### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 060, 31 pages      math.QA/0703893      https://doi.org/10.3842/SIGMA.2007.060
Contribution to the Vadim Kuznetsov Memorial Issue

### Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel

Evgeny Mukhin a, Vitaly Tarasov a, b and Alexander Varchenko c
a) Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 North Blackford St, Indianapolis, IN 46202-3216, USA
b) St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191023, Russia
c) Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA

Received March 28, 2007; Published online April 25, 2007

Abstract
Let M be the tensor product of finite-dimensional polynomial evaluation Y(glN)-modules. Consider the universal difference operator D = åN k=0(-1)k Tk(u) e-ku whose coefficients Tk(u): M ® M are the XXX transfer matrices associated with M. We show that the difference equation Df = 0 for an M-valued function f has a basis of solutions consisting of quasi-exponentials. We prove the same for the universal differential operator D = åNk=0(-1)kSk(u) uN-k whose coefficients Sk(u) : M® M are the Gaudin transfer matrices associated with the tensor product M of finite-dimensional polynomial evaluation glN[x]-modules.

Key words: Gaudin model; XXX model; universal differential operator.

pdf (432 kb)   ps (276 kb)   tex (30 kb)

References

1. Chervov A., Talalaev D., Universal G-oper and Gaudin eigenproblem, hep-th/0409007.
2. Drinfeld V., A new realization of Yangians and quantized affine algebras, Soviet Math. Dokl. 36 (1988), 212-216.
3. Kulish P., Reshetikhin N., Diagonalization of GL(n) invariant transfer-matrices and quantum N-wave system (Lee model), J. Phys. A: Math. Gen. 15 (1983), L591-L596.
4. Molev A., Nazarov M., Olshanski G., Yangians and classical Lie algebras, Russian Math. Surveys 51 (1996), no. 2, 205-282, hep-th/9409025.
5. Mukhin E., Tarasov V., Varchenko A., The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, math.AG/0512299.
6. Mukhin E., Tarasov V., Varchenko A., Bethe eigenvectors of higher transfer matrices, J. Stat. Mech. Theory Exp. (2006), no. 8, P08002, 44 pages, math.QA/0605015.
7. Mukhin E., Varchenko A., Critical points of master functions and flag varieties, Commun. Contemp. Math. 6 (2004), no. 1, 111-163, math.QA/0209017.
8. Mukhin E., Varchenko A., Solutions to the XXX type Bethe ansatz equations and flag varieties, Cent. Eur. J. Math. 1 (2003), no. 2, 238-271, math.QA/0211321.
9. Mukhin E., Varchenko A., Spaces of quasi-polynomials and the Bethe ansatz, math.QA/0604048.
10. Reshetikhin N., Varchenko A., Quasiclassical asymptotics of solutions to the KZ equations, in Geometry, Topology and Physics for R. Bott, Intern. Press, 1995, 293-322, hep-th/9402126.
11. Rimányi R., Stevens L., Varchenko A., Combinatorics of rational functions and Poincaré-Birkhoff-Witt expansions of the canonical U(n-)-valued differential form, Ann. Comb. 9 (2005), no. 1, 57-74, math.CO/0407101.
12. Schechtman V., Varchenko A., Arrangements of hyperplanes and Lie algebra homology, Invent. Math. 106 (1991), 139-194.
13. Talalaev D., Quantization of the Gaudin system, hep-th/0404153.
14. Tarasov V., Varchenko A., Combinatorial formulae for nested bethe vectors, math.QA/0702277.