Antti J. Niemi

Path integrals and geometry of trajectories

Abstract A geometrical interpretation of path integrals is developed in the space of trajectories. This yields a supersymmetric formulation of a generic path integral, with the supersymmetry resembling the BRST supersymmetry of a first class constrained system. If the classical equation of motion is a Killing vector field in the space of trajectories, the supersymmetry localizes the path integral to classical trajectories and the WKB approximation becomes exact. This can be viewed as a path integral generalization of the Duistermaat-Heckman theorem, which states the conditions for the exactness of the WKB approximation for integrals in a compact phase space.

Geometry of N=1/2 supersymmetry and the Atiyah-Singer index theorem

Abstract We present an explicit realization of a conjecture by Atiyah and Witten on the geometrical interpretation of N= 1 2 supersymmetry. We apply our formalism to compute exactly (for all β) the path integral which yields the index of a Dirac operator for arbitrary gauge and gravitational background fields. Our approach is different from the original one by Alvarez-Gaume, and by Friedan and Windey in the sense that we use the co-adjoint orbit method to realize the gauge symmetry. In this way we avoid the need to introduce a projection operator to the desired representation of the gauge group.

Orbit geometry, group representations and topological quantum field theories

Abstract We investigate the supersymmetric formulation of a generic phase space path integral in models where the supersymmetry action is supersymmetrically exact. Naive arguments suggest that the path integral is trivial but a more careful examination reveals that it defines a nontrivial topological quantum field theory. As an example we discuss the quantization of the group action on coadjoint orbits in the complex polarization corresponding to the coherent state representation.

Supersymplectic geometry of supersymmetric quantum field theories

Abstract We develop a conceptually new, geometric approach to supersymmetry. In particular, we argue that the construction of a generic supersymmetric theory entails only symplectic geometry either in a loop space parametrized by the bosonic degrees of freedom or in a superloop space parametrized by both bosonic and fermionic degrees of freedom. In the bosonic loop space a generic supersymmetry theory can be constructed using a model dependent loop space symplectic two-form, the corresponding symplectic one-form and a model independent vector field that determines circle action in the loop space. In the superloop space the construction of a generic supersymmetric theory employs a model independent symplectic two-form, the pertinent symplectic one-form, a model independent vector field that determines circle action in the superloop space, and the interaction is obtained by introducing a canonical transformation in the superloop space. A Poincare supersymmetric quantum field theory is a realization of our formalism in terms of space-time variables that admit a natural Lorentz-invariant interpretation. We expect that our geometric approach to supersymmetry opens a novel point of view to a large class of problems, including the mechanism of supersymmetry breaking, structure of topological field theories and even aspects of quantum integrability.

Pedagogical introduction to BRST

Abstract We review some recent results in the BRST quantization of constrained systems. In particular, we explain how the Parisi-Sourlas extension of BRST supersymmetry emerges and how it implies formal equivalence with reduced phase space quantization. We construct the pertinent generators of Parisi-Sourlas superrotations for both first and second class constraint algebras, and as an explicit example we consider open bosonic strings.

Geometric approach to supersymmetry

We argue that a generic supersymmetric theory can be characterized entirely in terms of loop space symplectic geometry, either in a loop space parametrized by the bosonic variables or in a superloop space parametrized by half of the bosonic and fermionic variables. A Poincare supersymmetric theory is a realization of our construction in terms of space-time variables that admit a natural Lorentz-invariant interpretation. Our approach opens a new, geometric point of view to a large number of problems, including the mechanism of supersymmetry breaking and the structure of topological quantum field theories.

Renormalization of topological actions and anyonic superconductivity

Abstract We argue the in a wide class of models relevant to anyonic superconductivity, the bare Chern-Simons term is renormalized to zero by instanton effects. This gives rise to a massless excitation, necessary for superconductivity to occur.

Four-dimensional topological Yang-Mills theory. Symmetries, local observables, Wilson loops and confinement

Abstract The BRST, antiBRST and ghost number operators in four-dimensional topological Yang-Mills theory are constructed from first principles. The results are compared with those obtained from the Batalin-Fradkin (BF) algorithm, a discrepancy is observed and explained. Ghost moments of the BRST operator are shown to reproduce the full Gauss law operator in a Parisi-Sourlas superspace Yang-Mills theory, and local BRST invariant observables are identified with superspace gauge invariant functionals. Planar Wilson loops are shown to exhibit an area law sufficient for confinement.

Gribov ambiguities and harmonic analysis

Abstract Vacuum structure in SU( N ) gauge theories is studied both in Coulomb and Landu gauge. It is shown that all Gribov copies are harmonic maps. Implications of harmonicity are discussed. Finally, a new SU(2) Landau gauge copy of the vacuum is presented.

On the vacuum structure in the Coulomb and Landau gauges

Vacuum structure in the SU(N) Coulomb and Landau gauges is studied by using the methods of harmonic maps. A systematic way for solving the Gribov vacuum copy equation is presented and many examples are discussed in both the Coulomb and Landau gauges as applications of the method. Finally, the physical interpretation of Gribov ambiguities is shortly reviewed from a topological point of view.