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

2. Quantum Fields in BF Theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011 Nevena Ilieva Introduction Two-dimensional BF theory gauge theory  star-product  Kontsevich approach  Torossian connection (correl. of B-exponentials)  star-product  Kontsevich approach  Torossian connection (correl. of B-exponentials)  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  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 (topological) Poisson  -model  VP of gauge theory  Sources at (z 1,…, z n )  Expectations of quantum A and B fields; quantum eqs.  regularization at (z 1,…, z n )  VP of gauge theory  Sources at (z 1,…, z n )  Expectations of quantum A and B fields; quantum eqs.  regularization at (z 1,…, z n ) A = ( A reg (z 1,), …, A reg (z n ) ) connection on the space of configs. of points (z 1,…, z n )

3. Quantum Fields in BF Theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011 Nevena Ilieva 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 M n B G -valued n-2 form  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  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

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

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

6. Quantum Fields in BF Theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011 Nevena Ilieva Feynman diagrams Short trees [T(l=1)]

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

8. Quantum Fields in BF Theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011 Nevena Ilieva Quantum equations of motion z1z1 zizi znzn u z1z1 zizi znzn u A reg - ½ [ A,A ] A sing

9. Quantum Fields in BF Theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011 Nevena Ilieva Quantum flat connection Dependence of the B -field correlators K η (z 1,…, z n ) on (z 1,…, z n ) Singularity → regularization by a new splitting no singularity at u = z i

10. Quantum Fields in BF Theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011 Nevena Ilieva Quantum flat connection Quantum equation of motion for B field Naïve expectation for K η (z 1,…, z n ) equation ( d z being the de Rham diff.) Contributing Feynman graphs

11. Quantum Fields in BF Theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011 Nevena Ilieva Quantum flat connection d - the total de Rham differential for all variables z 1,…, z n Consider functions α i (η 1,…, η n )  G, i = 1,…, n Lie algebra 1-forms (a 1,…, a n ) as components of a connection A (a 1,…, a n ), with values in this LA

12. Quantum Fields in BF Theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011 Nevena Ilieva Quantum flat connection j  i : A (u) → A (j) reg u zjzj does not contribute u = z j Similarly for the differential of gauge field A  holomorphic components  anti-holomorphic components  mixed components ( F ij corresponds to { z i, z j } )

13. Quantum Fields in BF Theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011 Nevena Ilieva Quantum flat connection The curvatute F of A vanishes. ( F ij ) k, k = 1,…, n (i) Components with k  i, j vanish identically (ii) (iii) The only non-vanishing component is ( F ii ) i with

14. Quantum Fields in BF Theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011 Nevena Ilieva Quantum flat connection vanish source term covariant-derivative term extra z i dependence due to the tree root

15. Quantum Fields in BF Theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011 Nevena Ilieva Quantum flat connection

16. Quantum Fields in BF Theory M. Mateev Memorial Conference, Sofia, 10-12.IV.2011 Nevena Ilieva Outlook Torossian connection Knizhnik-Zamolodchikov connection / WZW irreps of G play the role of T(l=1) propagator d ln (z i -z j )/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 A KZ → 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

17. Quantum Fields in BF Theory

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