# The Ising model: from elliptic curves to modular forms and Calabi–Yau equations

@article{Bostan2010TheIM, title={The Ising model: from elliptic curves to modular forms and Calabi–Yau equations}, author={Alin Bostan and Salah Boukraa and Saoud Hassani and Mark van Hoeij and J-M Maillard and Jacques-Arthur Weil and Nadjah Zenine}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2010}, volume={44}, pages={045204} }

We show that almost all the linear differential operators factors obtained in the analysis of the n-particle contributions 's of the susceptibility of the Ising model for n ⩽ 6 are linear differential operators associated with elliptic curves. Beyond the simplest differential operators factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral E, the second and third order differential operators Z2, F2, F3, can actually be…

## 33 Citations

### The Ising model and special geometries

- Mathematics
- 2014

We show that the globally nilpotent G-operators corresponding to the factors of the linear differential operators annihilating the multifold integrals χ(n) of the magnetic susceptibility of the Ising…

### Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

- Mathematics
- 2016

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations…

### Diagonals of Rational Functions: From Differential Algebra to Effective Algebraic Geometry

- MathematicsSymmetry
- 2022

We show that the results we had previously obtained on diagonals of 9- and 10-parameter families of rational functions in three variables x, y, and z, using creative telescoping, yielding modular…

### Differential algebra on lattice Green functions and Calabi–Yau operators

- Mathematics
- 2013

We revisit miscellaneous linear differential operators mostly associated with lattice Green functions in arbitrary dimensions, but also Calabi–Yau operators and order-7 operators corresponding to…

### Factorization of the Ising model form factors

- Mathematics
- 2011

We present a general method for analytically factorizing the n-fold form factor integrals f(n)N, N(t) for the correlation functions of the Ising model on the diagonal in terms of the hypergeometric…

### Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?

- Mathematics
- 2016

We study the class of non-holonomic power series with integer coefficients that reduce, modulo primes, or powers of primes, to algebraic functions. In particular we try to determine whether the…

### Ising n-fold integrals as diagonals of rational functions and integrality of series expansions

- Mathematics
- 2012

We show that the n-fold integrals χ(n) of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the ‘Ising class’, or n-fold integrals from enumerative…

### Differential algebra on lattice green and calabi-yau operators

- Mathematics
- 2018

We revisit miscellaneous linear differential operators mostly associated with lattice Green functions in arbitrary dimensions, but also Calabi-Yau operators and order-seven operators corresponding to…

### Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi–Yau equations

- Mathematics
- 2011

We give the exact expressions of the partial susceptibilities χ(3)d and χ(4)d for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi–Yau ODEs, and more specifically,…

### Schwarzian conditions for linear differential operators with selected differential Galois groups

- Mathematics
- 2017

We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions can be…

## References

SHOWING 1-10 OF 115 REFERENCES

### Globally nilpotent differential operators and the square Ising model

- Mathematics
- 2009

We recall various multiple integrals with one parameter, related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility,…

### High order Fuchsian equations for the square lattice Ising model:

- Mathematics
- 2009

We consider the Fuchsian linear differential equation obtained (modulo a prime) for , the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can…

### Singularities of n-fold integrals of the Ising class and the theory of elliptic curves

- Mathematics
- 2007

We introduce some multiple integrals that are expected to have the same singularities as the singularities of the n-particle contributions χ(n) to the susceptibility of the square lattice Ising…

### High-order Fuchsian equations for the square lattice Ising model: χ(6)

- Mathematics
- 2010

This paper deals with , the six-particle contribution to the magnetic susceptibility of the square lattice Ising model. We have generated, modulo a prime, series coefficients for . The length of the…

### Holonomy of the Ising model form factors

- Mathematics
- 2006

We study the Ising model two-point diagonal correlation function C(N, N) by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of…

### Picard–Fuchs Uniformization and Modularity¶of the Mirror Map

- Mathematics
- 2000

Abstract:Arithmetic properties of mirror symmetry (type IIA-IIB string duality) are studied. We give criteria for the mirror map q-series of certain families of Calabi–Yau manifolds to be automorphic…

### Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties

- Mathematics
- 1993

We formulate general conjectures about the relationship between the A-model connection on the cohomology of ad-dimensional Calabi-Yau complete intersectionV ofr hypersurfacesV1,...,Vr in a toric…

### Solving second order linear differential equations with Klein's theorem

- MathematicsISSAC
- 2005

A variation on the earlier formulas, namely the formulas will base the formulas on invariants of the differential Galois group instead of semi-invariants, to make the algorithm more easy to implement for various differential fields k.

### Landau singularities and singularities of holonomic integrals of the Ising class

- Mathematics
- 2007

We consider families of multiple and simple integrals of the ‘Ising class’ and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set…

### Fuchs versus Painlevé

- Mathematics
- 2007

We, briefly, recall the Fuchs–Painlevé elliptic representation of Painlevé VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the…