Quantum spectral curve

introduction

• A quantum curve is a Schrodinger operator-like noncommutative analogue of a plane curve which encodes (quantum) enumerative invariants in a new and interesting way.
• One says that a pair (P,Q) of ordinary differential operators specify a quantum curve if [P,Q]=const.
• If a pair of difference operators (K,L) obey the relation KL=const LK we say that they specify a discrete quantum curve.
• This terminology is prompted by well known results about commuting differential and difference operators, relating pairs of such operators with pairs of meromorphic functions on algebraic curves obeying some conditions.
• moduli spaces of quantum curves.
• how to quantize a pair of commuting differential or difference operators (i.e. to construct the corresponding quantum curve or discrete quantum curve)
• The Schrodinger operator annihilates a wave function which can be constructed using the WKB method, and conjecturally constructed in a rather different way via topological recursion.
• Quantum curves arise in various contexts in modern mathematical physics
  • in matrix models as quantization of spectral curves [1, 2]
  • in topological string theory as quantization of mirror curves [3]
  • in systems of intersecting branes and in Seiberg-Witten theory [4,5],
  • as quantum A-polynomials and their generalizations in knot theory and its physical realizations [6, 7]
  • in various enumerative problems related to moduli spaces of Riemann surfaces [8, 9]

related items

• Baxter TQ relations
• Determinant solutions of T-systems

expositions

• Dumitrescu, Olivia, and Motohico Mulase. “Lectures on the Topological Recursion for Higgs Bundles and Quantum Curves.” arXiv:1509.09007 [math-Ph], September 30, 2015. http://arxiv.org/abs/1509.09007.

computational resource

• https://arxiv.org/abs/1504.06640
• https://arxiv.org/abs/1411.4758

articles

• A. Alexandrov, D. Lewanski, S. Shadrin, Ramifications of Hurwitz theory, KP integrability and quantum curves, arXiv:1512.07026 [math-ph], December 22 2015, http://arxiv.org/abs/1512.07026
• Arpad Hegedus, Jozsef Konczer, Strong coupling results from the numerical solution of the quantum spectral curve, arXiv:1604.02346[hep-th], April 08 2016, http://arxiv.org/abs/1604.02346v1
• Manabe, Masahide, and Piotr Sułkowski. “Quantum Curves and Conformal Field Theory.” arXiv:1512.05785 [hep-Th, Physics:math-Ph], December 17, 2015. http://arxiv.org/abs/1512.05785.
• Izosimov, Anton. “Singularities of Integrable Systems and Algebraic Curves.” arXiv:1509.08996 [math-Ph, Physics:nlin], September 29, 2015. http://arxiv.org/abs/1509.08996.
• http://arxiv.org/abs/1509.06954
• Iwaki, Kohei, and Axel Saenz. “Quantum Curve and the First Painlev’e Equation.” arXiv:1507.06557 [math-Ph, Physics:nlin], July 23, 2015. http://arxiv.org/abs/1507.06557.
• Zenkevich, Yegor. “Quantum Spectral Curve for (q,t)-Matrix Model.” arXiv:1507.00519 [hep-Th, Physics:math-Ph], July 2, 2015. http://arxiv.org/abs/1507.00519.
• Gu, Jie, Albrecht Klemm, Marcos Marino, and Jonas Reuter. ‘Exact Solutions to Quantum Spectral Curves by Topological String Theory’. arXiv:1506.09176 [hep-Th], 30 June 2015. http://arxiv.org/abs/1506.09176.
• Gromov, Nikolay, Fedor Levkovich-Maslyuk, and Grigory Sizov. ‘Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS5/CFT4’. arXiv:1504.06640 [hep-Th], 24 April 2015. http://arxiv.org/abs/1504.06640.
• Luu, Martin, and Albert Schwarz. ‘Fourier Duality of Quantum Curves’. arXiv:1504.01582 [hep-Th, Physics:math-Ph], 7 April 2015. http://arxiv.org/abs/1504.01582.
• Norbury, Paul. ‘Quantum Curves and Topological Recursion’. arXiv:1502.04394 [math-Ph], 15 February 2015. http://arxiv.org/abs/1502.04394.
• Gromov, Nikolay, Vladimir Kazakov, Sebastien Leurent, and Dmytro Volin. “Quantum Spectral Curve for Arbitrary State/operator in AdS\(_5\)/CFT\(_4\).” arXiv:1405.4857 [hep-Th, Physics:math-Ph], May 19, 2014. http://arxiv.org/abs/1405.4857.
• Schwarz, Albert. ‘Quantum Curves’. arXiv:1401.1574 [hep-Th, Physics:math-Ph], 7 January 2014. http://arxiv.org/abs/1401.1574.
• [1] S. Gukov and P. Sulkowski, A-polynomial, B-model, and Quantization , JHEP 1202 (2012) 070, [arXiv:1108.0002].
• [2] M. Aganagic, M. C. N. Cheng, R. Dijkgraaf, D. Krefl, and C. Vafa, Quantum Geometry of Refined Topological Strings,JHEP 11 (2012) 019, [arXiv:1105.0630].
• [3] M. Aganagic, R. Dijkgraaf, A. Klemm, M. Marino, and C. Vafa, Topological Strings and Integrable Hierarchies, Commun. Math. Phys. 261 (2006) 451–516, [hep-th/0312085].
• [4] R. Dijkgraaf, L. Hollands, P. Sulkowski, and C. Vafa, Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions,JHEP0802 (2008) 106, [arXiv:0709.4446].
• [5] R. Dijkgraaf, L. Hollands, and P. Sulkowski, Quantum Curves and D-Modules, JHEP 0911(2009) 047, [arXiv:0810.4157].
• [6] R. Dijkgraaf and H. Fuji, The volume conjecture and topological strings ,Fortschr. Phys.57 (2009), no. 9 825–856.
• [7] R. Dijkgraaf, H. Fuji, and M. Manabe, The volume conjecture, perturbative knot invariants, and recursion relations for topological strings, Nuclear Phys. B849 (2011), no. 1 166–211.
• [8] P. Norbury, Quantum curves and topological recursion, arXiv:1502.0439.
• [9] O. Dumitrescu and M. Mulase, Lectures on the topological recursion for Higgs bundles and quantum curves, arXiv:1509.0900