Yang-Mills Connections on Nonorientable Surfaces

This article is a follow-up of ``Holonomy and Path Structures in General Relativity and Yang-Mills Theory" by Barrett, J. W. (Int.J.Theor.Phys., vol.30, No.9, 1991). Its main goal is to provide an alternative proof of this part of the reconstruction theorem which concerns the existence of a connection. A construction of connection 1-form is presented. The formula expressing the local coefficients of connection in terms of the holonomy map is obtained as an immediate consequen...

Let T be a compact torus and X a nice compact T-space (say a manifold or variety). We introduce a functor assigning to X a "GKM-sheaf" F_X over a "GKM-hypergraph" G_X. Under the condition that X is equivariantly formal, the ring of global sections of F_X are identified with the equivariant cohomology, H_T^*(X; C). We show that GKM-sheaves provide a general framework able to incorporate numerous constructions in the GKM-theory literature. In the second half of the paper we a...

We study the behavior of the Yang-Mills flow for unitary connections on compact and non-compact oriented surfaces with varying metrics. The flow can be used to define a one dimensional foliation on the space of SU(2) representations of a once punctured surface. This foliation universalizes over Teichm\"uller space and is equivariant with respect to the action of the mapping class group. It is shown how to extend the foliation as a singular foliation over the Strebel boundary ...

Refinements of the Yang-Mills stratifications of spaces of connections over a compact Riemann surface are investigated. The motivation for this study was the search for a complete set of relations between the standard generators for the cohomology of the moduli spaces of stable holomorphic bundles of rank n and degree d when n and d are coprime and n>2.

Sengupta's lower bound for the Yang-Mills action on smooth connections on a bundle over a Riemann surface generalizes to the space of connections whose action is finite. In this larger space the inequality can always be saturated. The Yang-Mills critical sets correspond to critical sets of the energy action on a space of paths. This may shed light on Atiyah and Bott's conjecture concerning Morse theory for the space of connections modulo gauge transformations.

We lay the foundations of a Morse homology on the space of connections on a principal $G$-bundle over a compact manifold $Y$, based on a newly defined gauge-invariant functional $\mathcal J$. While the critical points of $\mathcal J$ correspond to Yang-Mills connections on $P$, its $L^2$-gradient gives rise to a novel system of elliptic equations. This contrasts previous approaches to a study of the Yang-Mills functional via a parabolic gradient flow. We carry out the complet...

If we consider the moduli space of flat connections of a non trivial principal SO(3)-bundle over a surface, then we can define a map from the set of perturbed closed geodesics, below a given energy level, into families of perturbed Yang-Mills connections depending on a small parameter. In this paper we show that this map is a bijection and maps perturbed geodesics into perturbed Yang-Mills connections with the same Morse index.

Let G be a connected, compact, semisimple Lie group. It is known that for a compact closed orientable surface $\Sigma$ of genus $l >1$, the order of the group $H^2(\Sigma,\pi_1(G))$ is equal to the number of connected components of the space $Hom(\pi_1(\Sigma),G)/G$ which can also be identified with the moduli space of gauge equivalence classes of flat G-bundles over $\Sigma$. We show that the same statement for a closed compact nonorientable surface which is homeomorphic t...

In this paper we use the Morse theory of the Yang-Mills-Higgs functional on the singular space of Higgs bundles on Riemann surfaces to compute the equivariant cohomology of the space of semistable U(2,1) and SU(2,1) Higgs bundles with fixed Toledo invariant. In the non-coprime case this gives new results about the topology of the U(2,1) and SU(2,1) character varieties of surface groups. The main results are a calculation of the equivariant Poincare polynomials, a Kirwan surje...

In this article we present some probably unexpected (in our opinion) properties of representations of Yang-Mills algebras. We first show that any free Lie algebra with m generators is a quotient of the Yang-Mills algebra ym(n) on n generators, for n greater than or equal to 2m. We derive from this that any semisimple Lie algebra, and even any affine Kac-Moody algebra is a quotient of ym(n), for n greater than or equal to 4. Combining this with previous results on representati...