A family of \(\mathrm{GL}_r\) multiplicative Higgs bundles on rational base. (English) Zbl 1464.16029
Summary: In this paper we study a restricted family of holomorphic symplectic leaves in the Poisson-Lie group \(\mathrm{GL}_r(\mathcal{K}_{\mathbb{P}^1_x})\) 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).
MSC:
| 16T25 | Yang-Baxter equations |
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 53D30 | Symplectic structures of moduli spaces |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
Keywords:
symplectic leaves; Poisson-Lie group; Yang-Baxter equation; Sklyanin brackets; Coulomb branch; multiplicative Higgs bundlesReferences:
| [1] | Aganagic, M. and Frenkel, E. and Okounkov, A., Quantum {\(q\)}-{L}anglands correspondence, Transactions of the Moscow Mathematical Society, 79, 1-83, (2018) · Zbl 1163.90353 · doi:10.1090/mosc/278 |
| [2] | Alday, Luis F. and Gaiotto, Davide and Tachikawa, Yuji, Liouville correlation functions from four-dimensional gauge theories, Letters in Mathematical Physics, 91, 2, 167-197, (2010) · Zbl 1163.90353 · doi:10.1007/s11005-010-0369-5 |
| [3] | Babelon, Olivier and Bernard, Denis and Talon, Michel, Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics, xii+602, (2003), Cambridge University Press, Cambridge · Zbl 1163.90353 · doi:10.1017/CBO9780511535024 |
| [4] | Babich, M. V., Birational {D}arboux coordinates on (co)adjoint orbits of {\({\rm GL}(N,{\mathbb C})\)}, Functional Analysis and its Applications, 50, 1, 17-30, (2016) · Zbl 1163.90353 · doi:10.1007/s10688-016-0124-5 |
| [5] | Bazhanov, Vladimir V. and Frassek, Rouven and {\L}ukowski, Tomasz and Meneghelli, Carlo and Staudacher, Matthias, Baxter {Q}-operators and representations of {Y}angians, Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, 850, 1, 148-174, (2011) · Zbl 1163.90353 · doi:10.1016/j.nuclphysb.2011.04.006 |
| [6] | Bazhanov, Vladimir V. and Tsuboi, Zengo, Baxter’s {Q}-operators for supersymmetric spin chains, Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, 805, 3, 451-516, (2008) · Zbl 1163.90353 · doi:10.1016/j.nuclphysb.2008.06.025 |
| [7] | Beilinson, A. A. and Drinfel’d, V. G., Quantization of {H}itchin’s fibration and {L}anglands’ program, Algebraic and Geometric Methods in Mathematical Physics ({K}aciveli, 1993), Math. Phys. Stud., 19, 3-7, (1996), Kluwer Acad. Publ., Dordrecht · Zbl 1163.90353 |
| [8] | Belavin, A. A. and Drinfel’d, V. G., Solutions of the classical {Y}ang–{B}axter equation for simple {L}ie algebras, Functional Analysis and its Applications, 16, 3, 159-180, (1982) · Zbl 1163.90353 · doi:10.1007/BF01081585 |
| [9] | Birkhoff, George D., The generalized {R}iemann problem for linear differential equations and the allied problems for linear difference and \(q\)-difference equations, Proceedings of the American Academy of Arts and Sciences, 49, 9, 521-568, (1913) · Zbl 1163.90353 · doi:10.2307/20025482 |
| [10] | Boos, Herman and G\"{o}hmann, Frank and Kl\"{u}mper, Andreas and Nirov, Khazret S. and Razumov, Alexander V., Exercises with the universal {\(R\)}-matrix, Journal of Physics. A. Mathematical and Theoretical, 43, 41, 415208, 35 pages, (2010) · Zbl 1163.90353 · doi:10.1088/1751-8113/43/41/415208 |
| [11] | Bottacin, Francesco, Poisson structures on moduli spaces of sheaves over {P}oisson surfaces, Inventiones Mathematicae, 121, 2, 421-436, (1995) · Zbl 1163.90353 · doi:10.1007/BF01884307 |
| [12] | Bottacin, Francesco, Symplectic geometry on moduli spaces of stable pairs, Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 28, 4, 391-433, (1995) · Zbl 1163.90353 · doi:10.24033/asens.1719 |
| [13] | Braverman, Alexander and Finkelberg, Michael and Nakajima, Hiraku, Coulomb branches of \(3d \mathcal{N}=4\) quiver gauge theories and slices in the affine Grassmannian · Zbl 1163.90353 |
| [14] | Braverman, Alexander and Finkelberg, Michael and Nakajima, Hiraku, Towards a mathematical definition of {C}oulomb branches of \(3\)-dimensional \(\mathcal{N}=4\) gauge theories, II · Zbl 1163.90353 |
| [15] | Charbonneau, Benoit and Hurtubise, Jacques, Singular {H}ermitian–{E}instein monopoles on the product of a circle and a {R}iemann surface, International Mathematics Research Notices. IMRN, 2011, 1, 175-216, (2011) · Zbl 1163.90353 · doi:10.1093/imrn/rnq059 |
| [16] | Cherkis, Sergey and Kapustin, Anton, Nahm transform for periodic monopoles and {\({\mathcal N}=2\)} super {Y}ang–{M}ills theory, Communications in Mathematical Physics, 218, 2, 333-371, (2001) · Zbl 1163.90353 · doi:10.1007/PL00005558 |
| [17] | Cherkis, Sergey A. and Kapustin, Anton, Hyper-{K}\"{a}hler metrics from periodic monopoles, Physical Review. D. Third Series, 65, 8, 084015, 10 pages, (2002) · Zbl 1163.90353 · doi:10.1103/PhysRevD.65.084015 |
| [18] | Cherkis, Sergey A. and Kapustin, Anton, Periodic monopoles with singularities and {\({\mathcal N}=2\)} super-{QCD}, Communications in Mathematical Physics, 234, 1, 1-35, (2003) · Zbl 1163.90353 · doi:10.1007/s00220-002-0786-0 |
| [19] | Chervov, A. and Talalaev, D., Quantum spectral curves, quantum integrable systems and the geometric {L}anglands correspondence · Zbl 1163.90353 |
| [20] | Costello, K., Supersymmetric gauge theory and the {Y}angian · Zbl 1163.90353 |
| [21] | Costello, Kevin and Witten, Edward and Yamazaki, Masahito, Gauge theory and integrability, {I}, ICCM Notices. Notices of the International Congress of Chinese Mathematicians, 6, 1, 46-119, (2018) · Zbl 1163.90353 · doi:10.4310/ICCM.2018.v6.n1.a6 |
| [22] | Derkachov, Sergey \'{E}. and Manashov, Alexander N., {\({\mathcal R}\)}-matrix and {B}axter {\({\mathcal Q}\)}-operators for the noncompact {\({\rm SL}(N,{\mathbb C})\)} invariant spin chain, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 2, 084, 20 pages, (2006) · Zbl 1163.90353 · doi:10.3842/SIGMA.2006.084 |
| [23] | Di Francesco, Philippe and Mathieu, Pierre and S\'{e}n\'{e}chal, David, Conformal field theory, Graduate Texts in Contemporary Physics, xxii+890, (1997), Springer-Verlag, New York · Zbl 1163.90353 · doi:10.1007/978-1-4612-2256-9 |
| [24] | Donagi, Ron, Spectral covers, Current Topics in Complex Algebraic Geometry ({B}erkeley, {CA}, 1992/93), Math. Sci. Res. Inst. Publ., 28, 65-86, (1995), Cambridge University Press, Cambridge · Zbl 1163.90353 |
| [25] | Donagi, Ron and Markman, Eyal, Spectral covers, algebraically completely integrable, {H}amiltonian systems, and moduli of bundles, Integrable Systems and Quantum Groups ({M}ontecatini {T}erme, 1993), Lecture Notes in Math., 1620, 1-119, (1996), Springer, Berlin · Zbl 1163.90353 · doi:10.1007/BFb0094792 |
| [26] | Donagi, Ron Y., Principal bundles on elliptic fibrations, Asian Journal of Mathematics, 1, 2, 214-223, (1997) · Zbl 1163.90353 · doi:10.4310/AJM.1997.v1.n2.a1 |
| [27] | Donagi, Ron Y., Geometry and integrability, Geometry and Integrability, London Math. Soc. Lecture Note Ser., 295, 21-59, (2003), Cambridge Univerity Press, Cambridge · Zbl 1163.90353 · doi:10.1017/CBO9780511543135.004 |
| [28] | Donagi, R. Y. and Gaitsgory, D., The gerbe of {H}iggs bundles, Transformation Groups, 7, 2, 109-153, (2002) · Zbl 1163.90353 · doi:10.1007/s00031-002-0008-z |
| [29] | Drinfel’d, V. G., Quantum groups, Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ols. 1, 2 ({B}erkeley, {C}alif., 1986), 798-820, (1987), Amer. Math. Soc., Providence, RI · Zbl 1163.90353 |
| [30] | Elliott, Chris and Pestun, Vasily, Multiplicative {H}itchin systems and supersymmetric gauge theory · Zbl 1163.90353 |
| [31] | Faddeev, L. D. and Takhtajan, L. A., Hamiltonian methods in the theory of solitons, Springer Series in Soviet Mathematics, x+592, (1987), Springer-Verlag, Berlin · Zbl 1163.90353 · doi:10.1007/978-3-540-69969-9 |
| [32] | Finkelberg, Michael and Tsymbaliuk, Alexander, Multiplicative slices, relativistic {T}oda and shifted quantum affine algebras · Zbl 1163.90353 |
| [33] | Frenkel, Edward and Ng\^o, Bao Ch\^au, Geometrization of trace formulas, Bulletin of Mathematical Sciences, 1, 1, 129-199, (2011) · Zbl 1163.90353 · doi:10.1007/s13373-011-0009-0 |
| [34] | Frenkel, Edward and Reshetikhin, Nicolai, The {\(q\)}-characters of representations of quantum affine algebras and deformations of {\({\mathcal W}\)}-algebras, Recent Developments in Quantum Affine Algebras and related Topics ({R}aleigh, {NC}, 1998), Contemp. Math., 248, 163-205, (1999), Amer. Math. Soc., Providence, RI · Zbl 1163.90353 · doi:10.1090/conm/248/03823 |
| [35] | Friedman, Robert and Morgan, John and Witten, Edward, Vector bundles and {\({\rm F}\)} theory, Communications in Mathematical Physics, 187, 3, 679-743, (1997) · Zbl 1163.90353 · doi:10.1007/s002200050154 |
| [36] | Gerasimov, A. and Kharchev, S. and Lebedev, D. and Oblezin, S., On a class of representations of the {Y}angian and moduli space of monopoles, Communications in Mathematical Physics, 260, 3, 511-525, (2005) · Zbl 1163.90353 · doi:10.1007/s00220-005-1417-3 |
| [37] | Gorsky, A. and Gukov, S. and Mironov, A., Multiscale {\(N=2\)} {SUSY} field theories, integrable systems and their stringy/brane origin, Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, 517, 1-3, 409-461, (1998) · Zbl 1163.90353 · doi:10.1016/S0550-3213(98)00055-8 |
| [38] | Gorsky, A. and Gukov, S. and Mironov, A., S{USY} field theories in higher dimensions and integrable spin chains, Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, 518, 3, 689-713, (1998) · Zbl 1163.90353 · doi:10.1016/S0550-3213(98)00106-0 |
| [39] | Haouzi, Nathan and Schmid, Christian, Little string origin of surface defects, Journal of High Energy Physics, 2017, 5, no. 5, 082, 53 pages, (2017) · Zbl 1163.90353 · doi:10.1007/JHEP05(2017)082 |
| [40] | Hernandez, David and Jimbo, Michio, Asymptotic representations and {D}rinfeld rational fractions, Compositio Mathematica, 148, 5, 1593-1623, (2012) · Zbl 1163.90353 · doi:10.1112/S0010437X12000267 |
| [41] | Hitchin, Nigel, Stable bundles and integrable systems, Duke Mathematical Journal, 54, 1, 91-114, (1987) · Zbl 1163.90353 · doi:10.1215/S0012-7094-87-05408-1 |
| [42] | Hurtubise, J. C. and Markman, E., Elliptic {S}klyanin integrable systems for arbitrary reductive groups, Advances in Theoretical and Mathematical Physics, 6, 5, 873-978, (2002) · Zbl 1163.90353 · doi:10.4310/ATMP.2002.v6.n5.a4 |
| [43] | Jimbo, Michio, Introduction to the {Y}ang–{B}axter equation, Braid group, knot theory and statistical mechanics, Adv. Ser. Math. Phys., 9, 111-134, (1989), World Sci. Publ., Teaneck, NJ · Zbl 1163.90353 · doi:10.1142/9789812798350_0005 |
| [44] | Kamnitzer, Joel and Webster, Ben and Weekes, Alex and Yacobi, Oded, Yangians and quantizations of slices in the affine {G}rassmannian, Algebra & Number Theory, 8, 4, 857-893, (2014) · Zbl 1163.90353 · doi:10.2140/ant.2014.8.857 |
| [45] | Kapustin, Anton and Witten, Edward, Electric-magnetic duality and the geometric {L}anglands program · Zbl 1163.90353 |
| [46] | Kimura, Taro and Pestun, Vasily, Quiver {W}-algebras, Letters in Mathematical Physics, 108, 6, 1351-1381, (2018) · Zbl 1163.90353 · doi:10.1007/s11005-018-1072-1 |
| [47] | Looijenga, Eduard, Root systems and elliptic curves, Inventiones Mathematicae, 38, 1, 17-32, (1976) · Zbl 1163.90353 · doi:10.1007/BF01390167 |
| [48] | Macdonald, I. G., Symmetric functions and {H}all polynomials, Oxford Mathematical Monographs, viii+180, (1979), The Clarendon Press, Oxford University Press, New York · Zbl 1163.90353 |
| [49] | Markman, Eyal, Spectral curves and integrable systems, Compositio Mathematica, 93, 3, 255-290, (1994) · Zbl 1163.90353 |
| [50] | Geometry and integrability, London Mathematical Society Lecture Note Series, 295, xii+153, (2003), Cambridge University Press, Cambridge · Zbl 1163.90353 · doi:10.1017/CBO9780511543135 |
| [51] | Meneghelli, C., Superconformal gauge theory, {Y}angian symmetry and {B}axter’s Q-operator · Zbl 1163.90353 |
| [52] | Molev, Alexander, Yangians and classical {L}ie algebras, Mathematical Surveys and Monographs, 143, xviii+400, (2007), Amer. Math. Soc., Providence, RI · Zbl 1163.90353 · doi:10.1090/surv/143 |
| [53] | Nakajima, Hiraku, Towards a mathematical definition of {C}oulomb branches of 3-dimensional {\({\mathcal N}=4\)} gauge theories, {I}, Advances in Theoretical and Mathematical Physics, 20, 3, 595-669, (2016) · Zbl 1163.90353 · doi:10.4310/ATMP.2016.v20.n3.a4 |
| [54] | Nekrasov, Nikita, {BPS/CFT} correspondence: non-perturbative {D}yson–{S}chwinger equations and \(qq\)-characters, Journal of High Energy Physics, 2016, 3, no. 3, 181, 69 pages, (2016) · Zbl 1163.90353 · doi:10.1007/JHEP03(2016)181 |
| [55] | Nekrasov, Nikita and Pestun, Vasily, Seiberg–{W}itten geometry of four dimensional \({\mathcal N}=2\) quiver gauge theories · Zbl 1163.90353 |
| [56] | Nekrasov, Nikita and Pestun, Vasily and Shatashvili, Samson, Quantum geometry and quiver gauge theories, Communications in Mathematical Physics, 357, 2, 519-567, (2018) · Zbl 1163.90353 · doi:10.1007/s00220-017-3071-y |
| [57] | Nekrasov, Nikita and Witten, Edward, The omega deformation, branes, integrability and {L}iouville theory, Journal of High Energy Physics, 2010, 9, no. 9, 092, 83 pages, (2010) · Zbl 1163.90353 · doi:10.1007/JHEP09(2010)092 |
| [58] | Sauloy, Jacques, Isomonodromy for complex linear {\(q\)}-difference equations, Th\'{e}ories asymptotiques et \'{e}quations de {P}ainlev\'{e}, S\'{e}min. Congr., 14, 249-280, (2006), Soc. Math. France, Paris · Zbl 1163.90353 |
| [59] | Seiberg, N. and Witten, E., Electric-magnetic duality, monopole condensation, and confinement in {\(N=2\)} supersymmetric {Y}ang–{M}ills theory, Nuclear Physics. B. Theoretical, Phenomenological, and Experimental High Energy Physics. Quantum Field Theory and Statistical Systems, 426, 1, 19-52, (1994) · Zbl 1163.90353 · doi:10.1016/0550-3213(94)90124-4 |
| [60] | Shapiro, Alexander, Grothendieck resolution, affine {G}rassmannian, and {Y}angian · Zbl 1163.90353 |
| [61] | Sklyanin, E. K., B\"{a}cklund transformations and {B}axter’s {\(Q\)}-operator, Integrable Systems: from Classical to Quantum ({M}ontr\'{e}al, {QC}, 1999), CRM Proc. Lecture Notes, 26, 227-250, (2000), Amer. Math. Soc., Providence, RI · Zbl 1163.90353 |
| [62] | Sklyanin, E. K., On complete integrability of the {L}andau–{L}ifshitz equation · Zbl 1163.90353 |
| [63] | Sklyanin, E. K., Some algebraic structures connected with the {Y}ang–{B}axter equation, Functional Analysis and its Applications, 16, 4, 263-270, (1982) · Zbl 1163.90353 · doi:10.1007/BF01077848 |
| [64] | Torrielli, Alessandro, Classical integrability, Journal of Physics. A. Mathematical and Theoretical, 49, 32, 323001, 31 pages, (2016) · Zbl 1163.90353 · doi:10.1088/1751-8113/49/32/323001 |
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