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

SIGMA 14 (2018), 069, 48 pages      arXiv:1612.05361      https://doi.org/10.3842/SIGMA.2018.069
Contribution to the Special Issue on Combinatorics of Moduli Spaces: Integrability, Cohomology, Quantisation, and Beyond

Loop Models and $K$-Theory

Paul Zinn-Justin
School of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia

Received November 28, 2017, in final form June 27, 2018; Published online July 13, 2018

Abstract
This is a review/announcement of results concerning the connection between certain exactly solvable two-dimensional models of statistical mechanics, namely loop models, and the equivariant $K$-theory of the cotangent bundle of the Grassmannian. We interpret various concepts from integrable systems ($R$-matrix, partition function on a finite domain) in geometric terms. As a byproduct, we provide explicit formulae for $K$-classes of various coherent sheaves, including structure and (conjecturally) square roots of canonical sheaves and canonical sheaves of conormal varieties of Schubert varieties.

Key words: quantum integrability; loop models; $K$-theory.

pdf (760 kb)   tex (65 kb)

References

  1. Aganagic M., Okounkov A., Elliptic stable envelope, arXiv:1604.00423.
  2. Bender M., Perrin N., Singularities of closures of spherical $B$-conjugacy classes of nilpotent orbits, arXiv:1412.5654.
  3. Birman J.S., Wenzl H., Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), 249-273.
  4. Brauer R., On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937), 857-872.
  5. Brion M., Lectures on the geometry of flag varieties, in Topics in Cohomological Studies of Algebraic Varieties, Trends Math., Birkhäuser, Basel, 2005, 33-85, math.AG/0410240.
  6. Cantini L., Sportiello A., Proof of the Razumov-Stroganov conjecture, J. Combin. Theory Ser. A 118 (2011), 1549-1574, arXiv:1003.3376.
  7. Di Francesco P., Zinn-Justin P., The quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials, J. Phys. A: Math. Gen. 38 (2005), L815-L822, math-ph/0508059.
  8. Fomin S., Kirillov A.N., Yang-Baxter equation, symmetric functions and Grothendieck polynomials, hep-th/9306005.
  9. Fomin S., Kirillov A.N., The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math. 153 (1996), 123-143.
  10. Fonseca T., Zinn-Justin P., On some ground state components of the $O(1)$ loop model, J. Stat. Mech. Theory Exp. 2009 (2009), P03025, 29 pages, arXiv:0901.1679.
  11. Frenkel I.B., Reshetikhin N.Yu., Quantum affine algebras and holonomic difference equations, Comm. Math. Phys. 146 (1992), 1-60.
  12. Ginzburg V., Kapranov M., Vasserot E., Elliptic algebras and equivariant elliptic cohomology, q-alg/9505012.
  13. Gorbounov V., Korff C., Quantum integrability and generalised quantum Schubert calculus, Adv. Math. 313 (2017), 282-356, arXiv:1408.4718.
  14. Gorbounov V., Rimányi R., Tarasov V., Varchenko A., Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra, J. Geom. Phys. 74 (2013), 56-86, arXiv:1204.5138.
  15. Grayson D., Stillman M., Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.
  16. Knutson A., Miller E., Subword complexes in Coxeter groups, Adv. Math. 184 (2004), 161-176, math.CO/0309259.
  17. Knutson A., Miller E., Gröbner geometry of Schubert polynomials, Ann. of Math. 161 (2005), 1245-1318.
  18. Knutson A., Zinn-Justin P., A scheme related to the Brauer loop model, Adv. Math. 214 (2007), 40-77, math.AG/0503224.
  19. Knutson A., Zinn-Justin P., The Brauer loop scheme and orbital varieties, J. Geom. Phys. 78 (2014), 80-110, arXiv:1001.3335.
  20. Knutson A., Zinn-Justin P., Grassmann-Grassmann conormal varieties, integrability and plane partitions, Ann. Inst. Fourier (Grenoble), to appear, arXiv:1612.04465.
  21. Knutson A., Zinn-Justin P., Varieties associated to loop models on finite domains, in preparation.
  22. Lenart C., Zainoulline K., Zhong C., Parabolic Kazhdan-Lusztig basis, Schubert classes, and equivariant oriented cohomology, arXiv:1608.06554.
  23. Maulik D., Okounkov A., Quantum groups and quantum cohomology, Astérisque, to appear, arXiv:1211.1287.
  24. Miller E., Sturmfels B., Combinatorial commutative algebra, Graduate Texts in Mathematics, Vol. 227, Springer-Verlag, New York, 2005.
  25. Murakami J., The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), 745-758.
  26. Nienhuis B., Critical and multicritical ${\rm O}(n)$ models, Phys. A 163 (1990), 152-157.
  27. Okounkov A., Lectures on $K$-theoretic computations in enumerative geometry, arXiv:1512.07363.
  28. Ramanathan A., Schubert varieties are arithmetically Cohen-Macaulay, Invent. Math. 80 (1985), 283-294.
  29. Razumov A.V., Stroganov Yu.G., Combinatorial nature of the ground-state vector of the ${\rm O}(1)$ loop model, Theoret. and Math. Phys. 138 (2004), 333-337, math.CO/0104216.
  30. Rimányi R., Tarasov V., Varchenko A., Trigonometric weight functions as $K$-theoretic stable envelope maps for the cotangent bundle of a flag variety, J. Geom. Phys. 94 (2015), 81-119, arXiv:1411.0478.
  31. Rimányi R., Tarasov V., Varchenko A., Zinn-Justin P., Extended Joseph polynomials, quantized conformal blocks, and a $q$-Selberg type integral, J. Geom. Phys. 62 (2012), 2188-2207, arXiv:1110.2187.
  32. Rothbach B.D., Borel orbits of $X^2 = 0$ in $\mathfrak{gl}_n$, Ph.D. Thesis, University of California, Berkeley, 2009, available at http://search.proquest.com//docview/304845738.
  33. Stanley R.P., Hilbert functions of graded algebras, Adv. Math. 28 (1978), 57-83.
  34. Su C., Zhao G., Zhong C., On the $K$-theory stable bases of the Springer resolution, arXiv:1708.08013.
  35. Svanes T., Coherent cohomology on Schubert subschemes of flag schemes and applications, Adv. Math. 14 (1974), 369-453.
  36. Woo A., Yong A., When is a Schubert variety Gorenstein?, Adv. Math. 207 (2006), 205-220, math.AG/0409490.
  37. Zinn-Justin P., Six-vertex, loop and tiling models: integrability and combinatorics, Lambert Academic Publishing, 2009, arXiv:0901.0665.
  38. Zinn-Justin P., Lectures on geometry, quantum integrability and symmetric functions, Lecture notes from a course at HSE, Moscow, 2015, available at http://www.lpthe.jussieu.fr/~pzinn/summary.pdf.
  39. Zinn-Justin P., The geometry of crossing loop models, in preparation.
  40. Zinn-Justin P., Di Francesco P., Quantum Knizhnik-Zamolodchikov equation, totally symmetric self-complementary plane partitions, and alternating sign matrices, Theoret. and Math. Phys. 154 (2008), 331-348, math-ph/0703015.

Previous article  Next article   Contents of Volume 14 (2018)