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Various analytic results which combine fermionic Brownian motion with stochastic integration are described, and it is shown that a wide class of stochastic differential equations in superspace have solutions. Such solutions are then used to derive a Feynman-Kac formula for a supersymmetric system in terms of the supercharge whose square is the Hamiltonian of the system. This is achieved by introducing superpaths parametrized by a commuting and an anticommuting time variable.

Working from first principles, quantization of a class of Hamiltonian systems with reducible symmetry is carried out by constructing first the appropriate reduced phase space and then the BRST cohomology. The constraints of this system correspond to a first class set for a group G and a second class set for a subgroup H. The BRST operator constructed is equivariant with respect to H. Using algebraic techniques analogous to those of equivariant de Rham theory, the BRST operator is shown to correspond to that obtained by BV quantization of a class of systems with reducible symmetry. The 'ghosts for ghosts' correspond to the even degree two generators in the Cartan model of equivariant cohomology. As an example of the methods developed, a topological model is described whose BRST quantization relates to the equivariant cohomology of a manifold under a circle action., Comment: 23 pages LaTeX. Some refrences added and some clarification made to text

Conditions which must be satisfied by the gauge-fixing fermion $\chi$ used in the BRST quantisation of constrained systems are established. These ensure that the extension of the Hamiltonian by the gauge-fixing term $[\Omega, \chi]$ (where $\Omega$ is the BRST charge) gives the correct path integral. (Lecture given at the conference Constrained Dynamics and Quantum Gravity II, Santa Margherita, Italy, September 1996), Comment: 5 pages LaTeX

The supersymmetric model developed by Witten to study the equivariant cohomology of a manifold with an isometric circle action is derived from the BRST quantization of a simple classical model. The gauge-fixing process is carefully analysed, and demonstrates that different choices of gauge-fixing fermion can lead to different quantum theories., 18 pages LaTeX

An anticommuting analogue of Brownian motion, corresponding to fermionic quantum mechanics, is developed, and combined with classical Brownian motion to give a generalised Feynman-Kac-It\^o formula for paths in geometric supermanifolds. This formula is applied to give a rigorous version of the proofs of the Atiyah-Singer index theorem based on supersymmetric quantum mechanics. It is also shown how superpaths, parametrised by a commuting and an anticommuting time variable, lead to a manifestly supersymmetric approach to the index of the Dirac operator. After a discussion of the BFV approach to the quantization of theories with symmetry, it is shown how the quantization of the topological particle leads to the supersymmetric model introduced by Witten in his study of Morse theory., Comment: 49 pages

The canonical quantization of the topological particle is described; it is shown that BRST quantization of the model gives the supersymmetric quantum mechanical model considered by Witten when investigating Morse theory, and the rigorous path integral method appropriate for this model is discussed. Possibilities for the extension of this work to two dimensional models are briefly considered., LaTeX, 8 pages, talk given at QG99, Third Meeting on Constrained Dynamics and Quantum Gravity, Villasimius (Sardinia, Italy) September 13-17, 1999

Nonsingularity conditions are established for the BFV gauge-fixing fermion which are sufficient for it to lead to the correct path integral for a theory with constraints canonically quantized in the BFV approach. The conditions ensure that anticommutator of this fermion with the BRST charge regularises the path integral by regularising the trace over non-physical states in each ghost sector. The results are applied to the quantization of a system which has a Gribov problem, using a non-standard form of the gauge-fixing fermion., Comment: 14 pages

After briefly describing Hamiltonian BRST methods and the multisymplectic approach to field theory, a symmetric geometric Lagrangian is studied by extending the BRST method to the multisymplectic setting. This work uses ideas first introduced by Hrabak [On the multisymplectic origin of the nonabelian deformation algebra of pseudoholomorphic embeddings in strictly almost Kähler ambient manifolds, and the corresponding BRST algebra, preprint (1999), arXiv: math-ph/9904026].

(Replacement because mailer changed `hat' for supercript into something weird. The macro `\sp' has been used in place of the `hat' character in this revised version.) Fermionic Brownian paths are defined as paths in a space para\-metr\-ised by anticommuting variables. Stochastic calculus for these paths, in conjunction with classical Brownian paths, is described; Brownian paths on supermanifolds are developed and applied to establish a Feynman-Kac formula for the twisted Laplace-Beltrami operator on differential forms taking values in a vector bundle. This formula is used to give a proof of the Atiyah-Singer index theorem which is rigorous while being closely modelled on the supersymmetric proofs in the physics literature., Comment: 18 pages, KCL-TH-92-5

A rigorous theory of integration in the space of paths in super space is developed, by extending Berezin's method of integration to spaces of anticommuting variables with an uncountably high dimension. A Feynmam-Kac-Ito formula for the heat kernel of a wide class of superspace differential operators is established. This formula is then used to make rigorous the supersummetric proofs of the Gauss-Bonnet-Chern theorem [1, 2].

It is shown that the Berezin approach to integration on supermanifolds can be applied to cases where the supermanifold is a twisted extension of a real manifold. This is done by showing that supermanifolds admit a subatlas of coordinate charts with transition functions of a quite restricted kind.

A new approach to superdifferentiable functions of Grassmann variables is developed, which avoids ambiguities in odd derivatives. This is used to give an improved definition of supermanifold over a finite-dimensional Grassmann algebra. A natural embedding of super-manifolds over Grassmann algebras with increasing number (L) of generators is developed, and thus a limit asL tends to infinity is possible. A correspondence between graded manifolds and supermanifolds is constructed, extending results of [5] and [8].

The kernel of the evolution operator for the supersymmetric square root of the Schrodinger equation of a supersymmetric quantum mechanical system is expressed as an integral in the space of paths in superspace. The measure is defined so that the superchange rather than the hamiltonian is fundamental. It is shown that the odd and even parts of the paths in superspace are components of a single superpath. The Fourier mode expansion of the superpath is derived.

A mathematically rigorous definition of a global supermanifold is given. This forms an appropriate model for a global version of superspace, and a class of functions is defined which corresponds to superfields. This new construction is compared with several pre‐existing definitions of supermanifold and graded manifold; it is shown to include all these definitions and to go beyond them, particularly in admitting the possibility of nontrivial topology in the anticommuting sector. Local differential geometry and potential applications to supergravity are considered.

Off-shell N = 2 supergravity is formulated in terms of an action which is the integral of a simple scalar function over the full superspace of the theory. This is an extended superspace, with an extra complex bosonic dimension corresponding to the central charge, which gives the superspace measure the correct dimension for constructing the action.

Super contours are defined to be superconformal maps of real (1,1)-dimensional superspace into complex (1,1)-dimensional superspace. A method for integration along supercontours is defined in terms of a modified Berezin integral on (1,1)-dimensional real superspace. The natural transformation properties of this modified Berezin integral are inherited by the contour integral, so that the contour integral of a half form on a general super-Riemann surface is defined.

A general mathematical framework for the super Lie groups of supersymmetric theories is presented. The definition of super Lie group is given in terms of supermanifolds, and two theorems (analogous to theorems in classical Lie group theory) are proved. The relationship of the super Lie groups defined here to the formal groups of Berezin and Kac and the graded Lie groups of Kostant is analyzed.

Various topics in the theory and application of the geometrical approach to supermanifolds are discussed. The construction of the superspace used in supergravity over an arbitrary spacetime manifold is described. Super Lie groups and their relation to graded Lie algebras (and more general structures referred to as ‘graded Lie modules’) are discussed, with examples. Certain supermanifolds, allowed in the geometric approach (using the fine topology), but having no analogue in the algebraic approach, are discussed. Finally lattice supersymmetry, and its relation to the differential geometry of supermanifolds, is discussed.

A detailed construction is given of a supermanifold which is compact and has far from trivial topology. Higher dimensional examples are also described. The Grassmann algebra from which the supermanifolds are constructed may have finite or infinite dimension.