  The N=2 superconformal Ward identities and their anomalies are discussed in
N=2 superspace (including N=2 harmonic superspace), at the level of the
low-energy effective action (LEEA) in four-dimensional N=2 supersymmetric field
theories. The (first) chiral N=2 supergravity compensator is related to the
known N=2 anomalous Ward identity in the N=2 (abelian) vector mulitplet sector.
As regards the hypermultiplet LEEA given by the N=2 non-linear sigma-model
(NLSM), a new anomalous N=2 superconformal Ward identity is found, whose
existence is related to the (second) analytic compensator in N=2 supergravity.
The celebrated solution of Seiberg and Witten is known to obey the (first)
anomalous Ward identity in the Coulomb branch. We find a few solutions to the
new anomalous Ward identity, after making certain assumptions about unbroken
internal symmetries. Amongst the N=2 NLSM target space metrics governing the
hypermultiplet LEEA are the SU(2)-Yang-Mills-Higgs monopole moduli-space
metrics that can be encoded in terms of the spectral curves (Riemann surfaces),
similarly to the Seiberg-Witten-type solutions. After a dimensional reduction
to three spacetime dimensions (3d), our results support the mirror symmetry
between the Coulomb and Higgs branches in 3d, N=4 gauge theories.
