## Duke Mathematical Journal

### A Landau–Ginzburg mirror theorem without concavity

Jérémy Guéré

#### Abstract

We provide a mirror symmetry theorem in a range of cases where state-of-the-art techniques that rely on concavity or convexity do not apply. More specifically, we work on a family of FJRW potentials (named for the Fan, Jarvis, Ruan, and Witten quantum singularity theory) which is viewed as the counterpart of a nonconvex Gromov–Witten potential via the physical Landau–Ginzburg/Calabi–Yau (LG/CY) correspondence. The main result provides an explicit formula for Polishchuk and Vaintrob’s virtual cycle in genus zero. In the nonconcave case of the so-called chain invertible polynomials, it yields a compatibility theorem with the FJRW virtual cycle and a proof of mirror symmetry for FJRW theory.

#### Article information

Source
Duke Math. J., Volume 165, Number 13 (2016), 2461-2527.

Dates
Revised: 1 October 2015
First available in Project Euclid: 12 May 2016

https://projecteuclid.org/euclid.dmj/1463080786

Digital Object Identifier
doi:10.1215/00127094-3477235

Mathematical Reviews number (MathSciNet)
MR3546967

Zentralblatt MATH identifier
1354.14081

#### Citation

Guéré, Jérémy. A Landau–Ginzburg mirror theorem without concavity. Duke Math. J. 165 (2016), no. 13, 2461--2527. doi:10.1215/00127094-3477235. https://projecteuclid.org/euclid.dmj/1463080786

#### References

• [1] D. Abramovich and A. Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), 27–75.
• [2] P. Berglund and T. Hübsch, A generalized construction of mirror manifolds, Nuclear Phys. B 393 (1993), 377–391.
• [3] H.-L. Chang, J. Li, and W.-P. Li, Witten’s top Chern class via cosection localization, Invent. Math. 200 (2015), 1015–1063.
• [4] A. Chiodo, The Witten top Chern class via $K$-theory, J. Algebraic Geom. 15 (2006), 681–707.
• [5] A. Chiodo, Towards an enumerative geometry of the moduli space of twisted curves and $r$th roots, Compos. Math. 144 (2008), 1461–1496.
• [6] A. Chiodo, H. Iritani, and Y. Ruan, Landau–Ginzburg/Calabi–Yau correspondence, global mirror symmetry and Orlov equivalence, Publ. Math. Inst. Hautes Études Sci. 119 (2014), 127–216.
• [7] A. Chiodo and Y. Ruan, Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations, Invent. Math. 182 (2010), 117–165.
• [8] A. Chiodo and Y. Ruan, A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence, Ann. Inst. Fourier (Grenoble) 61 (2011), 2803–2864.
• [9] A. Chiodo and Y. Ruan, LG/CY correspondence: The state space isomorphism, Adv. Math. 227 (2011), 2157–2188.
• [10] A. Chiodo and D. Zvonkine, Twisted $r$-spin potential and Givental’s quantization, Adv. Theor. Math. Phys. 13 (2009), 1335–1369.
• [11] T. Coates, A. Corti, H. Iritani, and H.-H. Tseng, Computing genus-zero twisted Gromov–Witten invariants, Duke Math. J. 147 (2009), 377–438.
• [12] T. Coates, A. Gholampour, H. Iritani, Y. Jiang, P. Johnson, and C. Manolache, The quantum Lefschetz hyperplane principle can fail for positive orbifold hypersurfaces, Math. Res. Lett. 19 (2012), 997–1005.
• [13] T. Coates and A. Givental, Quantum Riemann–Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (2007), 15–53.
• [14] C. Faber, Program to compute intersections on the moduli space of pointed curves, preprint, math.stanford.edu/~vakil/programs/index.html (accessed 22 March 2016).
• [15] C. Faber, S. Shadrin, and D. Zvonkine, Tautological relations and the $r$-spin Witten conjecture, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 621–658.
• [16] H. Fan, T. Jarvis, and Y. Ruan, The Witten equation, mirror symmetry, and quantum singularity theory, Ann. of Math. (2) 178 (2013), 1–106.
• [17] H. Fan, T. Jarvis, and Y. Ruan, The Witten equation and its virtual fundamental cycle, preprint, arXiv:0712.4025v3 [math.AG].
• [18] H. Fan and Y. Shen, Quantum ring of singularity $X^{p}+XY^{q}$, Michigan Math. J. 62 (2013), 185–207.
• [19] W. Fulton and S. Lang, Riemann–Roch Algebra, Grundlehren Math. Wiss. 277 Springer, New York, 1985.
• [20] S. Gährs, “Picard–Fuchs equations of special one-parameter families of invertible polynomials” in Arithmetic and Geometry of K3 Surfaces and Calabi–Yau Threefolds, Fields Inst. Commun. 67, Springer, New York, 2013, 285–310.
• [21] A. Givental, “A mirror theorem for toric complete intersections” in Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progr. Math. 160, Birkhaüser, Boston, 1998, 141–175.
• [22] A. Givental, “Symplectic geometry of Frobenius structures” in Frobenius Manifolds (Bonn, 2002), Aspects Math. E36, Vieweg, Wiesbaden, 2004, 91–112.
• [23] M. L. Green, A new proof of the explicit Noether–Lefschetz theorem, J. Differential Geom. 27 (1988), 155–159.
• [24] M. A. Guest, From Quantum Cohomology to Integrable Systems, Oxf. Grad. Texts Math. 15, Oxford Univ. Press, Oxford, 2008.
• [25] M. Krawitz, FJRW rings and Landau–Ginzburg mirror symmetry, Ph.D. dissertation, University of Michigan, Ann Arbor, Mich., 2010.
• [26] M. Krawitz and Y. Shen, Landau–Ginzburg/Calabi–Yau correspondence of all genera for elliptic orbifold $p^{1}$, preprint, arXiv:1106.6270v1 [math.AG].
• [27] M. Kreuzer, The mirror map for invertible LG models, Phys. Lett. B 328 (1994), 312–318.
• [28] M. Kreuzer and H. Skarke, On the classification of quasihomogeneous functions, Comm. Math. Phys. 150 (1992), 137–147.
• [29] B. H. Lian, K. Liu, and S.-T. Yau, Mirror principle, I, Asian J. Math. 1 (1997), 729–763.
• [30] T. Mochizuki, The virtual class of the moduli stack of stable $r$-spin curves, Comm. Math. Phys. 264 (2006), 1–40.
• [31] D. R. Morrison, Picard–Fuchs equations and mirror maps for hypersurfaces, preprint, arXiv:hep-th/9111025v1.
• [32] A. Polishchuk and A. Vaintrob, Chern characters and Hirzebruch–Riemann–Roch formula for matrix factorizations, Duke Math. J. 161 (2012), 1863–1926.
• [33] A. Polishchuk and A. Vaintrob, Matrix factorizations and cohomological field theories, preprint, arXiv:1105.2903v4 [math.AG].
• [34] E. Witten, Phases of $N=2$ theories in two dimensions, Nuclear Phys. B 403 (1993), 159–222.
• [35] A. Zinger, Standard versus reduced genus-one Gromov–Witten invariants, Geom. Topol. 12 (2008), 1203–1241.