Examples of mirror partners arising from integrable systems (Q2748289)

This note is an announcement of the observation of an example of a mirror symmetry phenomenon [in the sense \textit{A. Strominger, S.-T. Yau} and \textit{E. Zaslow}, ``Mirror symmetry is T-duality'', Nucl. Phys. B 479, No. 1-2, 243-259 (1996; Zbl 0896.14024)]. The authors describe some hyper-Kähler varieties which are equipped with special Lagrangian torus fibrations, dual to each other. The main result of this paper claims that the stringy mixed Hodge polynomials of these varieties are equal. This provides mathematical evidence in support of mirror symmetry. NEWLINENEWLINENEWLINEOne of the hyper-Kähler manifolds is Simpson's moduli space of local systems over a smooth projective curve with structure group \(SL(n)\). Its mirror partner, which corresponds to the Langlands dual group \(PGL(n)\), is obtained as a quotient of such a moduli space. It is an orbifold. Both spaces are non-compact, which seems to be the reason for the equality of the Hodge numbers rather than their usual mirror relation. The torus fibrations arise from Hitchin systems on moduli spaces of stable Higgs bundles (such a moduli space is diffeomorphic to a moduli space of local systems). It is conjectured that similar results hold for any reductive algebraic group and its Langlands dual group. This would strongly relate mirror symmetry to Langlands duality. NEWLINENEWLINENEWLINEAt the moment this review was written, the full version, with proofs and improved results, was available as preprint only [math.AG/0205236 on arXiv.org: \textit{T. Hausel} and \textit{M. Thaddeus}: ``Mirror symmetry, Langlands duality, and the Hitchin system'']. In the full version, the meaning and function of the B-field have been clarified. Notation, background and results are formulated more clearly and proofs can be found there. Nevertheless, the paper under review might still serve as a bilingual introduction to the full version.