# General mirror pairs for gauged linear sigma models

- 109 Downloads
- 1 Citations

## Abstract

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.

## Keywords

Conformal Field Models in String Theory Superstring Vacua## Notes

### **Open Access**

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.

## References

- [1]E. Witten,
*Phases of N*= 2*theories in two-dimensions*,*Nucl. Phys.***B 403**(1993) 159 [hep-th/9301042] [INSPIRE].CrossRefADSGoogle Scholar - [2]P.S. Aspinwall, B.R. Greene and D.R. Morrison,
*Calabi-Yau moduli space, mirror manifolds and space-time topology change in string theory*,*Nucl. Phys.***B 416**(1994) 414 [hep-th/9309097] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [3]D.R. Morrison and M.R. Plesser,
*Summing the instantons: quantum cohomology and mirror symmetry in toric varieties*,*Nucl. Phys.***B 440**(1995) 279 [hep-th/9412236] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [4]V.V. Batyrev,
*Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties*,*J. Alg. Geom.***3**(1994) 493 [alg-geom/9310003] [INSPIRE].zbMATHMathSciNetGoogle Scholar - [5]L. Borisov,
*Towards the mirror symmetry for Calabi-Yau complete intersections in Gorenstein toric Fano varieties*, alg-geom/9310001. - [6]D.R. Morrison and M.R. Plesser,
*Towards mirror symmetry as duality for two-dimensional Abelian gauge theories*,*Nucl. Phys. Proc. Suppl.***46**(1996) 177 [hep-th/9508107] [INSPIRE].zbMATHMathSciNetCrossRefADSGoogle Scholar - [7]
- [8]P.S. Aspinwall and B.R. Greene,
*On the geometric interpretation of N*= 2*superconformal theories*,*Nucl. Phys.***B 437**(1995) 205 [hep-th/9409110] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [9]A.C. Avram, M. Kreuzer, M. Mandelberg and H. Skarke,
*The web of Calabi-Yau hypersurfaces in toric varieties*,*Nucl. Phys.***B 505**(1997) 625 [hep-th/9703003] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [10]I.M. Gelfand, M.M. Kapranov and A.V. Zelevinski,
*Discriminants, resultants and multidimensional determinants*, Birkhäuser, Boston MA U.S.A. (1994).zbMATHCrossRefGoogle Scholar - [11]P. Berglund and T. Hubsch,
*A generalized construction of mirror manifolds*,*Nucl. Phys.***B 393**(1993) 377 [hep-th/9201014] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [12]C. Vafa and E. Witten,
*On orbifolds with discrete torsion*,*J. Geom. Phys.***15**(1995) 189 [hep-th/9409188] [INSPIRE].zbMATHMathSciNetCrossRefADSGoogle Scholar - [13]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).Google Scholar - [14]P.S. Aspinwall, I.V. Melnikov and M.R. Plesser, (0
*,*2)*elephants*,*JHEP***01**(2012) 060 [arXiv:1008.2156] [INSPIRE]. - [15]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]. - [16]P.S. Aspinwall and M.R. Plesser,
*Decompactifications and massless D-branes in hybrid models*,*JHEP***07**(2010) 078 [arXiv:0909.0252] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [17]E. Witten,
*Mirror manifolds and topological field theory*, in*Essays on mirror manifolds*, S.-T. Yau ed., International Press, (1992) [hep-th/9112056] [INSPIRE]. - [18]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]. - [19]B.R. Greene, D.R. Morrison and C. Vafa,
*A geometric realization of confinement*,*Nucl. Phys.***B 481**(1996) 513 [hep-th/9608039] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [20]N. Addington and P.S. Aspinwall,
*Categories of massless D-branes and del Pezzo surfaces*,*JHEP***07**(2013) 176 [arXiv:1305.5767] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [21]P. Candelas, E. Derrick and L. Parkes,
*Generalized Calabi-Yau manifolds and the mirror of a rigid manifold*,*Nucl. Phys.***B 407**(1993) 115 [hep-th/9304045] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [22]P.S. Aspinwall, B.R. Greene and D.R. Morrison,
*The monomial divisor mirror map*,*Internat. Math. Res. Notices***1993**319 [alg-geom/9309007] [INSPIRE]. - [23]A. Strominger,
*Massless black holes and conifolds in string theory*,*Nucl. Phys.***B 451**(1995) 96 [hep-th/9504090] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [24]L.J. Billera, P. Filliman and B. Sturmfels,
*Constructions and complexity of secondary polytopes*,*Adv. Math.***83**(1990) 155.zbMATHMathSciNetCrossRefGoogle Scholar - [25]K. Hori and J. Walcher,
*D-branes from matrix factorizations*,*Comptes Rendus Physique***5**(2004) 1061 [hep-th/0409204] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [26]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]. - [27]A.G. Kuznetsov,
*Derived category of a cubic threefold and the variety V*_{14},*Tr. Mat. Inst. Steklova***246**(2004) 183 [math/0303037]. - [28]D. Gepner,
*Exactly solvable string compactifications on manifolds of*SU(*N*)*holonomy*,*Phys. Lett.***B 199**(1987) 380 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [29]N. Addington and P.S. Aspinwall, to appear.Google Scholar
- [30]M. Krawitz,
*FJRW rings and Landau-Ginzburg mirror symmetry*, arXiv:0906.0796. - [31]L.A. Borisov,
*Berglund-Hübsch mirror symmetry via vertex algebras*,*Commun. Math. Phys.***320**(2013) 73 [arXiv:1007.2633] [INSPIRE].zbMATHCrossRefADSGoogle Scholar - [32]
- [33]D. Favero and T.L. Kelly,
*Toric mirror constructions and derived equivalence*, arXiv:1412.1354 [INSPIRE]. - [34]B.R. Greene and M.R. Plesser,
*Mirror manifolds: a brief review and progress report*, hep-th/9110014 [INSPIRE]. - [35]P.S. Aspinwall, D.R. Morrison and M. Gross,
*Stable singularities in string theory*,*Commun. Math. Phys.***178**(1996) 115 [hep-th/9503208] [INSPIRE].zbMATHMathSciNetCrossRefADSGoogle Scholar - [36]A. Caldararu, J. Distler, S. Hellerman, T. Pantev and E. Sharpe,
*Non-birational twisted derived equivalences in Abelian GLSMs*,*Commun. Math. Phys.***294**(2010) 605 [arXiv:0709.3855] [INSPIRE].zbMATHMathSciNetCrossRefADSGoogle Scholar - [37]N. Addington,
*The derived category of the intersection of four quadrics*, arXiv:0904.1764. - [38]B.R. Greene and M.R. Plesser,
*Duality in Calabi-Yau moduli space*,*Nucl. Phys.***B 338**(1990) 15 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar - [39]A. Kuznetsov,
*Derived categories of quadric fibrations and intersections of quadrics*,*Adv. Math.***218**(2008) 1340 [math/0510670]. - [40]M. Kreuzer and H. Skarke,
*On the classification of quasihomogeneous functions*,*Commun. Math. Phys.***150**(1992) 137 [hep-th/9202039] [INSPIRE].zbMATHMathSciNetCrossRefADSGoogle Scholar - [41]M. Kreuzer and H. Skarke,
*Complete classification of reflexive polyhedra in four-dimensions*,*Adv. Theor. Math. Phys.***4**(2002) 1209 [hep-th/0002240] [INSPIRE].MathSciNetGoogle Scholar