Flopping and slicing: SO(4) and Spin(4)-models

Esole, Mboyo; Kang, Monica Jinwoo

doi:10.4310/ATMP.2019.v23.n4.a2

Flopping and slicing: SO(4) and Spin(4)-models

Feb 13, 2018

64 pages

Published in:

Adv.Theor.Math.Phys. 23 (2019) 4, 1003-1066

Published: 2019

e-Print:

1802.04802 [hep-th]

DOI:

10.4310/ATMP.2019.v23.n4.a2

View in:

pdf

reference search27 citations

Citations per year

Abstract: (International Press)

We study the geometric engineering of gauge theories with gauge group

\operatorname{Spin}(4)

and

\operatorname{SO}(4)

using crepant resolutions of Weierstrass models. The corresponding elliptic fibrations realize a collision of singularities corresponding to two fibers with dual graphs A1. There are eight different ways to engineer such collisions using decorated Kodaira fibers. The Mordell–Weil group of the elliptic fibration is required to be trivial for

\operatorname{Spin}(4)

and

\mathbb{Z} / 2 \mathbb{Z}

for

\operatorname{SO}(4)

.Each of these models has two possible crepant resolutions connected by a flop. We also compute a generating function for the Euler characteristic of such elliptic fibrations over a base of arbitrary dimensions. In the case of a threefold, we also compute the triple intersection numbers of the fibral divisors. In the case of Calabi–Yau threefolds, we also compute their Hodge numbers and check the cancellations of anomalies in a six-dimensional supergravity theory.

Note:

45 pages+references, 12 figures, and 4 tables

group: Spin(4)
engineering: geometrical
dimension: 6
fibre bundle
gauge field theory: SO(4)
supergravity
singularity
space-time: Calabi-Yau
anomaly

References(66)

Figures(0)

[1]

Chern classes of blow-ups

P. Aluffi

- Math.Proc.Cambridge Phil.Soc. 148 (2010) 227-242

[2]

Chern class identities from tadpole matching in type IIB and F-theory

Paolo Aluffi
(
- Florida State U.
)
,
Mboyo Esole
(
- Leuven U.
)

- JHEP 03 (2009) 032
•
e-Print:
- 0710.2544
•
DOI:
- 10.1088/1126-6708/2009/03/032

[3]

New Orientifold Weak Coupling Limits in F-theory

Paolo Aluffi
(
- Florida State U.
)
,
Mboyo Esole
(
- Harvard U., Phys. Dept.
)

- JHEP 02 (2010) 020
•
e-Print:
- 0908.1572
•
DOI:
- 10.1007/JHEP02(2010)020

[4]

Singular Geometry and Higgs Bundles in String Theory

- SIGMA 14 (2018) 037
•
e-Print:
- 1710.08453
•
DOI:
- 10.3842/SIGMA.2018.037

[5]

T-Branes at the Limits of Geometry

- JHEP 10 (2017) 058
•
e-Print:
- 1702.06137
•
DOI:
- 10.1007/JHEP10(2017)058

[6]

The SO(32) heterotic string on a K3 surface

Paul S. Aspinwall
(
- Cornell U., LNS
)
,
Mark Gross
(
- Cornell U., LNS
)

- Phys.Lett.B 387 (1996) 735-742
•
e-Print:
- hep-th/9605131
•
DOI:
- 10.1016/0370-2693(96)01095-7

[7]

Nonsimply connected gauge groups and rational points on elliptic curves

Paul S. Aspinwall
(
- Duke U.
)
,
David R. Morrison
(
- Duke U.
)

- JHEP 07 (1998) 012
•
e-Print:
- hep-th/9805206
•
DOI:
- 10.1088/1126-6708/1998/07/012

[8]

A Systematic search for anomaly-free supergravities in six dimensions

- JHEP 10 (2005) 052
•
e-Print:
- hep-th/0508172
•
DOI:
- 10.1088/1126-6708/2005/10/052

[9]

Birational Calabi-Yau n-folds have equal Betti numbers. In New trends in algebraic geometry (Warwick,), volume 264 of London Math. Soc. Lecture Note Ser., pages 1-11. Cambridge Univ. Press, Cambridge, 1999

V.V. Batyrev

[10]

Geometric singularities and enhanced gauge symmetries

M. Bershadsky
(
- Harvard U.
)
,
Kenneth A. Intriligator
(
- Princeton, Inst. Advanced Study
)
,
S. Kachru
(
- Harvard U.
)
,
David R. Morrison
(
- Rutgers U., Piscataway and
- Duke U.
)
,
V. Sadov
(
- Princeton, Inst. Advanced Study
)

et al.

- Nucl.Phys.B 481 (1996) 215-252
•
e-Print:
- hep-th/9605200
•
DOI:
- 10.1016/S0550-3213(96)90131-5

[11]

Colliding singularities in F theory and phase transitions

M. Bershadsky
(
- Harvard U.
)
,
A. Johansen
(
- Harvard U.
)

- Nucl.Phys.B 489 (1997) 122-138
•
e-Print:
- hep-th/9610111
•
DOI:
- 10.1016/S0550-3213(97)00027-8

[12]

F theory with quantized fluxes

- Adv.Theor.Math.Phys. 3 (1999) 727-773
•
e-Print:
- hep-th/9805056
•
DOI:
- 10.4310/ATMP.1999.v3.n3.a9

[13]

Six-dimensional (1,0) effective action of F-theory via M-theory on Calabi-Yau threefolds

- JHEP 05 (2012) 019
•
e-Print:
- 1112.1082
•
DOI:
- 10.1007/JHEP05(2012)019

[14]

Groups and Lie Algebras. Chap. 1-3, Translated from the 1971 French original. Reprint of theEnglish translation. Elements of Mathematics (Berlin) -Verlag, Berlin

N. Bourbaki

[15]

Groups and Lie Algebras. Chap. 7-9., Translated from the 1975 and 1982 French originals by A. Pressley Elements of Mathematics (Berlin) -Verlag, Berlin

N. Bourbaki

[16]

Eleven-dimensional supergravity compactified on Calabi-Yau threefolds

A.C. Cadavid
(
- Cal State, Northridge
)
,
Anna Ceresole
(
- Turin Polytechnic and
- INFN, Trieste
)
,
R. D'Auria
(
- Turin Polytechnic and
- INFN, Trieste
)
,
S. Ferrara
(
- CERN
)

- Phys.Lett.B 357 (1995) 76-80
•
e-Print:
- hep-th/9506144
•
DOI:
- 10.1016/0370-2693(95)00891-N

[17]

Supergravity Theory in 11 Dimensions

E. Cremmer
(
- Ecole Normale Superieure
)
,
B. Julia
(
- Ecole Normale Superieure
)
,
Joel Scherk
(
- Ecole Normale Superieure
)

- ,
- Phys.Lett.B 76 (1978) 409-412,
- IN *APPELQUIST, T. (ED.) ET AL.: MODERN KALUZA-KLEIN THEORIES*, 202-205. (PHYS. LETT. B76 (1978) 409-412) AND PARIS EC. NORM. SUP. - LPTENS 78-10 (78,REC.MAY) 9 P. (SEE BOOK INDEX),
•
DOI:
- 10.1016/0370-2693(78)90894-8

[18]

Courbes elliptiques: formulaire d’après J. Tate In Modular functions of one variable, IV (Proc. Internat. summer school, Univ. Antwerp, Antwerp, 1972), pages 53-73. Lecture Notes in Math., Vol. 476 Berlin

P. Deligne

[19]

Strings on Orbifolds. 2.

- Nucl.Phys.B 274 (1986) 285-314,
- In *Schellekens, A.N. (ed.): Superstring construction* 181-210. (Nucl. Phys. B274 (1986) 285-314) and Preprint - Dixon, L.J. (86,rec.Mar.) 55 p. (see Book Index)
•
DOI:
- 10.1016/0550-3213(86)90287-7