quantum equations of motion in bf theory with sources n. ilieva a. alekseev
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Quantum equations of motion in BF theory with
sources
N. Ilieva
A. Alekseev
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Introduction
Two-dimensional BF theorygauge 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 (z1,…, zn) Expectations of quantum A and B fields; quantum eqs. regularization at (z1,…, zn)
VP of gauge theory Sources at (z1,…, zn) Expectations of quantum A and B fields; quantum eqs. regularization at (z1,…, zn) A = (A reg (z1,), …, A reg
(zn) ) connection on the space of configs. of points (z1,…, zn)
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Classical action and equations of motion
G connected Lie group; Gtr(ab) inv scalar product on GA gauge field on the G-bundle P over
Mn
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
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Classical action and equations of motion
Propagator M; gauge choiceTriple vertex f abc
Connected Feynman diagrams:
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 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
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Feynman diagrams
Short trees [T(l=1)]
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Quantum equations of motion
B - [A,B]
Coincides with the classical eqn.
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Quantum equations of motion
z1
zi
zn
u
z1
zi
zn
uu
• •
•
• • •
Are
g
- ½[A,A]
Asing
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
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
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Quantum flat connection
Quantum equation of motion for B field
Naïve expectation for Kη (z1,…, zn) equation ( dz being the de Rham
diff.)
Contributing Feynman
graphs
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Quantum flat connection
d - the total de Rham differentialfor 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
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Quantum flat connection
j i : A(u) → A(j)reg
uzj does not contribute
u = zj
Similarly for the differential of gauge field A
holomorphic components anti-holomorphic components mixed components
(Fij corresponds to {zi, zj})
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Quantum flat connection
The curvatute F of A vanishes.
(Fij )k , k = 1,… , n
(i) Components with k i, j vanish identically
(ii)
(iii) The only non-vanishing component is (Fii )i
with
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Quantum flat connection
vani
sh source
term
covariant-derivative
term
extra zi dependence due to the tree
root
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Quantum flat connection
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Outlook
Torossian connection
Knizhnik-Zamolodchikov connection / WZW
jiji eet , 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
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Quantum Fields in BF Theory Quantum Fields in BF Theory
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Quantum Fields in BF Theory Quantum Fields in BF Theory M. Mateev Memorial Conference,M. Mateev Memorial Conference,Sofia, 10-12.IV.2011Sofia, 10-12.IV.2011
Nevena Ilieva
Feynman diagrams