Physics – Mathematical Physics
Scientific paper
2011-10-08
Physics
Mathematical Physics
35 pages
Scientific paper
We give the exact expressions of the partial susceptibilities $\chi^{(3)}_d$ and $\chi^{(4)}_d$ for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi-Yau ODEs, and more specifically, $_3F_2([1/3,2/3,3/2],\, [1,1];\, z)$ and $_4F_3([1/2,1/2,1/2,1/2],\, [1,1,1]; \, z)$ hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for $\chi^{(3)}_d$ and $\chi^{(4)}_d$. We also give new results for $\chi^{(5)}_d$. We see in particular, the emergence of a remarkable order-six operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the $n$-fold integrals of the Ising model are not only "Derived from Geometry" (globally nilpotent), but actually correspond to "Special Geometry" (homomorphic to their formal adjoint). This raises the question of seeing if these "special geometry" Ising-operators, are "special" ones, reducing, in fact systematically, to (selected, k-balanced, ...) $_{q+1}F_q$ hypergeometric functions, or correspond to the more general solutions of Calabi-Yau equations.
Assis Michael
Boukraa Salah
Hassani Samira
Hoeij Mark van
Maillard Jean-Marie
No associations
LandOfFree
Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-497256