We carefully analyze the conditions for an abelian gauged linear σ-model to exhibit nontrivial IR behavior described by a nonsingular superconformal field theory determining a superstring vacuum. This is done without reference to a geometric phase, by associating singular behavior to a noncompact space of (semi-)classical vacua. We find that models determined by reflexive combinatorial data are nonsingular for generic values of their parameters. This condition has the pleasant feature that the mirror of a nonsingular gauged linear σ-model is another such model, but it is clearly too strong and we provide an example of a non-reflexive mirror pair. We discuss a weaker condition inspired by considering extremal transitions, which is also mirror symmetric and which we conjecture to be sufficient. We apply these ideas to extremal transitions and to understanding the way in which both Berglund-Hübsch mirror symmetry and the Vafa-Witten mirror orbifold with discrete torsion can be seen as special cases of the general combinatorial duality of gauged linear σ-models. In the former case we encounter an example showing that our weaker condition is still not necessary.
L. Borisov, Towards the mirror symmetry for Calabi-Yau complete intersections in Gorenstein toric Fano varieties, alg-geom/9310001.
I.M. Gelfand, M.M. Kapranov and A.V. Zelevinski, Discriminants, resultants and multidimensional determinants, Birkhäuser, Boston MA U.S.A. (1994).
D.A. Cox, J.B. Little and H.K. Schenck, Toric varities, Graduate Studies in Mathematics 124, Amer. Math. Soc., Providence RI U.S.A. (2010).
L.A. Borisov, L. Chen and G.G. Smith, The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18 (2005) 193 [math/0309229].
V. Batyrev and B. Nill, Combinatorial aspects of mirror symmetry, in Integer points in polyhedra — geometry, number theory, representation theory, algebra, optimization, statistics, Contemp. Math. 452, Amer. Math. Soc., Providence RI U.S.A. (2008), pg. 35 [math/0703456] [INSPIRE].
L.J. Billera, P. Filliman and B. Sturmfels, Constructions and complexity of secondary polytopes, Adv. Math. 83 (1990) 155.
D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities, in Algebra, arithmetic, and geometry: in honor of Yu.I. Manin. Vol. II, Progr. Math. 270, Birkhäuser, Boston MA U.S.A. (2009), pg. 503 [math/0503632].
A.G. Kuznetsov, Derived category of a cubic threefold and the variety V 14, Tr. Mat. Inst. Steklova 246 (2004) 183 [math/0303037].
D. Gepner, Exactly solvable string compactifications on manifolds of SU(N ) holonomy, Phys. Lett. B 199 (1987) 380 [INSPIRE].
N. Addington and P.S. Aspinwall, to appear.
M. Krawitz, FJRW rings and Landau-Ginzburg mirror symmetry, arXiv:0906.0796.
N. Addington, The derived category of the intersection of four quadrics, arXiv:0904.1764.
B.R. Greene and M.R. Plesser, Duality in Calabi-Yau moduli space, Nucl. Phys. B 338 (1990) 15 [INSPIRE].
A. Kuznetsov, Derived categories of quadric fibrations and intersections of quadrics, Adv. Math. 218 (2008) 1340 [math/0510670].
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
ArXiv ePrint: 1507.00301
About this article
Cite this article
Aspinwall, P.S., Plesser, M.R. General mirror pairs for gauged linear sigma models. J. High Energ. Phys. 2015, 29 (2015). https://doi.org/10.1007/JHEP11(2015)029
- Conformal Field Models in String Theory
- Superstring Vacua