[Paper Review] Contact structures and supersymmetric mechanics

Andrew James Bruce|arXiv (Cornell University)|Aug 26, 2011

Geometric and Algebraic Topology14 references5 citations

TL;DR

This paper establishes a direct correspondence between the $d=1$, $N=2$ super-Poincaré algebra and contact geometry on the supermanifold $\mathbb{R}^{1|2}$, showing that the standard supersymmetry transformations act as strict contactomorphisms with respect to a canonical Grassmann-odd contact form $\alpha = dt + i(\theta d\bar{\theta} + \bar{\theta} d\theta)$. The key contribution is the identification of the supercharge vector fields $Q$ and $\bar{Q}$ as strict contact vector fields, and the Reeb vector field as $\partial/\partial t$, thereby embedding supersymmetric mechanics in the framework of contact geometry.

ABSTRACT

We reexamine the relation between contact structures on supermanifolds and supersymmetric mechanics in the superspace formulation. This allows one to use the language of contact geometry when dealing with the d = 1, N = 2 super-Poincare algebra.

Motivation & Objective

• To establish a geometric framework linking $d=1$, $N=2$ supersymmetric mechanics to contact structures on supermanifolds.
• To show that the standard supersymmetry transformations in superspace are strict contactomorphisms with respect to a specific super contact form.
• To identify the Reeb vector field in the supercontact setting and relate it to the time-translation generator.
• To clarify the geometric role of the supercovariant derivatives $\mathbbmss{D}$ and $\bar{\mathbbmss{D}}$ as spanning the contact distribution.
• To unify the coset space method and Maurer–Cartan formalism with contact geometry in the context of supersymmetric mechanics.

Proposed method

• Define a Grassmann-odd one-form $\alpha = dt + i(\theta d\bar{\theta} + \bar{\theta} d\theta)$ on $\mathbb{R}^{1|2}$, which serves as the super contact form.
• Verify that $\alpha$ is non-vanishing and defines a corank $(1|0)$ hyperplane distribution $\mathcal{D}_\alpha = \ker \alpha$, spanned by the odd vector fields $\mathbbmss{D}$ and $\bar{\mathbbmss{D}}$.
• Confirm the non-degeneracy of $d\alpha = 2i d\theta \wedge d\bar{\theta}$ on $\mathcal{D}_\alpha$ via contraction with $\mathbbmss{D}$ and $\bar{\mathbbmss{D}}$.
• Show that the SUSY transformations preserve $\alpha$ by computing $\alpha' = \alpha$, proving they are strict contactomorphisms.
• Compute the Lie derivatives $L_Q \alpha = 0$ and $L_{\bar{Q}} \alpha = 0$, confirming $Q$ and $\bar{Q}$ are strict contact vector fields.
• Identify the Reeb vector field $P = \partial/\partial t$ via the conditions $i_P \alpha = 1$ and $i_P d\alpha = 0$.

Experimental results

Research questions

• RQ1 How can the $d=1$, $N=2$ super-Poincaré algebra be geometrically realized using contact structures on supermanifolds?
• RQ2 What is the precise role of the supercontact form $\alpha = dt + i(\theta d\bar{\theta} + \bar{\theta} d\theta)$ in encoding supersymmetry?
• RQ3 Are the standard supercharge vector fields $Q$ and $\bar{Q}$ strict contact vector fields with respect to this form?
• RQ4 How does the Reeb vector field in this supercontact structure relate to the time-translation generator in supersymmetric mechanics?
• RQ5 Can the coset space method and Maurer–Cartan formalism be naturally linked to contact geometry in this context?

Key findings

• The supercontact form $\alpha = dt + i(\theta d\bar{\theta} + \bar{\theta} d\theta)$ on $\mathbb{R}^{1|2}$ defines a genuine contact structure, with $d\alpha = 2i d\theta \wedge d\bar{\theta}$ non-degenerate on the kernel distribution.
• The SUSY transformations act as strict contactomorphisms, as $\alpha$ is invariant under the transformation $t \to t + i(\epsilon \bar{\theta} - \theta \bar{\epsilon})$, $\theta \to \theta + \epsilon$, $\bar{\theta} \to \bar{\theta} + \bar{\epsilon}$.
• The supercharge vector fields $Q = \partial/\partial\theta + i\bar{\theta} \partial/\partial t$ and $\bar{Q} = \partial/\partial\bar{\theta} + i\theta \partial/\partial t$ are strict contact vector fields, satisfying $L_Q \alpha = 0$ and $L_{\bar{Q}} \alpha = 0$.
• The Reeb vector field is identified as $P = \partial/\partial t$, uniquely determined by $i_P \alpha = 1$ and $i_P d\alpha = 0$.
• The hyperplane distribution $\mathcal{D}_\alpha = \operatorname{Span}\{ \mathbbmss{D}, \bar{\mathbbmss{D}} \}$, where $\mathbbmss{D} = \partial/\partial\theta - i\bar{\theta} \partial/\partial t$, is not involutive, as $[\mathbbmss{D}, \bar{\mathbbmss{D}}] = -2i \partial/\partial t \notin \mathcal{D}_\alpha$.
• The construction is model-independent and provides a geometric interpretation of the $d=1$, $N=2$ super-Poincaré algebra using contact geometry, unifying supersymmetry with classical differential geometry.