Moduli spaces in string theory vs. soliton theory

Question

In both string theory and soliton theory, moduli spaces are frequently used.

As far as I known, for soliton theory, moduli spaces are something like collective coordinates for solitons, and for string theory, moduli spaces is the spaces of all metrices divided by all conformal rescalings and diffeomorphisms. It seems like these two definitions(?) of moduli spaces are quite different, but the same terminology is used in both cases. I also learned that the name 'moduli spaces' comes from abstract geometry, but I don't know if that's any help here.

My question is the following: Could anyone provide an intuitive connection between the two uses of moduli spaces, or highlight the differences?

Answer 1

This is a situation where knowing the history of the terminology can be helpful.

The QFT/string theory terminology comes from algebraic geometry, where the term moduli space is used for any space whose points correspond to some kind of geometric object. The projective space $\mathbb{P}(V)$, for example, is the moduli space of lines in the vector space $V$. Likewise, a moduli space of instantons is the space of solutions to a set of instanton equations. And the moduli space of complex curves is what you end up integrating over in perturbative string theory after accounting for the gauge symmetries acting on the worldsheet metric.

The word 'modulus' (plural 'moduli') just means 'parameter'. Moduli spaces were originally thought of as spaces of parameters, rather than as spaces of geometric objects; mathematicians were interested in how the various ways of parameterizing geometric objects were related and eventually realized these parameters were coordinates on a space. String theorists have resurrected this old terminology by using the term 'moduli field' to refer to a field which parametrizes a moduli space.