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Atiyah-Hitchin-Singer Ref 1 states that the number of virtual dimensions of the instanton moduli space for SU(N) Yang-Mills theory with topological charge $\mathcal{Q}$ over a manifold $X$ is given by $$ D=4N \mathcal{Q} - (N^2-1)\frac{\chi(X) + \sigma(X)}{2} \;, $$ where $\chi$ and $\sigma$ are the Euler number and the signature of the 4-manifold $X$.

Questions:

  1. If I understand correctly, the virtual dimensions coincide with the actual dimensions for irreducible solutions when the gauge group is completely broken by the solutions. What is the mathematical definition of virtual dimension, precisely, of the instanton moduli space?

  2. If I understand correctly, $\mathcal{Q}$ shall be related to the Pontryagin class $p_1(E)$? (in p.428 or in p.431) $$ \mathcal{Q} \sim p_1(E) = -\frac{1}{ 4 \pi^2}\int \text{tr}(\Omega^2)= \frac{1}{ 4 \pi^2}\int |\Omega|^2 \omega, $$ where $\Omega$ is the curvature 2-form and $\omega$ is the volume form?

  3. How to digest this dimension equality $ D=4N \mathcal{Q} - (N^2-1)\frac{\chi(X) + \sigma(X)}{2}$?

Since $D$ is a dimension, $D$ is an integer.

(1) Can $D=0$?

(2) $\chi(X) + \sigma(X)$ must be an even integer, for the $D$ to be an integer.

(3) Since $D \geq 0$, and the integer $D \in \mathbb{N}$, we have $$4N \mathcal{Q} > (N^2-1)\frac{\chi(X) + \sigma(X)}{2}$$ always. Why is this true intuitively? How do we see that increasing the $\frac{\chi(X) + \sigma(X)}{2}$, we obtain a smaller $D$?

Ref 1: M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Selfduality in Four-Dimensional Riemannian Geometry, Proc. Roy. Soc. Lond. A362 (1978) 425–461.

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    $\begingroup$ The virtual dimension is a Fredholm index of some elliptic complex. And a very small computation shows why $\chi+\sigma$ is even (see for example here: math.stackexchange.com/questions/1599833/…). Your questions will be generally answered when reading introductory notes on this stuff (see for example: Kronheimer-Donaldon's book "The geometry of four-manifolds"). $\endgroup$ – Chris Gerig Oct 17 '18 at 22:58
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    $\begingroup$ The claim that this coincides with the "actual dimension" of the space of solutions is only true under transversality conditions which do not necessarily hold. This is true on a simply connected manifold for generic choice of metric. There is no reason to believe that $D \geq 0$. You can make $\chi + \sigma$ an arbitrarily large number, say by connect summing copies of $\Bbb{CP}^2$. I don't think there's a reason to think that there should be intuitive content to an index theory calculation. $\endgroup$ – Mike Miller Oct 17 '18 at 23:23
  • $\begingroup$ Thank you very much for the helpful guidelines. Thanks! $\endgroup$ – wonderich Oct 18 '18 at 2:07

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