1 / 18

Quantum equations of motion in BF theory with sources

Quantum equations of motion in BF theory with sources. N. Ilieva. A. Alekseev. Introduction. gauge theory. Two-dimensional BF theory. (topological) Poisson -model.  star-product  Kontsevich approach  Torossian connection (correl. of B-exponentials).  VP of gauge theory

gavan
Download Presentation

Quantum equations of motion in BF theory with sources

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Quantum equations of motion in BF theory with sources N. Ilieva A. Alekseev

  2. Introduction gauge theory Two-dimensional BF theory (topological) Poisson -model  star-product  Kontsevich approach  Torossian connection (correl. of B-exponentials) •  VP of gauge theory • Sources at (z1,…, zn) • Expectations of quantum A and B fields; quantum eqs. • regularization at(z1,…, zn) A =(Areg (z1,), …, Areg (zn) ) connection on the space of configs. of points (z1,…, zn) • A. Cattaneo, J. Felder, Commun. Math. Phys. 212 (2000) 591–611 • M. Kontsevich, Lett. Math. Phys. 66 (2003) 157–216 • C. Torossian, J. Lie Theory 12 (2001) 597-616 M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  3. Classical action and equations of motion G connected Lie group; G tr(ab) inv scalar product on G A gauge field on the G-bundle P over Mn B G -valuedn-2form • E. Witten, CMP 117 (1988) 353; 118 (1988) 411; 121 (1989) 351 • A.S. Schwarz, Lett. Math. Phys. 2 (1978) 247–252 • D. Birmingham et al., Phys. Rep.209 (1991) 129-340 M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  4. Classical action and equations of motion Propagator  M; gauge choice Triple vertex  f abc Connected Feynman diagrams: M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  5. BF theory with sources Correlation function / theory without sources Partition function / theory with sources O:A(u), B(u)  not gauge-inv observables  source term breaks the gauge inv of the action M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  6. Feynman diagrams Short trees [T(l=1)] M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  7. Quantum equations of motion B -[A,B] Coincides with the classical eqn. M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  8. z1 z1 • •• u u zi zi • • • zn zn Quantum equations of motion Areg - ½[A,A] Asing M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  9. Quantum flat connection Dependence of the B-field correlators Kη (z1,…, zn) on (z1,…, zn) Singularity → regularization by a new splitting no singularity at u = zi M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  10. Quantum flat connection Quantum equation of motion for B field Naïve expectation for Kη (z1,…, zn) equation (dzbeing the de Rham diff.) Contributing Feynman graphs M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  11. Quantum flat connection d - the total de Rham differential for all variables z1,…, zn Consider functions αi (η1,…, ηn)  G, i = 1,…, n Lie algebra 1-forms (a1,…, an) as components of a connection A (a1,…, an), with values in this LA M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  12. Quantum flat connection Similarly for the differential of gauge field A zj u j i : A(u) → A(j)reg does not contribute u = zj • holomorphic components • anti-holomorphic components • mixed components • (Fij corresponds to {zi, zj}) M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  13. Quantum flat connection The curvatute FofA vanishes. (Fij)k , k = 1,… , n (i) Components withk  i, j vanish identically (ii) (iii) The only non-vanishing component is (Fii)i with M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  14. Quantum flat connection vanish source term covariant-derivative term extra zidependence due to the tree root M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  15. Quantum flat connection M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  16. Outlook Torossian connection Knizhnik-Zamolodchikov connection / WZW irreps of G play the role of T(l=1) propagator d ln (zi-zj)/2πi • KZ connection as eqn on the wave funtion of the CS TFT with n time-like Wilson lines (corr. primary fields) • holonomy matrices of the flat connection AKZ→ braiding of Wilson lines • 3D TFT with TC as a wave function eqn (?) • non-local observables → insertions of exp(tr ηB(z)) in 2D theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

  17. Thank you!

  18. Feynman diagrams M. Mateev Memorial Conference, Sofia, 10-12.IV.2011

More Related