435 research outputs found
A path integral derivation of -genus
The formula for the Hirzebruch -genus of complex manifolds is a
consequence of the Hirzebruch-Riemann-Roch formula. The classical index
formulae for Todd genus, Euler number, and Signature correspond to the case
when the complex variable 0, -1, and 1 respectively. Here we give a {\it
direct} derivation of this nice formula based on supersymmetric quantum
mechanics.Comment: 5 page
Equivariance, BRST and Superspace
The structure of equivariant cohomology in non-abelian localization formulas
and topological field theories is discussed. Equivariance is formulated in
terms of a nilpotent BRST symmetry, and another nilpotent operator which
restricts the BRST cohomology onto the equivariant, or basic sector. A
superfield formulation is presented and connections to reducible (BFV)
quantization of topological Yang-Mills theory are discussed.Comment: (24 pages, report UU-ITP and HU-TFT-93-65
Quantum cohomology of flag manifolds and Toda lattices
We discuss relations of Vafa's quantum cohomology with Floer's homology
theory, introduce equivariant quantum cohomology, formulate some conjectures
about its general properties and, on the basis of these conjectures, compute
quantum cohomology algebras of the flag manifolds. The answer turns out to
coincide with the algebra of regular functions on an invariant lagrangian
variety of a Toda lattice.Comment: 35 page
Localization and Diagonalization: A review of functional integral techniques for low-dimensional 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 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
then discuss some applications to path integrals and give an overview of the
relevant literature. The applications we deal with include supersymmetric
quantum mechanics, cohomological field theories, phase space path integrals,
and two-dimensional Yang-Mills theory.Comment: 72 pages (60 A4 pages), LaTeX (to appear in the Journal of
Mathematical Physics Special Issue on Functional Integration (May 1995)
Measures on Banach Manifolds and Supersymmetric Quantum Field Theory
We show how to construct measures on Banach manifolds associated to
supersymmetric quantum field theories. These measures are mathematically
well-defined objects inspired by the formal path integrals appearing in the
physics literature on quantum field theory. We give three concrete examples of
our construction. The first example is a family of measures on a
space of functions on the two-torus, parametrized by a polynomial (the
Wess-Zumino-Landau-Ginzburg model). The second is a family \mu_\cG^{s,t} of
measures on a space \cG of maps from to a Lie group (the
Wess-Zumino-Novikov-Witten model). Finally we study a family
of measures on the product of a space of connection s on the trivial principal
bundle with structure group on a three-dimensional manifold with a
space of \fg-valued three-forms on
We show that these measures are positive, and that the measures
\mu_\cG^{s,t} are Borel probability measures. As an application we show that
formulas arising from expectations in the measures \mu_\cG^{s,1} reproduce
formulas discovered by Frenkel and Zhu in the theory of vertex operator
algebras. We conjecture that a similar computation for the measures
where is a homology three-sphere, will yield the
Casson invariant of Comment: Minor correction
Topics on D-brane charges with B-fields
In this review we show how K-theory classifies RR-charges in type II string
theory and how the inclusion of the B-field modifies the general structure
leading to the twisted K-groups. Our main purpose is to give an expository
account of the physical relevance of K-theory and, in order to make it, we
consider different points of view: processes of tachyon condensation,
cancellation of global anomalies and gauge fixings. As a field to test the
proposals of K-theory, we concentrate on the study of the D6-brane, now seen as
a non-abelian monopole.Comment: 63 pages, no figures. To appear in the special issue of Int. J. Geom.
Meth. Mod. Phys., v.1, N4 (August 2004
N=2 Topological Yang-Mills Theory on Compact K\"{a}hler Surfaces
We study a topological Yang-Mills theory with fermionic symmetry. Our
formalism is a field theoretical interpretation of the Donaldson polynomial
invariants on compact K\"{a}hler surfaces. We also study an analogous theory on
compact oriented Riemann surfaces and briefly discuss a possible application of
the Witten's non-Abelian localization formula to the problems in the case of
compact K\"{a}hler surfaces.Comment: ESENAT-93-01 & YUMS-93-10, 34pages: [Final Version] to appear in
Comm. Math. Phy
Light-Ray Radon Transform for Abelianin and Nonabelian Connection in 3 and 4 Dimensional Space with Minkowsky Metric
We consider a real manifold of dimension 3 or 4 with Minkovsky metric, and
with a connection for a trivial GL(n,C) bundle over that manifold. To each
light ray on the manifold we assign the data of paralel transport along that
light ray. It turns out that these data are not enough to reconstruct the
connection, but we can add more data, which depend now not from lines but from
2-planes, and which in some sence are the data of parallel transport in the
complex light-like directions, then we can reconstruct the connection up to a
gauge transformation. There are some interesting applications of the
construction: 1) in 4 dimensions, the self-dual Yang Mills equations can be
written as the zero curvature condition for a pair of certain first order
differential operators; one of the operators in the pair is the covariant
derivative in complex light-like direction we studied. 2) there is a relation
of this Radon transform with the supersymmetry. 3)using our Radon transform, we
can get a measure on the space of 2 dimensional planes in 4 dimensional real
space. Any such measure give rise to a Crofton 2-density. The integrals of this
2-density over surfaces in R^4 give rise to the Lagrangian for maps of real
surfaces into R^4, and therefore to some string theory. 4) there are relations
with the representation theory. In particular, a closely related transform in 3
dimensions can be used to get the Plancerel formula for representations of
SL(2,R).Comment: We add an important discussion part, establishing the relation of our
Radon transform with the self-dual Yang-Mills, string theory, and the
represntation theory of the group SL(2,R
Fivebranes and 4-manifolds
We describe rules for building 2d theories labeled by 4-manifolds. Using the
proposed dictionary between building blocks of 4-manifolds and 2d N=(0,2)
theories, we obtain a number of results, which include new 3d N=2 theories
T[M_3] associated with rational homology spheres and new results for
Vafa-Witten partition functions on 4-manifolds. In particular, we point out
that the gluing measure for the latter is precisely the superconformal index of
2d (0,2) vector multiplet and relate the basic building blocks with coset
branching functions. We also offer a new look at the fusion of defect lines /
walls, and a physical interpretation of the 4d and 3d Kirby calculus as
dualities of 2d N=(0,2) theories and 3d N=2 theories, respectivelyComment: 81 pages, 18 figures. v2: misprints corrected, clarifications and
references added. v3: additions and corrections about lens space theory,
4-manifold gluing, smooth structure
Topological quantum field theory and invariants of graphs for quantum groups
On basis of generalized 6j-symbols we give a formulation of topological
quantum field theories for 3-manifolds including observables in the form of
coloured graphs. It is shown that the 6j-symbols associated with deformations
of the classical groups at simple even roots of unity provide examples of this
construction. Calculational methods are developed which, in particular, yield
the dimensions of the state spaces as well as a proof of the relation,
previously announced for the case of by V.Turaev, between these
models and corresponding ones based on the ribbon graph construction of
Reshetikhin and Turaev.Comment: 38 page
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