Results 1 - 10 of 14

J-holomorphic curves, moment maps, and invariants of Hamiltonian group actions

by Kai Cieliebak, Ana Rita Gaio, Dietmar A. Salamon , 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 Cauchy-Riemann operator, the curvature, and the moment map (see (17) belo ..."
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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 Cauchy-Riemann 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 Atiyah-Floer 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 anti-self-dual instantons over the 4-manifold 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 anti-self-duality 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...
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Duistermaat-Heckman measures and moduli spaces of flat bundles over surfaces

by A. Alekseev, E. Meinrenken, C. Woodward , 2001
"... We introduce Liouville measures and Duistermaat-Heckman measures for Hamiltonian group actions with group valued moment maps. The theory is illustrated by applications to moduli spaces of flat bundles on surfaces. ..."
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We introduce Liouville measures and Duistermaat-Heckman measures for Hamiltonian group actions with group valued moment maps. The theory is illustrated by applications to moduli spaces of flat bundles on surfaces.
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A spin decomposition of the Verlinde formulas for type A modular categories, preprint

by Christian Blanchet , 2001
"... additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We discuss reductions and refinements of these formulas for modular cate ..."
Abstract - Cited by 11 (2 self) - Add to MetaCart
additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We discuss reductions and refinements of these formulas for modular categories related with SU(N). Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer. We introduce here the notion of a spin modular category, and give the proof of the decomposition theorem in this general context. Given a simple, simply connected complex Lie group G, the Verlinde formula [37] is a combinatorial function VG: (K, g) ↦ → VG(K, g) associated with G (here the integers K and g are respectively the level and the genus). In conformal field theory this formula gives the dimension
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Heat kernels, symplectic geometry, moduli spaces and finite groups

by Kefeng Liu - Surveys in Differential Geometry 5 , 1999
"... In this note we want to discuss some applications of heat kernels in symplectic geometry, moduli spaces and finite groups. More precisely we will prove the nonabelian localization formula in symplectic geometry, derive formulas for the symplectic volume ..."
Abstract - Cited by 11 (3 self) - Add to MetaCart
In this note we want to discuss some applications of heat kernels in symplectic geometry, moduli spaces and finite groups. More precisely we will prove the nonabelian localization formula in symplectic geometry, derive formulas for the symplectic volume
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Trace functionals on non-commutative deformations of moduli spaces of flat connections

by Philippe Roche, András Szenes , 8
"... Let G be a compact connected and simply connected Lie group, and Σ be a compact topological Riemann surface with a point p marked on it. One can associate to this data the moduli space of flat G connections on the punctured Riemann surface Σ denoted by M G = M G [Σp]. This ..."
Abstract - Cited by 8 (1 self) - Add to MetaCart
Let G be a compact connected and simply connected Lie group, and Σ be a compact topological Riemann surface with a point p marked on it. One can associate to this data the moduli space of flat G connections on the punctured Riemann surface Σ denoted by M G = M G [Σp]. This
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A residue theorem for rational trigonometric sums and Verlinde’s formula

by András Szenes - Duke Math. J
"... We present a compact formula computing rational trigonometric sums. Such sums appeared in the work of E. Verlinde on the dimension of conformal blocks in Wess-Zumino-Witten (WZW) theory. As an application, we show that a formula of J.-M. Bismut and F. Labourie for the Riemann-Roch numbers of moduli ..."
Abstract - Cited by 6 (1 self) - Add to MetaCart
We present a compact formula computing rational trigonometric sums. Such sums appeared in the work of E. Verlinde on the dimension of conformal blocks in Wess-Zumino-Witten (WZW) theory. As an application, we show that a formula of J.-M. Bismut and F. Labourie for the Riemann-Roch numbers of moduli spaces of flat connections on a Riemann surface coincides with Verlinde’s expression. 1.
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Formulas of Verlinde type for non-simply connected groups

by A. Alekseev, E. Meinrenken, C. Woodward
"... Abstract. We derive Verlinde’s formula from the fixed point formula for loop groups proved in the companion paper [FP], and extend it to compact, connected groups that are not necessarily simply-connected. 1. ..."
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Abstract. We derive Verlinde’s formula from the fixed point formula for loop groups proved in the companion paper [FP], and extend it to compact, connected groups that are not necessarily simply-connected. 1.
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Counts of maps to Grassmannians and intersections on the moduli space of bundles, AG/0602335

by Alina Marian, Dragos Oprea
"... ABSTRACT. We show that intersection numbers on the moduli space of stable bundles of coprime rank and degree over a smooth complex curve can be recovered as highestdegree asymptotics in formulas of Vafa-Intriligator type. In particular, we explicitly evaluate all intersection numbers appearing in th ..."
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ABSTRACT. We show that intersection numbers on the moduli space of stable bundles of coprime rank and degree over a smooth complex curve can be recovered as highestdegree asymptotics in formulas of Vafa-Intriligator type. In particular, we explicitly evaluate all intersection numbers appearing in the Verlinde formula. Our results are in agreement with previous computations of Witten, Jeffrey-Kirwan and Liu. Moreover, we prove the vanishing of certain intersections on a suitable Quot scheme which can be interpreted as giving equations between counts of maps to the Grassmannian. 1.
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Mathematical results inspired by physics

by Kefeng Liu - Proc. ICM 2002
"... I will discuss results of three different types in geometry and topology. (1) General vanishing and rigidity theorems of elliptic genera proved by using modular forms, Kac-Moody algebras and vertex operator algebras. (2) The computations of intersection numbers of the moduli spaces of flat connectio ..."
Abstract - Cited by 2 (2 self) - Add to MetaCart
I will discuss results of three different types in geometry and topology. (1) General vanishing and rigidity theorems of elliptic genera proved by using modular forms, Kac-Moody algebras and vertex operator algebras. (2) The computations of intersection numbers of the moduli spaces of flat connections on a Riemann surface by using heat kernels. (3) The mirror principle about counting curves in Calabi-Yau and general projective manifolds by using hypergeometric series.
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ICM 2002 · Vol. III · 1–3 Mathematical Results Inspired by Physics

by Kefeng Liu
"... I will discuss results of three different types in geometry and topology. (1) General vanishing and rigidity theorems of elliptic genera proved by using modular forms, Kac-Moody algebras and vertex operator algebras. (2) The computations of intersection numbers of the moduli spaces of flat connectio ..."
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I will discuss results of three different types in geometry and topology. (1) General vanishing and rigidity theorems of elliptic genera proved by using modular forms, Kac-Moody algebras and vertex operator algebras. (2) The computations of intersection numbers of the moduli spaces of flat connections on a Riemann surface by using heat kernels. (3) The mirror principle about counting curves in Calabi-Yau and general projective manifolds by using hypergeometric series.
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