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

SIGMA 15 (2019), 031, 42 pages      arXiv:1808.00799

A Family of ${\rm GL}_r$ Multiplicative Higgs Bundles on Rational Base

Rouven Frassek and Vasily Pestun
Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France

Received September 09, 2018, in final form April 10, 2019; Published online April 25, 2019

In this paper we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group ${\rm GL}_r(\mathcal{K}_{\mathbb{P}^1_x})$ with rational quadratic Sklyanin brackets induced by a one-form with a single quadratic pole at $\infty \in \mathbb{P}_{1}$. The restriction of the family is that the matrix elements in the defining representation are linear functions of $x$. We study how the symplectic leaves in this family are obtained by the fusion of certain fundamental symplectic leaves. These symplectic leaves arise as minimal examples of (i) moduli spaces of multiplicative Higgs bundles on $\mathbb{P}^{1}$ with prescribed singularities, (ii) moduli spaces of $U(r)$ monopoles on $\mathbb{R}^2 \times S^1$ with Dirac singularities, (iii) Coulomb branches of the moduli space of vacua of 4d $\mathcal{N}=2$ supersymmetric $A_{r-1}$ quiver gauge theories compactified on a circle. While degree 1 symplectic leaves regular at $\infty \in \mathbb{P}^1$ (Coulomb branches of the superconformal quiver gauge theories) are isomorphic to co-adjoint orbits in $\mathfrak{gl}_{r}$ and their Darboux parametrization and quantization is well known, the case irregular at infinity (asymptotically free quiver gauge theories) is novel. We also explicitly quantize the algebra of functions on these moduli spaces by presenting the corresponding solutions to the quantum Yang-Baxter equation valued in Heisenberg algebra (free field realization).

Key words: symplectic leaves; Poisson-Lie group; Yang-Baxter equation; Sklyanin brackets; Coulomb branch; multiplicative Higgs bundles.

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  1. Aganagic M., Frenkel E., Okounkov A., Quantum $q$-Langlands correspondence, Trans. Moscow Math. Soc. 79 (2018), 1-83, arXiv:1701.03146.
  2. Alday L.F., Gaiotto D., Tachikawa Y., Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91 (2010), 167-197.
  3. Babelon O., Bernard D., Talon M., Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2003.
  4. Babich M.V., Birational Darboux coordinates on (co)adjoint orbits of ${\rm GL}(N,{\mathbb C})$, Funct. Anal. Appl. 50 (2016), 17-30.
  5. Bazhanov V.V., Frassek R., Łukowski T., Meneghelli C., Staudacher M., Baxter Q-operators and representations of Yangians, Nuclear Phys. B 850 (2011), 148-174, arXiv:1010.3699.
  6. Bazhanov V.V., Tsuboi Z., Baxter's Q-operators for supersymmetric spin chains, Nuclear Phys. B 805 (2008), 451-516, arXiv:0805.4274.
  7. Beilinson A.A., Drinfel'd V.G., Quantization of Hitchin's fibration and Langlands' program, in Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993), Math. Phys. Stud., Vol. 19, Kluwer Acad. Publ., Dordrecht, 1996, 3-7.
  8. Belavin A.A., Drinfel'd V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl. 16 (1982), 159-180.
  9. Birkhoff G.D., The generalized Riemann problem for linear differential equations and the allied problems for linear difference and $q$-difference equations, Proc. Amer. Acad. Arts Sci. 49 (1913), 521-568.
  10. Boos H., Göhmann F., Klümper A., Nirov K.S., Razumov A.V., Exercises with the universal $R$-matrix, J. Phys. A: Math. Theor. 43 (2010), 415208, 35 pages, arXiv:1004.5342.
  11. Bottacin F., Poisson structures on moduli spaces of sheaves over Poisson surfaces, Invent. Math. 121 (1995), 421-436.
  12. Bottacin F., Symplectic geometry on moduli spaces of stable pairs, Ann. Sci. 'Ecole Norm. Sup. (4) 28 (1995), 391-433.
  13. Braverman A., Finkelberg M., Nakajima H., Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian, arXiv:1604.03625.
  14. Braverman A., Finkelberg M., Nakajima H., Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II, arXiv:1601.03586.
  15. Charbonneau B., Hurtubise J., Singular Hermitian-Einstein monopoles on the product of a circle and a Riemann surface, Int. Math. Res. Not. 2011 (2011), 175-216, arXiv:0812.0221.
  16. Cherkis S., Kapustin A., Nahm transform for periodic monopoles and ${\mathcal N}=2$ super Yang-Mills theory, Comm. Math. Phys. 218 (2001), 333-371, arXiv:hep-th/0006050.
  17. Cherkis S.A., Kapustin A., Hyper-Kähler metrics from periodic monopoles, Phys. Rev. D 65 (2002), 084015, 10 pages, arXiv:hep-th/0109141.
  18. Cherkis S.A., Kapustin A., Periodic monopoles with singularities and ${\mathcal N}=2$ super-QCD, Comm. Math. Phys. 234 (2003), 1-35, arXiv:hep-th/0011081.
  19. Chervov A., Talalaev D., Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, arXiv:hep-th/0604128.
  20. Costello K., Supersymmetric gauge theory and the Yangian, arXiv:1303.2632.
  21. Costello K., Witten E., Yamazaki M., Gauge theory and integrability, I, ICCM Not. 6 (2018), 46-119, arXiv:1709.09993.
  22. Derkachov S.E., Manashov A.N., ${\mathcal R}$-matrix and Baxter ${\mathcal Q}$-operators for the noncompact ${\rm SL}(N,{\mathbb C})$ invariant spin chain, SIGMA 2 (2006), 084, 20 pages, arXiv:nlin.SI/0612003.
  23. Di Francesco P., Mathieu P., Sénéchal D., Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997.
  24. Donagi R.Y., Spectral covers, in Current Topics in Complex Algebraic Geometry (Berkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ., Vol. 28, Cambridge University Press, Cambridge, 1995, 65-86.
  25. Donagi R.Y., Principal bundles on elliptic fibrations, Asian J. Math. 1 (1997), 214-223, arXiv:alg-geom/9702002.
  26. Donagi R.Y., Geometry and integrability, in Geometry and Integrability, London Math. Soc. Lecture Note Ser., Vol. 295, Cambridge Univerity Press, Cambridge, 2003, 21-59.
  27. Donagi R.Y., Gaitsgory D., The gerbe of Higgs bundles, Transform. Groups 7 (2002), 109-153, arXiv:math.AG/0005132.
  28. Donagi R.Y., Markman E., Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 1-119, arXiv:alg-geom/9507017.
  29. Drinfel'd V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vols. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, 798-820.
  30. Elliott C., Pestun V., Multiplicative Hitchin systems and supersymmetric gauge theory, arXiv:1812.05516.
  31. Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1987.
  32. Finkelberg M., Tsymbaliuk A., Multiplicative slices, relativistic Toda and shifted quantum affine algebras, arXiv:1708.01795.
  33. Frenkel E., Ngô B.C., Geometrization of trace formulas, Bull. Math. Sci. 1 (2011), 129-199, arXiv:1004.5323.
  34. Frenkel E., Reshetikhin N., The $q$-characters of representations of quantum affine algebras and deformations of ${\mathcal W}$-algebras, in Recent Developments in Quantum Affine Algebras and related Topics (Raleigh, NC, 1998), Contemp. Math., Vol. 248, Amer. Math. Soc., Providence, RI, 1999, 163-205, arXiv:math.QA/9810055.
  35. Friedman R., Morgan J., Witten E., Vector bundles and ${\rm F}$ theory, Comm. Math. Phys. 187 (1997), 679-743, arXiv:hep-th/9701162.
  36. Gerasimov A., Kharchev S., Lebedev D., Oblezin S., On a class of representations of the Yangian and moduli space of monopoles, Comm. Math. Phys. 260 (2005), 511-525, arXiv:math.AG/0409031.
  37. Gorsky A., Gukov S., Mironov A., Multiscale $N=2$ SUSY field theories, integrable systems and their stringy/brane origin, Nuclear Phys. B 517 (1998), 409-461, arXiv:hep-th/9707120.
  38. Gorsky A., Gukov S., Mironov A., SUSY field theories in higher dimensions and integrable spin chains, Nuclear Phys. B 518 (1998), 689-713, arXiv:hep-th/9710239.
  39. Haouzi N., Schmid C., Little string origin of surface defects, J. High Energy Phys. 2017 (2017), no. 5, 082, 53 pages, arXiv:1608.07279.
  40. Hernandez D., Jimbo M., Asymptotic representations and Drinfeld rational fractions, Compos. Math. 148 (2012), 1593-1623, arXiv:1104.1891.
  41. Hitchin N., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91-114.
  42. Hurtubise J.C., Markman E., Elliptic Sklyanin integrable systems for arbitrary reductive groups, Adv. Theor. Math. Phys. 6 (2002), 873-978, arXiv:math.AG/0203031.
  43. Jimbo M., Introduction to the Yang-Baxter equation, in Braid group, knot theory and statistical mechanics, Adv. Ser. Math. Phys., Vol. 9, World Sci. Publ., Teaneck, NJ, 1989, 111-134.
  44. Kamnitzer J., Webster B., Weekes A., Yacobi O., Yangians and quantizations of slices in the affine Grassmannian, Algebra Number Theory 8 (2014), 857-893, arXiv:1209.0349.
  45. Kapustin A., Witten E., Electric-magnetic duality and the geometric Langlands program, arXiv:hep-th/0604151.
  46. Kimura T., Pestun V., Quiver W-algebras, Lett. Math. Phys. 108 (2018), 1351-1381, arXiv:1512.08533.
  47. Looijenga E., Root systems and elliptic curves, Invent. Math. 38 (1976), 17-32.
  48. Macdonald I.G., Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979.
  49. Markman E., Spectral curves and integrable systems, Compositio Math. 93 (1994), 255-290.
  50. Mason L., Nutku Y. (Editors), Geometry and integrability, London Mathematical Society Lecture Note Series, Vol. 295, Cambridge University Press, Cambridge, 2003.
  51. Meneghelli C., Superconformal gauge theory, Yangian symmetry and Baxter's Q-operator, Ph.D. Thesis, Humboldt University Berlin, 2011.
  52. Molev A., Yangians and classical Lie algebras, Mathematical Surveys and Monographs, Vol. 143, Amer. Math. Soc., Providence, RI, 2007.
  53. Nakajima H., Towards a mathematical definition of Coulomb branches of 3-dimensional ${\mathcal N}=4$ gauge theories, I, Adv. Theor. Math. Phys. 20 (2016), 595-669, arXiv:1503.03676.
  54. Nekrasov N., BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and $qq$-characters, J. High Energy Phys. 2016 (2016), no. 3, 181, 69 pages, arXiv:1512.05388.
  55. Nekrasov N., Pestun V., Seiberg-Witten geometry of four dimensional ${\mathcal N}=2$ quiver gauge theories, arXiv:1211.2240.
  56. Nekrasov N., Pestun V., Shatashvili S., Quantum geometry and quiver gauge theories, Comm. Math. Phys. 357 (2018), 519-567, arXiv:1312.6689.
  57. Nekrasov N., Witten E., The omega deformation, branes, integrability and Liouville theory, J. High Energy Phys. 2010 (2010), no. 9, 092, 83 pages, arXiv:1002.0888.
  58. Sauloy J., Isomonodromy for complex linear $q$-difference equations, in Théories asymptotiques et équations de Painlevé, S'emin. Congr., Vol. 14, Soc. Math. France, Paris, 2006, 249-280.
  59. Seiberg N., Witten E., Electric-magnetic duality, monopole condensation, and confinement in $N=2$ supersymmetric Yang-Mills theory, Nuclear Phys. B 426 (1994), 19-52, arXiv:hep-th/9407087.
  60. Shapiro A., Grothendieck resolution, affine Grassmannian, and Yangian,Ph.D. Thesis, University of California, Berkeley, 2016.
  61. Sklyanin E.K., On complete integrability of the Landau-Lifshitz equation, Tech. Rep. LOMI-E-79-3, 1979.
  62. Sklyanin E.K., Some algebraic structures connected with the Yang-Baxter equation, Funct. Anal. Appl. 16 (1982), 263-270.
  63. Sklyanin E.K., Bäcklund transformations and Baxter's $Q$-operator, in Integrable Systems: from Classical to Quantum (Montréal, QC, 1999), CRM Proc. Lecture Notes, Vol. 26, Amer. Math. Soc., Providence, RI, 2000, 227-250, arXiv:nlin.SI/0009009.
  64. Torrielli A., Classical integrability, J. Phys. A: Math. Theor. 49 (2016), 323001, 31 pages, arXiv:1606.02946.

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