Communications in Mathematical Physics

, Volume 371, Issue 3, pp 839–920 | Cite as

Reconstructing GKZ via Topological Recursion

  • Hiroyuki FujiEmail author
  • Kohei Iwaki
  • Masahide Manabe
  • Ikuo Satake


In this article, a novel description of the hypergeometric differential equation found from Gel’fand–Kapranov–Zelevinsky’s system (referred to as GKZ equation) for Givental’s J-function in the Gromov–Witten theory will be proposed. The GKZ equation involves a parameter \(\hbar \), and we will reconstruct it as a quantum curve from the classical limit \(\hbar \rightarrow 0\) via the topological recursion. In this analysis, the spectral curve (referred to as GKZ curve) plays a central role, and it can be described by the critical point set of the mirror Landau–Ginzburg potential. Our novel description is derived via the duality relations of the string theories, and various physical interpretations suggest that the GKZ equation is identified with the quantum curve for the brane partition function in the cohomological limit. As an application of our novel picture for the GKZ equation, we will discuss the Stokes phenomenon for the equivariant \({\mathbb {C}}\mathbf{P }^{1}\) model, and the wall-crossing formula for the total Stokes matrix will be examined. And as a byproduct of this analysis, we will study Dubrovin’s conjecture for this equivariant model.



The authors thank Prof.  Hiroshi Iritani who suggests his idea on the equivariant version of the Dubrovin’s conjecture. KI also thanks Dr.  Fumihiko Sanda for fruitful discussion. HF and MM thank Prof. Piotr Sułkowski for stimulating discussions and useful comments. The research of HF and IS is supported by the Grant-in-Aid for Challenging Research (Exploratory) [# 17K18781]. The research of HF is also supported by the Grant-in-Aid for Scientific Research(C) [# 17K05239], and Grant-in-Aid for Scientific Research(B) [# 16H03927] from the Japan Ministry of Education, Culture, Sports, Science and Technology, and Fund for Promotion of Academic Research from Department of Education in Kagawa University. The research of KI is supported by the Grant-in-Aid for JSPS KAKENHI KIBAN(S) [# 16H06337], Young Scientists Grant-in-Aid for (B) [# 16K17613] from the Japan Ministry of Education, Culture, Sports, Science and Technology. The work of MM is supported by the ERC Starting Grant no. 335739 “Quantum fields and knot homologies” funded by the European Research Council under the European Union’s Seventh Framework Programme. The work of MM is also supported by Max-Planck-Institut für Mathematik in Bonn.


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Authors and Affiliations

  1. 1.Faculty of EducationKagawa UniversityTakamatsuJapan
  2. 2.QGM, Department of MathematicsAarhus UniversityAarhus CDenmark
  3. 3.Graduate School of MathematicsNagoya UniversityNagoyaJapan
  4. 4.Faculty of PhysicsUniversity of WarsawWarsawPoland
  5. 5.Max-Planck-Institut für MathematikBonnGermany

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