[Paper Review] Superconnections and Parallel Transport

Florin Dumitrescu|ArXiv.org|Nov 18, 2007

Graphene research and applications5 references5 citations

TL;DR

This paper introduces a geometric notion of parallel transport along superpaths—paths in a manifold equipped with an odd vector field—using superconnections on Z/2-graded vector bundles. It establishes that a superconnection gives rise to a unique superparallel transport via a differential equation involving a connection and an odd endomorphism-valued form, and shows this transport uniquely reconstructs the original superconnection.

ABSTRACT

This note addresses the construction of a notion of parallel transport along superpaths arising from the concept of a superconnection on a vector bundle over a manifold $M$. A superpath in $M$ is, loosely speaking, a path in $M$ together with an odd vector field in $M$ along the path. We also develop a notion of parallel transport associated with a connection (a.k.a. covariant derivative) on a vector bundle over a \emph{supermanifold} which is a direct generalization of the classical notion of parallel transport for connections over manifolds.

Motivation & Objective

• To develop a geometric framework for parallel transport along superpaths in supermanifolds, generalizing classical parallel transport for connections on manifolds.
• To interpret superconnections—odd differential operators on graded bundles—as geometric objects via a new notion of parallel transport.
• To establish a correspondence between superconnections and their associated superparallel transport, showing the latter uniquely recovers the former.
• To provide a geometric realization of 1|1-dimensional supersymmetric field theories via superparallel transport, linking to differential K-theory and index theory.

Proposed method

• Defines a superpath as a map from R^{1|1} to a manifold M, where R^{1|1} carries an odd vector field D = ∂_θ + θ∂_t.
• Lifts a superpath c: R^{1|1} → M to a superpath c̃: R^{1|1} → ΠTM, the odd tangent bundle of M.
• Introduces the condition for a section ψ along c to be parallel: (c*∇)_D ψ − (c̃*A)ψ = 0, where A is an odd endomorphism-valued form.
• Uses the pullback of the connection ∇ and the form A via the lifted path to define a superconnection-compatible transport operator.
• Applies an inverse adiabatic limit process to recover the classical parallel transport of the connection ∇ from the superparallel transport.
• Reconstructs the superconnection (∇, A) from the parallel transport by analyzing the transport operator on the total space ΠTM × R^{1|1}.

Experimental results

Research questions

• RQ1 How can parallel transport be generalized from ordinary manifolds to supermanifolds using superconnections?
• RQ2 What is the geometric meaning of a superconnection in terms of parallel transport along superpaths?
• RQ3 Can the superconnection be uniquely reconstructed from its associated superparallel transport?
• RQ4 How does the superparallel transport relate to the classical parallel transport of the underlying connection?
• RQ5 What is the role of the odd vector field D = ∂_θ + θ∂_t in defining superparallel sections?

Key findings

• A superconnection A = A₁ + A₂ + ... on a Z/2-graded vector bundle E over M induces a unique superparallel transport via the differential equation (c*∇)_D ψ = (c̃*A)ψ.
• The superparallel transport is invariant under reparametrization and gluing of superpaths, generalizing the classical invariance of parallel transport.
• The inverse adiabatic limit of the superparallel transport recovers the classical parallel transport of the connection ∇ = A₁.
• The odd endomorphism-valued form A ∈ Ω*(M, End E)^odd is uniquely recoverable from the superparallel transport via the pullback map on ΠTM × R^{1|1}.
• In the case M = pt, the superparallel transport corresponds to the supergroup homomorphism R^{1|1} → GL(V), given explicitly by (t,θ) ↦ e^{-tA² + θA}, where A is an odd endomorphism on V.
• The superparallel transport defines a supergroup homomorphism, confirming its compatibility with the supergroup structure of R^{1|1}.