Soliton Moduli Spaces and Homotopy Theory

Question

The four-dimensional $SU(N)$ Yang-Mills Lagrangian is given by $$\mathcal{L}=\frac{1}{2e^2}\mathrm{Tr}F_{\mu\nu}F^{\mu\nu}$$

and gives rise to the Euclidean equations of motion $\mathcal{D}_\mu F^{\mu\nu}=0$ with covariant derivative $\mathcal{D}_\mu$. Finite action solutions $A_\mu$ satisfy the condition that,

$$A \to ig^{-1}\partial_\mu g$$

as we approach $\partial \mathbb{R}^4 \cong \mathbb{S}^3_{\infty}$, with $g$ an element of $SU(N)$. These provide a map from $\mathbb{S}^3_\infty$ to $SU(N)$, and are classified by homotopy theory. In Tong's lecture notes on solitons, he states without proof that the second Chern class, or Pontryagin number $k \in \mathbb{Z}$ is given by

$$k = \frac{1}{24\pi^2}\int_{S^3_\infty}\mathrm{d}S^3_\mu \, \, \mathrm{Tr} \,(\partial_\nu g)g^{-1}(\partial_\rho g)g^{-1} (\partial_\sigma g)g^{-1}\epsilon^{\mu \nu \rho \sigma}$$

As I understand them, Chern classes are characteristic classes of bundles on manifolds; in this case what bundle and manifold is $k$ associated with? How does one obtain $k$ in this case? Tong states "The integer... counts how many times the group wraps itself around spatial $\mathbb{S}^3_\infty$."

In addition, Tong states without rigorous proof the metric of the moduli space (the space of all solutions to the equations of motion which are self-dual):

$$g_{\alpha \beta} = \frac{1}{2e^2} \int \mathrm{d}^4 x \mathrm{Tr} \, \, (\delta_\alpha A_\mu)(\delta_\beta A_\mu)$$

with $\delta_\alpha A_\mu = \partial A_\mu / \partial X^{\alpha} + \mathcal{D}_\mu \Omega_\alpha$ where $\Omega_\alpha$ is an infinitesimal transformation, and $X^{\alpha}$ are the collective coordinates. How does one compute such a metric of a moduli space? Why should it be given by the sum of all zero modes?

I would prefer an answer which utilizes arguments from differential geometry, and topology.

Answer 1

First let me refer you to Eric Weinberg's book where the instanton moduli space is described in more detail.

Principal bundles over 4-dimensional Riemannian manifolds are classified by the second Chern class = Instanton number and the t' Hooft discrete Abelian magnetic fluxes. Please see the following Lecture notes by Måns Henningson.

t' Hooft fluxes are present only when the gauge group has a nontrivial center, thus in the case of $SU(N)$, the classification is according to the instanton numbers.

A compactified Minkowski space, can be thought of as a four dimensional ball $B^4$, with the boundary points ( at infinity) $S^3$ identified. Thus by the Stokes theorem, the instanton number is given by:

$$ k = \int_{B^4}*F \wedge F =\int_{B^4}dCS(A) = \int_{S^3_{\infty}} CS(A) = \int_{S^3_{\infty}} WZW(g) $$

Where $CS(A)$ and $WZW(g)$ are the Chern-Simons and the Wess-Zumino-Novikov-Witten functionals respectively and the last step is derived from the substitution of the pure gauge condition at infinity.

The local geometry of the moduli space can be understood as follows: The instanton solutions are of the form:$ A_{\mu} = A_{\mu}(X^\alpha)$, where $X^\alpha$ are the coordinates of the moduli space. These solutions are local minima of the action, for all constant values of the moduli but the action is not extremal when these coordinates are made to depend on time. This time dependence is introduced to study the dynamics of the moduli near the classical solution.

The difference between the action with time varying moduli and time invariant ones is due to the time dependence of the moduli coordinates, assuming the time derivatives are small(i.e., by substituting $X^{\alpha}= X_0^{\alpha} + t \dot{X}_0^{\alpha}$, the leading terms have the least number of time derivatives thus the varied action must have the form:

$$ I = \frac{1}{2g^2} \int d^4x F_{\mu\nu}(A) F^{\mu\nu}(A) = \frac{8\pi^2}{g^2} k + \int dt B_{\alpha}(X)\dot{X}^{\alpha} +g_{\alpha \beta}(X)\dot{X}^{\alpha}\dot{X}^{\beta} + ...$$

Where the last step is obtained after the integration of the "known" solution over the spatial coordinates.

This action has just the form of a particle moving on a Riemannian manifold having a metric $g$ in a magnetic field $B$. The exterior derivative of the magnetic field can be interpreted as the symplectic structure of the moduli space. The form of the metric taken by David Tong will give exactly the same metric, because, the leading terms in the moduli time derivatives of the Yang Mills action will include a time derivative, thus we are left with the variation of the gauge fields themselves.

This is only the local structure of the moduli space. This analysis will not tell us, for example, if the symplectic structure is exact or closed. Of course, the global structure of the moduli space requires deeper analysis.

One global property that can be "relatively" easily computed is the moduli space dimension, even if a simple closed form of the solution is not available: The dimension is the count perturbation $A_{\mu}+a_{\mu}$ of a self dual solution $A_{\mu}$ which is also self dual modulo gauge transformations. Inserting the gauge potential in the self duality equation, The following condition is obtained (Weinberg equation 10.112):

$$ \eta^{\mu \nu} D_{A \mu} a_{\nu} = 0$$

where $D_{A}$ is the covariant derivative of the classical solution, and $ \eta^{\mu \nu} $ is a self dual matrix defined in Weinberg (equation 10.74). To remove the pure gauge direction, there exist another orthogonality condition to all pure gauge directions:

$$D_{A\mu} a^{\mu} = 0$$.

These two conditions can be combined together into a single Dirac equation:

$$\not{D}_{A} \Psi = 0$$.

where, the relation of the gauge field perturbation and the spinor $\Psi$ are given by: $a_{\mu} = \sigma_{\mu}^{\alpha \dot{\alpha}} \Psi_{\alpha \dot{\alpha}}$

This construction converts the problem of counting the number of moduli to counting the number of a Dirac equation zero modes. For the Dirac equation, the number of zero modes is given by the Atiyah-Singer index theorem.