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How to Wick rotate the instanton number of Yang-Mills theory? (Related to the earlier question Wick rotate the Yang-Mills $SU(N)$ gauge theory's field strength?)

My question is particularly about the statements in Weinberg, how to establish from Euclidean (23.6.4) to Minkowski (23.6.5)?

How to justify the conventions chosen for:

  • $\epsilon^{ijkl}$ vs. $\epsilon^E_{ijkl}$?

  • $F_{34}$ vs. $F_{30}$?

$$ v=\frac{1}{64\pi^2}\int (d^4x)_E \ \epsilon^E_{ijkl}F_{\alpha ij}F_{\alpha kl}\tag{23.6.4} $$ This can be expressed in terms of a Minkowskian path integral; since $(d^4x)_E = i\ d^4x$; $F_{\alpha 34}=-iF_{\alpha 30}$; and $\epsilon^{1230}=-1$, Eq. (23.6.4) may be written $$ v=-\frac{1}{64\pi^2}\int d^4x \ \epsilon^{\kappa\lambda\rho\sigma}F_{\alpha \kappa\lambda}F_{\alpha \rho\sigma} \tag{23.6.5} $$

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  • $\begingroup$ Back to the question -- It looks to me that the $$\epsilon^{0123} F_{01} F_{23} d^4x =\epsilon^{0123} (- i F_{41}) F_{23} (i d^4x_E) = \epsilon^{0123} F_{41} F_{23} d^4x_E.$$ $\endgroup$
    – ann marie cœur
    Commented Mar 7, 2021 at 2:07
  • $\begingroup$ So the question really becomes how $\epsilon^{0123}$ and $\epsilon^{4123}_E$ are related by a +, -, sign or not? It seems that Weinberg had to use $$\epsilon^{4123}_E=- \epsilon^{0123}$$ but why is that? (Is this a common or convenient convention?) $\endgroup$
    – ann marie cœur
    Commented Mar 7, 2021 at 2:09

1 Answer 1

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Differential forms and the Levi-Civita symbol/tensor are in principle invariant under Wick rotation itself, cf. e.g. my Phys.SE post here.

However, the orientation changes $(0,1,2,3)\to (1,2,3,4)$ due to the relabeling $x^0_E=x^4_E$. This cost a sign in the Levi-Civita symbol/tensor [and a sign in the winding number formulas (23.6.4+5)], since $$ \epsilon^{1234}~=~1~=~\epsilon^{0123}~=~-\epsilon^{1230}, $$ cf. Ref. 1 below eq. (23.5.4) and above eq. (23.6.5).

References:

  1. S. Weinberg, Quantum Theory of Fields, Vol. 2, 1996; chapter 23.

  2. A. Bilal & S. Metzger, arXiv:hep-th/0307152; text between eqs. (2.20) & (2.21).

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