nLab integration over supermanifolds

Idea

There are different ways to define a differential volume element on a smooth manifold. Some of these definitions can be carried over to supergeometry, others cannot. The possibly most familiar way of talking about differential volume elements, in terms of top-degree differential forms, does not carry over to supermanifolds.

No top-degree forms in supergeometry

In supergeometry the notion of top-degree form does not in general make sense, since there are no top-degree wedge powers of “odd 1-forms”: if for instance $\theta_1$ and $\theta_2$ are odd functions on some super Cartesian space $\mathbb{R}^{p|q}$ and $d \theta_1$ and $d \theta_2$ are their differential 1-forms, then the wedge product of these is symmetric in that

$$d\theta_1 \wedge d \theta_2 = + d\theta_2 \wedge d \theta_1 \,.$$

Notice the plus sign on the right, which is the product of one minus sign for interchanging $\theta_1$ and $\theta_2$, and another minus sign for interchanging the two differentials. See at signs in supergeometry for more on this.

Accordingly, the wedge product of the differential of an odd function $\theta$ with itself does not in general vanish:

$$(d \theta \wedge d\theta = 0) \Leftrightarrow (\theta = 0) \,.$$

On the cartesian supermanifold $\mathbb{R}^{n|m}$ with canonical even coordinate functions $\{x^i\}_1^n$ and canonical odd coordinate functions $\{\theta^j\}_1^m$ the differential form which one would want to regard as the canonical volume form is

$$\omega := d x^1 \wedge \cdots \wedge d x^n \wedge d\theta^1 \wedge \cdots \wedge d\theta^m \,.$$

Due to the above, this is not a top form, since for instance

$$\omega \wedge d\theta^1 \neq 0 \,.$$

But this example also indicates the solution: apparently for integration it is not really essential that a form is a top power, what is rather essential is that it is, locally, the wedge product of a basis of 1-forms. This perspective then does lead to a sensible definition of volume forms (and more generally “integrable forms”) on supermanifolds, described below.

Solution

Therefore the naïve identification of differential volume measures with top degree forms has to be refined. The idea is to characterize a volume form by other means, in particular as an equivalence class of choices of bases for the space of 1-forms, and then to define integrable forms to be pairs consisting of such a generalized volume form and a multivector: this pair is supposed to represent the differential form one would obtain could one contract the multivector with the volume form, as in ordinary differential geometry.

The definition of integration of integrable forms in supergeometry in terms of multivector fields leads, in the case that the supermanifold in an NQ-supermanifold to the BV formalism.

Related concepts

• integration

• Berezin integral

References

• Pierre Deligne, John Morgan, Ch 3 in: Notes on Supersymmetry (following Joseph Bernstein), in: Quantum Fields and Strings, A course for mathematicians, 1, Amer. Math. Soc. Providence (1999) 41-97 [ISBN:978-0-8218-2014-8, web version, pdf]

• Ivan Mirković, Sec 4-6 in: Notes on Super Math, in Quantum Field Theory Seminar, lecture notes (2004) [pdf, pdf]

Exposition:

• Urs Schreiber, Integration over supermanifolds&

A general abstract discussion in terms of D-module theory:

• Frédéric Paugam, Homotopical Poisson Reduction of gauge theories (pdf)

Geometric discussion of picture number appearing in the context of integration over supermanifolds (and originally seen in the quantization of the NSR superstring, crucial in superstring field theory) is due to

• Alexander Belopolsky, Picture changing operators in supergeometry and superstring theory (arXiv:hep-th/9706033)

and further amplified in

• Edward Witten, appendix D of Notes On Super Riemann Surfaces And Their Moduli (arXiv:1209.2459)

In this perspective picture number is an extra grading on differential forms on supermanifolds induced from a choice of integral top-form needed to define integration over supermanifolds:

• Roberto Catenacci, Pietro Grassi, Simone Noja, Superstring Field Theory, Superforms and Supergeometry, Journal of Geometry and Physics Volume 148, February 2020, 103559 (arXiv:1807.09563)

  (with an eye towards superstring field theory)

• C. A. Cremonini, Pietro Grassi, Pictures from Super Chern-Simons Theory (arXiv:1907.07152)

  (with an eye towards super Chern-Simons theory)

• C. A. Cremonini, Pietro Grassi, S. Penati, Supersymmetric Wilson Loops via Integral Forms (arXiv:2003.01729)

  (for super-Wilson lines)

Review:

• Pietro Grassi, Integral Forms and Applications, Sestri Levante 2015 (pdf, pdf)

See also:

• Sergio L. Cacciatori, Simone Noja, Riccardo Re, The Unifying Double Complex on Supermanifolds (arXiv:2004.10906)