Higher gauge theory -- differential versus integral formulation

The term higher gauge theory refers to the generalization of gauge theory to a theory of connections at two levels, essentially given by 1- and 2-forms. So far, there have been two approaches to this subject. The differential picture uses non-Abelian 1- and 2-forms in order to generalize the connection 1-form of a conventional gauge theory to the next level. The integral picture makes use of curves and surfaces labeled with elements of non-Abelian groups and generalizes the formulation of gauge theory in terms of parallel transports. We recall how to circumvent the classic no-go theorems in order to define non-Abelian surface ordered products in the integral picture. We then derive the differential picture from the integral formulation under the assumption that the curve and surface labels depend smoothly on the position of the curves and surfaces. We show that some aspects of the no-go theorems are still present in the differential (but not in the integral) picture. This implies a substantial structural difference between non-perturbative and perturbative approaches to higher gauge theory. We finally demonstrate that higher gauge theory provides a geometrical explanation for the extended topological symmetry of BF-theory in both pictures.Comment: 26 pages, LaTeX with XYPic diagrams; v2: typos corrected and presentation improve

Path Integral for Space-time Noncommutative Field Theory

The path integral for space-time noncommutative theory is formulated by means of Schwinger's action principle which is based on the equations of motion and a suitable ansatz of asymptotic conditions. The resulting path integral has essentially the same physical basis as the Yang-Feldman formulation. It is first shown that higher derivative theories are neatly dealt with by the path integral formulation, and the underlying canonical structure is recovered by the Bjorken-Johnson-Low (BJL) prescription from correlation functions defined by the path integral. A simple theory which is non-local in time is then analyzed for an illustration of the complications related to quantization, unitarity and positive energy conditions. From the view point of BJL prescription, the naive quantization in the interaction picture is justified for space-time noncommutative theory but not for the simple theory non-local in time. We finally show that the perturbative unitarity and the positive energy condition, in the sense that only the positive energy flows in the positive time direction for any fixed time-slice in space-time, are not simultaneously satisfied for space-time noncommutative theory by the known methods of quantization.Comment: 21 page

Seiberg-Witten map for noncommutative super Yang-Mills theory

In this letter we derive the Seiberg-Witten map for noncommutative super Yang-Mills theory in Wess-Zumino gauge. Following (and using results of) hep-th/0108045 we split the observer Lorentz transformations into a covariant particle Lorentz transformation and a remainder which gives directly the Seiberg-Witten differential equations. These differential equations lead to a theta-expansion of the noncommutative super Yang-Mills action which is invariant under commutative gauge transformations and commutative observer Lorentz transformation, but not invariant under commutative supersymmetry transformations: The theta-expansion of noncommutative supersymmetry leads to a theta-dependent symmetry transformation. For this reason the Seiberg-Witten map of super Yang-Mills theory cannot be expressed in terms of superfields.Comment: 9 page

On membrane interaction in matrix theory

We compute the interaction potential between two parallel transversely boosted wrapped membranes (with fixed momentum $p_-$) in D=11 supergravity with compact light-like direction. We show that the supergravity result is in exact agreement with the potential following from the all-order Born-Infeld-type action conjectured to be the leading planar infra-red part of the quantum super Yang-Mills effective action. This provides a non-trivial test of consistency of the arguments relating Matrix theory to a special limit of type II string theory. We also find the potential between two (2+0) D-brane bound states in D=10 supergravity (corresponding to the case of boosted membrane configuration in 11-dimensional theory compactified on a space-like direction). We demonstrate that the result reduces to the SYM expression for the potential in the special low-energy (\a'\to 0) limit, in agreement with previous suggestions. In appendix we derive the action obtained from the D=11 membrane action by the world-volume duality transformation of the light-like coordinate $x^-$ into a 3-vector.Comment: 18 pages, latex. v2: Some clarifying remarks and references added. v3: Further minor corrections and reference

Hard Non-commutative Loops Resummation

The non-commutative version of the euclidean $g^2\phi^4$ theory is considered. By using Wilsonian flow equations the ultraviolet renormalizability can be proved to all orders in perturbation theory. On the other hand, the infrared sector cannot be treated perturbatively and requires a resummation of the leading divergencies in the two-point function. This is analogous to what is done in the Hard Thermal Loops resummation of finite temperature field theory. Next-to-leading order corrections to the self-energy are computed, resulting in $O(g^3)$ contributions in the massless case, and $O(g^6\log g^2)$ in the massive one.Comment: 4 pages, 3 figures. The resummation procedure is now discussed also at finite ultraviolet cut-off. Minor changes in abstract and references. Final version to be published in Physical Review Letter

Parallel transport on non-Abelian flux tubes

I propose a way of unambiguously parallel transporting fields on non-Abelian flux tubes, or strings, by means of two gauge fields. One gauge field transports along the tube, while the other transports normal to the tube. Ambiguity is removed by imposing an integrability condition on the pair of fields. The construction leads to a gauge theory of mathematical objects known as Lie 2-groups, which are known to result also from the parallel transport of the flux tubes themselves. The integrability condition is also shown to be equivalent to the assumption that parallel transport along nearby string configurations are equal up to arbitrary gauge transformations. Attempts to implement this condition in a field theory leads to effective actions for two-form fields.Comment: significant portions of text rewritten, references adde

Noncommutative QFT and Renormalization

Field theories on deformed spaces suffer from the IR/UV mixing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this disease by adding one more marginal operator. We review these ideas, show the application to $\phi^3$ models and use the heat kernel expansion methods for a scalar field theory coupled to an external gauge field on a $\theta$-deformed space and derive noncommutative gauge field actions.Comment: To appear in the proceedings of the Workshop "Noncommutative Geometry in Field and String Theory", Corfu, 2005 (Greece