## Advances in Theoretical and Mathematical Physics

- Adv. Theor. Math. Phys.
- Volume 10, Number 5 (2006), 713-745.

### Topological membrane theory from Mathai-Quillen formalism

Lilia Anguelova, Annamaria Sinkovics, and Paul de Medeiros

#### Abstract

It is suggested that topological membranes play a fundamental role in the recently proposed topological $M$-theory. We formulate a topological theory of membranes wrapping associative three-cycles in a sevendimensional target space with $G_2$ holonomy. The topological BRST rules and BRST invariant action are constructed via the Mathai–Quillen formalism. In a certain gauge, we show this theory to be equivalent to a membrane theory with two BRST charges found by Beasley and Witten. We argue that at the quantum level, an additional topological term should be included in the action, which measures the contributions of membrane instantons. We construct a set of local and non-local observables for the topological membrane theory. As the BRST cohomology of local operators turns out to be isomorphic to the de Rham cohomology of the $G_2$ manifold, our observables agree with the spectrum of $d = 4$, $N = 1 G_2$ compactifications of $M$-theory.

#### Article information

**Source**

Adv. Theor. Math. Phys., Volume 10, Number 5 (2006), 713-745.

**Dates**

First available in Project Euclid: 5 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.atmp/1175791015

**Mathematical Reviews number (MathSciNet)**

MR2281546

**Zentralblatt MATH identifier**

1115.81373

**Subjects**

Primary: 53Dxx: Symplectic geometry, contact geometry [See also 37Jxx, 70Gxx, 70Hxx]

Secondary: 81T45: Topological field theories [See also 57R56, 58Dxx]

#### Citation

Anguelova, Lilia; de Medeiros, Paul; Sinkovics, Annamaria. Topological membrane theory from Mathai-Quillen formalism. Adv. Theor. Math. Phys. 10 (2006), no. 5, 713--745. https://projecteuclid.org/euclid.atmp/1175791015