Abelian Varieties, Theta Functions and the Fourier Transform

Gauss Optics and Gauss Sum on an Optical Phenomena

“…Further as in survey of Polishchuk [32], the Gauss sum and the symplectic structure determine the structure of the Abelian variety though the theta functions. Due to the properties of the Abelian structure, i.e., theorem of cube, the Gauss sum is connected with another physical problem, Chern-Simons-Witten theory of the three-manifold related to some Riemann surfaces [10,18,39].…”

“…In fact, n −Â B n is written by n −τ n which shows the periodic structure in the Abelian variety of genus one [32] 4 . Oskolkov [28] and Berry and Bodenschatz [1] dealt with different τ as time development and showed interesting patterns.…”

“…Then we realize A I 2 as in (32). Using the reciprocity for Gauss sums (6) or the reciprocity corresponding to the wave-particle complementarity (22), we also realize A II 2 (25).…”

Lectures on Bridgeland Stability

“…In the case of P 1 semistable vector bundles are simply direct sums of line bundles, and stable vector bundles are simply line bundles. Vector bundles on elliptic curves can also be classified (see [Ati57] for the original source and [Pol03] for a modern proof):…”

Raynaud’s vector bundles and base points of the generalized Theta divisor

“…Having a Θ-divisor implies semi stability immediately. Moreover, Polishchuk shows in [9] that the Fourier-Mukai transform FM P gives an equivalence between semistable bundles of rank r and degree zero and torsion sheaves of length r. See [6] for a presentation of Atiyah's results on vector bundles on elliptic curves (cf. [1]) in terms of Fourier-Mukai transforms.…”