Skip to main content
SpringerLink
Log in

Derived Categories and Stacks in Physics

  • Chapter
  • First Online:
Book cover Homological Mirror Symmetry

Part of the book series: Lecture Notes in Physics ((LNP,volume 757))

  • 2217 Accesses

Abstract

We review how both derived categories and stacks enter physics. The physical realization of each has many formal similarities. For example, in both the cases, equivalences are realized via renormalization group flow: in the case of derived categories, (boundary) renormalization group flow realizes the mathematical procedure of localization on quasi-isomorphisms and, in the case of stacks, worldsheet renormalization group flow realizes presentation-independence. For both, we outline current technical issues and applications.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. C. Weibel, An Introduction to Homological Algebra, Cambridge studies in advanced mathematics 38, Cambridge University Press, 1994.

    Google Scholar 

  2. R. Hartshorne, Residues and Duality, Lecture Notes in Mathematics 20, Springer-Verlag, Berlin, 1966.

    Google Scholar 

  3. R. Thomas, “Derived categories for the working mathematician,” math.AG/0001045.

    Google Scholar 

  4. E. Sharpe, “Lectures on D-branes and sheaves,” lectures given at the twelfth Oporto meeting on “Geometry, topology, and physics,” hep-th/0307245.

    Google Scholar 

  5. A. Vistoli, “Intersection theory on algebraic stacks and on their moduli spaces,” Inv. Math. 97 (1989) 613–670.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. T. Gomez, “Algebraic stacks,” Proc. Indian Acad. Sci. Math. Sci. 111 (2001) 1–31, math.AG/9911199.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Kontsevich, “Homological algebra of mirror symmetry,” in Proceedings of the International Congress of Mathematicians, pp. 120–139, Birkhäuser, (1995), alg-geom/9411018.

    Google Scholar 

  8. J. Polchinski, “Dirichlet branes and Ramond-Ramond charges,” Phys. Rev. Lett. 75 (1995) 4724–4727, hep-th/9510017.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. J. Harvey and G. Moore, “On the algebras of BPS states,” Comm. Math. Phys. 197 (1998) 489–519, hep-th/9609017.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. S. Katz and E. Sharpe, “D-branes, open string vertex operators, and Ext groups,” Adv. Theor. Math. Phys. 6 (2003) 979–1030, hep-th/0208104.

    Google Scholar 

  11. S. Katz, T. Pantev, and E. Sharpe, “D-branes, orbifolds, and Ext groups,” Nucl. Phys. B673 (2003) 263–300, hep-th/0212218.

    Article  ADS  MathSciNet  Google Scholar 

  12. A. Caldararu, S. Katz, and E. Sharpe, “D-branes, B fields, and Ext groups,” Adv. Theor. Math. Phys. 7 (2004) 381–404, hep-th/0302099.

    MathSciNet  Google Scholar 

  13. A. Sen, “Tachyon condensation on the brane-antibrane system,” J. High Energy Phys. 08 (1998) 012, hep-th/9805170.

    Article  ADS  Google Scholar 

  14. E. Witten, “D-branes and K theory,” J. High Energy Phys. 9812 (1998) 019, hep-th/9810188.

    Article  MathSciNet  ADS  Google Scholar 

  15. E. Sharpe, “D-branes, derived categories, and Grothendieck groups,” Nucl. Phys. B561 (1999) 433–450, hep-th/9902116.

    Article  ADS  MathSciNet  Google Scholar 

  16. M. Douglas, “D-branes, categories, and cal n=1 supersymmetry,” J. Math. Phys. 42 (2001) 2818–2843, hep-th/0011017.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. T. Bridgeland, “Stability conditions on triangulated categories,” math.Annals of Math. 166 (2007) 317–345.

    Google Scholar 

  18. D. Freed and E. Witten, “Anomalies in string theory with D-branes,” hep-th/9907189.

    Google Scholar 

  19. A. Abouelsaood, C. Callan, C. Nappi, and S. Yost, “Open strings in background gauge fields,” Nucl. Phys. B280 (1987) 599–624.

    Article  ADS  MathSciNet  Google Scholar 

  20. T. Gomez and E. Sharpe, “D-branes and scheme theory,” hep-th/0008150.

    Google Scholar 

  21. R. Donagi, S. Katz, and E. Sharpe, “Spectra of D-branes with Higgs vevs,” Adv. Theor. Math. Phys. 8 (2005) 813–259, hep-th/0309270.

    MathSciNet  Google Scholar 

  22. R. Donagi, L. Ein, and R. Lazarsfeld, “A non-linear deformation of the Hitchin dynamical system,” alg-geom/9504017, a.k.a. “Nilpotent cones and sheaves on K3 surfaces,” pp. 51–61 in Birational Algebraic Geometry, Contemp. Math. 207, Amer. Math. Soc, Providence, Rhode Island, (1997).

    Google Scholar 

  23. P. Aspinwall and A. Lawrence, “Derived categories and zero-brane stability,” J. High Energy Phys. 0108 (2001) 004, hep-th/0104147.

    Article  ADS  MathSciNet  Google Scholar 

  24. A. Bondal and M. Kapranov, “Enhanced triangulated categories,” Math. USSR Sbornik 70 (1991) 93–107.

    Article  MATH  MathSciNet  Google Scholar 

  25. C. Lazaroiu, “Generalized complexes and string field theory,” JHEP 0106 (2001) 052, hep-th/0102122.

    Google Scholar 

  26. C. Lazaroiu, “Graded lagrangians, exotic topological D-branes and enhanced triangulated categories,” JHEP 0106 (2001) 064, hep-th/0105063.

    Google Scholar 

  27. E. Diaconescu, “Enhanced D-brane categories from string field theory,” JHEP 0106 (2001) 016, hep-th/0104200.

    Article  ADS  MathSciNet  Google Scholar 

  28. E. Witten, “Some comments on string dynamics,” contribution to proceedings of Strings ’95, hep-th/9507121.

    Google Scholar 

  29. M. Douglas, B. Greene, and D. Morrison, “Orbifold resolution by D-branes,” Nucl. Phys. B506 (1997) 84–106, hep-th/9704151.

    Article  MathSciNet  ADS  Google Scholar 

  30. E. Sharpe, “String orbifolds and quotient stacks,” Nucl. Phys. B627 (2002) 445–505, hep-th/0102211.

    Article  ADS  MathSciNet  Google Scholar 

  31. E. Witten, “Two-dimensional models with (0,2) supersymmetry: perturbative aspects,” hep-th/0504078.

    Google Scholar 

  32. E. Witten, “The N matrix model and gauged WZW models,” Nucl. Phys. B371 (1992) 191–245.

    Article  ADS  MathSciNet  Google Scholar 

  33. T. Pantev and E. Sharpe, “Notes on gauging noneffective group actions,” hep-th/0502027.

    Google Scholar 

  34. T. Pantev and E. Sharpe, “String compactifications on Calabi-Yau stacks,” Nucl. Phys. B733 (2006) 233–296, hep-th/0502044.

    Article  ADS  MathSciNet  Google Scholar 

  35. T. Pantev and E. Sharpe, “GLSM’s for gerbes (and other toric stacks),” Adv. Theor. Math. Phys. 10 (2006) 77–121, hep-th/0502053.

    MATH  MathSciNet  Google Scholar 

  36. S. Hellerman, A. Henriques, T. Pantev, E. Sharpe, and M. Ando, “Cluster decomposition, T-duality, and gerby CFT’s,” Adv. Theor. Math. Phys. 11 (2007) 751–818, hep-th/0606034.

    Google Scholar 

  37. E. Sharpe, “Discrete torsion,” Phys. Rev. D68 (2003) 126003, hep-th/0008154.

    Google Scholar 

  38. E. Sharpe, “Recent developments in discrete torsion,” Phys. Lett. B498 (2001) 104–110, hep-th/0008191.

    ADS  MathSciNet  Google Scholar 

  39. J. Distler and R. Plesser, private communication.

    Google Scholar 

  40. A. Kapustin, E. Witten, “Electric-magnetic duality and the geometric Langlands program,” hep-th/0604151.

    Google Scholar 

  41. M. Strassler, “Duality, phases, spinors, and monopoles in SO(n) and Spin(n) gauge theories,” hep-th/9709081.

    Google Scholar 

  42. D. Morrison and R. Plesser, “Towards mirror symmetry as duality for two-dimensional abelian gauge theories,” Nucl. Phys. Proc. Suppl. 46 (1996) 177–186, hep-th/9508107.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  43. K. Hori and C. Vafa, “Mirror symmetry,” hep-th/0002222.

    Google Scholar 

  44. L. Borisov, L. Chen, and G. Smith, “The orbifold Chow ring of toric Deligne-Mumford stacks,” math.AG/0309229.

    Google Scholar 

  45. R. Donagi and T. Pantev, “Langlands duality for Hitchin systems,” math.AG/0604617.

    Google Scholar 

  46. D. Abramovich, T. Graber, and A. Vistoli, “Algebraic orbifold quantum products,” math.AG/0112004.

    Google Scholar 

  47. T. Graber and J. Bryan, private communication.

    Google Scholar 

  48. D. Morrison and R. Plesser, “Summing the instantons: quantum cohomology and mirror symmetry in toric varieties,” Nucl. Phys. B440 (1995) 279–354, hep-th/9412236.

    Article  ADS  MathSciNet  Google Scholar 

  49. R. Karp, “C2/Z_n fractional branes and monodromy,” Comm. Math. Phys. 270 (2007) 163–196, hep-th/0510047.

    Google Scholar 

  50. R. Karp, “On the Cn/Z_m fractional branes,” hep-th/0602165.

    Google Scholar 

  51. C. Herzog and R. Karp, “On the geometry of quiver gauge theories (stacking exceptional collections),” hep-th/0605177.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departments of Physics, Mathematics, University of Utah Salt Lake City, UT 84112, USA

    E. Sharpe

Authors
  1. E. Sharpe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Sharpe .

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Sharpe, E. (2008). Derived Categories and Stacks in Physics. In: Homological Mirror Symmetry. Lecture Notes in Physics, vol 757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68030-7_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-68030-7_8

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-68029-1

  • Online ISBN: 978-3-540-68030-7

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation