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)?
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$\begingroup$ 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 $M$). $\endgroup$– EmilyJan 22, 2019 at 17:38
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6$\begingroup$ This is a tangent but why gauge theory on schemes? $\endgroup$– AlexArvanitakisJan 22, 2019 at 18:52
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1$\begingroup$ @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... $\endgroup$– EmilyJan 22, 2019 at 19:30
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1$\begingroup$ ...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. $\endgroup$– EmilyJan 22, 2019 at 19:30
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3$\begingroup$ 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 $\endgroup$– AlexArvanitakisJan 22, 2019 at 19:40
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Gauge Theory on $C^\infty$-(Super)schemes:
Chien-Hao Liu and Shing-Tung Yau have investigated this question (see [1] and [2]) in the context of (super) $C^\infty$-algebraic geometry.
(Super) $C^\infty$-algebraic geometry is a generalization of algebraic geometry in which rings are substituted by ($\mathbb{Z}/2$-graded) $C^\infty$-rings, defined as $\mathbb{R}$-algebras having all smooth functions as operations, instead of just the ordinary algebra ones. It was introduced by Dominic Joyce in [3]. (See [4] and [5] for surveys and [6] for further development.)
More specifically, Liu and Yau construct in [1, Section 1.2] a $C^\infty$-superscheme $\widehat{X}=(X,\widehat{\mathscr{O}}_X)$, called the $d=4$, $\mathcal{N}=1$ superspace. In [2], they extend this construction and introduce the concept of towered superspace $\widehat{X}^\widehat{\boxplus}$, which is a $C^\infty$-superscheme. From $\widehat{X}^\widehat{\boxplus}$, the authors construct another $C^\infty$-superscheme $X^\mathrm{physics}$. Finally, they define two quantum field theories on $X^\mathrm{physics}$, which are analogues of the Wess-Zumino model and supersymmetric $\mathrm{U}(1)$ gauge theory with matter.
References
[1] Chien-Hao Liu and Shing-Tung Yau. $N=1$ fermionic D3-branes in RNS formulation I. $C^\infty$-Algebrogeometric foundations of $d=4,N=1$ supersymmetry, SUSY-rep compatible hybrid connections, and $\widehat{D}$-chiral maps from a $d=4$ $N=1$ Azumaya/matrix superspace. arXiv:1808.05011 [math.DG]
[2] Chien-Hao Liu and Shing-Tung Yau. Physicists' $d=3+1$, $N=1$ superspace-time and supersymmetric QFTs from a tower construction in complexified $\mathbb{Z}/2$-graded $C^\infty$-Algebraic Geometry and a purge-evaluation/index-contracting map. arXiv:1902.06246 [hep-th]
[3] Algebraic Geometry over $C^\infty$-rings, Dominic Joyce. arXiv:1001.0023 [math.AG]
[4] An introduction to C-infinity schemes and C-infinity algebraic geometry, Dominic Joyce. arXiv:1001.0023 [math.AG]
[5] $C^\infty$-Algebraic Geometry, Elana Kalashnikov. Available at https://people.maths.ox.ac.uk/joyce/theses/KalashnikovMSc.pdf.
[6] D-manifolds and d-orbifolds: a theory of derived differential geometry, Dominic Joyce. Available at https://people.maths.ox.ac.uk/joyce/dmanifolds.html.