## Friday, September 30, 2005

### Finiteness of moduli spaces

Zhiqin Lu and Mike Douglas propose a physics proof of the finiteness of the moduli spaces - where the volume is measured by the Zamolodchikov metric - that was recently promoted by Cumrun Vafa in his Swampland. It is not quite clear how general their proof is but it has essentially the following parts:
• argue that your non-linear sigma-model may be constructed from a gauged linear sigma-model (GLSM) and RG flow
• show that the volume of the moduli space is finite in the GLSM - it's because the moduli space is something like CP^{125} with a non-singular metric
• demonstrate that the finiteness does not change by the RG flow: although the total "time" of the flow is infinite, most of the changes appear in a particular finite interval where the RG scale is comparable to the typical scales of the GLSM given by its coupling(s)
They need to assume a gap, making the connection between the discrete spectrum and the finiteness of the volume a bit more comprehensible. It's a bit confusing that the existence of dualities - that seem essential for the finiteness - does not seem to play too much of a role. It's probably because the GLSM does not have most of these dualities and it directly describes a fundamental domain. The dualities appear in the IR limit only. Do I understand it well?