Searcharxiv⌕ Search

arXiv subjects

P. Zinn-Justin

Publications and source records attributed to P. Zinn-Justin.

At least 19 recordsLinked to original sources
arXiv2016

Grassmann-Grassmann conormal varieties, integrability, and plane partitions

We give a conjectural formula for sheaves supported on (irreducible) conormal varieties inside the cotangent bundle of the Grassmannian, such that their equivariant $K$-class is given by the partition function of an integrable loop model, and furthermore their $K$-theoretic pushforward to a point is a solution of the level $1$ quantum Knizhnik-Zamolodchikov equation. We prove these results in the case that the Lagrangian is smooth (hence is the conormal bundle to a subGrassmannian). To compute the pushforward to a point, or equivalently to the affinization, we simultaneously degenerate the Lagrangian and sheaf (over the affinization); the sheaf degenerates to a direct sum of cyclic modules over the geometric components, which are in bijection with plane partitions, giving a geometric interpretation to the Razumov-Stroganov correspondence satisfied by the loop model.

math.AG↗
arXiv2015

Quiver varieties and quantum Knizhnik--Zamolodchikov equation

We show how equivariant volumes of tensor product quiver varieties of type A are given by matrix elements of vertex operators of centrally extended doubles of Yangians, and how they satisfy in some cases the rational, level 1, quantum Knizhnik--Zamolodchikov equation.

math.RT↗
arXiv2014

Refined Cauchy/Littlewood identities and six-vertex model partition functions: II. Proofs and new conjectures

We prove two identities of Hall-Littlewood polynomials, which appeared recently in a paper by two of the authors. We also conjecture, and in some cases prove, new identities which relate infinite sums of symmetric polynomials and partition functions associated with symmetry classes of alternating sign matrices. These identities generalize those already found in our earlier paper, via the introduction of additional parameters. The left hand side of each of our identities is a simple refinement of a relevant Cauchy or Littlewood identity. The right hand side of each identity is (one of the two factors present in) the partition function of the six-vertex model on a relevant domain.

math.CO↗
arXiv2013

Higher spin polynomial solutions of quantum Knizhnik--Zamolodchikov equation

We provide explicit formulae for highest-weight to highest-weight correlation functions of perfect vertex operators of $U_q(\hat{\mathfrak{sl}(2)})$ at arbitrary integer level $\ell$. They are given in terms of certain Macdonald polynomials. We apply this construction to the computation of the ground state of higher spin vertex models, spin chains (spin $\ell/2$ XXZ) or loop models in the root of unity case $q=-e^{-iπ/(\ell+2)}$.

math-ph↗
arXiv2013

Discrete holomorphicity and quantized affine algebras

We consider non-local currents in the context of quantized affine algebras, following the construction introduced by Bernard and Felder. In the case of $U_q(A_1^{(1)})$ and $U_q(A_2^{(2)})$, these currents can be identified with configurations in the six-vertex and Izergin--Korepin nineteen-vertex models. Mapping these to their corresponding Temperley--Lieb loop models, we directly identify non-local currents with discretely holomorphic loop observables. In particular, we show that the bulk discrete holomorphicity relation and its recently derived boundary analogue are equivalent to conservation laws for non-local currents.

math-ph↗
arXiv2013

Path representation of maximal parabolic Kazhdan-Lusztig polynomials

We provide simple rules for the computation of Kazhdan--Lusztig polynomials in the maximal parabolic case. They are obtained by filling regions delimited by paths with "Dyck strips" obeying certain rules. We compare our results with those of Lascoux and Schützenberger.

math.CO↗
arXiv2012

Extended Joseph polynomials, quantized conformal blocks, and a q-Selberg type integral

We consider the tensor power $V=(C^N)^{\otimes n}$ of the vector representation of $gl_N$ and its weight decomposition $V=\oplus_{λ=(λ_1,...,λ_N)}V[λ]$. For $λ= (λ_1 \geq ... \geq λ_N)$, the trivial bundle $V[λ]\times \C^n\to\C^n$ has a subbundle of q-conformal blocks at level l, where $l = λ_1-λ_N$ if $λ_1-λ_N> 0$ and l=1 if $λ_1-λ_N=0$. We construct a polynomial section $I_λ(z_1,...,z_n,h)$ of the subbundle. The section is the main object of the paper. We identify the section with the generating function $J_λ(z_1,...,z_n,h)$ of the extended Joseph polynomials of orbital varieties, defined in [DFZJ05,KZJ09]. For l=1, we show that the subbundle of q-conformal blocks has rank 1 and $I_λ(z_1,...,z_n,h)$ is flat with respect to the quantum Knizhnik-Zamolodchikov discrete connection. For N=2 and l=1, we represent our polynomial as a multidimensional q-hypergeometric integral and obtain a q-Selberg type identity, which says that the integral is an explicit polynomial.

math-ph↗
arXiv2012

Sum rule for the eight-vertex model on its combinatorial line

We investigate the conjectured ground state eigenvector of the 8-vertex model inhomogeneous transfer matrix on its combinatorial line, i.e., at $η=π/3$, where it acquires a particularly simple form. We compute the partition function of the model on an infinite cylinder with certain restrictions on the inhomogeneities, and taking the homogeneous limit, we obtain an expression for the squared norm of the ground state of the XYZ spin chain as a solution of a differential recurrence relation.

math-ph↗
arXiv2012

Loop models, random matrices and planar algebras

We define matrix models that converge to the generating functions of a wide variety of loop models with fugacity taken in sets with an accumulation point. The latter can also be seen as moments of a non-commutative law on a subfactor planar algebra. We apply this construction to compute the generating functions of the Potts model on a random planar map.

math.OA↗
arXiv2010

Knot theory and matrix integrals

The large size limit of matrix integrals with quartic potential may be used to count alternating links and tangles. The removal of redundancies amounts to renormalizations of the potential. This extends into two directions: higher genus and the counting of "virtual" links and tangles; and the counting of "coloured" alternating links and tangles. We discuss the asymptotic behavior of the number of tangles as the number of crossings goes to infinity.

math-ph↗
arXiv2009

Combinatorial point for higher spin loop models

Integrable loop models associated with higher representations (spin k/2) of U_q(sl(2)) are investigated at the point q=-e^{iπ/(k+2)}. The ground state eigenvalue and eigenvectors are described. Introducing inhomogeneities into the models allows to derive a sum rule for the ground state entries.

math-ph↗
arXiv2009

Spectral analysis of the free orthogonal matrix

We compute the spectral measure of the standard generators $u_{ij}$ of the Wang algebra $A_o(n)$. We show in particular that this measure has support $[-2/\sqrt{n+2},2/\sqrt{n+2}]$, and that it has no atoms. The computation is done by using various techniques, involving the general Wang algebra $A_o(F)$, a representation of $SU^q_2$ due to Woronowicz, and several calculations with orthogonal polynomials.

math.OA↗
arXiv2009

On some ground state components of the O(1) loop model

We address a number of conjectures about the ground state O(1) loop model, computing in particular two infinite series of partial sums of its entries and relating them to the enumeration of plane partitions. Our main tool is the use of integral formulae for a polynomial solution of the quantum Knizhnik-Zamolodchikov equation.

math-ph↗
arXiv2009

Littlewood--Richardson coefficients and integrable tilings

We provide direct proofs of product and coproduct formulae for Schur functions where the coefficients (Littlewood--Richardson coefficients) are defined as counting puzzles. The product formula includes a second alphabet for the Schur functions, allowing in particular to recover formulae of [Molev--Sagan '99] and [Knutson--Tao '03] for factorial Schur functions. The method is based on the quantum integrability of the underlying tiling model.

math-ph↗