Matvei Libine
In this article we give a totally new proof of the integral localization formula for equivariantly closed differential forms (Theorem 7.11 in [BGV]). We restate it here as Theorem 2. This localization formula is very well known, but the author hopes to adapt this proof to obtain a more general result in the future.
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Let G_R be a Lie group acting on an oriented manifold M, and let $ω$ be an equivariantly closed form on M. If both G_R and M are compact, then the integral $\int_M ω$ is given by the fixed point integral localization formula (Theorem 7.11 in [BGV]). Unfortunately, this formula fails when the acting Lie group G_R is not compact: there simply may not be enough fixed points present. A proposed remedy is to modify the action of G_R in such a way that all fixed points are accounted for. Let G_R be a real semisimple Lie group, possibly noncompact. One of the most important examples of equivariantly …
The classical integral localization formula for equivariantly closed forms (Theorem 7.11 in [BGV]) is well-known and requires the acting Lie group to be compact. It is restated here as Theorem 2. In this article we extend this result to NONcompact groups. The main result is Theorem 20. Then, using this generalization, we prove an analogue of the Gauss-Bonnet theorem for constructible sheaves (Theorem 43). These results can be used to obtain a generalization of the Riemann-Roch-Hirzebruch integral formula.
For a finite-dimensional (but possibly noncompact) symplectic manifold with a compact group acting with a proper moment map, we show that the square of the moment map is an equivariantly perfect Morse function in the sense of Kirwan, and that the set of critical points of the square of the moment map is a countable discrete union of compact sets. We show that certain integrals of equivariant cohomology classes localizes as a sum of contributions from these compact critical sets, and we bound the contribution from each critical set. In the case (1) that the contribution from higher critical set…
This article provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed by W.Schmid in [Sch]. A corresponding problem in the compact group setting was solved by N.Berline, E.Getzler and M.Vergne in [BGV] by an application of the theory of equivariant forms and particularly the fixed point integral localization formula. This article (besides its representation-theoretical significance) provides a whole family of examples where it is possible to localize integrals to fixed points with respect to an action of a NONcompact …
In this article I describe my recent geometric localization argument dealing with actions of NONcompact groups which provides a geometric bridge between two entirely different character formulas for reductive Lie groups and answers the question posed in [Sch]. A corresponding problem in the compact group setting was solved by N.Berline, E.Getzler and M.Vergne in [BGV] by an application of the theory of equivariant forms and, particularly, the fixed point integral localization formula. This localization argument seems to be the first successful attempt in the direction of building a similar the…
Brane space localization and genus-0 T-duality with $H$-flux
该研究在膜空间构建几何理论，证明膜空间定域化分两阶段实现，通过定域化运输Hori态射得到分次亏格0 T对偶，相关态射在非零水平为同构且与膜特征等兼容。
In this note we study the localization of Futaki-Morita integrals at isolated degenerate zeros by giving a streamlined exposition in the spirit of Bott and implement the localization procedure for a holomorphic vector field on $CP^n$ with a maximally degenerate zero, giving an essentially unique formula for the Futaki-Morita integral invariants without using a summation over multiple points. In a coming paper we will apply similar calculations to the Calabi-Futaki invariant of a Kähler blowup.
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 low-dimensional gauge theories. These are the functional integral counterparts of the Mathai-Quillen formalism, the Duistermaat-Heckman 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 th…
We prove an analogue of the Atiyah-Bott-Berline-Vergne localization formula in the setting of equivariant basic cohomology of $K$-contact manifolds. As a consequence, we deduce analogues of Witten's nonabelian localization and the Jeffrey-Kirwan residue formula, which relate equivariant basic integrals on a contact manifold $M$ to basic integrals on the contact quotient $M_0 := μ^{-1}(0)/G$, where $μ$ denotes the contact moment map for the action of a torus $G$. In the special case that $M \to N$ is an equivariant Boothby-Wang fibration, our formulae reduce to the usual ones for the symplectic…
This paper generalizes Bismut's equivariant Chern character to the setting of abelian gerbes. In particular, associated to an abelian gerbe with connection, an equivariantly closed differential form is constructed on the space of maps of a torus into the manifold. These constructions are made explicit using a new local version of the higher Hochschild complex, resulting in differential forms given by iterated integrals. Connections to two dimensional topological field theories are indicated. Similarly, this local higher Hochschild complex is used to calculate the 2-holonomy of an abelian gerbe…
Given an odd vector field $Q$ on a supermanifold $M$ and a $Q$-invariant density $μ$ on $M$, under certain compactness conditions on $Q$, the value of the integral $\int_{M}μ$ is determined by the value of $μ$ on any neighborhood of the vanishing locus $N$ of $Q$. We present a formula for the integral in the case where $N$ is a subsupermanifold which is appropriately non-degenerate with respect to $Q$. In the process, we discuss the linear algebra necessary to express our result in a coordinate independent way. We also extend stationary phase approximation and the Morse-Bott Lemma to supermani…
Equivariant localization techniques give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In particular, we identify a candidate target for the elliptic Bismut-Chern character.
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