Elliptic theta functions and the fractional quantum Hall effect

Phases of dual superconductivity and confinement in softly broken N=2 supersymmetric Yang-Mills theories

“…. , n i , and Θ[ a b ](z|τ ) are the Riemann Theta functions with characteristics, see [35] and references quoted therein. Notice that the solution describes n 1 chromoelectric vortices, located at the zeroes of ϕ n1 1 , and n 2 vortices of the other kind centered around the zeroes of ϕ n2 2 .…”

Higher order addition laws on Abelian varieties and the fractional quantum Hall effect

“…For applications to the study of the quantum Hall effect under periodic conditions, it is very important to compute explicitly the homomorphism ξ * N (see [4] and the last Section of this paper). Observe that the kernel of ξ N is ∆(X N ), where X N is the N−torsion subgroup of X and ∆ : X…”

“…We shall apply the results of the first section to compute some vector spaces of theta functions which are relevant in the study of the fractional quantum Hall effect, ( for a similar discussion for elliptic curves see [4]) .…”

“…From a physical point of view, this means that we are considering a system of many electrons moving on g two-dimensional layers with different periodic conditions defined by the period matrix τ ij . As in our earlier paper [4] , we are naturally lead to study the isogenies…”

“…On the algebraic torus, Σ, this change of coordinates becomes the isogeny mentioned above. In the second quantization formalism, to extend the Laughlin variational principle for the fractional quantum Hall effect to the periodic case thus requires the use of generalized addition formulas for elliptic theta functions, see [4].…”