Letters in Mathematical Physics

, Volume 85, Issue 2–3, pp 163–171

Mastering the Master Space

  • Davide Forcella
  • Amihay Hanany
  • Yang-Hui he
  • Alberto Zaffaroni
Article

Abstract

Supersymmetric gauge theories have an important but perhaps under-appreciated notion of a master space, which controls the full moduli space. For world-volume theories of D-branes probing a Calabi-Yau singularity \({\mathcal X}\) the situation is particularly illustrative. In the case of one physical brane, the master space \({\mathcal F^b}\) is the space of F-terms and a particular quotient thereof is \({\mathcal X}\) itself. We study various properties of \({\mathcal F^b}\) which encode such physical quantities as Higgsing, BPS spectra, hidden global symmetries, etc. Using the plethystic program we also discuss what happens at higher number N of branes.

Mathematics Subject Classification (2000)

81T60 81T30 

Keywords

supersymmetric gauge theory D-branes plethystics global symmetries counting gauge invariants 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beasley C., Greene B.R., Lazaroiu C.I., Plesser M.R.: D3-branes on partial resolutions of abelian quotient singularities of Calabi-Yau threefolds. Nucl. Phys. B 566, 599 (2000) arXiv:hep-th/9907186MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Benvenuti, S., Feng, B., Hanany, A., He, Y.H.: Counting BPS operators in gauge theories: quivers, syzygies and plethystics. arXiv:hep-th/0608050Google Scholar
  3. 3.
    Berenstein D.: Reverse geometric engineering of singularities. JHEP 0204, 052 (2002) arXiv:hep-th/0201093CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Butti A., Forcella D., Zaffaroni A.: Counting BPS baryonic operators in CFTs with Sasaki-Einstein duals. JHEP 0706, 069 (2007) arXiv:hep-th/0611229CrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Butti, A., Forcella, D., Hanany, A., Vegh, D., Zaffaroni, A.: Counting chiral operators in quiver gauge theories. arXiv:0705.2771 [hep-th]Google Scholar
  6. 6.
    Craw A., Maclagan D., Thomas R.R.: Moduli of McKay quiver representations I: the coherent component. Proc. Lond. Math. Soc. 95(1), 179–198 (2007) arXiv:0505115MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Craw A., Maclagan D., Thomas R.R.: Moduli of McKay quiver representations II: Grobner basis techniques. J. Algebra 316(2), 514–535 (2007) arXiv:0611840MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Feng B., Hanany A., He Y.H.: D-brane gauge theories from toric singularities and toric duality. Nucl. Phys. B 595, 165 (2001) arXiv:hep-th/0003085MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Feng B., Hanany A., He Y.H., Uranga A.M.: Toric duality as Seiberg duality and brane diamonds. JHEP 0112, 035 (2001) arXiv:hep-th/0109063CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Feng B., Franco S., Hanany A., He Y.H.: Unhiggsing the del Pezzo. JHEP 0308, 058 (2003) arXiv:hep-th/0209228CrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Feng B., Hanany A., He Y.H.: Counting gauge invariants: the plethystic program. JHEP 0703, 090 (2007) arXiv:hep-th/0701063CrossRefADSMathSciNetGoogle Scholar
  12. 12.
    Forcella, D., Hanany, A., He, Y.-H., Zaffaroni, A.: The Master Space of \({\mathcal{N}=1}\) Gauge Theories. arXiv:0801.1585 [hep-th]Google Scholar
  13. 13.
    Forcella, D., Hanany, A., Zaffaroni, A.: Baryonic generating functions. arXiv:hep-th/0701236Google Scholar
  14. 14.
    Forcella, D.: BPS Partition Functions for Quiver Gauge Theories: Counting Fermionic Operators. arXiv:0705.2989 [hep-th]Google Scholar
  15. 15.
    Franco S., Hanany A., Kazakopoulos P.: Hidden exceptional global symmetries in 4d CFTs. JHEP 0407, 060 (2004) arXiv:hep-th/0404065CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Franco S., Hanany A., Kennaway K.D., Vegh D., Wecht B.: Brane dimers and quiver gauge theories. JHEP 0601, 096 (2006) arXiv:hep-th/0504110CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Gray J., He Y.H., Jejjala V., Nelson B.D.: Exploring the vacuum geometry of N = 1 gauge theories. Nucl. Phys. B 750, 1 (2006) arXiv:hep-th/0604208MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Grayson, D., Stillman, M.: Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
  19. 19.
    Hanany, A., Kennaway, K.D.: Dimer models and toric diagrams. arXiv:hep-th/0503149Google Scholar
  20. 20.
    Hanany, A., Romelsberger, C.: Counting BPS operators in the chiral ring of N = 2 supersymmetric gauge theories or N = 2 braine surgery. arXiv:hep-th/0611346Google Scholar
  21. 21.
    Kennaway, K.D.: Brane tilings. Int. J. Mod. Phys. A 22, 2977 (2007). arXiv:0706.1660 [hep-th], cf. also refs thereinGoogle Scholar
  22. 22.
    Morrison D.R., Plesser M.R.: Non-spherical horizons. I. Adv. Theor. Math. Phys. 3, 1 (1999) arXiv:hep-th/9810201MATHMathSciNetGoogle Scholar
  23. 23.
    Okonek, C., Teleman, A.: Master spaces and the coupling principle: from geometric invariant theory to gauge theory. Commun. Math. Phys. 205, 437–458 (1999); and cf. references thereinGoogle Scholar
  24. 24.
    Stanley R.: Hilbert functions of graded algebras. Adv. Math. 28, 57–83 (1978)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  • Davide Forcella
    • 1
    • 2
  • Amihay Hanany
    • 3
    • 4
  • Yang-Hui he
    • 5
    • 6
    • 7
  • Alberto Zaffaroni
    • 8
  1. 1.International School for Advanced Studies (SISSA/ISAS) and INFN-Sezione di TriesteTriesteItaly
  2. 2.PH-TH DivisionCERN, 1211Geneva 23Switzerland
  3. 3.Department of PhysicsTechnion, Israel Institute of TechnologyHaifaIsrael
  4. 4.Theoretical Physics Group, Blackett LaboratoryImperial CollegeLondonUK
  5. 5.Merton CollegeOxfordUK
  6. 6.Mathematical InstituteOxfordUK
  7. 7.Rudolf Peierls Centre for Theoretical PhysicsUniversity of OxfordOxfordUK
  8. 8.Università di Milano-Bicocca and INFNsezione di Milano-BicoccaMilanoItaly

Personalised recommendations