Gauge theory on schemes

Question

Gauge theories are traditionally formulated in space-time, superspace, manifolds, or supermanifolds.

Are there formulations of gauge theory in the context of algebraic geometry (e.g. defining fields on affine varieties, schemes, or stacks)?

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user Untitled

Comments

For instance, one particularly algebraic approach for manifolds (which could be promptly applied to schemes) is given in [Section 2.3, DEF$^+$99], where a Lagrangian on a manifold is defined as an element of a particular double chain complex (which is a subcomplex of a complex that is very similar to the de Rham complex of $\mathcal{F}\times M$, with $\mathcal{F}$ the space of fields (see there) on $M$).

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user Untitled

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[DEF$^+$99] Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, and Edward Witten, editors. Quantum fields and strings: a course for mathematicians. Vol. 1, 2. American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 1999. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997.

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user Untitled

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This is a tangent but why gauge theory on schemes?

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user AlexArvanitakis

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@AlexArvanitakis I don't have a particular objective in mind, but I think it might lead to interesting algebraic results, as gauge theory on manifolds has lead to (for example) Donaldson invariants. Also, it might be a natural way to study supersymmetry: in an affine scheme, nilpotents are elements of the underlying ring (also seem as functions on the scheme), whereas the sheaf of superspace $\mathbb{R}^{p|q}$ is the freely generated sheaf $\mathscr{C}^\infty[\theta^1,\dots,\theta^q]$ by nilpotent...

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user Untitled

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...generators $\theta^1,\dots,\theta^q$ of the sheaf $\mathscr{C}^\infty$ of smooth real-valued functions on ordinary space. That is, schemes include nilpotents by themselves, while superspace adds them a bit artificially.

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user Untitled

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fair enough. I was really looking for context that would jog my memory... You could look into papers by Eric Sharpe e.g. the notes arxiv.org/pdf/hep-th/0307245.pdf

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user AlexArvanitakis

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@AlexArvanitakis These notes (and his papers) are great! Thank you.

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user Untitled