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

SIGMA 17 (2021), 036, 53 pages      arXiv:1702.00511
Contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mulase for his 65th birthday

Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles

Olivia Dumitrescu ab and Motohico Mulase cd
a) Department of Mathematics, University of North Carolina at Chapel Hill, 340 Phillips Hall, CB 3250, Chapel Hill, NC 27599-3250 USA
b) Simion Stoilow Institute of Mathematics, Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania
c) Department of Mathematics, University of California, Davis, CA 95616-8633, USA
d) Kavli Institute for Physics and Mathematics of the Universe, The University of Tokyo, Kashiwa, Japan

Received December 31, 2019, in final form March 12, 2021; Published online April 09, 2021

Quantum curves were introduced in the physics literature. We develop a mathematical framework for the case associated with Hitchin spectral curves. In this context, a quantum curve is a Rees $\mathcal{D}$-module on a smooth projective algebraic curve, whose semi-classical limit produces the Hitchin spectral curve of a Higgs bundle. We give a method of quantization of Hitchin spectral curves by concretely constructing one-parameter deformation families of opers. We propose a variant of the topological recursion of Eynard-Orantin and Mirzakhani for the context of singular Hitchin spectral curves. We show that a PDE version of topological recursion provides all-order WKB analysis for the Rees $\mathcal{D}$-modules, defined as the quantization of Hitchin spectral curves associated with meromorphic ${\rm SL}(2,\mathbb{C})$-Higgs bundles. Topological recursion can be considered as a process of quantization of Hitchin spectral curves. We prove that these two quantizations, one via the construction of families of opers, and the other via the PDE recursion of topological type, agree for holomorphic and meromorphic ${\rm SL}(2,\mathbb{C})$-Higgs bundles. Classical differential equations such as the Airy differential equation provides a typical example. Through these classical examples, we see that quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and quantum invariants, such as Gromov-Witten invariants.

Key words: quantum curve; Hitchin spectral curve; Higgs field; Rees $\mathcal{D}$-module; opers; non-Abelian Hodge correspondence; mirror symmetry; Airy function; quantum invariants; WKB approximation; topological recursion.

pdf (750 kb)   tex (73 kb)  


  1. Aganagic M., Dijkgraaf R., Klemm A., Mariño M., Vafa C., Topological strings and integrable hierarchies, Comm. Math. Phys. 261 (2006), 451-516, arXiv:hep-th/0312085.
  2. Andersen J.E., Borot G., Orantin N., Modular functors, cohomological field theories, and topological recursion, in Topological Recursion and its Influence in Analysis, Geometry, and Topology, Proc. Sympos. Pure Math., Vol. 100, Amer. Math. Soc., Providence, RI, 2018, 1-58, arXiv:1711.04729.
  3. Arinkin D., On $\lambda$-connections on a curve where $\lambda$ is a formal parameter, Math. Res. Lett. 12 (2005), 551-565.
  4. Atiyah M.F., Bott R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523-615.
  5. Beauville A., Narasimhan M.S., Ramanan S., Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398 (1989), 169-179.
  6. Beilinson A., Drinfeld V., Opers, arXiv:math.AG/0501398.
  7. Bertola M., Dubrovin B., Yang D., Simple Lie algebras, Drinfeld-Sokolov hierarchies, and multi-point correlation functions, arXiv:1610.07534.
  8. Biquard O., Boalch P., Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004), 179-204, arXiv:math.DG/0111098.
  9. Bloch S., The dilogarithm and extensions of Lie algebras, in Algebraic $K$-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Math., Vol. 854, Springer, Berlin - New York, 1981, 1-23.
  10. Boalch P., Hyperkähler manifolds and nonabelian Hodge theory of (irregular) curves, arXiv:1203.6607.
  11. Boalch P., Symplectic manifolds and isomonodromic deformations, Adv. Math. 163 (2001), 137-205, arXiv:2002.00052.
  12. Bouchard V., Ciosmak P., Hadasz L., Osuga K., Ruba B., Sułkowski P., Super quantum Airy structures, Comm. Math. Phys. 380 (2020), 449-522, arXiv:1907.08913.
  13. Bouchard V., Hernández Serrano D., Liu X., Mulase M., Mirror symmetry for orbifold Hurwitz numbers, J. Differential Geom. 98 (2014), 375-423, arXiv:1301.4871.
  14. Bouchard V., Klemm A., Mariño M., Pasquetti S., Remodeling the B-model, Comm. Math. Phys. 287 (2009), 117-178, arXiv:0709.1453.
  15. Bouchard V., Klemm A., Mariño M., Pasquetti S., Topological open strings on orbifolds, Comm. Math. Phys. 296 (2010), 589-623, arXiv:0807.0597.
  16. Chapman K.M., Mulase M., Safnuk B., The Kontsevich constants for the volume of the moduli of curves and topological recursion, Commun. Number Theory Phys. 5 (2011), 643-698, arXiv:1009.2055.
  17. Chekhov L., Eynard B., Orantin N., Free energy topological expansion for the 2-matrix model, J. High Energy Phys. 2006 (2006), no. 12, 053, 31 pages, arXiv:math-ph/0603003.
  18. Collier B., Wentworth R., Conformal limits and the Białynicki-Birula stratification of the space of $\lambda$-connections, Adv. Math. 350 (2019), 1193-1225, arXiv:1808.01622.
  19. Dijkgraaf R., Fuji H., Manabe M., The volume conjecture, perturbative knot invariants, and recursion relations for topological strings, Nuclear Phys. B 849 (2011), 166-211, arXiv:1010.4542.
  20. Dijkgraaf R., Hollands L., Sułkowski P., Quantum curves and $\mathcal D$-modules, J. High Energy Phys. 2009 (2009), no. 11, 047, 59 pages, arXiv:0810.4157.
  21. Dijkgraaf R., Hollands L., Sułkowski P., Vafa C., Supersymmetric gauge theories, intersecting branes and free fermions, J. High Energy Phys. 2008 (2008), no. 2, 106, 57 pages, arXiv:0709.4446.
  22. Dijkgraaf R., Verlinde H., Verlinde E., Loop equations and Virasoro constraints in nonperturbative two-dimensional quantum gravity, Nuclear Phys. B 348 (1991), 435-456.
  23. Do N., Manescu D., Quantum curves for the enumeration of ribbon graphs and hypermaps, Commun. Number Theory Phys. 8 (2014), 677-701, arXiv:1312.6869.
  24. Donaldson S.K., A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. 18 (1983), 269-277.
  25. Dumitrescu O., A journey from the Hitchin section to the oper moduli, in String-Math 2016, Proc. Sympos. Pure Math., Vol. 98, Amer. Math. Soc., Providence, RI, 2018, 107-138, arXiv:1701.00155.
  26. Dumitrescu O., Fredrickson L., Kydonakis G., Mazzeo R., Mulase M., Neitzke A., Opers versus nonabelian Hodge, arXiv:1607.02172.
  27. Dumitrescu O., Mulase M., Quantum curves for Hitchin fibrations and the Eynard-Orantin theory, Lett. Math. Phys. 104 (2014), 635-671, arXiv:1310.6022.
  28. Dumitrescu O., Mulase M., Edge contraction on dual ribbon graphs and 2D TQFT, J. Algebra 494 (2018), 1-27, arXiv:1508.05922.
  29. Dumitrescu O., Mulase M., Lectures on the topological recursion for Higgs bundles and quantum curves, in The Geometry, Topology and Physics of Moduli Spaces of Higgs bundles, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., Vol. 36, World Sci. Publ., Hackensack, NJ, 2018, 103-198, arXiv:1509.09007.
  30. Dumitrescu O., Mulase M., Quantization of spectral curves for meromorphic Higgs bundles through topological recursion, in Topological Recursion and its Influence in Analysis, Geometry, and Topology, Proc. Sympos. Pure Math., Vol. 100, Amer. Math. Soc., Providence, RI, 2018, 179-229, arXiv:1411.1023.
  31. Dumitrescu O., Mulase M., Safnuk B., Sorkin A., The spectral curve of the Eynard-Orantin recursion via the Laplace transform, in Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, Contemp. Math., Vol. 593, Amer. Math. Soc., Providence, RI, 2013, 263-315, arXiv:1202.1159.
  32. Dunin-Barkowski P., Mulase M., Norbury P., Popolitov A., Shadrin S., Quantum spectral curve for the Gromov-Witten theory of the complex projective line, J. Reine Angew. Math. 726 (2017), 267-289, arXiv:1312.5336.
  33. Eynard B., Topological expansion for the 1-Hermitian matrix model correlation functions, J. High Energy Phys. 2004 (2004), no. 11, 031, 35 pages, arXiv:hep-th/0407261.
  34. Eynard B., Invariants of spectral curves and intersection theory of moduli spaces of complex curves, Commun. Number Theory Phys. 8 (2014), 541-588, arXiv:1104.0176.
  35. Eynard B., Mulase M., Safnuk B., The Laplace transform of the cut-and-join equation and the Bouchard-Mariño conjecture on Hurwitz numbers, Publ. Res. Inst. Math. Sci. 47 (2011), 629-670, arXiv:0907.5224.
  36. Eynard B., Orantin N., Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1 (2007), 347-452, arXiv:math-ph/0702045.
  37. Eynard B., Orantin N., Algebraic methods in random matrices and enumerative geometry, arXiv:0811.3531.
  38. Eynard B., Orantin N., Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture, Comm. Math. Phys. 337 (2015), 483-567, arXiv:1205.1103.
  39. Fang B., Liu C.-C.M., Zong Z., All-genus open-closed mirror symmetry for affine toric Calabi-Yau 3-orbifolds, Algebr. Geom. 7 (2020), 192-239, arXiv:1310.4818.
  40. Fang B., Liu C.-C.M., Zong Z., On the remodeling conjecture for toric Calabi-Yau 3-orbifolds, J. Amer. Math. Soc. 33 (2020), 135-222, arXiv:1604.07123.
  41. Ginzburg V., Lectures on $\mathcal D$-modules, University of Chicago, 1998.
  42. Goulden I.P., Jackson D.M., Transitive factorisations into transpositions and holomorphic mappings on the sphere, Proc. Amer. Math. Soc. 125 (1997), 51-60, arXiv:math.CO/9903094.
  43. Grassi A., Hatsuda Y., Mariño M., Topological strings from quantum mechanics, Ann. Henri Poincaré 17 (2016), 3177-3235, arXiv:1410.3382.
  44. Gukov S., Sułkowski P., A-polynomial, B-model, and quantization, J. High Energy Phys. 2012 (2012), no. 2, 070, 57 pages, arXiv:1108.0002.
  45. Gunning R.C., Special coordinate coverings of Riemann surfaces, Math. Ann. 170 (1967), 67-86.
  46. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  47. Hitchin N.J., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91-114.
  48. Hodge A.R., Mulase M., Hitchin integrable systems, deformations of spectral curves, and KP-type equations, in New Developments in Algebraic Geometry, Integrable Systems and Mirror Symmetry (RIMS, Kyoto, 2008), Adv. Stud. Pure Math., Vol. 59, Math. Soc. Japan, Tokyo, 2010, 31-77, arXiv:0801.0015.
  49. Hollands L., Topological strings and quantum curves, Ph.D. Thesis, University of Amsterdam, 2009, arXiv:0911.3413.
  50. Hollands L., Kidwai O., Higher length-twist coordinates, generalized Heun's opers, and twisted superpotentials, Adv. Theor. Math. Phys. 22 (2018), 1713-1822, arXiv:1710.04438.
  51. Huybrechts D., Lehn M., Stable pairs on curves and surfaces, J. Algebraic Geom. 4 (1995), 67-104, arXiv:alg-geom/9211001.
  52. Iwaki K., Marchal O., Saenz A., Painlevé equations, topological type property and reconstruction by the topological recursion, J. Geom. Phys. 124 (2018), 16-54, arXiv:1601.02517.
  53. Iwaki K., Saenz A., Quantum curve and the first Painlevé equation, SIGMA 12 (2016), 011, 24 pages, arXiv:1507.06557.
  54. Kapustin A., Witten E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1-236, arXiv:hep-th/0604151.
  55. Kontsevich M., Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1-23.
  56. Kontsevich M., Soibelman Y., Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and mirror symmetry, in Homological Mirror Symmetry and Tropical Geometry, Lect. Notes Unione Mat. Ital., Vol. 15, Springer, Cham, 2014, 197-308, arXiv:1303.3253.
  57. Kontsevich M., Soibelman Y., Airy structures and symplectic geometry of topological recursion, in Topological Recursion and its Influence in Analysis, Geometry, and Topology, Proc. Sympos. Pure Math., Vol. 100, Amer. Math. Soc., Providence, RI, 2018, 433-489, arXiv:1701.09137.
  58. Kostant B., The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973-1032.
  59. Maisonobe P., Sabbah C., Aspects of the theory of $\mathcal D$-modules, Kaiserslautern Lecture Notes, 2002.
  60. Mariño M., Open string amplitudes and large order behavior in topological string theory, J. High Energy Phys. 2008 (2008), no. 3, 060, 34 pages, arXiv:hep-th/0612127.
  61. Mariño M., Spectral theory and mirror symmetry, in String-Math 2016, Proc. Sympos. Pure Math., Vol. 98, Amer. Math. Soc., Providence, RI, 2018, 259-294, arXiv:1506.07757.
  62. Mochizuki T., Wild harmonic bundles and wild pure twistor $D$-modules, Astérisque 340 (2011), x+607 pages, arXiv:0803.1344.
  63. Mulase M., Penkava M., Topological recursion for the Poincaré polynomial of the combinatorial moduli space of curves, Adv. Math. 230 (2012), 1322-1339, arXiv:1009.2135.
  64. Mulase M., Shadrin S., Spitz L., The spectral curve and the Schrödinger equation of double Hurwitz numbers and higher spin structures, Commun. Number Theory Phys. 7 (2013), 125-143, arXiv:1301.5580.
  65. Mulase M., Sułkowski P., Spectral curves and the Schrödinger equations for the Eynard-Orantin recursion, Adv. Theor. Math. Phys. 19 (2015), 955-1015, arXiv:1210.3006.
  66. Mulase M., Zhang N., Polynomial recursion formula for linear Hodge integrals, Commun. Number Theory Phys. 4 (2010), 267-293, arXiv:0908.2267.
  67. Mumford D., Tata lectures on theta. II. Jacobian theta functions and differential equations, Progress in Mathematics, Vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984.
  68. Mumford D., Fogarty J., Kirwan F., Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2), Vol. 34, Springer-Verlag, Berlin, 1994.
  69. Narasimhan M.S., Seshadri C.S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540-567.
  70. Norbury P., Quantum curves and topological recursion, in String-Math 2014, Proc. Sympos. Pure Math., Vol. 93, Amer. Math. Soc., Providence, RI, 2016, 41-65, arXiv:1502.04394.
  71. Petit F., Quantization of spectral curves and DQ-modules, J. Noncommut. Geom. 13 (2019), 161-191, arXiv:1507.04315.
  72. Schwarz A., Quantum curves, Comm. Math. Phys. 338 (2015), 483-500, arXiv:1401.1574.
  73. Simpson C.T., Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992), 5-95.
  74. Teschner J., Supersymmetric gauge theories, quantization of ${\mathcal M}_{\rm flat}$, and conformal field theory, in New Dualities of Sypersymmetric Gauge Theories, Math. Phys. Stud., Springer, Cham, 2016, 375-417, arXiv:1412.7140.
  75. Vakil R., Enumerative geometry of curves via degeneration methods, Ph.D. Thesis, Harvard University, 1997.
  76. Witten E., Two-dimensional gravity and intersection theory on moduli space, in Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh University, Bethlehem, PA, 1991, 243-310.
  77. Witten E., Gauge theory and wild ramification, Anal. Appl. (Singap.) 6 (2008), 429-501, arXiv:0710.0631.
  78. Witten E., Mirror symmetry, Hitchin's equations, and Langlands duality, in The Many Facets of Geometry, Oxford University Press, Oxford, 2010, 113-128, arXiv:0802.0999.
  79. Witten E., Analytic continuation of Chern-Simons Theory, in Chern-Simons Gauge Theory: 20 Years After, AMS/IP Stud. Adv. Math., Vol. 50, Amer. Math. Soc., Providence, RI, 2011, 347-446, arXiv:1001.2933.

Previous article  Next article  Contents of Volume 17 (2021)