Abstract
We investigate the dynamics of space-time filling five-branes wrapped on curves in heterotic and orientifold Calabi-Yau compactifications. We first study the leading \( \mathcal{N} = 1 \) scalar potential on the infinite deformation space of the brane-curve around a supersymmetric configuration. The higher order potential is also determined by a brane superpotential which we compute for a subset of light deformations. We argue that these deformations map to new complex structure deformations of a non-Calabi-Yau manifold which is obtained by blowing up the brane-curve into a four-cycle and by replacing the brane by background fluxes. This translates the original brane-bulk system into a unifying geometrical formulation. Using this blow-up geometry we compute the complete set of open-closed Picard-Fuchs differential equations and identify the brane superpotential at special points in the field space for five-branes in toric Calabi-Yau hypersurfaces. This has an interpretation in open mirror symmetry and enables us to list compact disk instanton invariants. As a first step towards promoting the blow-up geometry to a supersymmetric heterotic background we propose a non-Kähler SU(3) structure and an identification of the three-form flux.
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R. Blumenhagen, M. Cvetič, P. Langacker and G. Shiu, Toward realistic intersecting D-brane models, Ann. Rev. Nucl. Part. Sci. 55 (2005) 71 [hep-th/0502005] [SPIRES].
D. Lüst, Intersecting brane worlds: A path to the standard model?, Class. Quant. Grav. 21 (2004) S1399 [hep-th/0401156] [SPIRES].
R. Blumenhagen, B. Körs, D. Lüst and S. Stieberger, Four-dimensional String Compactifications with D-branes, Orientifolds and Fluxes, Phys. Rept. 445 (2007) 1 [hep-th/0610327] [SPIRES].
M.R. Douglas and S. Kachru, Flux compactification, Rev. Mod. Phys. 79 (2007) 733 [hep-th/0610102] [SPIRES].
F. Denef, Les Houches Lectures on Constructing String Vacua, arXiv:0803.1194 [SPIRES].
J. Simons, Minimal varieties in Riemannian manifolds, Ann. Math. 88 (1968) 62.
R. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998) 705.
E. Witten, Branes and the dynamics of QCD, Nucl. Phys. B 507 (1997) 658 [hep-th/9706109] [SPIRES].
M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041 [SPIRES].
M. Aganagic, A. Klemm and C. Vafa, Disk instantons, mirror symmetry and the duality web, Z. Naturforsch. A 57 (2002) 1 [hep-th/0105045] [SPIRES].
P. Mayr, N = 1 mirror symmetry and open/closed string duality, Adv. Theor. Math. Phys. 5 (2002) 213 [hep-th/0108229] [SPIRES].
W. Lerche and P. Mayr, On N = 1 mirror symmetry for open type-II strings, hep-th/0111113 [SPIRES].
W. Lerche, P. Mayr and N. Warner, Holomorphic N = 1 special geometry of open-closed type-II strings, hep-th/0207259 [SPIRES].
W. Lerche, P. Mayr and N. Warner, N = 1 special geometry, mixed Hodge variations and toric geometry, hep-th/0208039 [SPIRES].
J. Walcher, Calculations for Mirror Symmetry with D-branes, JHEP 09 (2009) 129 [arXiv:0904.4905] [SPIRES].
D. Krefl and J. Walcher, Real Mirror Symmetry for One-parameter Hypersurfaces, JHEP 09 (2008) 031 [arXiv:0805.0792] [SPIRES].
D.R. Morrison and J. Walcher, D-branes and Normal Functions, arXiv:0709.4028 [SPIRES].
J. Walcher, Opening mirror symmetry on the quintic, Commun. Math. Phys. 276 (2007) 671 [hep-th/0605162] [SPIRES].
J. Knapp and E. Scheidegger, Towards Open String Mirror Symmetry for One-Parameter Calabi-Yau Hypersurfaces, arXiv:0805.1013 [SPIRES].
J. Knapp and E. Scheidegger, Matrix Factorizations, Massey Products and F-terms for Two-Parameter Calabi-Yau Hypersurfaces, arXiv:0812.2429 [SPIRES].
M. Baumgartl, I. Brunner and M.R. Gaberdiel, D-brane superpotentials and RG flows on the quintic, JHEP 07 (2007) 061 [arXiv:0704.2666] [SPIRES].
M. Baumgartl and S. Wood, Moduli Webs and Superpotentials for Five-Branes, JHEP 06 (2009) 052 [arXiv:0812.3397] [SPIRES].
M. Baumgartl, I. Brunner and M. Soroush, D-brane Superpotentials: Geometric and Worldsheet Approaches, Nucl. Phys. B 843 (2011) 602 [arXiv:1007.2447] [SPIRES].
H. Jockers and M. Soroush, Effective superpotentials for compact D5-brane Calabi-Yau geometries, Commun. Math. Phys. 290 (2009) 249 [arXiv:0808.0761] [SPIRES].
H. Jockers and M. Soroush, Relative periods and open-string integer invariants for a compact Calabi-Yau hypersurface, Nucl. Phys. B 821 (2009) 535 [arXiv:0904.4674] [SPIRES].
T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, The D5-brane effective action and superpotential in N = 1 compactifications, Nucl. Phys. B 816 (2009) 139 [arXiv:0811.2996] [SPIRES].
M. Alim, M. Hecht, P. Mayr and A. Mertens, Mirror Symmetry for Toric Branes on Compact Hypersurfaces, JHEP 09 (2009) 126 [arXiv:0901.2937] [SPIRES].
M. Alim et al., Hints for Off-Shell Mirror Symmetry in type-II/F-theory Compactifications, Nucl. Phys. B 841 (2010) 303 [arXiv:0909.1842] [SPIRES].
M. Alim et al., Type II/F-theory Superpotentials with Several Deformations and N = 1 Mirror Symmetry, arXiv:1010.0977 [SPIRES].
T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Computing Brane and Flux Superpotentials in F-theory Compactifications, JHEP 04 (2010) 015 [arXiv:0909.2025] [SPIRES].
M. Aganagic and C. Beem, The Geometry of D-brane Superpotentials, arXiv:0909.2245 [SPIRES].
S. Li, B.H. Lian and S.-T. Yau, Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces, arXiv:0910.4215 [SPIRES].
T.W. Grimm, T.-W. Ha, A. Klemm and D. Klevers, Five-Brane Superpotentials and Heterotic/F-theory Duality, Nucl. Phys. B 838 (2010) 458 [arXiv:0912.3250] [SPIRES].
H. Fuji, S. Nakayama, M. Shimizu and H. Suzuki, A Note on Computations of D-brane Superpotential, arXiv:1011.2347 [SPIRES].
M. Shimizu and H. Suzuki, Open mirror symmetry for Pfaffian Calabi-Yau 3-folds, JHEP 03 (2011) 083 [arXiv:1011.2350] [SPIRES].
A. Strominger, Superstrings with Torsion, Nucl. Phys. B 274 (1986) 253 [SPIRES].
M. Larfors, D. Lüst and D. Tsimpis, Flux compactification on smooth, compact three-dimensional toric varieties, JHEP 07 (2010) 073 [arXiv:1005.2194] [SPIRES].
F. Chen, K. Dasgupta, P. Franche, S. Katz and R. Tatar, Supersymmetric Configurations, Geometric Transitions and New Non-Kähler Manifolds, arXiv:1007.5316 [SPIRES].
F. Chen, K. Dasgupta, P. Franche and R. Tatar, Toward the Gravity Dual of Heterotic Small Instantons, Phys. Rev. D 83 (2011) 046006 [arXiv:1010.5509] [SPIRES].
M. Becker, K. Dasgupta, A. Knauf and R. Tatar, Geometric transitions, flops and non-Kähler manifolds. I, Nucl. Phys. B 702 (2004) 207 [hep-th/0403288] [SPIRES].
K. Becker, M. Becker, P.S. Green, K. Dasgupta and E. Sharpe, Compactifications of heterotic strings on non-Kähler complex manifolds. II, Nucl. Phys. B 678 (2004) 19 [hep-th/0310058] [SPIRES].
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. B 751 (2006) 108 [hep-th/0604137] [SPIRES].
M. Becker, L.-S. Tseng and S.-T. Yau, New Heterotic Non-Kähler Geometries, arXiv:0807.0827 [SPIRES].
K. Behrndt and S. Gukov, Domain walls and superpotentials from M-theory on Calabi-Yau three-folds, Nucl. Phys. B 580 (2000) 225 [hep-th/0001082] [SPIRES].
G. Lopes Cardoso, G. Curio, G. Dall’Agata and D. Lüst, BPS action and superpotential for heterotic string compactifications with fluxes, JHEP 10 (2003) 004 [hep-th/0306088] [SPIRES].
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 [arXiv:0802.0410] [SPIRES].
P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Vacuum Configurations for Superstrings, Nucl. Phys. B 258 (1985) 46 [SPIRES].
R. Friedman, J. Morgan and E. Witten, Vector bundles and F-theory, Commun. Math. Phys. 187 (1997) 679 [hep-th/9701162] [SPIRES].
D. Freed, J.A. Harvey, R. Minasian and G.W. Moore, Gravitational anomaly cancellation for M-theory fivebranes, Adv. Theor. Math. Phys. 2 (1998) 601 [hep-th/9803205] [SPIRES].
P. Griffiths and J. Harris, Principles of Algebraic Geometry’, John Wiley and Sons Inc. (1978).
R. Bott and L.W. Tu, Differential Forms In Algebraic Topology, Springer-Verlag, Germany (1982).
D. Huybrechts, Complex Geometry, Springer-Verlag Berlin Heidelberg, Germany (2005).
K. Kodaira and D.C. Spencer, On deformations of complex analytic structures I, Ann. Math. 67 (1958) 328.
K. Kodaira and D.C. Spencer, On deformations of complex analytic structures II, Ann. Math. 67 (1958) 403.
K. Kodaira, A Theorem of Completeness of Characteristic Systems for Analytic Families of Compact Submanifolds of Compact Manifolds, Ann. Math. 75 (1962) 146.
S. Kachru, S.H. Katz, A.E. Lawrence and J. McGreevy, Open string instantons and superpotentials, Phys. Rev. D 62 (2000) 026001 [hep-th/9912151] [SPIRES].
S. Kachru, S.H. Katz, A.E. Lawrence and J. McGreevy, Mirror symmetry for open strings, Phys. Rev. D 62 (2000) 126005 [hep-th/0006047] [SPIRES].
S. Gukov, C. Vafa and E. Witten, CFT’s from Calabi-Yau four-folds, Nucl. Phys. B 584 (2000) 69 [hep-th/9906070] [SPIRES].
J. Polchinski and A. Strominger, New Vacua for Type II String Theory, Phys. Lett. B 388 (1996) 736 [hep-th/9510227] [SPIRES].
S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Commun. Math. Phys. 167 (1995) 301 [hep-th/9308122] [SPIRES].
S.H. Katz, Rational curves on Calabi-Yau threefolds, hep-th/9202017 [SPIRES].
K. Hori et al., Mirror symmetry, Oxford University Press, Oxforx U.K. (2003).
S.H. Katz, A. Klemm and C. Vafa, M-theory, topological strings and spinning black holes, Adv. Theor. Math. Phys. 3 (1999) 1445 [hep-th/9910181] [SPIRES].
C. Voisin, Hodge Theory and Complex Algebraic Geometry, I, Cambridge University Press, Cambridge U.K. (2002).
V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493 [alg-geom/9310003].
D. Cox and S. Katz, Mirror symmetry and algebraic geometry, Oxford University Press, Oxford U.K. (1999).
E. Witten, Phases of N = 2 theories in two dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [SPIRES].
K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [SPIRES].
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. B 359 (1991) 21 [SPIRES].
S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nucl. Phys. B 433 (1995) 501 [hep-th/9406055] [SPIRES].
P. Candelas, A. Font, S.H. Katz and D.R. Morrison, Mirror symmetry for two parameter models. 2, Nucl. Phys. B 429 (1994) 626 [hep-th/9403187] [SPIRES].
P. Koerber, Lectures on Generalized Complex Geometry for Physicists, Fortsch. Phys. 59 (2011) 169 [arXiv:1006.1536] [SPIRES].
N.J. Hitchin, The geometry of three-forms in six and seven dimensions, math/0010054.
S. Chiossi and S. Salamon, The intrinsic torsion of SU(3) and G 2 structures, math/0202282.
G. Lopes Cardoso et al., Non-Kähler string backgrounds and their five torsion classes, Nucl. Phys. B 652 (2003) 5 [hep-th/0211118] [SPIRES].
M. Gualtieri, Generalized Kähler geometry, arXiv:1007.3485 [SPIRES].
V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Remodeling the B-model, Commun. Math. Phys. 287 (2009) 117 [arXiv:0709.1453] [SPIRES].
T.W. Grimm and T. Weigand, On Abelian Gauge Symmetries and Proton Decay in Global F-theory GUTs, Phys. Rev. D 82 (2010) 086009 [arXiv:1006.0226] [SPIRES].
T.W. Grimm, The N = 1 effective action of F-theory compactifications, Nucl. Phys. B 845 (2011) 48 [arXiv:1008.4133] [SPIRES].
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Grimm, T.W., Klemm, A. & Klevers, D. Five-brane superpotentials, blow-up geometries and SU(3) structure manifolds. J. High Energ. Phys. 2011, 113 (2011). https://doi.org/10.1007/JHEP05(2011)113
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DOI: https://doi.org/10.1007/JHEP05(2011)113