Holonomic Quantum Fields IV

On classification ofN=2 supersymmetric theories

“…It is well known that these equations describe isomonodromic deformation. In fact this is precisely what the Kyoto school mean when they talk of 'holonomic field theory' [23]. 16 Indeed for LG models Ψ(x) is related by a linear integral transform to the usual SQM wave function.…”

Geometry of 2D topological field theories

“…The standard machinery of integration of scaling reductions of integrable systems [60,76,127] suggests to add a di erential equation in the spectral parameter z for the auxiliary function = (t; z). Proposition 3.1.…”

“…They can be represented in the form of compatibility conditions of (3.146) with linear system @ i = A i u i + B i ; i = 1 ; : : : ; n (3:147) for some matrices B i . T o represent (3.74) as a reduction of the Schlesinger equations one put A i = E i V; B i = ad E i ad 1 U V: (3:148) Observe that the hamiltonian structure (3.107) of the equations (3.74) is obtained by the reduction (3.148) of the hamiltonian structure of general Schlesinger equations found in [127]. The Painleve property of the equations (3.74) follows thus also from the results of Malgrange (see in [102].…”

“…The rotation coe cients ij (u) and^i j (u) of two Frobenius manifolds with rational invertible matrix valued function g(z) preserving the form of the operator are called Schlesinger transformations of the operator [127]. They preserve unchanged the monodromy property of the operator.…”

Spin–spin correlation functions of the XXZ- Heisenberg chain in a magnetic field

“…Although the method to compute eigenstates and energy levels goes back to H. Bethe in 1931 [1,2,3,4], the knowledge of its spin correlation functions has been for a long time restricted to the free fermion point ∆ = 0, a case for which nevertheless tremendous works have been necessary to obtain full answers [5,6,7,8,9,10].…”