Iterated integrals on affine curves
Motivated by amplitude calculations in string theory we establish basic properties of homotopy invariant iterated integrals on affine curves.
math.AG↗
arXiv subjects
Martin Luu
Publications and source records attributed to Martin Luu.
Rigidity of Kac-Schwarz operators
In his work on the mathematical formulation of 2d quantum gravity Schwarz established a rigidity result for Kac-Schwarz operators for the n-KdV hierarchies. Later on, Adler and van Moerbeke as well as Fastr\'{e} obtained different proofs of this result. We give yet another proof of the rigidity, one that in fact holds for all Drinfeld-Sokolov hierarchies.
math-ph↗
Spectral curve duality beyond the two-matrix model
We describe a simple algebraic approach to several spectral duality results for integrable systems and illustrate the method for two types of examples: The Bertola-Eynard-Harnad spectral duality of the two-matrix model as well as the various dual descriptions of minimal model conformal field theories coupled to gravity.
math-ph↗
Fourier duality of quantum curves
There are two different ways to deform a quantum curve along the flows of the KP hierarchy. We clarify the relation between the two KP orbits: In the framework of suitable connections attached to the quantum curve they are related by a local Fourier duality. As an application we give a conceptual proof of duality results in 2D quantum gravity.
math-ph↗
Langlands parameters of quivers in the Sato Grassmannian
Motivated by quantum field theoretic partition functions that can be expressed as products of tau functions of the KP hierarchy we attach several types of local geometric Langlands parameters to quivers in the Sato Grassmannian. We study related questions of Virasoro constraints, of moduli spaces of relevant quivers, and of classical limits of the Langlands parameters.
math-ph↗
Local Langlands Duality and a Duality of Conformal Field Theories
We show that the numerical local Langlands duality for GL_n and the T - duality of two-dimensional quantum gravity arise from one and the same symmetry principle. The unifying theme is that the local Fourier transform in both its l-adic and complex incarnation gives rise to symmetries of arithmetic and geometric local Langlands parameters.
hep-th↗
Duality of 2D gravity as a local Fourier duality
The p - q duality is a relation between the (p,q) model and the (q,p) model of two-dimensional quantum gravity. Geometrically this duality corresponds to a relation between the two relevant points of the Sato Grassmannian. Kharchev and Marshakov have expressed such a relation in terms of matrix integrals. Some explicit formulas for small p and q have been given in the work of Fukuma-Kawai-Nakayama. Already in the duality between the (2,3) model and the (3,2) model the formulas are long. In this work a new approach to p - q duality is given: It can be realized in a precise sense as a local Fourier duality of D-modules. This result is obtained as a special case of a local Fourier duality between irregular connections associated to Kac-Schwarz operators. Therefore, since these operators correspond to Virasoro constraints, this allows to view the p - q duality as a consequence of the duality of the relevant Virasoro constraints.
math-ph↗