A Comparison of Landau-Ginzburg Models for Odd Dimensional Quadrics
by
Clelia Pech
Konstanze Rietsch
Vol. 13 No. 3 (2018) P.249~P.291
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DOI: | https://doi.org/10.21915/BIMAS.2018301 |
| | 10.21915/BIMAS.2018301 |
ABSTRACT
In [25], the second author defined a Landau-Ginzburg model for homogeneous spaces $G/P$. In this paper,
we reformulate this LG model in the case of the odd-dimensional quadric $X=Q_{2m-1}$. Namely we introduce a regular
function $\mathcal{W}_{\mathrm{can}}$ on a variety $\check{X}_{\mathrm{can}}\times\mathbb{C}^*$, where $\check{X}_{\mathrm{can}}$ is the complement of a particular anticanonical divisor in
the projective space $\mathbb{C}\mathbb{P}^{2m-1}=\mathbb{P}(H^*(X,\mathbb{C})^*)$. Firstly we prove that the Jacobi ring associated to $\mathcal{W}_{\mathrm{can}}$ is
isomorphic to the quantum cohomology ring of the quadric, and that this isomorphism is compatible with the identification of homogeneous coordinates on
$\check{X}_{\mathrm{can}}\subset \mathbb{C}\mathbb{P}^{2m-1}$ with elements of $H^*(X,\mathbb{C})$. Secondly we find a very natural Laurent polynomial formula
for $\mathcal{W}_{\mathrm{can}}$ by restricting it to a 'Lusztig torus' in $\check{X}_{\mathrm{can}}$. Thirdly we show that the Dubrovin connection on
$H^*(X,\mathbb{C}[q])$ embeds into the Gauss-Manin system associated to $\mathcal{W}_{\mathrm{can}}$ and deduce a flat section formula in terms of oscillating integrals.
Finally, we compare $(\check{X}_{\mathrm{can}},\mathcal{W}_{\mathrm{can}})$ with previous Landau-Ginzburg models defined for odd quadrics. Namely, we prove that it is a partial
compactification of Givental's original LG model [10]. We show that our LG model is isomorphic to the Lie-theoretic LG
model from [25]. Moreover it is birationally equivalent to an LG model introduced by Gorbounov and Smirnov [13],
and it is algebraically isomorphic to Gorbounov and Smirnov's mirror for $Q_3$, implying a tameness property in that case.
KEYWORDS
Mirror Symmetry, quadrics, Lie theory, Gromov-Witten theory, quantum cohomology, Landau-Ginzburg model, Gauss-Manin system.
MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 14N35, 14M17, 14J33, 57T15.
MILESTONES
Received: 2016-09-12
Revised : 2017-03-27
Accepted: 2017-10-22
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