Yukawa couplings in heterotic compactification. (English) Zbl 1203.81130

Yukawa couplings – more precisely, their \(F\)-term contribution coming from the superpotential – play a prominent role in the characterization of phenomenologically viable models arising from superstring compactifications. An important task of string phenomenology is to provide an effective way of computing them for a given string background. In the heterotic context, most of the results were restricted to the case when the gauge bundle \(V\to X\) on the internal Calabi-Yau manifold \(X\) is its holomorphic tangent bundle.
This paper improves the situation by giving an effective, algorithmic way of computing Yukawas when \(V\) is a polystable holomorphic vector bundle constructed via the monad method, and \(X\) is a (favourable) Calabi-Yau complete intersection in \(\mathbb{P}^{n_1}\times \dots \times \mathbb{P}^{n_k}\). The power of the method stems from the fact, for these configurations, Cech cohomology groups of \(V\) can be presented as quotients of polynomial rings, and the computation of Yukawa couplings is reduced to the operation of multiplying polynomials and to a projection on a distinguished one-dimensional subspace. This results in a purely algebraic formalism, which the authors exploit in the case with no anti-generations and with GUT gauge group \(E_6\), \(SO(10)\) and \(SU(5)\). Some remarks on the relation of this method and the heterotic engineering of \(SO(10)\) one-Higgs models are also included.


81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J81 Relationships between surfaces, higher-dimensional varieties, and physics
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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[1] Greene B.R., Kirklin K.H., Miron P.J., Ross G.G.: 27**3 Yukawa Couplings For A Three Generation Superstring Model. Phys. Lett. B 192, 111 (1987)
[2] Candelas P.: Yukawa Couplings Between (2,1) Forms. Nucl. Phys. B 298, 458 (1988) · Zbl 0659.53059
[3] Candelas P., Kalara S.: Yukawa couplings for a three generation superstring compactification. Nucl. Phys. B 298, 357 (1988)
[4] McOrist J., Melnikov I.V.: Summing the Instantons in Half-Twisted Linear Sigma Models. JHEP 0902, 026 (2009) · Zbl 1245.81249
[5] Donagi, R., Reinbacher, R., Yau, S.T.: Yukawa couplings on quintic threefolds. http://arxiv.org/abs/hep-th/0605203v1 , 2006
[6] Donagi R., He Y.H., Ovrut B.A., Reinbacher R.: The particle spectrum of heterotic compactifications. JHEP 0412, 054 (2004) · Zbl 1247.14044
[7] Berglund P., Parkes L., Hubsch T.: The Complete Matter Sector In A Three Generation Compactification. Commun. Math. Phys. 148, 57 (1992) · Zbl 0759.53044
[8] Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology. Cambridge: Cambridge Univ. Pr., 1987 · Zbl 0619.53002
[9] Gabella M., He Y.H., Lukas A.: An Abundance of Heterotic Vacua. JHEP 0812, 027 (2008) · Zbl 1329.81313
[10] Anderson L.B., He Y.H., Lukas A.: Heterotic compactification, an algorithmic approach. JHEP 0707, 049 (2007)
[11] Candelas P., Dale A.M., Lutken C.A., Schimmrigk R.: Complete Intersection Calabi-Yau Manifolds. Nucl. Phys. B 298, 493 (1988)
[12] Okonek C., Schneider M., Spindler H.: Vector Bundles on Complex Projective Spaces. Birkhäuser Verlag, Basel (1988)
[13] Anderson L.B., He Y.H., Lukas A.: Monad Bundles in Heterotic String Compactifications. JHEP 0807, 104 (2008)
[14] Anderson, L.B.: Heterotic and M-theory Compactifications for String Phenomenology. Oxford University DPhil Thesis, 2008, http://arxiv.org/abs/0808.3621v1[hep-th] , 2008
[15] Anderson, L.B., He, Y.H., Lukas, A.: Vector bundle stability in heterotic monad models. In preparation
[16] Donaldson, S.K.: Some numerical results in complex differential geometry. http://arxiv.org/abs/math/0512625v1[math.DG] , 2005. Douglas, M.R., Karp, R.L., Lukic, S., Reinbacher, R.: Numerical solution to the hermitian Yang-Mills equation on the Fermat quintic. JHEP 0712, 083 (2007); Douglas, M.R., Karp, R.L., Lukic, S., Reinbacher, R.: Numerical Calabi-Yau metrics. J. Math. Phys. 49, 032302 (2008). Braun, V., Brelidze, T., Douglas, M.R., Ovrut, B.A.: Calabi-Yau Metrics for Quotients and Complete Intersections. JHEP 0805, 080 (2008) · Zbl 1153.81351
[17] Blumenhagen R., Moster S., Weigand T.: Heterotic GUT and standard model vacua from simply connected Calabi-Yau manifolds. Nucl. Phys. B 751, 186 (2006) · Zbl 1192.81257
[18] Blumenhagen R., Honecker G., Weigand T.: Loop-corrected compactifications of the heterotic string with line bundles. JHEP 0506, 020 (2005)
[19] Distler J., Greene B.R.: Aspects of (2,0) String Compactifications. Nucl. Phys. B 304, 1 (1988)
[20] Lukas A., Ovrut B.A., Waldram D.: On the four-dimensional effective action of strongly coupled heterotic string theory. Nucl. Phys. B 532, 43 (1998) · Zbl 1047.81551
[21] Lukas A., Ovrut B.A., Stelle K.S., Waldram D.: The universe as a domain wall. Phys. Rev. D 59, 086001 (1999)
[22] Hubsch T.: Calabi-Yau manifolds: A Bestiary for physicists. World Scientific, Singapore (1992) · Zbl 0771.53002
[23] Hartshorne, R.: Algebraic Geometry, Springer. GTM 52, Springer-Verlag, 1977; Griffith, P., Harris, J., Principles of algebraic geometry. New York: Wiley-Interscience, 1978
[24] Grayson, D., Stillman, M.: Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
[25] Greuel, G.-M., Pfister, G., Schönemann, H.: Singular: a computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserslautern (2001). Available at http://www.singular.uni-kl.de/ · Zbl 1344.13002
[26] Gray, J., He, Y.H., Ilderton, A., Lukas, A.: ”STRINGVACUA: A Mathematica Package for Studying Vacuum Configurations in String Phenomenology.” Comput. Phys. Commun. 180, 107–119 (2009); arXiv:0801.1508 [hep-th]. Gray, J., He, Y.H., Ilderton, A., Lukas, A.: ”A new method for finding vacua in string phenomenology,” JHEP 0707 (2007) 023; Gray, J., He, Y.H., Lukas, A.: ”Algorithmic algebraic geometry and flux vacua.” JHEP 0609 (2006) 031; The Stringvacua Mathematica package is available at: http://www-thphys.physics.ox.ac.uk/projects/Stringvacua/ · Zbl 1198.81156
[27] Braun, V., He, Y.H., Ovrut, B.A., Pantev, T.: ”A heterotic standard model.” Phys. Lett. B 618, 252 (2005); ”The exact MSSM spectrum from string theory.” JHEP 0605, 043 (2006) · Zbl 1247.81349
[28] Donagi R., He Y.H., Ovrut B.A., Reinbacher R.: Moduli dependent spectra of heterotic compactifications. Phys. Lett. B 598, 279 (2004) · Zbl 1247.14044
[29] Bouchard V., Donagi R.: An SU(5) heterotic standard model. Phys. Lett. B 633, 783 (2006) · Zbl 1247.81348
[30] Buchberger, B.: ”An Algorithm for Finding the Bases Elements of the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal” (German), Phd thesis, Univ. of Innsbruck (Austria), 1965; B. Buchberger, ”An Algorithmical Criterion for the Solvability of Algebraic Systems of Equations” (German), Aequationes Mathematicae 4(3), 374–383,1970; English translation can be found in: Buchberger, B., Winkler, F., eds.: ”Gröbner Bases and Applications.” Volume 251 of the L.M.S. series, Cambridge: Cambridge University Press, 1998; Proc. of the International Conference ”33 Years of Gröbner bases”; See B. Buchberger, ”Gröbner Bases: A Short Introduction for Systems Theorists.” p1-19 Lecture Notes in Computer Science, Computer Aided Systems Theory - EUROCAST 2001, Berlin-Heidelberg: Springer, 2001, pp. 1–19 · Zbl 1245.13020
[31] Gray, J.: A Simple Introduction to Grobner Basis Methods in String Phenomenology. http://arxiv.org/abs/0901.1662v1[hep-th] , 2009
[32] Anderson L.B., Gray J., Lukas A., Ovrut B.: The Edge Of Supersymmetry: Stability Walls in Heterotic Theory. Phys. Lett B 677, 190–194 (2009)
[33] Anderson L.B., Gray J., Lukas A., Ovrut B.: Stability Walls in Heterotic Theories. JHEP 0909, 026 (2009)
[34] Avramov, L.L., Grayson, D.R.: Resolutions and cohomology over complete intersections, In: Computations in algebraic geometry with Macaulay 2, Algorithms Comput. Math., Vol. 8, Berlin: Springer, 2002, pp. 131–178 · Zbl 0994.13006
[35] Boardman J.M.: The principle of signs. Enseignement Math. (2) 12, 191–194 (1966) · Zbl 0147.27803
[36] Bourbaki, N.: Éléments de mathématique. Algèbre. Chapitre 10. Algèbre homologique, Berlin: Springer-Verlag, 2007, (Reprint of the 1980 original [Paris: Masson]) · Zbl 1105.18001
[37] Cartan H., Eilenberg S.: Homological algebra. Princeton University Press, Princeton, N. J. (1956) · Zbl 0075.24305
[38] Godement, R.: Topologie algébrique et théorie des faisceaux, Actualit’es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Paris: Hermann, 1964
[39] Grayson D.R.: Adams operations on higher K-theory. K-Theory 6(2), 97–111 (1992) · Zbl 0776.19001
[40] Swan R.G.: Cup products in sheaf cohomology, pure injectives, and a substitute for projective resolutions. J. Pure Appl. Algebra 144(2), 169–211 (1999) · Zbl 0942.18004
[41] Weibel, C.A.: An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, Vol. 38, Cambridge: Cambridge University Press, 1994 · Zbl 0797.18001
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