Integration and Global Aspects of Supermanifolds

Abstract

The concept of supermanifold enables one to extend the idea of a conventional manifold to include anticommuting objects. Much of the original work in this field was motivated by a desire to clarify the mathematics of fermi field quantisation; in the approach of Berezin and Leîtes¹ and Kostant², the sheaf of C functions on a manifold X is extended to a sheaf A(·) of graded commutative algebras; the key idea is that there is an open cover $X = \mathop{ \cup }\limits_{{\alpha \in \Lambda }} {U_{\alpha }}$ such that

$$ A\left( {{U_{\alpha }}} \right) \simeq {C_{\infty }}\left( {{U_{\alpha }}} \right) \otimes \Lambda \left( {{K^{n}}} \right) $$

((1.1.1))

where ∧(Kⁿ) denotes the Grassmann algebra over Kⁿ (with K the real or complex field). If X has dimension m the supermanifold is said to be (m, n) dimensional. At this point it is useful to define these Grassmann algebras in detail: ∧(Kⁿ) is the algebra over K with generators 1 (the unit) and θ¹, ..., θⁿ and with relations

$$ {\theta ^{i}}{\theta ^{j}} = - {\theta ^{j}}{\theta ^{i}}\quad i,j = 1,...,n$$

((1.1.2))

A typical element may be written

$$ b = {\lambda _{\phi }} + \sum\limits_{{i = 1}}^{n} {{\lambda _{{\left( i \right)}}}{\theta ^{i}} + \sum\limits_{{i < j = 1}}^{n} {{\lambda _{{\left( {i,j} \right)}}}{\theta ^{i}}{\theta ^{j}}} + ....,}$$

((1.1.3))

where the λ coefficients are elements of K. A more compact nation (due Kostant²) is to let M_n denote the set of sequences of intergers (μ₁, ...,μ_k) with 1 ≤ μ₁ < ... < μ_k ≤ n, and

$$ {\theta ^{\mu }}: = {\theta ^{{{\mu _{1}}}}}...{\theta ^{{{\mu _{k}}}}} $$

((1.1.4))

.

SERC advanced research fellow