# nLab large volume limit

### Context

#### Riemannian geometry

Riemannian geometry

## Basic definitions

• Riemannian manifold

• moduli space of Riemannian metrics

• pseudo-Riemannian manifold

• geodesic

• Levi-Civita connection

• ## Theorems

• Poincaré conjecture-theorem
• ## Applications

• gravity

• # Contents

## Idea

In Riemannian geometry given any function/functional on the space of Riemannian metrics $g$ on some manifold $X$, then its large volume limit is, if it exists, the limit of the functional evaluated on a sequence $t g$ of metrics as $t \to \infty$.

This plays a role in particular in the studies of sigma-model quantum field theory with target space $(X,g)$. Here the large volume limit may equivalently be thought of as the limit in which the extension of the brane described by the $\sigma$-model vanishes.

One example is the Witten genus, which is the large volume limit of the partition function of the superstring $\sigma$-model (Witten 87, p. 4)

## References

• Edward Witten, Elliptic Genera And Quantum Field Theory , Commun.Math.Phys. 109 525 (1987) (Euclid)

Last revised on March 12, 2014 at 10:15:23. See the history of this page for a list of all contributions to it.