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117
Equivariant Cohomology, Koszul Duality, and the Localization Theorem
 Invent. Math
, 1998
"... This paper concerns three aspects of the action of a compact group K on a space ..."
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Cited by 147 (4 self)
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This paper concerns three aspects of the action of a compact group K on a space
SasakiEinstein manifolds and volume minimisation
, 2006
"... We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian ..."
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Cited by 53 (2 self)
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We study a variational problem whose critical point determines the Reeb vector field for a Sasaki–Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein–Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi–Yau cone M, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaat– Heckman formula and also to a limit of a certain equivariant index on M that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of any Sasaki–Einstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension n = 3 these results provide, via AdS/CFT, the geometric counterpart of a–maximisation in four dimensional superconformal field theories. We also show that our variational problem dynamically sets to zero the Futaki
On RiemannRoch formulas for multiplicities
 J. Amer. Math. Soc
"... A Theorem due to Guillemin and Sternberg [17] about geometric quantization of Hamiltonian actions of compact Lie groups G on compact Kähler manifolds says that the dimension of the Ginvariant subspace is equal to the RiemannRoch number of the symplectically reduced space. Combined with the shiftin ..."
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Cited by 38 (2 self)
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A Theorem due to Guillemin and Sternberg [17] about geometric quantization of Hamiltonian actions of compact Lie groups G on compact Kähler manifolds says that the dimension of the Ginvariant subspace is equal to the RiemannRoch number of the symplectically reduced space. Combined with the shiftingtrick, this gives explicit formulas for the multiplicities of the various irreducible components. One of the assumptions of the Theorem is that the reduction is regular, so that the reduced space is a smooth symplectic manifold. In this paper, we prove a generalization of this result to the case where the reduced space may have orbifold singularities. Our proof uses localization techniques from equivariant cohomology, and relies in particular on recent work of JeffreyKirwan [20] and Guillemin [14]. Since there are no complex geometry arguments involved, the result also extends to non Kählerian settings. 1
The Spectrum of Coupled Random Matrices
, 1999
"... this paper, we will use the following operators e ..."
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Cited by 37 (10 self)
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this paper, we will use the following operators e
Periodic Hamiltonian flows on four dimensional manifolds
, 1998
"... We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian S 1spaces. Additionally, we show that all these spaces are Kähler, that every such space is obtained from a simple model by a sequence of symplectic blowups, and that if th ..."
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Cited by 34 (7 self)
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We classify the periodic Hamiltonian flows on compact four dimensional symplectic manifolds up to isomorphism of Hamiltonian S 1spaces. Additionally, we show that all these spaces are Kähler, that every such space is obtained from a simple model by a sequence of symplectic blowups, and that if the fixed points are isolated then the space is a toric variety.
Dual giant gravitons in SasakiEinstein backgrounds
"... We study the dynamics of BPS D3–branes wrapped on a three–sphere in AdS5 × L, so–called dual giant gravitons, where L is a Sasakian five–manifold. The phase space of these configurations is the symplectic cone X over L, and geometric quantisation naturally produces a Hilbert space of L 2 –normalisab ..."
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Cited by 22 (4 self)
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We study the dynamics of BPS D3–branes wrapped on a three–sphere in AdS5 × L, so–called dual giant gravitons, where L is a Sasakian five–manifold. The phase space of these configurations is the symplectic cone X over L, and geometric quantisation naturally produces a Hilbert space of L 2 –normalisable holomorphic functions on X, whose states are dual to scalar chiral BPS operators in the dual superconformal field theory. We define classical and quantum partition functions and relate them to earlier mathematical constructions by the authors and S.–T. Yau, hepth/0603021. In particular, a Sasaki–Einstein metric then minimises an entropy function associated with the D3–brane. Finally, we introduce a grand canonical partition function that counts multiple dual giant gravitons. This is related simply to the index–character of the above reference, and provides an elegant method for counting multi–trace scalar BPS operators in the dual superconformal field theory. Contents
DuistermaatHeckman measures and moduli spaces of flat bundles over surfaces, Geom
 Funct. Anal
"... Abstract. We introduce Liouville measures and DuistermaatHeckman measures for Hamiltonian group actions with group valued moment maps. The theory is illustrated by applications to moduli spaces of flat bundles on surfaces. 1. ..."
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Cited by 21 (5 self)
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Abstract. We introduce Liouville measures and DuistermaatHeckman measures for Hamiltonian group actions with group valued moment maps. The theory is illustrated by applications to moduli spaces of flat bundles on surfaces. 1.
Localization and Diagonalization  A Review of Functional Integral Techniques for LowDimensional Gauge Theories and Topological Field Theories
"... We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and lowdimensional gauge theories. These are the functional integral counterparts of the MathaiQuillen formal ..."
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Cited by 20 (1 self)
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We review localization techniques for functional integrals which have recently been used to perform calculations in and gain insight into the structure of certain topological field theories and lowdimensional gauge theories. These are the functional integral counterparts of the MathaiQuillen formalism, the DuistermaatHeckman theorem, and the Weyl integral formula respectively. In each case, we first introduce the necessary mathematical background (Euler classes of vector bundles, equivariant cohomology, topology of Lie groups), and describe the finite dimensional integration formulae. We then discuss some applications to path integrals and give an overview of the relevant literature. The applications we deal with include supersymmetric quantum mechanics, co...