Low--dimensional sisters of Seiberg-Witten effective theory

We consider the theories obtained by dimensional reduction to D=1,2,3 of 4D supersymmetric Yang--Mills theories and calculate there the effective low-energy lagrangia describing moduli space dynamics -- the low-dimensional analogs of the Seiberg--Witten effective lagrangian. The effective theories thus obtained are rather beautiful and interesting from mathematical viewpoint. In addition, their study allows one to understand better some essential features of 4D supersymmetric theories, in particular -- the nonrenormalisation theorems.Comment: 39 pages. A contribution to Ian Kogan memorial volume. Minor corrections, a reference adde

Exact Results in Gauge Theories: Putting Supersymmetry to Work. The 1999 Sakurai Prize Lecture

Powerful methods based on supersymmetry allow one to find exact solutions to certain problems in strong coupling gauge theories. The inception of some of these methods (holomorphy in the gauge coupling and other chiral parameters, in conjunction with instanton calculations) dates back to the 1980's. I describe the early exact results -- the calculation of the beta function and the gluino condensate -- and their impact on the subsequent developments. A brief discussion of the recent breakthrough discoveries where these results play a role is given.Comment: Based on the talk at the Centennial Meeting of The American Physical Society, March 20-26, Atlanta, GA. LaTex (uses sprocl.sty), 36 pages, 5 eps figures include

Constraints on Higher Derivative Operators in Maximally Supersymmetric Gauge Theory

Following the work of Dine and Seiberg for SU(2), we study the leading irrelevant operators on the moduli space of N=4 supersymmetric SU(N) gauge theory. These operators are argued to be one-loop exact, and are explicitly computed.Comment: 6 pages, harvmac. Note added. (Only a subset of the leading irrelevant operators have been shown to be one-loop exact.

Non-holomorphic Corrections from Threebranes in F Theory

We construct solutions of type IIB supergravity dual to N=2 super Yang-Mills theories. By considering a probe moving in a background with constant coupling and an AdS_{5} component in its geometry, we are able to reproduce the exact low energy effective action for the theory with gauge group SU(2) and N_{f}=4 massless flavors. After turning on a mass for the flavors we find corrections to the AdS_{5} geometry. In addition, the coupling shows a power law dependence on the energy scale of the theory. The origin of the power law behaviour of the coupling is traced back to instanton corrections. Instanton corrections to the four derivative terms in the low energy effective action are correctly obtained from a probe analysis. By considering a Wilson loop in this geometry we are also able to compute the instanton effects on the quark-antiquark potential. Finally we consider a solution corresponding to an asymptotically free field theory. Again, the leading form of the four derivative terms in the low energy effective action are in complete agreement with field theory expectations.Comment: 23 pages, uses harvmac, References added, typos corrected and minor improvements to discussion of N dependence, to appear in Phys. Rev.

N=(0, 2) Deformation of (2, 2) Sigma Models: Geometric Structure, Holomorphic Anomaly and Exact Beta Functions

We study N=(0,2) deformed (2,2) two-dimensional sigma models. Such heterotic models were discovered previously on the world sheet of non-Abelian strings supported by certain four-dimensional N=1 theories. We study geometric aspects and holomorphic properties of these models, and derive a number of exact expressions for the beta functions in terms of the anomalous dimensions analogous to the NSVZ beta function in four-dimensional Yang-Mills. Instanton calculus provides a straightforward method for the derivation. The anomalous dimensions are calculated up to two loops implying that one of the beta functions is explicitly known up to three loops. The fixed point in the ratio of the couplings found previously at one loop is not shifted at two loops. We also consider the N=(0,2) supercurrent supermultiplet (the so-called hypercurrent) and its anomalies, as well as the "Konishi anomaly." This gives us another method for finding exact $\beta$ functions. We prove that despite the chiral nature of the models under consideration quantum loops preserve isometries of the target space.Comment: 38 pages, 6 figures, minor changes in the text and references, the journal versio