A COMPARISON OF LANDAU-GINBURG MODELS FOR ODD DIMENSIONAL QUADRICS

TY - JOUR

T1 - A COMPARISON OF LANDAU-GINBURG MODELS FOR ODD DIMENSIONAL QUADRICS

AU - Pech, C.

AU - Rietsch, K.

PY - 2018/9/27

Y1 - 2018/9/27

N2 - 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 = Q2m−1. Namely we introduce a regular function Wcan on a variety ˇ Xcan ×C∗, where ˇ Xcan is the complement of a particular anticanonical divisor in the projective space CP2m−1 = P(H∗(X,C)∗). Firstly we prove that the Jacobi ring associated to Wcan is isomorphic to the quantum cohomology ring of the quadric, and that this isomorphism is compatible with the identification of homogeneous coordinates on ˇ Xcan ⊂ CP2m−1 with elements of H∗(X,C). Secondly we find a very natural Laurent polynomial formula for Wcan by restricting it to a ‘Lusztig torus’ in ˇ Xcan. Thirdly we show that the Dubrovin connection on H∗(X,C[q]) embeds into the Gauss-Manin system associated to Wcan and deduce a flat section formula in terms of oscillating integrals. Finally, we compare ( ˇ Xcan,Wcan) 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 Q3, implying a tameness property in that case.

AB - 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 = Q2m−1. Namely we introduce a regular function Wcan on a variety ˇ Xcan ×C∗, where ˇ Xcan is the complement of a particular anticanonical divisor in the projective space CP2m−1 = P(H∗(X,C)∗). Firstly we prove that the Jacobi ring associated to Wcan is isomorphic to the quantum cohomology ring of the quadric, and that this isomorphism is compatible with the identification of homogeneous coordinates on ˇ Xcan ⊂ CP2m−1 with elements of H∗(X,C). Secondly we find a very natural Laurent polynomial formula for Wcan by restricting it to a ‘Lusztig torus’ in ˇ Xcan. Thirdly we show that the Dubrovin connection on H∗(X,C[q]) embeds into the Gauss-Manin system associated to Wcan and deduce a flat section formula in terms of oscillating integrals. Finally, we compare ( ˇ Xcan,Wcan) 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 Q3, implying a tameness property in that case.

KW - math.AG

KW - 14M17, 14J33, 20G20, 14N35

U2 - https://doi.org/10.21915/BIMAS.2018301

DO - https://doi.org/10.21915/BIMAS.2018301

M3 - Article

SN - 2304-7895

VL - 13

SP - 249

EP - 291

JO - Bulletin of the Institute of Mathematics Academia Sinica

JF - Bulletin of the Institute of Mathematics Academia Sinica

IS - 3

ER -