Advances in Theoretical and Mathematical Physics

On $\mathcal{N} = 1$ 4d effective couplings for F-theory and heterotic vacua

Hans Jockers, Peter Mayr, and Johannes Walcher

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Abstract

We show that certain superpotential and Kähler potential couplings of $\mathcal{N} = 1$ supersymmetric compactifications with branes or bundles can be computed from Hodge theory and mirror symmetry. This applies to F-theory on a Calabi–Yau four-fold and three-fold compactifications of type II and heterotic strings with branes. The heterotic case includes a class of bundles on elliptic manifolds constructed by Friedmann, Morgan and Witten. Mirror symmetry of the four-fold computes non-perturbative corrections to mirror symmetry on the three-folds, including D-instanton corrections. We also propose a physical interpretation for the observation byWarner that relates the deformation spaces of certain matrix factorizations and the periods of non-compact four-folds that are ALE fibrations.

Article information

Source
Adv. Theor. Math. Phys., Volume 14, Number 5 (2010), 1433-1514.

Dates
First available in Project Euclid: 21 September 2011

Permanent link to this document
https://projecteuclid.org/euclid.atmp/1316638443

Mathematical Reviews number (MathSciNet)
MR2826186

Zentralblatt MATH identifier
1251.81076

Citation

Jockers, Hans; Mayr, Peter; Walcher, Johannes. On $\mathcal{N} = 1$ 4d effective couplings for F-theory and heterotic vacua. Adv. Theor. Math. Phys. 14 (2010), no. 5, 1433--1514. https://projecteuclid.org/euclid.atmp/1316638443


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References

  • [1] E. Witten, New issues in manifolds of SU(3) holonomy, Nucl. Phys. B268 (1986), 79.
  • [2] E.Witten, Chern–Simons gauge theory as a string theory, Prog. Math. 133 (1995), 637.
  • [3] T. R. Taylor and C. Vafa, RR flux on Calabi–Yau and partial supersymmetry breaking, Phys. Lett. B474 (2000), 130.
  • [4] A. Strominger, Superstrings with torsion, Nucl. Phys. B274 (1986), 253.
  • [5] K. Hori and C. Vafa, Mirror symmetry.
  • [6] C. Vafa, Extending mirror conjecture to Calabi-Yau with bundles.
  • [7] M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs.
  • [8] W. Lerche, P. Mayr and N. Warner, N = 1 special geometry, mixed Hodge variations and toric geometry. Holomorphic N = 1 special geometry of open–closed type II strings, arXiv:hep-th/0207259.
  • [9] H. Jockers and M. Soroush, Effective superpotentials for compact D5-brane Calabi-Yau geometries, Commun. Math. Phys. 290 (2009), 249.
  • [10] M. Alim, M. Hecht, P. Mayr and A. Mertens, Mirror symmetry for toric branes on compact hypersurfaces, JHEP 0909 (2009), 126.
  • [11] M. Alim, M. Hecht, H. Jockers, P. Mayr, A. Mertens and M. Soroush, Hints for off-shell mirror symmetry in type II/F-theory compactifications, Nucl. Phys. B841 (2010), 303–338.
  • [12] R. Friedman, J. Morgan and E. Witten, Vector bundles and F theory, Commun. Math. Phys. 187 (1997), 679.
  • [13] M. Bershadsky, A. Johansen, T. Pantev and V. Sadov, On fourdimensional compactifications of F-theory, Nucl. Phys. B505 (1997), 165.
  • [14] C. Vafa, Evidence for F-theory, Nucl. Phys. B469 (1996), 403.
  • [15] D. R. Morrison and C. Vafa, Compactifications of F-theory on Calabi–Yau threefolds – I, Nucl. Phys. B473 (1996), 74. Compactifications of F-theory on Calabi–Yau threefolds – II, Nucl. Phys. B476 (1996), 437.
  • [16] P. Mayr, N = 1 mirror symmetry and open/closed string duality, Adv. Theor. Math. Phys. 5 (2002), 213.
  • [17] M. Aganagic and C. Beem, The geometry of D-brane superpotentials,.
  • [18] W. Lerche and P. Mayr, On N = 1 mirror symmetry for open type II strings.
  • [19] T. W. Grimm, T. W. Ha, A. Klemm and D. Klevers, Computing brane and flux superpotentials in F-theory compactifications, JHEP 1004 (2010), 015.
  • [20] S. Li, B. H. Lian and S. T. Yau, Picard–Fuchs equations for relative periods and Abel–Jacobi map for Calabi–Yau hypersurfaces. [math.AG].
  • [21] H. Ooguri and C. Vafa, Two-dimensional black hole and singularities of CY manifolds, Nucl. Phys. B463 (1996), 55.
  • [22] S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4D N = 2 gauge theories. I, Adv. Theor. Math. Phys. 1 (1998), 53.
  • [23] P. Berglund and P. Mayr, Heterotic string/F-theory duality from mirror symmetry, Adv. Theor. Math. Phys. 2 (1999), 1307.
  • [24] S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau fourfolds, Nucl. Phys. B 584 (2000), 69 [Erratum-ibid. B608 (2001), 477].
  • [25] T. Eguchi, N. P. Warner and S. K. Yang, ADE singularities and coset models, Nucl. Phys. B607 (2001), 3.
  • [26] J. Knapp and H. Omer, Matrix factorizations, minimal models and Massey products, JHEP 0605 (2006), 064.
  • [27] E. Witten, Branes and the dynamics of QCD, Nucl. Phys. B507 (1997), 658.
  • [28] S. Kachru, S. H. Katz, A. E. Lawrence and J. McGreevy, Open string instantons and superpotentials, Phys. Rev. D62 (2000), 026001. Mirror symmetry for open strings, Phys. Rev. D62 (2000), 126005.
  • [29] J. Walcher, Opening mirror symmetry on the quintic, Commun. Math. Phys. 276 (2007), 671.
  • [30] D. R. Morrison and J. Walcher, D-branes and normal functions,.
  • [31] G. Curio and R. Y. Donagi, Moduli in N = 1 heterotic/F-theory duality, Nucl. Phys. B518 (1998), 603.
  • [32] P. S. Aspinwall, Aspects of the hypermultiplet moduli space in string duality, JHEP 9804 (1998), 019.
  • [33] D. Lüst, P. Mayr, S. Reffert and S. Stieberger, F-theory flux, destabilization of orientifolds and soft terms on D7-branes, Nucl. Phys. B732 (2006), 243.
  • [34] H. Jockers and J. Louis, D-terms and F-terms from D7-brane fluxes, Nucl. Phys. B718 (2005), 203.
  • [35] R. Donagi and M. Wijnholt, Model building with F-theory,.
  • [36] P. Berglund and P. Mayr, Non-perturbative superpotentials in F-theory and string duality.
  • [37] A. Grothendieck, La théorie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137–154.
  • [38] R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom. 54(2) (2000), 367.
  • [39] P. S. Aspinwall and D. R. Morrison, Point-like instantons on K3 orbifolds, Nucl. Phys. B503 (1997), 533.
  • [40] C. Vafa, Superstrings and topological strings at large N, J. Math. Phys. 42 (2001), 2798.
  • [41] S. B. Giddings, S. Kachru and J. Polchinski, Hierarchies from fluxes in string compactifications, Phys. Rev. D66 (2002), 106006.
  • [42] K. Becker, M. Becker, C. Vafa and J. Walcher, Moduli stabilization in non-geometric backgrounds, Nucl. Phys. B770 (2007), 1.
  • [43] M. R. Douglas, Effective potential and warp factor dynamics,.
  • [44] M. Haack, J. Louis and M. Marquart, Type IIA and heterotic string vacua in D = 2, Nucl. Phys. B598 (2001), 30.
  • [45] C. Vafa and E.Witten, Dual string pairs with N = 1 and N = 2 supersymmetry in four dimensions, Nucl. Phys. Proc. Suppl. 46 (1996), 225.
  • [46] A. Sen, Orientifold limit of F-theory vacua, Nucl. Phys. Proc. Suppl. 68 (1998), 92 [Nucl. Phys. Proc. Suppl. 67 (1998), 81].
  • [47] A. Sen and S. Sethi, The mirror transform of type I vacua in six dimensions, Nucl. Phys. B499 (1997), 45.
  • [48] B. Andreas, G. Curio, D. Hernandez Ruiperez and S. T. Yau, Fibrewise T-duality for D-branes on elliptic Calabi-Yau, JHEP 0103 (2001), 020.
  • [49] N. Seiberg and E.Witten, Comments on string dynamics in six dimensions, Nucl. Phys. B471 (1996), 121.
  • [50] S. Gukov and M. Haack, IIA string theory on Calabi-Yau fourfolds with background fluxes, Nucl. Phys. B639 (2002), 95.
  • [51] N. Halmagyi, I. V. Melnikov and S. Sethi, Instantons, hypermultiplets and the heterotic string, JHEP 0707 (2007), 086.
  • [52] L. Andrianopoli, R. D’Auria, S. Ferrara and M. A. Lledo, JHEP 0303 (2003), 044. C. Angelantonj, R. D’Auria, S. Ferrara and M. Trigiante, K3 × T2/Z2 orientifolds with fluxes, open string moduli and critical points, Phys. Lett. B583 (2004), 331.
  • [53] N. Nekrasov, H. Ooguri and C. Vafa, S-duality and topological strings, JHEP 0410 (2004), 009.
  • [54] R. D’Auria, S. Ferrara, M. Trigiante and S. Vaula, Gauging the Heisenberg algebra of special quaternionic manifolds, Phys. Lett. B610 (2005), 147.
  • [55] M. Graña, J. Louis and D. Waldram, SU(3) x SU(3) compactification and mirror duals of magnetic fluxes, JHEP 0704 (2007), 101.
  • [56] P. S. Aspinwall and J. Louis, On the ubiquity of K3 fibrations in string duality, Phys. Lett. B369 (1996), 233.
  • [57] P. S. Aspinwall, An analysis of fluxes by duality.
  • [58] E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B443 (1995), 85.
  • [59] E. Perevalov and G. Rajesh, Mirror symmetry via deformation of bundles on K3 surfaces, Phys. Rev. Lett. 79 (1997), 2931.
  • [60] C. M. Hull, Compactifications of the heterotic superstring, Phys. Lett. B178 (1986), 357.
  • [61] I. Bars, D. Nemeschansky and S. Yankielowicz, Compactified superstrings and torsion, Nucl. Phys. B278 (1986), 632.
  • [62] K. Becker, M. Becker, K. Dasgupta and P. S. Green, Compactifications of heterotic theory on non-Kähler complex manifolds. I, JHEP 0304 (2003), 007. K. Becker, M. Becker, P. S. Green, K. Dasgupta and E. Sharpe, Compactifications of heterotic strings on non-Kähler complex manifolds. II, Nucl. Phys. B678 (2004), 19.
  • [63] G. Lopes Cardoso, G. Curio, G. Dall’Agata and D. Lüst, Heterotic string theory on non-Kähler manifolds with H-flux and gaugino condensate, Fortsch. Phys. 52 (2004), 483.
  • [64] S. Gurrieri, A. Lukas and A. Micu, Heterotic on half-flat, Phys. Rev. D 70 (2004), 126009. Heterotic string compactifications on half-flat manifolds II, JHEP 0712 (2007), 081.
  • [65] I. Benmachiche, J. Louis and D. Martinez-Pedrera, The effective action of the heterotic string compactified on manifolds with SU(3) structure, Class. Quant. Grav. 25 (2008), 135006.
  • [66] K. Becker, M. Becker, K. Dasgupta and S. Prokushkin, Properties of heterotic vacua from superpotentials, Nucl. Phys. B666 (2003), 144.
  • [67] G. Lopes Cardoso, G. Curio, G. Dall’Agata and D. Lüst, BPS action and superpotential for heterotic string compactifications with fluxes, JHEP 0310 (2003), 004.
  • [68] D. Andriot, R. Minasian and M. Petrini, Flux backgrounds from Twist duality, JHEP 0912 (2009), 028.
  • [69] K. Dasgupta, G. Rajesh and S. Sethi, M theory, orientifolds and Gflux, JHEP 9908 (1999), 023.
  • [70] J. X. Fu and S. T. Yau, Existence of supersymmetric Hermitian metrics with torsion on non-Kaehler manifolds, arXiv:hep-th/ 0509028.
  • [71] K. Becker, M. Becker, J. X. Fu, L. S. Tseng and S. T. Yau, Anomaly cancellation and smooth non-Kähler solutions in heterotic string theory, Nucl. Phys. B751 (2006), 108.
  • [72] R. Bott and L. W. Tu, Differential forms in algebraic topology, Springer-Verlag, Berlin.
  • [73] G. Rajesh, Toric geometry and F-theory/heterotic duality in four dimensions, JHEP 9812 (1998), 018.
  • [74] R. Donagi, A. Lukas, B. A. Ovrut and D. Waldram, Non-perturbative vacua and particle physics in M-theory, JHEP 9905 (1999), 018.
  • [75] S. J. Gates, M. T. Grisaru and M. E. Wehlau, A study of general 2D, N = 2 matter coupled to supergravity in superspace, Nucl. Phys. B460 (1996), 579.
  • [76] E. Witten, Phases of N = 2 theories in two dimensions, Nucl. Phys. B403 (1993), 159.
  • [77] B. de Wit, M. T. Grisaru, E. Rabinovici and H. Nicolai, Two loop finiteness of D = 2 supergravity, Phys. Lett. B286 (1992), 78.
  • [78] W. Lerche, Fayet-Iliopoulos potentials from four-folds, JHEP 9711 (1997), 004.
  • [79] P. Mayr, Mirror symmetry, N = 1 superpotentials and tensionless strings on Calabi-Yau four-folds, Nucl. Phys. B494 (1997), 489.
  • [80] H. Jockers and J. Louis, The effective action of D7-branes in N = 1 Calabi-Yau orientifolds, Nucl. Phys. B705 (2005), 167.
  • [81] R. Pandharipande, J. Solomon and J. Walcher, Disk enumeration on the quintic 3-fold, J. Amer. Math. Soc. 21 (2008), 1169–1209 [arXiv.org:math/0610901].
  • [82] V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebr. Geom. 3 (1994), 493.
  • [83] B. R. Greene, D. R. Morrison and M. R. Plesser, Mirror manifolds in higher dimension, Commun. Math. Phys. 173 (1995), 559.
  • [84] A. Klemm, B. Lian, S. S. Roan and S. T. Yau, Calabi-Yau fourfolds for M- and F-theory compactifications, Nucl. Phys. B518 (1998), 515.
  • [85] P. Candelas, X. C. De La Ossa, P. S. Green and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B359 (1991), 21.
  • [86] H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B577 (2000), 419.
  • [87] B. de Wit, V. Kaplunovsky, J. Louis and D. Lüst, Perturbative couplings of vector multiplets in N = 2 heterotic string vacua, Nucl. Phys. B451 (1995), 53. I. Antoniadis, S. Ferrara, E. Gava, K. S. Narain and T. R. Taylor, Perturbative prepotential and monodromies in N = 2 heterotic superstring, Nucl. Phys. B447 (1995), 35.
  • [88] P. S. Aspinwall and M. R. Plesser, T-duality can fail, JHEP 9908 (1999), 001.
  • [89] M. Aganagic, A. Klemm and C. Vafa, Disk instantons, mirror symmetry and the duality web, Z. Naturforsch. A57 (2002), 1.
  • [90] M. Bershadsky, K. A. Intriligator, S. Kachru, D. R. Morrison, V. Sadov and C. Vafa, Geometric singularities and enhanced gauge symmetries, Nucl. Phys. B481 (1996), 215.
  • [91] T. Graber and E. Zaslow, Open string Gromov–Witten invariants: calculations and a mirror ‘theorem’.
  • [92] M. Roček, C. Vafa and S. Vandoren, Hypermultiplets and topological strings, JHEP 0602 (2006), 062. D. Robles-Llana, F. Saueressig and S. Vandoren, String loop corrected hypermultiplet moduli spaces, JHEP 0603 (2006), 081. D. Robles-Llana, M. Roček, F. Saueressig, U. Theis and S. Vandoren, Nonperturbative corrections to 4D string theory effective actions from SL(2, Z) duality and supersymmetry, Phys. Rev. Lett. 98 (2007), 211602. S. Alexandrov, B. Pioline, F. Saueressig and S. Vandoren, D-instantons and twistors, JHEP 0903 (2009), 044.
  • [93] E. Witten, Heterotic string conformal field theory and A-D-E singularities, JHEP 0002 (2000), 025.
  • [94] P. S. Aspinwall and M. R. Plesser, Heterotic string corrections from the dual type II string, JHEP 0004 (2000), 025.
  • [95] P. Mayr, Conformal field theories on K3 and three-dimensional gauge theories, JHEP 0008 (2000), 042.
  • [96] K. Hori, H. Ooguri and C. Vafa, Non-Abelian conifold transitions and N = 4 dualities in three dimensions, Nucl. Phys. B504 (1997), 147.
  • [97] W. Lerche, C. Vafa and N. P. Warner, Chiral rings in N = 2 superconformal theories, Nucl. Phys. B324 (1989), 427.
  • [98] Y. Kazama and H. Suzuki, New N = 2 superconformal field theories and superstring compactification, Nucl. Phys. B321 (1989), 232.
  • [99] M. Herbst, C. I. Lazaroiu and W. Lerche, Superpotentials, A(infinity) relations and WDVV equations for open topological strings, JHEP 0502 (2005), 071.
  • [100] C. Curto, D. R. Morrison, Threefold flops via matrix factorizations.
  • [101] J. Distler and S. Kachru, (0,2) Landau–Ginzburg theory, Nucl. Phys. B413 (1994), 213.
  • [102] S. Govindarajan, T. Jayaraman and T. Sarkar, Worldsheet approaches to D-branes on supersymmetric cycles, Nucl. Phys. B580 (2000), 519.
  • [103] H. Jockers and M. Soroush, Relative periods and open-string integer invariants for a compact Calabi-Yau hypersurface, Nucl. Phys. B821 (2009), 535.
  • [104] T. W. Grimm, T. W. Ha, A. Klemm and D. Klevers, The D5-brane effective action and superpotential in N = 1 compactifications, Nucl. Phys. B816 (2009), 139.
  • [105] S. Govindarajan and T. Jayaraman, Boundary fermions, coherent sheaves and D-branes on Calabi-Yau manifolds, Nucl. Phys. B618 (2001), 50.
  • [106] L. B. Anderson, J. Gray, D. Grayson, Y. H. He and A. Lukas, Yukawa couplings in heterotic compactification, Commun. Math. Phys. 297 (2010), 95–127. Y. H. He, S. J. Lee and A. Lukas, Heterotic models from vector bundles on toric Calabi-Yau manifolds, JHEP 1005 (2010), 071. L. B. Anderson, J. Gray, Y. H. He and A. Lukas, Exploring positive monad bundles and a new heterotic standard model, JHEP 1002 (2010), 054.
  • [107] K. Hori and J. Walcher, F-term equations near Gepner points, JHEP 0501 (2005), 008.
  • [108] S. K. Ashok, E. Dell’Aquila, D. E. Diaconescu and B. Florea, Obstructed D–branes in Landau–Ginzburg orbifolds, Adv. Theor. Math. Phys. 8 (2004), 427.
  • [109] P. S. Aspinwall and S. H. Katz, Computation of superpotentials for D-branes, Commun. Math. Phys. 264 (2006), 227.
  • [110] M. Baumgartl, I. Brunner and M. R. Gaberdiel, D-brane superpotentials and RG flows on the quintic, JHEP 0707 (2007), 061.
  • [111] H. Jockers and W. Lerche, Matrix factorizations, D-branes and their deformations, Nucl. Phys. Proc. Suppl. 171 (2007), 196.
  • [112] J. Knapp and E. Scheidegger, Matrix factorizations, massey products and F-terms for two-parameter Calabi-Yau hypersurfaces,.
  • [113] A. C. Avram, M. Kreuzer, M. Mandelberg and H. Skarke, Searching for K3 fibrations, Nucl. Phys. B494 (1997), 567. M. Kreuzer and H. Skarke, Calabi-Yau 4-folds and toric fibrations, J. Geom. Phys. 26 (1998), 272.
  • [114] G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 3.0.1 — a computer algebra system for polynomial computations, Center for Computer Algebra, University of Kaiserslautern, 2006; http://www.singular.uni-kl.de.
  • [115] P. Candelas and A. Font, Duality between the webs of heterotic and type II vacua, Nucl.Phys. B511 (1998), 295.
  • [116] P. Candelas, E. Perevalov and G. Rajesh, Toric geometry and enhanced gauge symmetry of F-theory/heterotic vacua, Nucl. Phys. B507 (1997), 445.