A Topological Lagrangian for Monopoles on Four-manifolds

We present a topological quantum field theory which corresponds to the moduli problem associated to Witten’s monopole equations for four-manifolds. The construction of the theory is carried out in purely geometrical terms using the Mathai-Quillen formalism, and the corresponding observables are described. These provide a rich set of new topological quantites. ⋆ E-mail: [email protected] In a recent work Witten [1] has shown that Donaldson theory [2,3] with gauge group SU(2) is equivalent to a new moduli problem which involves an abelian Yang-Mills connection and a spinor coupled in a pair of “monopole equations”. This result is a consecuence of previous work on N = 2 and N = 4 Yang-Mills theory [6,4,5]. The equivalence discovered by Witten is very powerfull and allows to write explicit expressions for Donaldson polynomials. An inmediate task which arises from his work is the search for a topological quantum field theory related to the new moduli problem presented in [1]. The observables of such topological quantum field theory could provide new topological invariants which could contain important topological information. The aim of this paper is to construct the topological quantum field theory corresponding to the new moduli problem proposed in [1]. This will be done using the Mathai-Quillen formalism [7]. The resulting theory turns out to be an abelian Donaldson-Witten theory, which as it is widely known can be obtained from the twisting of N = 2 Yang-Mills theory, coupled to a twisted version of the N = 2 hypermultiplet [8,9,10]. The resulting type of topological model has been studied previously in [11, 12]. Related topological quantum field theories have been analyzed in [13], and their connection to the moduli problem presented in [1] has been recently considered in [14]. The Mathai-Quillen formalism allows one to construct the action of a topological quantum field theory starting from moduli problems formulated in purely geometrical terms. Moduli problems are often stated in the following form: given a moduli space M and a vector bundle over M, V, one defines the basic equations of the problem as sections of this vector bundle. Typically one is interested in computing the Euler characteristic of this bundle, or, equivalently, its Thom class. In the case at hand, because of the gauge symmetry of the theory, one also has the action of a group G on both, the manifold M and the vector bundle. Rather than compute the Euler characteristic of the bundle itself one wants to get rid of the gauge degrees of freedom and compute the Euler characteristic of the quotient bundle obtained “dividing by G”: V/G −→ M/G. In the same way, the