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Topological \\\"observables\\\" in semiclassical field theories

TitleTopological \\\"observables\\\" in semiclassical field theories
Publication TypeJournal Article
Year of Publication1992
AuthorsNolasco, M, Reina, C
JournalPhys. Lett. B 297 (1992) 82-88

We give a geometrical set up for the semiclassical approximation to euclidean field theories having families of minima (instantons) parametrized by suitable moduli spaces ${\\\\cal M}$. The standard examples are of course Yang-Mills theory and non-linear $\\\\sigma$-models. The relevant space here is a family of measure spaces $\\\\tilde {\\\\cal N} \\\\ra {\\\\cal M}$, with standard fibre a distribution space, given by a suitable extension of the normal bundle to ${\\\\cal M}$ in the space of smooth fields. Over $\\\\tilde {\\\\cal N}$ there is a probability measure $d\\\\mu$ given by the twisted product of the (normalized) volume element on ${\\\\cal M}$ and the family of gaussian measures with covariance given by the tree propagator $C_\\\\phi$ in the background of an instanton $\\\\phi \\\\in {\\\\cal M}$. The space of \\\"observables\\\", i.e. measurable functions on ($\\\\tilde {\\\\cal N}, \\\\, d\\\\mu$), is studied and it is shown to contain a topological sector, corresponding to the intersection theory on ${\\\\cal M}$. The expectation value of these topological \\\"observables\\\" does not depend on the covariance; it is therefore exact at all orders in perturbation theory and can moreover be computed in the topological regime by setting the covariance to zero.


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