Contents

Idea

A $\sigma$-model is a quantum field theory which is induced from a target space that carries some geometric structure, usually that of an n-bundle with connection representing a gauge background field.

The fields of a $\sigma$-model on parameter space $\Sigma$ are maps from $\Sigma$ to target space $X$.

One way to make this precise for topological $\sigma$-models is to say that target space $X$, or possibly the gauge bundle $P\to X$ over it, represents an functor that sends cobordism cospans to spans which in turn are taken to act by pull-push on the quantum states, which are objects in geometric infinity-function theory living over the mapping spaces $[\Sigma ,P]$.

Physical interpretation

$\sigma$-model quantum field theories on parameter spaces $\Sigma$ of top dimension $n+1$ are to be thought of as encoding the quantum mechanics of the propagation of an $n$-dimensional particle, called an $n$-brane, in target space $X$, subject to the forces imposed on it by the backgreound field, under which it is said to be charged.

These cases describe non-topological quantum field theories. Here the formalization of the notion of $\sigma$-model is not entirely complete. Yet

Examples

Non-topological $\sigma$-models

• The canonical textbook example of a quantum mechanical system is of this form for $n=1$: A line bundle with connection $E\to X$ on a Riemannian manifold $X$ induces the 1-dimensional quantum field theory which is the quantum mechanics of a point particle which propagates on $X$, subject to the forces of gravitation (given by the metric on $X$) and electromagnetism (given by the line bundle with connection). The Hamilton operator encoding this quantum dynamics in this case is the Laplace-operator of $TX$ twisted by the line bundle $E$.

• Generalizing in the above example the line bundle $E$ by an abelian bundle gerbe with a connection yields a background for a 2-dimensional $\sigma$-model which mayb be thought of as describing the propgation of a string. The best-studied version of this is the case where $X=G$ is a Lie group, in which case this $\sigma$-model is known as the Wess–Zumino–Witten model.

Topological $\sigma$-models

• Dijkgraaf-Witten theory is the (2+1)-dimensioanl $\sigma$-model induced from an abelian 2-gerbe on $BG$, for $G$ a finite group.

• Chern-Simons theory is supposed to be analogously the $\sigma$-model induced from an abelian 2-gerbe with connection on $BG$, but now for $G$ a Lie group.

• the Poisson sigma-model is a model whose target is a Poisson Lie algebroid.

• in AKSZ theory this is generalized to a large class of sigma models with Lie infinity-algebroids as target.

• Rozansky–Witten theory is essentially the $\sigma$-model for $X$ a smooth projective variety.

References

First indications on how to formalize $\sigma$-models in a higher categorical context were given in

• Dan Freed, Higher algebraic structures and quantization (arXiv)

A more complete formalization along the lines of the above operation was indicated in

• Marco Grandis, Cospans in Algebraic Topology

and

• David Ben-Zvi, John Francis, David Nadler, Integral transforms and Drinfeld centers in derived geometry (arXiv) .

More discussion of the latter is at geometric infinity-function theory.