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How should a geometer think about quotienting out by a Fermionic symmetry? Is this a formal concept? A strictly linear concept? A sheaf theoretic concept?

How does symplectic reduction work with odd symmetries within geometric quantization? Are there moduli or instanton-like deformation complexes where the space of symmetries acting on the space of fields includes both even and odd symmetries?

For example, can the familiar instanton moduli problem whose deformations are governed by the linearized complex: $$\Omega^0(Ad_0) \stackrel{d_A}\longrightarrow \Omega^1(Ad_0 ) \stackrel{\Pi \circ d_A} \longrightarrow \bar\Omega^2(Ad_0 )$$ be enhanced by fermionic symmetries to some structure whose linearized deformation complex looks like: $$\Omega^0(Ad_0 \oplus Ad_1) \stackrel{d_A}\longrightarrow \Omega^1(Ad_0 \oplus Ad_1) \stackrel{\Pi \circ d_A} \longrightarrow \bar\Omega^2(Ad_0 \oplus Ad_1)$$ in a geometrically meaningful way?

In some sense the question is trying to understand how 'odd symmetries' or Fermionic coordinates can be meaningfully carried through a familiar non-linear problem in geometry which makes use of a non-linear group action. Are these odd symmetries true geometric symmetries to be exponentiated in some way, or are they more properly formal symmetry-like operators analogous to true symmetries? Thanks in advance.

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1 Answer 1

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This answer will address the two questions about symplectic reduction and the instanton moduli space separately.

First question:

An accepted method to generalize symplectic reduction to include fermionic symmetries is by means the theory of symplectic supermanifolds (and more generally Poisson supermanifolds). Please see the following exposition by Tilmann Glimm.

Symplectic supermanifolds can be equipped with two types of symplectic structures, odd and even. In Glim's article the reduction theorem is proved for the two types of symplectic structures.

An example of a symplectic supermanifold is the supercotangent bundle (see for example the following article by: J. P. Michel) of a spin manifold. Locally this bundle is covered by charts $\{p_i, q^i, \theta_i\}$, consisting of position, momentum and Grassmann coordinates.

The symmetries divided by in the supersymplectic reduction consist of Lie supergroups with Hamiltonian action on the supersymplectic manifold. In Glimm's article, there is a detailed description of the symplectic reduction of the Bose-Fermi oscillator by its Hamiltonian supersymmetry group (which is a subgroup of the orthosymplectic group).

The theory of geometric quantization can be extended to supergeometry. The quantization of odd symplectic structures lead to Batalin-Vilkovisky theory. In its simplest application, this quantization results quantization spaces isomorphic to the de-Rham complex of the manifold.

The classical mechanical theory associated with the even symplectic particle is the Berezin-Marinov classical description of the spin by means of Grassmann variables. Its quantization corresponds to superparticles, where the fermions consist of sections of the spinor bundle over the base manifold.

Second question:

Basically, the instanton moduli space for super Yang-Mills theories consists of the usual instanton moduli space given by Atiyah Hitchin Singer deformation complex together with the zero modes of a twisted Dirac operator (see for example the following article by: Mainiero and Walter Tangarife in the context of the Seiberg-Witten theory.

A supersymmetric generalization of the deformation complex actually exists at least for the N= 4 super Yang-Mills as given in Labastida and Lozano's article (equation (3.6).)

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