we develop the (extended) analogue of Schlessin ..."
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Cited by 17 (9 self)
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We introduce a precise notion, in terms of some Schlessinger’s type conditions, of extended deformation functors which is compatible with most of recent ideas in the Derived Deformation Theory (DDT) program and with geometric examples. With this notion we develop the (extended) analogue of Schlessinger and obstruction theories. The inverse mapping theorem holds for natural transformations of extended deformation functors and all such functors with finite dimensional tangent space are prorepresentable in the homotopy category. Finally we prove that the primary obstruction map induces a structure of graded Lie algebra on the tangent space. Mathematics Subject Classification (1991): 13D10, 14B10, 14D15.
Witten’s top Chern class on the moduli space of higher spin curves, math.AG/0208112, to appear
 in Proceedings of the Workshop on Frobenius manifolds
"... Abstract. We prove that the algebraic Witten’s “top Chern class ” constructed in [9] satisfies the axioms for the spin virtual class formulated in [5]. This paper is a sequel to [9]. Its goal is to verify that the virtual top Chern class c1/r in the Chow group of the moduli space of higher spin curv ..."
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Cited by 14 (1 self)
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Abstract. We prove that the algebraic Witten’s “top Chern class ” constructed in [9] satisfies the axioms for the spin virtual class formulated in [5]. This paper is a sequel to [9]. Its goal is to verify that the virtual top Chern class c1/r in the Chow group of the moduli space of higher spin curves M 1/r g,n, constructed in [9], satisfies all the axioms of spin virtual class formulated in [5]. Hence, according to [5], it gives rise to a cohomological field theory in the sense of KontsevichManin [7]. As was observed in [9], the only nontrivial axioms that have to be checked for the class c1/r are two axioms that we call Vanishing axiom and Ramond factorization axiom. The first of them requires c 1/r to vanish on all the components of the moduli space M 1/r g,n, where one of the markings is equal to r − 1. The second demands vanishing of the pushforward of c 1/r restricted to the components of the moduli space corresponding to the so called Ramond sector, under some natural finite maps. Recall that the virtual top Chern class is a crucial ingredient in the generalized Witten’s conjecture formulated in [10], [11]. The original indextheoretic construction of this
Semiinfinite flags. II. Local and global intersection cohomology of quasimaps’ spaces. Differential topology, infinitedimensional Lie algebras, and applications
, 1999
"... 1.1. This paper is a sequel to [FM]. We will make a free use of notations, conventions and results of loc. cit. One of the main results of the present work is a computation of local IC stalks of the Schubert strata closures in the spaces Z α. We prove that the generating functions of these stalks ar ..."
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Cited by 14 (2 self)
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1.1. This paper is a sequel to [FM]. We will make a free use of notations, conventions and results of loc. cit. One of the main results of the present work is a computation of local IC stalks of the Schubert strata closures in the spaces Z α. We prove that the generating functions of these stalks are given by the generic
On equivariant quantum cohomology
, 1995
"... There is exactly one straight line passing through any two given distinct points; there is exactly one quadratic curve on the complex projective plane ..."
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Cited by 14 (2 self)
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There is exactly one straight line passing through any two given distinct points; there is exactly one quadratic curve on the complex projective plane