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.

related items

• Baxter TQ relations
• Determinant solutions of T-systems

articles

• 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.