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Register a new userCoherent states, Path integral, and Semiclassical approximation
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Using the generalized coherent states we argue that the path integral formulae for $SU(2)$ and $SU(1,1)$ (in the discrete series) are WKB exact,if the starting point is expressed as the trace of $e^{-iThat H}$ with $hat H$ being given by a linear combination of generators. In our case,WKB approximation is achieved by taking a large ``spin limit: $J,Krightarrow infty$. The result is obtained directly by knowing that the each coefficient vanishes under the $J^{-1}$($K^{-1}$) expansion and is examined by another method to be legitimated. We also point out that the discretized form of path integral is indispensable, in other words, the continuum path integral expression leads us to a wrong result. Therefore a great care must be taken when some geometrical action would be adopted, even if it is so beautiful, as the starting ingredient of path integral.
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We provide a semiclassical description of framed BPS states in four-dimensional N = 2 super Yang-Mills theories probed by t Hooft defects, in terms of a supersymmetric quantum mechanics on the moduli space of singular monopoles. Framed BPS states, like their ordinary counterparts in the theory without defects, are associated with the L^2 kernel of certain Dirac operators on moduli space, or equivalently with the L^2 cohomology of related Dolbeault operators. The Dirac/Dolbeault operators depend on two Cartan-valued Higgs vevs. We conjecture a map between these vevs and the Seiberg-Witten special coordinates, consistent with a one-loop analysis and checked in examples. The map incorporates all perturbative and nonperturbative corrections that are relevant for the semiclassical construction of BPS states, over a suitably defined weak coupling regime of the Coulomb branch. We use this map to translate wall crossing formulae and the no-exotics theorem to statements about the Dirac/Dolbeault operators. The no-exotics theorem, concerning the absence of nontrivial SU(2)_R representations in the BPS spectrum, implies that the kernel of the Dirac operator is chiral, and further translates into a statement that all L^2 cohomology of the Dolbeault operator is concentrated in the middle degree. Wall crossing formulae lead to detailed predictions for where the Dirac operators fail to be Fredholm and how their kernels jump. We explore these predictions in nontrivial examples. This paper explains the background and arguments behind the results announced in a short accompanying note.
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