Skip to main content

Advertisement

SpringerLink
Go to cart
  1. Home
  2. Communications in Mathematical Physics
  3. Article
On the Extension of Stringlike Localised Sectors in 2+1 Dimensions
Download PDF
Your article has downloaded

Similar articles being viewed by others

Slider with three articles shown per slide. Use the Previous and Next buttons to navigate the slides or the slide controller buttons at the end to navigate through each slide.

Tube algebras, excitations statistics and compactification in gauge models of topological phases

21 October 2019

Alex Bullivant & Clement Delcamp

SW 3 2 2 $$ \mathcal{SW}\left(\frac{3}{2},2\right) $$ subsymmetry in G2, Spin(7) and N $$ \mathcal{N} $$ = 2 CFTs

28 July 2020

Marc-Antoine Fiset

From torus bundles to particle–hole equivariantization

18 February 2022

Shawn X. Cui, Paul Gustafson, … Qing Zhang

Unifying the 6D N $$ \mathcal{N} $$ = (1, 1) string landscape

20 February 2023

Bernardo Fraiman & Héctor Parra De Freitas

Construction of two-dimensional topological field theories with non-invertible symmetries

07 December 2021

Tzu-Chen Huang, Ying-Hsuan Lin & Sahand Seifnashri

Log-Modular Quantum Groups at Even Roots of Unity and the Quantum Frobenius I

22 February 2021

Cris Negron

Extended Quantum Field Theory, Index Theory, and the Parity Anomaly

12 June 2018

Lukas Müller & Richard J. Szabo

Cardy Algebras, Sewing Constraints and String-Nets

24 January 2022

Matthias Traube

Recent developments in topological string theory

24 April 2019

Min-Xin Huang

Download PDF
  • Open Access
  • Published: 02 February 2011

On the Extension of Stringlike Localised Sectors in 2+1 Dimensions

  • Pieter Naaijkens1 

Communications in Mathematical Physics volume 303, pages 385–420 (2011)Cite this article

  • 401 Accesses

  • 2 Citations

  • Metrics details

Abstract

In the framework of algebraic quantum field theory, we study the category \({\Delta_{{\rm BF}}^{\mathfrak{A}}}\) of stringlike localised representations of a net of observables \({\mathcal{O} \mapsto \mathfrak{A}(\mathcal{O})}\) in three dimensions. It is shown that compactly localised (DHR) representations give rise to a non-trivial centre of \({\Delta_{{\rm BF}}^{\mathfrak{A}}}\) with respect to the braiding. This implies that \({\Delta_{{\rm BF}}^{\mathfrak{A}}}\) cannot be modular when non-trivial DHR sectors exist. Modular tensor categories, however, are important for topological quantum computing. For this reason, we discuss a method to remove this obstruction to modularity.

Indeed, the obstruction can be removed by passing from the observable net \({\mathfrak{A}(\mathcal{O})}\) to the Doplicher-Roberts field net \({\mathfrak{F}(\mathcal{O})}\). It is then shown that sectors of \({\mathfrak{A}}\) can be extended to sectors of the field net that commute with the action of the corresponding symmetry group. Moreover, all such sectors are extensions of sectors of \({\mathfrak{A}}\). Finally, the category \({\Delta_{{\rm BF}}^{\mathfrak{F}}}\) of sectors of \({\mathfrak{F}}\) is studied by investigating the relation with the categorical crossed product of \({\Delta_{{\rm BF}}^{\mathfrak{A}}}\) by the subcategory of DHR representations. Under appropriate conditions, this completely determines the category \({\Delta_{{\rm BF}}^{\mathfrak{F}}}\).

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Araki H.: von Neumann algebras of local observables for free scalar field. J. Math. Phys. 5, 1–13 (1964)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Bakalov, B., Kirillov, A. Jr.: Lectures on tensor categories and modular functors. Volume 21 of University Lecture Series. Providence, RI: Amer. Math. Soc., 2001

  3. Böckenhauer J., Evans D.E.: Modular invariants, graphs and α-induction for nets of subfactors. I. Commun. Math. Phys. 197(2), 361–386 (1998)

    Article  MATH  ADS  Google Scholar 

  4. Borchers H.-J.: A remark on a theorem of B. Misra. Commun. Math. Phys. 4, 315–323 (1967)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Bruguières A.: Catégories prémodulaires, modularisations et invariants des variétés de dimension 3. Math. Ann. 316(2), 215–236 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Buchholz D.: The physical state space of quantum electrodynamics. Commun. Math. Phys. 85(1), 49–71 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Buchholz D., Fredenhagen K.: Locality and the structure of particle states. Commun. Math. Phys. 84(1), 1–54 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Buchholz D., Haag R.: The quest for understanding in relativistic quantum physics. J. Math. Phys. 41(6), 3674–3697 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  9. Conti R., Doplicher S., Roberts J.E.: Superselection theory for subsystems. Commun. Math. Phys. 218(2), 263–281 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. D’Antoni, C.: Technical properties of the quasi-local algebra. In: Kastler [29], pp. 248–258

  11. Deligne, P.: Catégories tannakiennes. In: The Grothendieck Festschrift, Vol. II, Volume 87 of Progr. Math., Boston, MA: Birkhäuser Boston, 1990, pp. 111–195

  12. Doplicher S., Haag R., Roberts J.E.: Fields, observables and gauge transformations. I. Commun. Math. Phys. 13, 1–23 (1969)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics. I. Commun. Math. Phys. 23, 199–230 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  14. Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics. II. Commun. Math. Phys. 35, 49–85 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  15. Doplicher S., Roberts J.E.: Fields, statistics and non-abelian gauge groups. Commun. Math. Phys. 28, 331–348 (1972)

    Article  ADS  MathSciNet  Google Scholar 

  16. Doplicher S., Roberts J.E.: Endomorphisms of C*-algebras, cross products and duality for compact groups. Ann. Math. (2) 130(1), 75–119 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Doplicher S., Roberts J.E.: A new duality theory for compact groups. Invent. Math. 98(1), 157–218 (1989)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. Doplicher S., Roberts J.E.: Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131(1), 51–107 (1990)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. Fredenhagen, K.: Generalizations of the theory of superselection sectors. In: Kastler [29], pp. 379–387

  20. Fredenhagen K., Rehren K.-H., Schroer B.: Superselection sectors with braid group statistics and exchange algebras. I. General theory. Commun. Math. Phys. 125(2), 201–226 (1989)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. Fredenhagen K., Rehren K.-H., Schroer B.: Superselection sectors with braid group statistics and exchange algebras. II. Geometric aspects and conformal covariance. Rev. Math. Phys. 4(Special Issue), 113–157 (1992)

    Article  MathSciNet  Google Scholar 

  22. Freedman, M.H., Kitaev, A., Larsen, M.J., Wang, Z.: Topological quantum computation. From: Mathematical challenges of the 21st century (Los Angeles, CA, 2000) Bull. Amer. Math. Soc. (N.S.), 40(1), 31–38 (2003)

  23. Freedman M.H., Larsen M., Wang Z.: A modular functor which is universal for quantum computation. Commun. Math. Phys. 227(3), 605–622 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. Fröhlich J., Gabbiani F.: Braid statistics in local quantum theory. Rev. Math. Phys. 2(3), 251–353 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  25. Guido D., Longo R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys. 148(3), 521–551 (1992)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  26. Haag, R.: Local quantum physics: Fields, particles, algebras. Texts and Monographs in Physics. Berlin: Springer-Verlag, Second edition, 1996

  27. Halvorson, H.: Algebraic quantum field theory. In: Butterfield, J., Earman, J. (eds.) Philosophy of Physics, London: Elsevier, 2006, pp. 731–922

  28. Hewitt, E., Ross, K.A.: Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups. Die Grundlehren der mathematischen Wissenschaften, Band 152. New York: Springer-Verlag, 1970

  29. Kastler, D. (ed.): The algebraic theory of superselection sectors: Introduction and recent results, River Edge, NJ: World Scientific Publishing Co. Inc., 1990

  30. Kawahigashi Y., Longo R., Müger M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219(3), 631–669 (2001)

    Article  MATH  ADS  Google Scholar 

  31. Kitaev A.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303(1), 2–30 (2003)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. Kitaev A.: Anyons in an exactly solved model and beyond. Ann. Phys 321(1), 2–111 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. Kowalzig, N.: Hopf Algebroids and Their Cyclic Theory. PhD thesis, Universiteit van Amsterdam and Universiteit Utrecht, 2009

  34. Longo R.: Index of subfactors and statistics of quantum fields. I. Commun. Math. Phys. 126(2), 217–247 (1989)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  35. Longo R.: Index of subfactors and statistics of quantum fields. II. Correspondences, braid group statistics and Jones polynomial. Commun. Math. Phys. 130(2), 285–309 (1990)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. Longo R., Roberts J.E.: A theory of dimension. K-Theory 11(2), 103–159 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  37. Mac Lane, S.: Categories for the working mathematician, Volume 5 of Graduate Texts in Mathematics. New York: Springer-Verlag, second edition, 1998

  38. Müger, M.: Abstract duality for symmetric tensor *-categories. Appendix to [27]

  39. Müger M.: On charged fields with group symmetry and degeneracies of Verlinde’s matrix S. Ann. Inst. H. Poincaré Phys. Théor. 71(4), 359–394 (1999)

    MATH  Google Scholar 

  40. Müger M.: Galois theory for braided tensor categories and the modular closure. Adv. Math. 150(2), 151–201 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  41. Müger M.: Conformal orbifold theories and braided crossed G-categories. Commun. Math. Phys. 260(3), 727–762 (2005)

    Article  MATH  ADS  Google Scholar 

  42. Mund J.: Borchers’ commutation relations for sectors with braid group statistics in low dimensions. Ann. Henri Poincaré 10(1), 19–34 (2009)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  43. Mund J.: The spin-statistics theorem for anyons and plektons in d = 2+1. Commun. Math. Phys. 286(3), 1159–1180 (2009)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  44. Naaijkens, P.: Localized endomorphisms in Kitaev’s toric code on the plane. Preprint arXiv:1012.3857

  45. Nayak C., Simon S.H., Stern A., Freedman M., Das Sarma S.: Non-abelian anyons and topological quantum computation. Rev. Mod. Phys. 80(3), 1083–1159 (2008)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  46. Panangaden, P., Paquette, É.: A categorical presentation of quantum computation with anyons. In: Coecke, B. (ed.), New structures for Physics, Lecture Notes in Physics. Berlin-Heidelberg-New York: Springer (2011)

  47. Rehren, K.-H.: Braid group statistics and their superselection rules. In: Kastler [29], pp. 333–355

  48. Rehren, K.-H.: Markov traces as characters for local algebras. From Recent advances in field theory (Annecy-le-Vieux, 1990), Nucl. Phys. B Proc. Suppl., 18B, 259–268 (1991)

  49. Rehren K.-H.: Field operators for anyons and plektons. Commun. Math. Phys. 145(1), 123–148 (1992)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  50. Roberts, J.E.: Cross products of von Neumann algebras by group duals. In: Symposia Mathematica, Volume XX, London: Academic Press, 1976, pp. 335–363

  51. Roberts J.E.: Local cohomology and superselection structure. Commun. Math. Phys. 51(2), 107–119 (1976)

    Article  MATH  ADS  Google Scholar 

  52. Roberts, J.E.: Lectures on algebraic quantum field theory. In: Kastler [29], pp. 1–112

  53. Saavedra Rivano, N.: Catégories Tannakiennes. Lecture Notes in Mathematics, Vol. 265. Berlin: Springer-Verlag, 1972

  54. Sutherland C.E.: Cohomology and extensions of von Neumann algebras. II. Publ. Res. Inst. Math. Sci. 16(1), 135–174 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  55. Szlachányi K., Vecsernyés P.: Quantum symmetry and braid group statistics in G-spin models. Commun. Math. Phys. 156(1), 127–168 (1993)

    Article  MATH  ADS  Google Scholar 

  56. Turaev, V.G.: Quantum invariants of knots and 3-manifolds. Volume 18 of de Gruyter Studies in Mathematics. Berlin: Walter de Gruyter & Co., 1994

  57. Verlinde E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300(3), 360–376 (1988)

    Article  MATH  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research is funded by NWO grant no. 613.000.608, which is gratefully acknowledged. I would also like to thank Michael Müger for valuable discussions and suggestions, and Klaas Landsman for a critical reading of the manuscript.

Open Access

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

Author information

Authors and Affiliations

  1. Radboud University Nijmegen, Institute for Mathematics, Astrophysics and Particle Physics, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands

    Pieter Naaijkens

Authors
  1. Pieter Naaijkens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pieter Naaijkens.

Additional information

Communicated by Y. Kawahigashi

Rights and permissions

Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Reprints and Permissions

About this article

Cite this article

Naaijkens, P. On the Extension of Stringlike Localised Sectors in 2+1 Dimensions. Commun. Math. Phys. 303, 385–420 (2011). https://doi.org/10.1007/s00220-011-1200-6

Download citation

  • Received: 30 April 2010

  • Accepted: 09 September 2010

  • Published: 02 February 2011

  • Issue Date: April 2011

  • DOI: https://doi.org/10.1007/s00220-011-1200-6

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Tensor Category
  • Double Cone
  • Algebraic Quantum
  • Superselection Sector
  • Tensor Functor
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Over 10 million scientific documents at your fingertips

Switch Edition
  • Academic Edition
  • Corporate Edition
  • Home
  • Impressum
  • Legal information
  • Privacy statement
  • Your US state privacy rights
  • How we use cookies
  • Your privacy choices/Manage cookies
  • Accessibility
  • FAQ
  • Contact us
  • Affiliate program

Not affiliated

Springer Nature

© 2023 Springer Nature Switzerland AG. Part of Springer Nature.