nLab linearized Yang-Mills equation

Idea

The linearized Yang-Mills equation is a linearization of the non-linear Yang-Mills equation, hence the best approximation by a linear equation at a given solution, called Yang-Mills connections. It is obtained as the differential of the Yang-Mills map, which maps the space of principal connections to their evaluation of the Yang-Mills equation term, so that its solutions are exactly those mapped to the origin. (Due to the affinity of the space of principal connections and the non-linearity, this is not the kernel.)

Since the tangent space has the same dimension as its underlying manifold, the local virtual dimension of the Yang-Mills moduli space can be obtained from the linearization using the Atiyah-Singer index theorem.

Yang-Mills map

Consider the setup described in detail in Yang-Mills equation. The Yang-Mills map is given by:

$$f\colon \mathcal{A}\rightarrow\Omega^1(M,Ad(P)), A\mapsto\delta_A F_A.$$

$A\in\mathcal{A}$ is a solution of the Yang-Mills equations if and only if $f(A)=0$. The differential of the Yang-Mills map in it is given by:

$$D_A f =\delta_A d_A+\star[-,\star F_A]\colon \Omega^1(M,Ad(P))\rightarrow\Omega^1(M,Ad(P)).$$

Linearized Yang-Mills equations

Let $B\in\Omega^1(M,Ad(P))$, then the linearized Yang-Mills equations at a solution $A\in\mathcal{A}$ are given by $D_A f(B)=0$ or equivalently by:

$$\delta_A d_A B+\star[B,\star F_A] =0.$$

Elliptic complex

Let $act\colon\mathcal{G}\rightarrow\mathcal{A}$ be the action of the gauge group $\mathcal{G}=Aut(P)$ on a solution $A$. Since the Yang-Mills equations are gauge invariant, one has $f\circ act=0$ for the composition and therefore $D_A f\circ D_1 act=0$ for their differential, which yields an elliptic complex $\mathcal{E}(A,\phi)$ given by:

$$1 \rightarrow End(Ad(P)) \xrightarrow{D_1 act}\Omega^1(M,Ad(P)) \xrightarrow{D_A f}\Omega^1(M,Ad(P)) \rightarrow 1.$$

It has the cohomology vector spaces:

$$H^0(\mathcal{E}(A)) =ker(D_1 act);$$

$$H^1(\mathcal{E}(A)) =ker(D_A f)/img(D_1 act);$$

$$H^2(\mathcal{E}(A)) =\Omega^1(M,Ad(P))/img(D_A f).$$

$H^0(\mathcal{E}(A))$ can be identified with the tangent space $T_1 Stab_A(\mathcal{G})$. $H^1(\mathcal{E}(A))$ can be identified with the tangent space $T_{[A]}\mathcal{M}$ of the Yang-Mills moduli space. If $H^0(\mathcal{E}(A))$ and $H^2(\mathcal{E}(A))$ vanish, then the analytic index is:

$$ind\mathcal{E}(A,\psi) =-dim H^0(\mathcal{E}(A)) +dim H^1(\mathcal{E}(A)) -dim H^2(\mathcal{E}(A)) =dim H^1(\mathcal{E}(A)) =dim T_{[A]}\mathcal{M}$$

(For finite-dimensional vector spaces in the complex, these can replace the cohomology vector spaces in the index due to the homomorphism theorem?.) The Atiyah-Singer index theorem assures this analytic index to be equal to the topological index, which depends only on topological invariants.

Related concepts

On ordinary Yang-Mills theory (YM):

• D=2 YM, D=3 YM, D=4 YM, D=5 YM, D=6 YM, D=7 YM, D=8 YM

• Maxwell theory/electromagnetism (U(1) YM), Donaldson theory (SU(2) YM), quantum chromodynamics (SU(3) YM)

• Yang-Mills equation, linearized Yang-Mills equation, Yang-Mills instanton, Yang-Mills field, stable Yang-Mills connection, Yang-Mills moduli space, Yang-Mills flow, F-Yang-Mills equation, Bi-Yang-Mills equation

• Uhlenbeck's singularity theorem, Uhlenbeck's compactness theorem

On variants of Yang-Mills theory and on super Yang-Mills theory (SYM):

• Yang-Mills-Higgs equations, stable Yang-Mills-Higgs pair, Yang-Mills-Higgs flow

• Einstein-Yang-Mills theory, Einstein-Yang-Mills-Dirac theory, Einstein-Yang-Mills-Dirac-Higgs theory

• 3D Yang-Mills-Chern-Simons theory

• 3d superconformal gauge field theory: D=3 N=1 SYM, D=3 N=2 SYM, D=3 N=4 SYM

• 4d superconformal gauge field theory: D=4 N=1 SYM, D=4 N=2 SYM, D=4 N=4 SYM

• D=5 SYM, D=6 SYM, D=7 SYM, D=10 SYM

• topological Yang-Mills theory, topologically twisted D=4 super Yang-Mills theory