Moduli Stacks and Derived Geometry of Gauge Fields

A moduli stack retains gauge automorphisms; its derived enhancement also retains obstruction and excess-intersection data. For flat $G$-connections on a closed surface, the tangent complex at a local system has cohomology $H^0$ for infinitesimal automorphisms, $H^1$ for deformations, and $H^2$ for obstructions. Replacing this object by its closed points or coarse character variety loses exactly the data that become nontrivial at reducible representations.

Required background. Orbit strata and stabilizers supplies isotropy and singular quotients. Derived critical loci and gauge quotients supplies homotopy quotients and tangent complexes. Elliptic gauge complexes supplies the deformation–obstruction interpretation. Helpful background. Derived and higher local-to-global structures supplies homotopy-coherent gluing.

From the character variety to the derived stack

Let $\Sigma$ be a closed oriented surface and $G$ a reductive group with Lie algebra $\mathfrak g$. A flat connection determines a representation $\rho:\pi_1(\Sigma)\to G$, well defined up to conjugation. Three related objects must be distinguished:

$$\operatorname{Hom}(\pi_1\Sigma,G)/G, \qquad [\operatorname{Hom}(\pi_1\Sigma,G)/G], \qquad \mathbf R\operatorname{Loc}_G(\Sigma) =\mathbf R\operatorname{Map}(\Sigma,BG).$$

The first is the coarse character variety. The second is the quotient stack, which retains the centralizer $Z_G(\rho)$ as the automorphism group of $\rho$. The third is derived: it also retains the homological failure of the flatness equations and conjugation action to meet transversely.

At $\rho$, the derived tangent complex is

$$\mathbb T_\rho\mathbf R\operatorname{Loc}_G(\Sigma) \simeq\mathbf R\Gamma(\Sigma,\operatorname{ad}\rho)[1].$$

With cohomological grading, this gives

$$H^{-1}(\mathbb T_\rho)=H^0(\Sigma,\operatorname{ad}\rho), \quad H^0(\mathbb T_\rho)=H^1(\Sigma,\operatorname{ad}\rho), \quad H^1(\mathbb T_\rho)=H^2(\Sigma,\operatorname{ad}\rho).$$

These are respectively infinitesimal stabilizers, first-order deformations modulo gauge, and obstructions. The shift is essential: the tangent object is a complex, not only its middle cohomology.

Goldman derives the cocycle tangent space $Z^1$, the conjugation directions $B^1$, and the quotient tangent $H^1$ on the simple locus in Goldman 1984, §1.2, pp. 202–205. His construction also shows why the ordinary quotient becomes singular where centralizers grow.

Irreducible and reducible surface local systems

Suppose $G$ is semisimple and $\rho$ is irreducible modulo a finite center. Then the Lie algebra of its centralizer vanishes, so $H^0=0$. Poincaré duality with an invariant nondegenerate pairing on $\mathfrak g$ gives $H^2\cong(H^0)^*=0$. The derived tangent complex is concentrated in degree zero, and the moduli stack is classically smooth near $\rho$. Euler characteristic then gives

$$\dim H^1(\Sigma,\operatorname{ad}\rho) =(2g-2)\dim G$$

for genus $g\ge2$ under these hypotheses.

At a reducible $\rho$, $H^0$ is nonzero and, by duality, so is $H^2$. The same coarse point then carries both automorphisms and obstruction directions. Keeping only $H^1$ can falsely suggest a smooth tangent space of the wrong dimension. Keeping only closed points loses even the centralizer.

If $BG$ carries its canonical $2$-shifted symplectic structure from an invariant nondegenerate form and $\Sigma$ supplies a two-dimensional orientation, transgression gives a $0$-shifted symplectic structure on the derived mapping stack, provided the mapping stack is derived Artin and locally of finite presentation. The precise existence theorem and its orientability hypotheses are Pantev, Toën, Vaquié, and Vezzosi 2013, Theorem 2.5, pp. 307–308. On the smooth simple locus, this recovers Goldman’s symplectic form. It does not assert that every coarse singular variety is an ordinary symplectic manifold.

Boundaries change this conclusion. Transgression over a closed oriented surface has no boundary term, whereas a surface with boundary generally produces boundary evaluation data and a shifted Poisson or relative symplectic structure rather than an absolute symplectic form on the unrestricted mapping stack. Fixing boundary holonomy conjugacy classes, or choosing compatible Lagrangian boundary conditions, can recover symplectic leaves. One therefore cannot apply the closed-surface theorem by silently discarding boundary contributions.

Flat connections as the first gauge application

The first application returns to Gauge Orbits, Gauss Constraints, and Stabilizers. A flat connection on $\Sigma$ has the elliptic deformation complex

$$\Omega^0(\Sigma,\operatorname{ad}P) \xrightarrow{d_A}\Omega^1(\Sigma,\operatorname{ad}P) \xrightarrow{d_A}\Omega^2(\Sigma,\operatorname{ad}P).$$

Flatness makes $d_A^2=[F_A,\cdot]=0$. Its cohomology is the de Rham model of the three groups above. Thus the stack and elliptic descriptions agree independently: covariantly constant sections are stabilizers, harmonic one-forms are infinitesimal moduli, and two-form cohomology contains obstructions.

The derived enhancement does not by itself quantize the theory, construct a gauge-invariant measure, compactify the moduli space, or solve global gauge fixing. It supplies the correct deformation object. Additional analytic or quantum input is still required.

It also distinguishes an obstruction space from an actual obstruction. A nonzero $H^2$ says that the nonlinear curvature equation may have a nontrivial Kuranishi map with values there; it does not say that every vector in $H^1$ is obstructed. Conversely, $H^2=0$ licenses formal unobstructedness near the chosen point only after the deformation problem and its analytic completion have been specified. It says nothing about distant components, compactness, or the existence of a measure on the resulting moduli object.

Failure test

The adversarial replacement sends $\mathbf R\operatorname{Loc}_G(\Sigma)$ to the set of closed conjugacy classes. At the trivial $SU(2)$ representation, the automorphism group $SU(2)$ disappears, and so do $H^0(\Sigma,\mathfrak{su}(2))$ and its dual obstruction group $H^2$. The retained point set cannot recover them. This is a concrete failure, not a preference for more elaborate language.

The converse boundary is that a derived enhancement does not make an obstructed point smooth. It records the obstruction in degree $+1$; it does not make that class vanish.

Exercises

For the trivial $SU(2)$ local system on a genus-$g$ closed surface, compute the dimensions of $H^0$, $H^1$, and $H^2$.

Solution

The adjoint local system is constant of rank three. Hence $H^i(\Sigma,\operatorname{ad}\rho)\cong H^i(\Sigma,\mathbb R)\otimes\mathfrak{su}(2)$. Therefore $\dim H^0=3$, $\dim H^1=2g\cdot3=6g$, and $\dim H^2=3$. The nonzero outer groups show that the trivial representation is reducible and derived, not a smooth point with tangent dimension $6g$ alone.

References

• Goldman, William M. “The Symplectic Nature of Fundamental Groups of Surfaces.” Advances in Mathematics 54 (1984): 200–225. DOI; Open PDF.
• Pantev, Tony, Bertrand Toën, Michel Vaquié, and Gabriele Vezzosi. “Shifted Symplectic Structures.” Publications Mathématiques de l’IHÉS 117 (2013): 271–328. DOI; Open PDF.