# Superconformal algebras for twisted connected sums and *G*_{2} mirror symmetry

Open Access

Regular Article - Theoretical PhysicsFirst Online:

- 45 Downloads
- 2 Citations

## Abstract

We realise the Shatashvili-Vafa superconformal algebra for *G*_{2} string compactifications by combining Odake and free conformal algebras following closely the recent mathematical construction of twisted connected sum *G*_{2} holonomy manifolds. By considering automorphisms of this realisation, we identify stringy analogues of two mirror maps proposed by Braun and Del Zotto for these manifolds.

## Keywords

Conformal Field Models in String Theory Conformal and W Symmetry Differential and Algebraic Geometry Download
to read the full article text

## 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]A. Kovalev,
*Twisted connected sums and special Riemannian holonomy*,*J. Reine Angew. Math.***565**(2003) 125.MathSciNetzbMATHGoogle Scholar - [2]A. Corti, M. Haskins, J. Nordström and T. Pacini,
*Asymptotically cylindrical Calabi-Yau 3-folds from weak Fano 3-folds*,*Geom. Topol.***17**(2013) 1955.MathSciNetCrossRefzbMATHGoogle Scholar - [3]A. Corti, M. Haskins, J. Nordström and T. Pacini, G
_{2}*-manifolds and associative submanifolds via semi-Fano*3*-folds*,*Duke Math. J.***164**(2015) 1971 [arXiv:1207.4470] [INSPIRE]. - [4]J. Halverson and D.R. Morrison,
*The landscape of M-theory compactifications on seven-manifolds with G*_{2}*holonomy*,*JHEP***04**(2015) 047 [arXiv:1412.4123] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [5]J. Halverson and D.R. Morrison,
*On gauge enhancement and singular limits in G*_{2}*compactifications of M-theory*,*JHEP***04**(2016) 100 [arXiv:1507.05965] [INSPIRE]. - [6]A.P. Braun,
*Tops as building blocks for G*_{2}*manifolds*,*JHEP***10**(2017) 083 [arXiv:1602.03521] [INSPIRE].ADSCrossRefGoogle Scholar - [7]T.C. da C. Guio, H. Jockers, A. Klemm and H.-Y. Yeh,
*Effective action from m-theory on twisted connected sum G*_{2}*-manifolds*,*Commun. Math. Phys.***359**(2018) 535 [arXiv:1702.05435] [INSPIRE]. - [8]A.P. Braun and M. Del Zotto,
*Mirror symmetry for G*_{2}*-manifolds: twisted connected sums and dual tops*,*JHEP***05**(2017) 080 [arXiv:1701.05202] [INSPIRE]. - [9]A.P. Braun and M. Del Zotto,
*Towards generalized mirror symmetry for twisted connected sum G*_{2}*manifolds*,*JHEP***03**(2018) 082 [arXiv:1712.06571] [INSPIRE]. - [10]A.P. Braun and S. Schäfer-Nameki,
*Compact, singular G*_{2}*-holonomy manifolds and M/Heterotic/F-theory duality*,*JHEP***04**(2018) 126 [arXiv:1708.07215] [INSPIRE]. - [11]A.P. Braun and S. Schäfer-Nameki, Spin(7)
*-manifolds as generalized connected sums and*3*D*\( \mathcal{N}=1 \)*theories*,*JHEP***06**(2018) 103 [arXiv:1803.10755] [INSPIRE]. - [12]A.P. Braun et al.,
*Infinitely many M*2*-instanton corrections to M -theory on G*_{2}*-manifolds*,*JHEP***09**(2018) 077 [arXiv:1803.02343] [INSPIRE]. - [13]D.D. Joyce,
*Compact Riemannian*7*-manifolds with holonomy G*_{2}*. I, II*,*J. Differential Geom.***43**(1996) 291.Google Scholar - [14]D.D. Joyce,
*Compact manifolds with special holonomy*, Oxford University Press, Oxford U.K. (2000).Google Scholar - [15]S.L. Shatashvili and C. Vafa,
*Superstrings and manifold of exceptional holonomy*,*Selecta Math.***1**(1995) 347 [hep-th/9407025] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar - [16]G. Papadopoulos and P.K. Townsend,
*Compactification of D*= 11*supergravity on spaces of exceptional holonomy*,*Phys. Lett.***B 357**(1995) 300 [hep-th/9506150] [INSPIRE]. - [17]K. Becker et al.,
*Supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau*4*folds*,*Nucl. Phys.***B 480**(1996) 225 [hep-th/9608116] [INSPIRE]. - [18]B.S. Acharya,
*Dirichlet Joyce manifolds, discrete torsion and duality*,*Nucl. Phys.***B 492**(1997) 591 [hep-th/9611036] [INSPIRE]. - [19]J.M. Figueroa-O’Farrill,
*A note on the extended superconformal algebras associated with manifolds of exceptional holonomy*,*Phys. Lett.***B 392**(1997) 77 [hep-th/9609113] [INSPIRE]. - [20]B.S. Acharya,
*On mirror symmetry for manifolds of exceptional holonomy*,*Nucl. Phys.***B 524**(1998) 269 [hep-th/9707186] [INSPIRE]. - [21]R. Blumenhagen and V. Braun,
*Superconformal field theories for compact G*_{2}*manifolds*,*JHEP***12**(2001) 006 [hep-th/0110232] [INSPIRE]. - [22]S. Gukov, S.-T. Yau and E. Zaslow,
*Duality and fibrations on G*_{2}*manifolds*, hep-th/0203217 [INSPIRE]. - [23]R. Roiban, C. Romelsberger and J. Walcher,
*Discrete torsion in singular G*_{2}*manifolds and real LG*,*Adv. Theor. Math. Phys.***6**(2003) 207 [hep-th/0203272] [INSPIRE]. - [24]T. Eguchi and Y. Sugawara,
*String theory on G*_{2}*manifolds based on Gepner construction*,*Nucl. Phys.***B 630**(2002) 132 [hep-th/0111012] [INSPIRE]. - [25]B. Noyvert,
*Unitary minimal models of SW*(3*/*2*,*3*/*2*,*2)*superconformal algebra and manifolds of G*_{2}*holonomy*,*JHEP***03**(2002) 030 [hep-th/0201198] [INSPIRE]. - [26]K. Sugiyama and S. Yamaguchi,
*Cascade of special holonomy manifolds and heterotic string theory*,*Nucl. Phys.***B 622**(2002) 3 [hep-th/0108219] [INSPIRE]. - [27]K. Sugiyama and S. Yamaguchi,
*Coset construction of noncompact*Spin(7)*and G*_{2}*CFTs*,*Phys. Lett.***B 538**(2002) 173 [hep-th/0204213] [INSPIRE]. - [28]T. Eguchi, Y. Sugawara and S. Yamaguchi,
*Supercoset CFT’s for string theories on noncompact special holonomy manifolds*,*Nucl. Phys.***B 657**(2003) 3 [hep-th/0301164] [INSPIRE]. - [29]M.R. Gaberdiel and P. Kaste,
*Generalized discrete torsion and mirror symmetry for G*_{2}*manifolds*,*JHEP***08**(2004) 001 [hep-th/0401125] [INSPIRE]. - [30]J. de Boer, A. Naqvi and A. Shomer,
*The topological G*_{2}*string*,*Adv. Theor. Math. Phys.***12**(2008) 243 [hep-th/0506211] [INSPIRE]. - [31]S. Odake,
*Extension of N*= 2*superconformal algebra and Calabi-Yau compactification*,*Mod. Phys. Lett.***A 4**(1989) 557 [INSPIRE]. - [32]S. Grigorian,
*Moduli spaces of G*_{2}*manifolds*,*Rev. Math. Phys.***22**(2010) 1061 [arXiv:0911.2185] [INSPIRE]. - [33]R. Blumenhagen,
*Covariant construction of N*= 1*superW algebras*,*Nucl. Phys.***B 381**(1992) 641 [INSPIRE]. - [34]R. Blumenhagen, W. Eholzer, A. Honecker and R. Hubel,
*New N*= 1*extended superconformal algebras with two and three generators*,*Int. J. Mod. Phys.***A 7**(1992) 7841 [hep-th/9207072] [INSPIRE]. - [35]P. Bouwknegt and K. Schoutens,
*W symmetry in conformal field theory*,*Phys. Rept.***223**(1993) 183 [hep-th/9210010] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [36]P.S. Howe and G. Papadopoulos,
*Holonomy groups and W symmetries*,*Commun. Math. Phys.***151**(1993) 467 [hep-th/9202036] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [37]X. De La Ossa and M.-A. Fiset, G
*-structure symmetries and anomalies in*(1*,*0)*non-linear σ-models*, arXiv:1809.01138 [INSPIRE]. - [38]K. Becker, D. Robbins and E. Witten,
*The α*^{′}*Expansion On A Compact Manifold Of Exceptional Holonomy*,*JHEP***06**(2014) 051 [arXiv:1404.2460] [INSPIRE]. - [39]P. Deligne et al.,
*Quantum fields and strings: a course for mathematicians. Volume 2*, American Mathematical Society, U.S.A. (1999).Google Scholar - [40]A. Ali,
*Classification of two-dimensional N*= 4*superconformal symmetries*, hep-th/9906096 [INSPIRE]. - [41]M. Ademollo et al.,
*Dual string models with nonabelian color and flavor symmetries*,*Nucl. Phys.***B 114**(1976) 297 [INSPIRE]. - [42]M. Ademollo et al.,
*Supersymmetric strings and color confinement*,*Phys. Lett.***B 62**(1976) 105.Google Scholar - [43]W. Lerche, C. Vafa and N.P. Warner,
*Chiral rings in N*= 2*superconformal theories*,*Nucl. Phys.***B 324**(1989) 427 [INSPIRE]. - [44]C. Vafa and E. Witten,
*On orbifolds with discrete torsion*,*J. Geom. Phys.***15**(1995) 189 [hep-th/9409188] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [45]A. Strominger, S.-T. Yau and E. Zaslow,
*Mirror symmetry is T duality*,*Nucl. Phys.***B 479**(1996) 243 [hep-th/9606040] [INSPIRE]. - [46]I.V. Melnikov, R. Minasian and S. Sethi,
*Spacetime supersymmetry in low-dimensional perturbative heterotic compactifications*,*Fortsch. Phys.***66**(2018) 1800027 [arXiv:1707.04613] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [47]M.-A. Fiset, C. Quigley and E.E. Svanes,
*Marginal deformations of heterotic G*_{2}*σ-models*,*JHEP***02**(2018) 052 [arXiv:1710.06865] [INSPIRE]. - [48]R. Roiban and J. Walcher,
*Rational conformal field theories with G*_{2}*holonomy*,*JHEP***12**(2001) 008 [hep-th/0110302] [INSPIRE]. - [49]V.G. Kac,
*Vertex algebras for beginners*, American Mathematical Society, U.S.A. (1997).Google Scholar - [50]E. Frenkel and D. Ben-Zvi,
*Vertex algebras and algebraic curves*, American Mathematical Society, U.S.A. (2001).Google Scholar - [51]K. Thielemans,
*An algorithmic approach to operator product expansions, W algebras and W strings*, Ph.D. thesis, Leuven U., 1994. hep-th/9506159 [INSPIRE]. - [52]K. Thielemans,
*A Mathematica package for computing operator product expansions*,*Int. J. Mod. Phys.***C 2**(1991) 787 [INSPIRE]. - [53]C. Quigley,
*Mirror symmetry in physics: the basics*,*Fields Inst. Monogr.***34**(2015) 211 [arXiv:1412.8180].MathSciNetCrossRefzbMATHGoogle Scholar - [54]
- [55]B.R. Greene,
*String theory on Calabi-Yau manifolds*, in the proceedings of*Fields, strings and duality. Theoretical Advanced Study Institute in Elementary Particle Physics (TASI’96)*, June 2-28, Boulder, U.S.A. (1996), hep-th/9702155 [INSPIRE].

## Copyright information

© The Author(s) 2018