# Open $r$-spin theory II: The analogue of Witten's conjecture for $r$-spin disks

@article{Buryak2018OpenT, title={Open \$r\$-spin theory II: The analogue of Witten's conjecture for \$r\$-spin disks}, author={Alexandr Buryak and Emily Clader and Ran J. Tessler}, journal={arXiv: Symplectic Geometry}, year={2018} }

We conclude the construction of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define open $r$-spin intersection numbers, and we prove that their generating function is closely related to the wave function of the $r$th Gelfand--Dickey integrable hierarchy. This provides an analogue of Witten's $r$-spin conjecture in the open setting and a first step toward the construction of an open version of Fan--Jarvis--Ruan--Witten theory. As an unexpected consequence…

## 14 Citations

### Open r-spin theory I: Foundations

- Mathematics
- 2020

We lay the foundation for a version of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define the notion of $r$-spin disks, their moduli space, and the Witten…

### Open $r$-spin theory III: a prediction for higher genus

- Mathematics
- 2022

. In our previous two papers, we constructed an r -spin theory in genus zero for Riemann surfaces with boundary and fully determined the corresponding intersection numbers, providing an analogue of…

### Higher Airy structures and topological recursion for singular spectral curves

- Mathematics
- 2020

We give elements towards the classification of quantum Airy structures based on the $W(\mathfrak{gl}_r)$-algebras at self-dual level based on twisted modules of the Heisenberg VOA of…

### Mirror Symmetry for open r-spin invariants

- Mathematics
- 2022

We show that a generating function for open r-spin enumerative invariants produces a universal unfolding of the polynomial x. Further, the coordinates parametrizing this universal unfolding are flat…

### A Construction of Open Descendant Potentials in All Genera

- MathematicsInternational Mathematics Research Notices
- 2022

We present a construction of an open analogue of total descendant and total ancestor potentials via an “open version” of Givental’s action. Our construction gives a genus expansion for an arbitrary…

### Flat F-Manifolds, F-CohFTs, and Integrable Hierarchies

- MathematicsCommunications in Mathematical Physics
- 2021

We define the double ramification hierarchy associated to an F-cohomological field theory and use this construction to prove that the principal hierarchy of any semisimple (homogeneous) flat…

### Open topological recursion relations in genus 1 and integrable systems

- Mathematics
- 2020

The paper is devoted to the open topological recursion relations in genus 1, which are partial differential equations that conjecturally control open Gromov-Witten invariants in genus 1. We find an…

### Genus expansion of open free energy in 2d topological gravity

- MathematicsJournal of High Energy Physics
- 2021

We study open topological gravity in two dimensions, or, the intersection theory on the moduli space of open Riemann surfaces initiated by Pandharipande, Solomon and Tessler. The open free energy,…

### TOPOLOGICAL RECURSION RELATIONS IN GENUS 1 AND INTEGRABLE SYSTEMS

- Mathematics
- 2020

The paper is devoted to the open topological recursion relations in genus 1, which are partial differential equations that conjecturally control open Gromov–Witten invariants in genus 1. We find an…

## References

SHOWING 1-10 OF 38 REFERENCES

### Open r-spin theory I: Foundations

- Mathematics
- 2020

We lay the foundation for a version of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define the notion of $r$-spin disks, their moduli space, and the Witten…

### Moduli Spaces of Higher Spin Curves and Integrable Hierarchies

- MathematicsCompositio Mathematica
- 2001

We prove the genus zero part of the generalized Witten conjecture, relating moduli spaces of higher spin curves to Gelfand–Dickey hierarchies. That is, we show that intersection numbers on the moduli…

### Matrix Models and A Proof of the Open Analog of Witten’s Conjecture

- Mathematics
- 2015

In a recent work, R. Pandharipande, J. P. Solomon and the second author have initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. They conjectured that…

### Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair

- Mathematics
- 2002

Let $(X,\omega)$ be a symplectic manifold, $J$ be an $\omega$-tame almost complex structure, and $L$ be a Lagrangian submanifold. The stable compactification of the moduli space of parametrized…

### GEOMETRY OF THE MODULI OF HIGHER SPIN CURVES

- Mathematics
- 1998

This article treats various aspects of the geometry of the moduli of r-spin curves and its compactification . Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves…

### The Witten equation, mirror symmetry and quantum singularity theory

- Mathematics
- 2007

For any non-degenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the…

### Open intersection numbers and the wave function of the KdV hierarchy

- Mathematics
- 2014

Recently R. Pandharipande, J. Solomon and R. Tessler initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. They conjectured that the generating series of…

### The combinatorial formula for open gravitational descendents

- Mathematics
- 2015

In recent works, [20],[21], descendent integrals on the moduli space of Riemann surfaces with boundary were defined. It was conjectured in [20] that the generating function of these integrals…

### Tautological relations and the r-spin Witten conjecture

- Mathematics
- 2010

A geometric interpretation of Y.P. Lee’s algorithm leads to a much simpler proof of the fact that every tautological relation gives rise to a universal relation, and implies that in any semi-simple Gromov–Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautology relations, the formal and the geometric Gronov– Witten potentials coincide.

### The Witten top Chern class via -theory

- Mathematics
- 2006

The Witten top Chern class is the crucial cohomology class needed to state a conjecture by Witten relating the Gelfand–Dikĭı hierarchies to higher spin curves. In [PV01], Polishchuk and Vaintrob…