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Extra info for Communications in Mathematical Physics - Volume 199
They are given by: Aχ = αχ det Bσ = βσ χ/2 ∂uI ∂aJ σ/8 , . 6) This may be proved by a modification of the argument of [21, 9]. The twisted theory with gauge group G has a gravitational contribution to the anomaly given by −(dimG)(χ + σ)/2. In the semiclassical regime the effective U (1)r theory gives the anomaly −r(χ+σ)/2. The remaining anomaly should be carried by the measure factor in the semiclassical region. On the other hand, near the divisor where a single hypermultiplet becomes massless, there is an accidental low-energy R-symmetry given by −σ/4 which should also show up in the measure factor in this region.
This is because the theta function involved in the expression never has singularities, and the only possible singularities come from −1/8 . But these must be cancelled by zeros of the theta function, as follows from monodromy invariance. In the case of the SU (2) theory, the explicit expression for these polynomials was obtained in  using the expansion of the theta functions in terms of Eisenstein series, but in the higher rank case these expansions are not available. However, these expressions can probably be obtained using the relation between Seiberg–Witten theory and integrable systems.
17) is finite-dimensional, and not a path integral. This expression can be formally considered as the partition function of a finite-dimensional topological “field” theory, obtained from the original one after restriction to the sector of harmonic forms. 18) − b ). 17) as the (euclidean) action SE of this topological field theory. It is Q-closed. 5. The contact term. As explained in , when 2-observables are taken into account there are possible contact terms in the low-energy description. 19) is duality invariant.