Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A proof of the Gaudin Bethe Ansatz conjecture   Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$
generated by higher integrals of the quantum Gaudin magnet chain attached to a
semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of
pairwise distinct complex numbers $z_1,\ldots,z_N$. We prove that this
subalgebra has a cyclic vector in the space of singular vectors of the tensor
product of any finite-dimensional irreducible $\mathfrak{g}$-modules, for all
values of the parameters $z_1,\ldots,z_N$. We deduce from this result the Bethe
Ansatz conjecture in the Feigin-Frenkel form which states that the joint
eigenvalues of the higher Gaudin Hamiltonians on the tensor product of
irreducible finite-dimensional $\mathfrak{g}$-modules are in 1-1 correspondence
with monodromy-free ${}^LG$-opers on the projective line with regular
singularities at the points $z_1,\ldots,z_N,\infty$ and the prescribed residues
at the singular points.

27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
19
The categories of this paper are: eess
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
17
The categories of this paper are: q-fin
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio