When one quantizes a scalar in the 1+1 dimensions in the kink background of a double well potential, one finds a spectrum that includes: (1) a zero mode corresponding to the classical particle position, (2) a bound state localized on the kink, and (3) the continuum of states analogous to those found in the trivial background.

When we quantize a non-Abelian gauge theory on an instanton background, are there analogs of the bound state (#2 above)? If so, how do they appear perturbatively in my Feynman rules? If not, how can we see that the instanton is really very different from the kink?



For the pure e.g. $d=4$ gauge theory instanton and gauge field perturbations around it, there is no negative mode – the counterpart of the bound state. It's the only one among the 3 classes that is absent here.

One may see this absence by noticing that the gauge theory may be embedded into a supersymmetric theory with the same gauge-field degrees of freedom, the instanton itself may be embedded as a solution that still preserves 1/2 of the supercharges, and supersymmetry implies that there can't be negative-energy solutions. A bound-state-like solution would yield creation operators for a tachyonic particle which may be proven not to exist in a SUSY theory, and because of the isomorphism between the Yang-Mills field equations, it doesn't exist in the pure gauge theory, either.

Similar questions are easily answered if one uses some string theory visualization. For example, an instanton solution on a D4-brane produces a dissolved D0-brane and the D0-D4 system in type IIA string theory still preserves some SUSY so it's the mininum-energy state among those with the same charges and there can't be any neutral particles at the D0-brane locus that would reduce the energy further.

However, one must be careful about more general perturbations in more general theories that may yield some negative modes in general.


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