# BPS open strings and *A*-*D*-*E*-singularities in F-theory on K3

- 153 Downloads
- 2 Citations

## Abstract

We improve on a recently constructed graphical representation of the supergravity 7-brane solution and apply this refined representation to re-study the open string description of the *A*-*D*-*E*-singularities in F-theory on K3. A noteworthy feature of the graphical representation is that it provides the complete global branch cut structure of the 7-brane solution which plays an important role in our analysis. We first identify those groups of branes which when made to coincide lead to the *A*-*D*-*E*-gauge groups. We next show that there is always a sufficient number of open BPS strings to account for all the generators of the gauge group. However, as we will show, there is in general no one-to-one relation between BPS strings and gauge group generators.

For the \( {D_{n + {4^{-} }}} \) and E-singularities, in order to relate BPS strings with gauge group generators, we make an SU(*n* + 4), respectively SU(5) subgroup of the \( {D_{n + {4^{-} }}} \) and E-gauge groups manifest. We find that only for the *D*-series (and for the standard *A*-series) this is sufficient to identify, in a one-to-one manner, which BPS strings correspond to which gauge group generators.

## Keywords

F-Theory D-branes Discrete and Finite Symmetries## References

- [1]C. Beasley, J.J. Heckman and C. Vafa,
*GUTs and exceptional branes in F-theory — I*,*JHEP***01**(2009) 058 [arXiv:0802.3391] [SPIRES].CrossRefMathSciNetADSGoogle Scholar - [2]C. Beasley, J.J. Heckman and C. Vafa,
*GUTs and exceptional branes in F-theory — II: experimental predictions*,*JHEP***01**(2009) 059 [arXiv:0806.0102] [SPIRES].CrossRefMathSciNetADSGoogle Scholar - [3]E.A. Bergshoeff, J. Hartong, T. Ortín and D. Roest,
*Seven-branes and supersymmetry*,*JHEP***02**(2007) 003 [hep-th/0612072] [SPIRES].CrossRefADSGoogle Scholar - [4]C. Vafa,
*Evidence for F-theory*,*Nucl. Phys.***B 469**(1996) 403 [hep-th/9602022] [SPIRES].CrossRefMathSciNetADSGoogle Scholar - [5]A. Johansen,
*A comment on BPS states in F-theory in*8*dimensions*,*Phys. Lett.***B 395**(1997) 36 [hep-th/9608186] [SPIRES].MathSciNetADSGoogle Scholar - [6]M.R. Gaberdiel and B. Zwiebach,
*Exceptional groups from open strings*,*Nucl. Phys.***B 518**(1998) 151 [hep-th/9709013] [SPIRES].CrossRefMathSciNetADSGoogle Scholar - [7]M.R. Gaberdiel, T. Hauer and B. Zwiebach,
*Open string-string junction transitions*,*Nucl. Phys.***B 525**(1998) 117 [hep-th/9801205] [SPIRES].CrossRefMathSciNetADSGoogle Scholar - [8]B.R. Greene, A.D. Shapere, C. Vafa and S.-T. Yau,
*Stringy cosmic strings and noncompact Calabi-Yau manifolds*,*Nucl. Phys.***B 337**(1990) 1 [SPIRES].CrossRefMathSciNetADSGoogle Scholar - [9]G.W. Gibbons, M.B. Green and M.J. Perry,
*Instantons and seven-branes in type IIB superstring theory*,*Phys. Lett.***B 370**(1996) 37 [hep-th/9511080] [SPIRES].MathSciNetADSGoogle Scholar - [10]W. Barth, C. Peters and A. Van de Ven,
*Compact complex surfaces*, Springer Verlag, Germany (1984).MATHGoogle Scholar - [11]J. Tate,
*Algorithm for determining the type of a singular fiber in an elliptic pencil. Modular functions of one variable IV*, Lecture Notes in Mathematics volume 476, Springer Verlag, Germany (1975).Google Scholar - [12]M. Bershadsky et al.,
*Geometric singularities and enhanced gauge symmetries*,*Nucl. Phys.***B 481**(1996) 215 [hep-th/9605200] [SPIRES].CrossRefMathSciNetADSGoogle Scholar - [13]P.C. Argyres and M.R. Douglas,
*New phenomena in*SU(3)*supersymmetric gauge theory*,*Nucl. Phys.***B 448**(1995) 93 [hep-th/9505062] [SPIRES].CrossRefMathSciNetADSGoogle Scholar - [14]P.C. Argyres, M. Ronen Plesser, N. Seiberg and E. Witten,
*New N*= 2*superconformal field theories in four dimensions*,*Nucl. Phys.***B 461**(1996) 71 [hep-th/9511154] [SPIRES].CrossRefADSGoogle Scholar - [15]A. Sen,
*Orientifold limit of F-theory vacua*,*Phys. Rev.***D 55**(1997) 7345 [hep-th/9702165] [SPIRES].ADSGoogle Scholar - [16]K. Dasgupta and S. Mukhi,
*F-theory at constant coupling*,*Phys. Lett.***B 385**(1996) 125 [hep-th/9606044] [SPIRES].MathSciNetADSGoogle Scholar - [17]E. Bergshoeff, J. Hartong and D. Sorokin,
*Q*7*-branes and their coupling to IIB supergravity*,*JHEP***12**(2007) 079 [arXiv:0708.2287] [SPIRES].CrossRefMathSciNetADSGoogle Scholar - [18]I.A. Bandos,
*On superembedding approach to type IIB*7*-branes*,*JHEP***04**(2009) 085 [arXiv:0812.2889] [SPIRES].CrossRefMathSciNetADSGoogle Scholar - [19]E.A. Bergshoeff, J. Hartong, A. Ploegh and D. Sorokin,
*Q-instantons*,*JHEP***06**(2008) 028 [arXiv:0801.4956] [SPIRES].CrossRefADSGoogle Scholar - [20]O. DeWolfe, T. Hauer, A. Iqbal and B. Zwiebach,
*Uncovering the symmetries on*(*p*,*q*) 7*-branes: beyond the Kodaira classification*,*Adv. Theor. Math. Phys.***3**(1999) 1785 [hep-th/9812028] [SPIRES].MATHMathSciNetGoogle Scholar - [21]O. DeWolfe, T. Hauer, A. Iqbal and B. Zwiebach,
*Uncovering infinite symmetries on*(*p*,*q*) 7*-branes: Kac-Moody algebras and beyond*,*Adv. Theor. Math. Phys.***3**(1999) 1835 [hep-th/9812209] [SPIRES].MATHMathSciNetGoogle Scholar - [22]E. Witten,
*Bound states of strings and p-branes*,*Nucl. Phys.***B 460**(1996) 335 [hep-th/9510135] [SPIRES].CrossRefMathSciNetADSGoogle Scholar - [23]E.G. Gimon and J. Polchinski,
*Consistency conditions for orientifolds and D-manifolds*,*Phys. Rev.***D 54**(1996) 1667 [hep-th/9601038] [SPIRES].MathSciNetADSGoogle Scholar - [24]A. Sen,
*F-theory and orientifolds*,*Nucl. Phys.***B 475**(1996) 562 [hep-th/9605150] [SPIRES].CrossRefADSGoogle Scholar - [25]R. Slansky,
*Group theory for unified model building*,*Phys. Rept.***79**(1981) 1 [SPIRES].CrossRefMathSciNetADSGoogle Scholar

## Copyright information

**Open Access** This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.