Group representations and the Euler characteristic of elliptically fibered Calabi–Yau threefolds

Authors:
Antonella Grassi and David R. Morrison

Journal:
J. Algebraic Geom. **12** (2003), 321-356

DOI:
https://doi.org/10.1090/S1056-3911-02-00337-5

Published electronically:
December 17, 2002

MathSciNet review:
1949647

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Abstract |
References |
Additional Information

Abstract: To every elliptic Calabi–Yau threefold with a section $X$ there can be associated a Lie group $G$ and a representation $\rho$ of that group, determined from the Weierstrass model and the types of singular fibers. We explain this construction, which first arose in physics. The requirement of anomaly cancellation in the associated physical theory makes some surprising predictions about the connection between $X$ and $\rho$, including an explicit formula (in terms of $\rho$) for the Euler characteristic of $X$. We give a purely mathematical proof of that formula in this paper, introducing along the way a new invariant of elliptic Calabi–Yau threefolds. We also verify the other geometric predictions which are consequences of anomaly cancellation, under some mild hypotheses. As a byproduct we discover a novel relation between the Coxeter number and the rank in the case of the simply laced groups in the “exceptional series” studied by Deligne.

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AKM P. S. Aspinwall, S. Katz, and D. R. Morrison, *Lie groups, Calabi–Yau threefolds and $F$-theory*, Adv. Theor. Math. Phys. **4** (2000) 95–126, hep-th/0002012v2.
BKKM P. Berglund, S. Katz, A. Klemm, and P. Mayr, *New Higgs transitions between dual $N{=}2$ string models*, Nucl. Phys. B **483** (1997) 209–228, hep-th/9605154.
BIKMSV M. Bershadsky, K. Intriligator, S. Kachru, D. R. Morrison, V. Sadov, and C. Vafa, *Geometric singularities and enhanced gauge symmetries*, Nucl. Phys. B **481** (1996) 215–252, hep-th/9605200.
Bri:Nice E. Brieskorn, *Singular elements of semi-simple algebraic groups*, Actes Congrès intern. Math. 1970, tome 2, Gauthier-Villars, Paris, 1971, pp. 279–284.
CPR P. Candelas, E. Perevalov, and G. Rajesh, *Matter from toric geometry*, Nucl. Phys. **B519** (1998) 225–238, hep-th/9707049.
coxeter:annals H. S. M. Coxeter, *Discrete groups generated by reflections*, Annals of Math. (2) **35** (1934) 588–621.
coxeter:weyl H. S. M. Coxeter, *Discrete groups generated by reflections*, in: “The Structure and Representation of Continuous Groups,” by Hermann Weyl, Institute for Advanced Study, 1935, pp. 186–210.
deligne P. Deligne, *Le série exceptionnelle de groupes de Lie*, C.R. Acad. Sci. Paris **t. 322, Série I** (1996) 321–326.
DE D.-E. Diaconescu and R. Entin, *Calabi–Yau spaces and five-dimensional field theories with exceptional gauge symmetry*, Nucl. Phys. **B538** (1999) 451–484, hep-th/9807170.
DV P. Du Val, *On isolated singularities which do not affect the condition of adjunction, Part I*, Proc. Cambridge Phil. Soc **30** (1934) 453–465.
DV:book P. Du Val, “Homographies, Quaternions, and Rotations,” Clarendon Press, Oxford, 1964.
Erler J. Erler, *Anomaly cancellation in six dimensions*, J. Math. Phys. **35** (1994) 1819–1833, hep-th/9304104.
FW. Fulton, “Intersection Theory,” Ergebn. Math. Grenzegeb. (3) **2**, Springer-Verlag, Berlin, 1984.
G A. Grassi, *Divisors on elliptic Calabi–Yau 4-folds and the superpotential in F-theory, I*, J. Geom. Phys. **28** (1998) 289–319, alg-geom/9704008.
AD A. Grassi and D. R. Morrison, *Anomalies and the Euler characteristic of elliptic Calabi–Yau threefolds*, to appear.
GS M. B. Green and J. H. Schwarz, *Anomaly cancellations in supersymmetric $D = 10$ gauge theory and superstring theory*, Phys. Lett. B **149** (1984) 117–122.
IMS K. Intriligator, D. R. Morrison, and N. Seiberg, *Five-dimensional supersymmetric gauge theories and degenerations of Calabi–Yau spaces*, Nucl. Phys. B **497** (1997) 56–100, hep-th/9702198.
IN Y. Ito and H. Nakajima, *McKay correspondence and Hilbert schemes in dimension three*, Topology **39** (2000) 1155–1191, alg-geom/9803120.
IR Y. Ito and M. Reid, *The McKay correspondence for finite subgroups of $SL(3, \mathbb C$)*, in: “Higher Dimensional Varieties (Trento 1994),” de Gruyter, Berlin, 1996, pp. 221–240, alg-geom/9411010.
KM S. Katz and D. R. Morrison, *Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups*, J. Algebraic Geom. **1** (1992) 449-530, alg-geom/9202002.
KMP S. Katz, D. R. Morrison, and M. R. Plesser, *Enhanced gauge symmetry in type II string theory*, Nucl. Phys. B **477** (1996) 105–140, hep-th/9601108.
KV S. Katz and C. Vafa, *Matter from geometry*, Nucl. Phys. B **497** (1997) 146–154, hep-th/9606086.
kod K. Kodaira, *On compact analytic surfaces, II, III*, Ann. of Math. **77** (1963) 563–626, **78** (1963) 1–40.
MP J. W. G. McKay and J. Patera “Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras,” M. Dekker, New York, 1981.
Mi R. Miranda, *Smooth models for elliptic threefolds,* in: “The Birational Geometry of Degenerations,” Progr. Math. **29**, Birkhäuser, Boston, 1983, pp. 85–133.
MVII D. R. Morrison and C. Vafa, *Compactifications of F-theory on Calabi–Yau threefolds, II*, Nucl. Phys. B **476** (1996) 437–469, hep-th/9603161.
Na1 N. Nakayama, *On Weierstrass models*, in: “Algebraic Geometry and Commutative Algebra,” Vol. II, Kinokuniya, Tokyo, 1988, pp. 405–431.
Na2 N. Nakayama, *Elliptic fibrations over surfaces, I*, in: “Algebraic Geometry and Analytic Geometry (Tokyo, 1990),” Springer, Tokyo, 1991, pp. 126–137.
R M. Reid, *McKay correspondence*, alg-geom/9702016.
Sadov V. Sadov, *Generalized Green–Schwarz mechanism in $F$ theory*, Phys. Lett. B **388** (1996) 45–50, hep-th/9606008.
Sagnotti A. Sagnotti, *A note on the Green–Schwarz mechanism in open-string theories*, Phys. Lett. B **294** (1992) 196–203, hep-th/9210127.
Schwarz J. H. Schwarz, *Anomaly-free supersymmetric models in six dimensions*, Phys. Lett. B **371** (1996) 223–230, hep-th/9512053.
SVW S. Sethi, C. Vafa, and E. Witten, *Constraints on low-dimensional string compactifications*, Nucl. Phys. B **480** (1996) 213–224, hep-th/9606122.
witten:MF E. Witten, *Phase transitions in M-theory and F-theory*, Nucl. Phys. **B471** (1996) 195–216, hep-th/9603150.

Asp P. S. Aspinwall and M. Gross, *The $SO(32)$ heterotic string on a K3 surface*, Phys. Lett. B **387** (1996) 735–742, hep-th/9605131.
AKM P. S. Aspinwall, S. Katz, and D. R. Morrison, *Lie groups, Calabi–Yau threefolds and $F$-theory*, Adv. Theor. Math. Phys. **4** (2000) 95–126, hep-th/0002012v2.
BKKM P. Berglund, S. Katz, A. Klemm, and P. Mayr, *New Higgs transitions between dual $N{=}2$ string models*, Nucl. Phys. B **483** (1997) 209–228, hep-th/9605154.
BIKMSV M. Bershadsky, K. Intriligator, S. Kachru, D. R. Morrison, V. Sadov, and C. Vafa, *Geometric singularities and enhanced gauge symmetries*, Nucl. Phys. B **481** (1996) 215–252, hep-th/9605200.
Bri:Nice E. Brieskorn, *Singular elements of semi-simple algebraic groups*, Actes Congrès intern. Math. 1970, tome 2, Gauthier-Villars, Paris, 1971, pp. 279–284.
CPR P. Candelas, E. Perevalov, and G. Rajesh, *Matter from toric geometry*, Nucl. Phys. **B519** (1998) 225–238, hep-th/9707049.
coxeter:annals H. S. M. Coxeter, *Discrete groups generated by reflections*, Annals of Math. (2) **35** (1934) 588–621.
coxeter:weyl H. S. M. Coxeter, *Discrete groups generated by reflections*, in: “The Structure and Representation of Continuous Groups,” by Hermann Weyl, Institute for Advanced Study, 1935, pp. 186–210.
deligne P. Deligne, *Le série exceptionnelle de groupes de Lie*, C.R. Acad. Sci. Paris **t. 322, Série I** (1996) 321–326.
DE D.-E. Diaconescu and R. Entin, *Calabi–Yau spaces and five-dimensional field theories with exceptional gauge symmetry*, Nucl. Phys. **B538** (1999) 451–484, hep-th/9807170.
DV P. Du Val, *On isolated singularities which do not affect the condition of adjunction, Part I*, Proc. Cambridge Phil. Soc **30** (1934) 453–465.
DV:book P. Du Val, “Homographies, Quaternions, and Rotations,” Clarendon Press, Oxford, 1964.
Erler J. Erler, *Anomaly cancellation in six dimensions*, J. Math. Phys. **35** (1994) 1819–1833, hep-th/9304104.
FW. Fulton, “Intersection Theory,” Ergebn. Math. Grenzegeb. (3) **2**, Springer-Verlag, Berlin, 1984.
G A. Grassi, *Divisors on elliptic Calabi–Yau 4-folds and the superpotential in F-theory, I*, J. Geom. Phys. **28** (1998) 289–319, alg-geom/9704008.
AD A. Grassi and D. R. Morrison, *Anomalies and the Euler characteristic of elliptic Calabi–Yau threefolds*, to appear.
GS M. B. Green and J. H. Schwarz, *Anomaly cancellations in supersymmetric $D = 10$ gauge theory and superstring theory*, Phys. Lett. B **149** (1984) 117–122.
IMS K. Intriligator, D. R. Morrison, and N. Seiberg, *Five-dimensional supersymmetric gauge theories and degenerations of Calabi–Yau spaces*, Nucl. Phys. B **497** (1997) 56–100, hep-th/9702198.
IN Y. Ito and H. Nakajima, *McKay correspondence and Hilbert schemes in dimension three*, Topology **39** (2000) 1155–1191, alg-geom/9803120.
IR Y. Ito and M. Reid, *The McKay correspondence for finite subgroups of $SL(3, \mathbb C$)*, in: “Higher Dimensional Varieties (Trento 1994),” de Gruyter, Berlin, 1996, pp. 221–240, alg-geom/9411010.
KM S. Katz and D. R. Morrison, *Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups*, J. Algebraic Geom. **1** (1992) 449-530, alg-geom/9202002.
KMP S. Katz, D. R. Morrison, and M. R. Plesser, *Enhanced gauge symmetry in type II string theory*, Nucl. Phys. B **477** (1996) 105–140, hep-th/9601108.
KV S. Katz and C. Vafa, *Matter from geometry*, Nucl. Phys. B **497** (1997) 146–154, hep-th/9606086.
kod K. Kodaira, *On compact analytic surfaces, II, III*, Ann. of Math. **77** (1963) 563–626, **78** (1963) 1–40.
MP J. W. G. McKay and J. Patera “Tables of Dimensions, Indices, and Branching Rules for Representations of Simple Lie Algebras,” M. Dekker, New York, 1981.
Mi R. Miranda, *Smooth models for elliptic threefolds,* in: “The Birational Geometry of Degenerations,” Progr. Math. **29**, Birkhäuser, Boston, 1983, pp. 85–133.
MVII D. R. Morrison and C. Vafa, *Compactifications of F-theory on Calabi–Yau threefolds, II*, Nucl. Phys. B **476** (1996) 437–469, hep-th/9603161.
Na1 N. Nakayama, *On Weierstrass models*, in: “Algebraic Geometry and Commutative Algebra,” Vol. II, Kinokuniya, Tokyo, 1988, pp. 405–431.
Na2 N. Nakayama, *Elliptic fibrations over surfaces, I*, in: “Algebraic Geometry and Analytic Geometry (Tokyo, 1990),” Springer, Tokyo, 1991, pp. 126–137.
R M. Reid, *McKay correspondence*, alg-geom/9702016.
Sadov V. Sadov, *Generalized Green–Schwarz mechanism in $F$ theory*, Phys. Lett. B **388** (1996) 45–50, hep-th/9606008.
Sagnotti A. Sagnotti, *A note on the Green–Schwarz mechanism in open-string theories*, Phys. Lett. B **294** (1992) 196–203, hep-th/9210127.
Schwarz J. H. Schwarz, *Anomaly-free supersymmetric models in six dimensions*, Phys. Lett. B **371** (1996) 223–230, hep-th/9512053.
SVW S. Sethi, C. Vafa, and E. Witten, *Constraints on low-dimensional string compactifications*, Nucl. Phys. B **480** (1996) 213–224, hep-th/9606122.
witten:MF E. Witten, *Phase transitions in M-theory and F-theory*, Nucl. Phys. **B471** (1996) 195–216, hep-th/9603150.

Additional Information

**Antonella Grassi**

Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104

Email:
grassi@math.upenn.edu

**David R. Morrison**

Affiliation:
School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540;
Department of Mathematics, Duke University, Durham, North Carolina 27708-0320

MR Author ID:
189764

Email:
drm@math.duke.edu

Received by editor(s):
November 1, 2000

Published electronically:
December 17, 2002

Additional Notes:
Research partially supported by the Harmon Duncombe Foundation, by the Institute for Advanced Study, and by National Science Foundation grants DMS-9401447, DMS-9401495, DMS-9627351 and DMS-9706707. We thank the Institute for Advanced Study, the Mathematisches Forschunginstitut Oberwolfach, and the Institute for Theoretical Physics, Santa Barbara, for hospitality during various stages of this project.