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I'm studying the moduli space of vacua for some supersymmetric gauge theory and I want to know specifically why it is important to know the geometry of this space. I know everything about the division between Higgs branch and Coulomb branch for a gauge theory, I know that the moduli space is a Kahler manifold and I know you can see the moduli space as a complex algebraic variety (or better affine variety). What I would like to better understand is what is the physical reason that brings us to think to the geometry of the moduli space? Why don't we just try to find out the properties of the moduli space of vacua just looking at a Lagrangian density? Is it then if a theory has no Lagrangian you can understand it anyway just looking at the geometry of the space?

Where can I find some good review or book where is explicitly written the importance of the geometry of the moduli space in order to understand the physics of a supersymmetric gauge theory?

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  • $\begingroup$ The geometry of the moduli space is intimately related to the physics of the low-energy theory flowing into a vacuum on the moduli space. For example, the low-energy effective action on the moduli space always includes a sigma-model for the moduli scalars, $\mathcal{L} \supset g_{IJ}(\phi) \partial_\mu \phi^I \partial^\mu \phi^J$, where $g_{IJ}(\phi)$ is the moduli space metric. $\endgroup$
    – Elliot Schneider
    Commented Apr 6, 2017 at 15:46

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Broadly speaking Langland's conjecture establishes a relationship between Geometry (and topology), Number theory (quantum eigenvalues) and Functional Analysis (eigenfunctions). So when we explore the geometry of the moduli, we expect to be able to understand what are the vacuum states (low energy physics).

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