Sigma model solitons and their moduli space metrics

We discuss the supersymmetric model and its soliton solutions in 2+1 dimensions. We classify supersymmetric maps and derive Bogomolny bounds. We also give the modified superalgebra and describe the metric on the parameter space of solitons.Laboratoratoire propre du Centre National de la Recherche Scientifique, associé á l'Ecole Normale Supérieure et à l'Université de Paris Sud     

One-vortex moduli space and Ricci flow

The metric on the moduli space of one abelian Higgs vortex on a surface has a natural geometrical evolution as the Bradlow parameter, which determines the vortex size, varies. It is shown by various arguments, and by calculations in special cases, that this geometrical flow has many similarities to Ricci flow.     

Electrodynamics on matrix space: non-Abelian by coordinates

Faddeev and Niemi have proposed a decomposition of SU(N) Yang–Mills theory in terms of new variables, appropriate for describing the theory in the infrared limit. We extend this method to SO(2N) Yang–Mills theory. We find that the SO(2N) connection decomposes according to irreducible representations of SO(N). The low-energy limit of the decomposed theory is expected to describe soliton-like configurations with nontrivial topological numbers. How the method of decomposition generalizes for SO(2N+1) Yang–Mills theory is also discussed. Received: 22 November 2000 / Published online: 8 June 2001     

Simple type and the boundary of moduli space

We measure, in two distinct ways, the extent to which the boundary region of moduli space contributes to the “simple type” condition of Donaldson theory. Using the natural geometric representative of μ(pt) defined in [L. Sadun, Commun. Math. Phys. 178 (1996) 107–113], the boundary region of moduli space contributes of the homology required for simple type, regardless of the topology or geometry of the underlying 4-manifold. The simple type condition thus reduces to the interior of the (k+1)th ASD moduli space, intersected with two representatives of (4 times) the point class, being homologous to 58 copies of the kth moduli space. This is peculiar, since the only known embeddings of the kth moduli space into the (k+1)th involve Taubes gluing, and the images of such embeddings lie entirely in the boundary region.When using the natural de Rham representatives of μ(pt) considered by Witten [Commun. Math. Phys. 117 (1988) 353], the boundary region contributes of what is needed for simple type, again regardless of the topology or geometry of the underlying 4-manifold. The difference between this and the geometric representative answer is not contradictory, as the contribution of a fixed region to the Donaldson invariants is geometric, not topological.     

Roaming moduli space using dynamical triangulations

In critical as well as in non-critical string theory the partition function reduces to an integral over moduli space after integration over matter fields. For non-critical string theory this moduli integrand is known for genus one surfaces. The formalism of dynamical triangulations provides us with a regularization of non-critical string theory. We show how to assign in a simple and geometrical way a moduli parameter to each triangulation. After integrating over possible matter fields we can thus construct the moduli integrand. We show numerically for c=0 and c=−2 non-critical strings that the moduli integrand converges to the known continuum expression when the number of triangles goes to infinity.     

Black holes and critical points in moduli space

We study the stabilization of scalars near a supersymmetric black hole horizon using the equation of motion of a particle moving in a potential and background metric. When the relevant 4-dimensional theory is described by special geometry, the generic properties of the critical points of this potential can be studied. We find that the extremal value of the central charge provides the minimal value of the BPS mass and of the potential under the condition that the moduli space metric is positive at the critical point. This is a property of a regular special geometry. We also study the critical points in all N 2 supersymmetric theories. We relate these ideas to the Weinhold and Ruppeiner metrics introduced in the geometric approach to thermodynamics and used for the study of critical phenomena.     

Axiomatic holonomy maps and generalized Yang-Mills moduli space

This Letter is a follow-up of Barrett, J. W.,Internat. J. Theoret. Phys. 30(9), (1991). Its main goal is to provide an alternative proof of that part of the reconstruction theorem which concerns the existence of a connection. A construction of a connection 1-form is presented. The formula expressing the local coefficients of the connection in terms of the holonomy map is obtained as an immediate consequence of that construction. Thus, the derived formula coincides with that used in Chan, H.-M., Scharbach, P., and Tsou, S. T.,Ann. Physics 166, 396–421 (1986). The reconstruction and representation theorems form a generalization of the fact that the pointed configuration space of the classical Yang-Mills theory is equivalent to the set of all holonomy maps. The point of this generalization is that there is a one-to-one correspondence not only between the holonomy maps and the orbits in the space of connections, but also between all maps M G fulfilling some axioms and all possible equivalence classes ofP(M, G) bundles with connections, where the equivalence relation is defined by a bundle isomorphism in a natural way.     

Construction of non-Abelian gauge theories on noncommutative spaces

We present a formalism to explicitly construct non-Abelian gauge theories on noncommutative spaces (induced via a star product with a constant Poisson tensor) from a consistency relation. This results in an expansion of the gauge parameter, the noncommutative gauge potential and fields in the fundamental representation, in powers of a parameter of the noncommutativity. This allows the explicit construction of actions for these gauge theories. Received: 13 June 2001 / Published online: 19 July 2001     

Integration over moduli space in superstring theory

A calculation of multiloop superstring amplitudes is considered, the equivalence of popular approaches to determining these amplitudes being discussed. A calculation of poorly defined integrals over singular configurations is clarified. Amplitudes obtained by a correct method do not involve divergences in any order of perturbation theory.     

Fermionic strings on the compactified moduli space

On the compactified moduli space we consider theN=2,N=4 local supersymmetric string theories. It would be proven that theN=2,N=4 fermionic string theories might not develop any tachyon pole, which might imply theg-loop partition functions forN=2,N=4 fermionic string would be finite.     

SuperW n gravity on compactified moduli space

We study the divergent behavior ofW gravity theories. As a tool, we use the Grothendieck-Riemann-Roch theorem on the compactified moduli space. We show thatW _n gravity has severe divergences caused by negative masses. However, for superextension ofW _n gravity the divergences by negative masses are miraculously cured by the counterpart contribution of superpartners.     

The Riemannian geometry of the Yang-Mills moduli space

The moduli space of self-dual connections over a Riemannian 4-manifold has a natural Riemannian metric, inherited from theL ² metric on the space of connections. We give a formula for the curvature of this metric in terms of the relevant Green operators. We then examine in great detail the moduli space ₁ ofk=1 instantons on the 4-sphere, and obtain an explicit formula for the metric in this case. In particular, we prove that ₁ is rotationally symmetric and has finite geometry: it is an incomplete 5-manifold with finite diameter and finite volume.Partially supported by Horace Rackham Faculty Research Grant from the University of MichiganPartially supported by N.S.F. Grant DMS-8603461     

The Hilbert series of the one instanton moduli space

The moduli space of k G-instantons on \( {\mathbb{R}^4} \) for a classical gauge group G is known to be given by the Higgs branch of a supersymmetric gauge theory that lives on Dp branes probing D(p + 4) branes in Type II theories. For p = 3, these (3 + 1) dimensional gauge theories have \( \mathcal{N} = 2 \) supersymmetry and can be represented by quiver diagrams. The F and D term equations coincide with the ADHM construction. The Hilbert series of the moduli spaces of one instanton for classical gauge groups is easy to compute and turns out to take a particularly simple form which is previously unknown. This allows for a G invariant character expansion and hence easily generalisable for exceptional gauge groups, where an ADHM construction is not known. The conjectures for exceptional groups are further checked using some new techniques like sewing relations in Hilbert Series. This is applied to Argyres-Seiberg dualities.