# Gauge theory on schemes

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)?

• 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 $M$). – Théo Jan 22 '19 at 17:38
• This is a tangent but why gauge theory on schemes? – AlexArvanitakis Jan 22 '19 at 18:52
• @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... – Théo Jan 22 '19 at 19:30
• ...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. – Théo Jan 22 '19 at 19:30
• 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 – AlexArvanitakis Jan 22 '19 at 19:40

Gauge Theory on $$C^\infty$$-(Super)schemes:

Chien-Hao Liu and Shing-Tung Yau have investigated this question (see  and ) 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 . (See  and  for surveys and  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 , 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

 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]

 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]

 Algebraic Geometry over $$C^\infty$$-rings, Dominic Joyce. arXiv:1001.0023 [math.AG]

 An introduction to C-infinity schemes and C-infinity algebraic geometry, Dominic Joyce. arXiv:1001.0023 [math.AG]

 $$C^\infty$$-Algebraic Geometry, Elana Kalashnikov. Available at https://people.maths.ox.ac.uk/joyce/theses/KalashnikovMSc.pdf.

 D-manifolds and d-orbifolds: a theory of derived differential geometry, Dominic Joyce. Available at https://people.maths.ox.ac.uk/joyce/dmanifolds.html.

• (Since I am not familiar with their work, this answer is probably only a decorated pointer to the literature. (At best!)) – Théo Feb 19 '19 at 20:34