On the link between the structure of A-branes observed in the homological mirror symmetry . . .

This paper is a review of some interesting results that has been obtained in the study of the categories of A-branes on the dual Hitchin fibers and some interesting phenomena associated with the endoscopy in the geometric Langlands correspondence of various authoritative theoretical physicists and mathematicians. The geometric Langlands correspondence has been interpreted as the mirror symmetry of the Hitchin fibrations for two dual reductive groups. This mirror symmetry reduces to T-duality on the generic Hitchin fibers. Also from this work we’ve showed that can be obtained interesting and new mathematical connections with some sectors of Number Theory and String Theory, principally with the modular elliptic curves, p-adic and adelic numbers and p-adic and adelic strings. In the Section 1, we have described some equations regarding the Galois group and Abelian class field theory, automorphic representations of GL2(AQ) and modular forms, adèles and vector bundles. In the Section 2, we have showed some equations regarding the moduli spaces of SL2 and SO3 Higgs bundles on an elliptic curve with tame ramification at one point. In the Section 3, we have showed some equations regarding the action of the Wilson and ‘t Hooft/Hecke operators on the electric and magnetic branes relevant to geometric endoscopy. In the Section 4, we have described the Hecke eigensheaves and the notion of fractional Hecke eigensheaves. In the Section 5, we have described some equations concerning the local and global Langlands correspondence. In the Section 6, we have described some equations regarding the automorphic functions associated to the fractional Hecke eigensheaves. In the Section 7, we have showed some equations concerning the modular elliptic curves belonging at the proof of Fermat’s Last Theorem. In the Section 8, we have showed some equations concerning the p-adic and adelic numbers and the p-adic and adelic strings. In the Section 9, we have described the P-N Model (Palumbo-Nardelli model) and the Ramanujan identities, solution applied to ten dimensional IIB supergravity (uplifted 10-dimensional solution) and connections with some equations concerning the Riemann zeta function.