Donaldson-Witten invariants for flows on four manifolds

Journal of Symplectic Geometry, 2019

This is an exposition of the Donaldson geometric flow on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. The original work appeared in [1].

1995

and Sezione INFN di Pavia. We study the first-order formalism of pure four-dimensional SU(2) Yang–Mills theory with theta-term. We describe the Green functions associated to electric and magnetic flux operators à la ’t Hooft by means of gauge-invariant non-local operators. These Green functions are related to Witten’s invariants of four-manifolds. 1

Communications in Mathematical Physics, 2014

In this paper we identify the problem of equivariant vortex counting in a (2, 2) supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov-Witten invariants of the GIT quotient target space determined by the quiver. We provide new contour integral formulae for the I and J -functions encoding the equivariant quantum cohomology of the target space. Its chamber structure is shown to be encoded in the analytical properties of the integrand. This is explained both via general arguments and by checking several key cases. We show how several results in equivariant Gromov-Witten theory follow just by deforming the integration contour. In particular we apply our formalism to compute Gromov-Witten invariants of the C 3 /Z n orbifold and of the Uhlembeck (partial) compactification of the moduli space of instantons on C 2 . Moreover, we analyse dualities of quantum cohomology rings of holomorphic vector bundles over Grassmannians, which are relevant to BPS Wilson loop algebrae.

2012

We present a method for computing the 3-point genus zero Gromov-Witten invariants of the complex flag manifold G/B from the relations of the small quantum cohomology algebra QH ∗ G/B (G is a complex semisimple Lie group and B is a Borel subgroup). In [Fo-Ge-Po] and [Ki-Ma], at least in the case G = GLnC, two algebraic/combinatoric methods have been proposed, based on suitably designed axioms. Our method is quite different, being differential geometric in nature; it is based on the approach to quantum cohomology described in [Gu], which is in turn based on the integrable systems point of view of Dubrovin and Givental. In §1 we shall review briefly the method of [Gu]. In §2 we discuss the special properties of G/B which lead to a computational algorithm. In fact the same method works for any Fano manifold whose cohomology is generated by two-dimensional classes, so our approach is more general than those of [Fo-Ge-Po] and [Ki-Ma]. In §3 we present explicit results for the case G = GLn...

Nuclear Physics B, 1991

A theory of topological gravity for smooth 4-manifolds is constructed. For a given 4-manifold, M, the measure in the functional integral over metrics on M is concentrated at metrics for which the self-dual part of the Weyl tensor vanishes, and the Ricci scalar is constant. A cohomology theory is presented, nontrivial observables are found, and identified with closed forms on the moduli space of such metrics on M. These observables are topological invariants of M, the analogues of Mumford classes for 4-manifolds admitting such metrics. An explicit analysis is provided for T 4.

Physics Letters A, 2019

Properties of non-barotropic flows are described using Lie derivatives of differential forms in a Euclidean four dimensional space-time manifold. Vanishing of the Lie derivative implies that the corresponding physical quantity remains invariant along the integral curves of the flow. Integral invariants of nonbarotropic perfect and viscous flows are studied using the concepts of relative and absolute invariance of forms. The four dimensional expressions for the rate of change of the generalized circulation, generalized vorticity flux, generalized helicity and generalized parity in the case of ideal and viscous non-barotropic flows are thereby obtained.

Journal of Mathematical Physics, 1995

For any Lie algebra 𝔤 and integral level k, there is defined an invariant Zk*(M, L) of embeddings of links L in 3-manifolds M, known as the Witten–Reshetikhin–Turaev invariant. It is known that for links in S3, Zk*(S3, L) is a polynomial in q=exp (2πi/(k+c𝔤v), namely, the generalized Jones polynomial of the link L. This paper investigates the invariant Zr−2*(M,○/) when 𝔤 =𝔰𝔩2 for a simple family of rational homology 3-spheres, obtained by integer surgery around (2, n)-type torus knots. In particular, we find a closed formula for a formal power series Z∞(M)∈Q[[h]] in h=q−1 from which Zr−2*(M,○/) may be derived for all sufficiently large primes r. We show that this formal power series may be viewed as the asymptotic expansion, around q=1, of a multivalued holomorphic function of q with 1 contained on the boundary of its domain of definition. For these particular manifolds, most of which are not Z-homology spheres, this extends work of Ohtsuki and Murakami in which the existence of pow...

Communications in Mathematical Physics, 2007

We construct a Topological Quantum Field Theory (in the sense of Atiyah [1]) associated to the universal finite-type invariant of 3-dimensional manifolds, as a functor from the category of 3-dimensional manifolds with parametrized boundary, satisfying some additional conditions, to an algebraic-combinatorial category. It is built together with its truncations with respect to a natural grading, and we prove that these TQFTs are non-degenerate and anomaly-free. The TQFT(s) induce(s) a (series of) representation(s) of a subgroup Lg of the Mapping Class Group that contains the Torelli group. The N = 1 truncation produces a TQFT for the Casson-Walker-Lescop invariant.

Topology, 1998

1999

Using the u-plane integral of Moore and Witten, we derive a simple expression for the Donaldson invariants of $\Sigma_g \times S^2$, where $\Sigma_g$ is a Riemann surface of genus g. This expression generalizes a theorem of Morgan and Szabo for g=1 to any genus g. We give two applications of our results: (1) We derive Thaddeus' formulae for the intersection pairings on the moduli space of rank two stable bundles over a Riemann surface. (2) We derive the eigenvalue spectrum of the Fukaya-Floer cohomology of $\Sigma_g \times S^1$.

Journal of Mathematical Physics, 2008

We introduce a method that generates invariant functions from perturbative classical field theories depending on external parameters. Applying our methods to several field theories such as abelian BF , Chern-Simons and 2-dimensional Yang-Mills theory, we obtain, respectively, the linking number for embedded submanifolds in compact varieties, the Gauss' and the second Milnor's invariant for links in S 3 , and invariants under area-preserving diffeomorphisms for configurations of immersed planar curves.

Topology, 2005

New relations among the genus-zero Gromov-Witten invariants of a complex projective manifold X are exhibited. When the cohomology of X is generated by divisor classes and classes "with vanishing one-point invariants," the relations determine many-point invariants in terms of one-point invariants.

Communications in Analysis and Geometry, 2001

Quantum Topology, 2014

We prove that the SU.2/ Witten-Reshetikhin-Turaev invariant of any 3-manifold with any colored link inside at any root of unity is an algebraic integer. As a byproduct, we get a new proof of the integrality of the SO.3/ Witten-Reshetikhin-Turaev invariant for any 3-manifold with any colored link inside at any root of unity of odd order.

Physics Letters B