14 research outputs found
On factorization hierarchy of equations for banana Feynman amplitudes
We present a review of the relations between various equations for maximal
cut banana Feynman diagrams, i.e. amplitudes with propagators substituted with
-functions. We consider both equal and generic masses. There are three
types of equation to consider: those in coordinate space, their Fourier
transform and Picard-Fuchs equations originating from the parametric
representation. First, we review the properties of the corresponding
differential operators themselves, mainly their factorization properties at the
equal mass locus and their form at special values of the dimension. Then we
study the relation between the Fourier transform of the coordinate space
equations and the Picard-Fuchs equations and show that they are related by
factorization as well. The equations in question are the counterparts of the
Virasoro constraints in the much-better studied theory of eigenvalue matrix
models and are the first step towards building a full-fledged theory of Feynman
integrals, which will reveal their hidden integrable structure.Comment: 13 page
Commutative subalgebras from Serre relations
We demonstrate that commutativity of numerous one-dimensional subalgebras in
algebra, i.e. the existence of many non-trivial integrable
systems described in recent arXiv:2303.05273 follows from the subset of
relations in algebra known as Serre relations. No other relations are needed
for commutativity. The Serre relations survive the deformation to the affine
Yangian , hence the commutative subalgebras do as
well. A special case of the Yangian parameters corresponds to the
-deformation. The preservation of Serre relations can be thought of a
selection rule for proper systems of commuting -deformed Hamiltonians.
On the contrary, commutativity in the extended family associated with
``rational (non-integer) rays" is {\it not} reduced to the Serre relations, and
uses also other relations in the algebra. Thus their
-deformation is less straightforward.Comment: 13 page
Commutative families in , integrable many-body systems and hypergeometric -functions
We explain that the set of new integrable systems generalizing the Calogero
family and implied by the study of WLZZ models, which was described in
arXiv:2303.05273, is only the tip of the iceberg. We provide its wide
generalization and explain that it is related to commutative subalgebras
(Hamiltonians) of the algebra. We construct many such
subalgebras and explain how they look in various representations. We start from
the even simpler contraction, then proceed to the one-body
representation in terms of differential operators on a circle, further
generalizing to matrices and in their eigenvalues, in finally to the bosonic
representation in terms of time-variables. Moreover, we explain that some of
the subalgebras survive the -deformation, an intermediate step from
to the affine Yangian. The very explicit formulas for the
corresponding Hamiltonians in these cases are provided. Integrable many-body
systems generalizing the rational Calogero model arise in the representation in
terms of eigenvalues. Each element of algebra gives rise to
KP/Toda -functions. The hidden symmetry given by the families of
commuting Hamiltonians is in charge of the special, (skew) hypergeometric
-functions among these.Comment: 43 page
Superintegrability in
This paper is devoted to the phenomenon of superintegrability. This phenomenon is manifested in the existence of a formula for character averages, expressed through the same characters at special points and of its various generalization. In this paper we develop a method of proving such formulas from first principle from Virasoro constraints and W-representation. We apply it to prove the formula for the Jack functions averages – appropriate analogue of characters for the -deformed Hermitian Gaussian matrix model. We also sketch the construction of W-operators from Calogero–Ruijsenaars Hamiltonians