arXiv subjects

Martin T. Luu

Publications and source records attributed to Martin T. Luu.

4 recordsLinked to original sources

arXiv2026

The KdV vacuum operator and its Katz extension

We define a connection on the formal disc that can be used to single out the vacuum of the Drinfeld-Sokolov KdV hierarchy associated to a simple complex finite-dimensional Lie algebra. As a connection, it has a canonical Katz extension from the disc to the sphere. We express this Katz extension in terms of the Kac coordinates of a suitable Weyl group conjugacy class. As a consequence, we show that the Katz extension has meaning in the context of the integrable hierarchy: It describes an additional symmetry.

nlin.SI↗

arXiv2026

Weyl orbit particles

The mass spectrum of affine Toda theory is known to be expressible in terms of a suitable eigenvector of the relevant Cartan matrix. The particles correspond in a precise manner to the Coxeter element orbits in the set of roots. Recently, variants of affine Toda theory have been constructed for many different Weyl group elements. Again, the particles correspond to orbits in the set of roots and this allows the calculation of the classical mass spectrum. We show how these spectral calculations generalize the affine Toda relation with the Cartan matrix. As an example, we calculate the spectrum for the unique non-Coxeter infinite family $\textrm{D}_{2n}(a_{n-1})$ of primitive regular conjugacy classes in the Weyl groups of complex simple Lie algebras.

math-ph↗

arXiv2025

Kirillov polynomials for the exceptional Lie algebra $\mathfrak g_2$

As part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of $q$ elements and fixed Jordan type. One obtains polynomials with respect to $q$ with many interesting properties and close relation to type A representation theory. In the present work we develop the corresponding theory for the exceptional Lie algebra $\mathfrak g_2$. In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.

math.RT↗

arXiv2025

The Toda-Weyl mass spectrum

The masses of affine Toda theories are known to correspond to the entries of a Perron-Frobenius eigenvector of the relevant Cartan matrix. The Lagrangian of the theory can be expressed in terms of a suitable eigenvector of a Coxeter element in the Weyl group. We generalize this set-up by formulating Lagrangians based on eigenvectors of arbitrary elements in the Weyl group. Under some technical conditions (that hold for many Weyl group elements), we calculate the classical mass spectrum. In particular, we indicate the relation to the relative geometry of special roots, generalizing the affine Toda mass spectrum description in terms of the Cartan matrix. Related questions of three point coupling and integrability are left to be addressed on a future occasion.

hep-th↗