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Solitons and Integrable Field Theories
Luiz A. Ferreira Instituto de F´ısica de S˜ ao Carlos, USP, Brazil
[email protected]
Cambridge, 11th December 2007
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Searching for (hidden) symmetries • Exact results require high level of symmetries • Local field theories are subjected to Coleman-Mandula • Supersymmetry is an improvement • Flat connections on loop spaces may be a possibility Look for operators that evolve as ( isospectral evolution ) W (t) = U W (0) U −1 Eigenvalues are conserved charges W (t) | λ i = λ | λ i
→
dλ =0 dt
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A standard result Aµ : Flat connection on space-time ( Fµν = 0 ) dW d xµ + Aµ W =0 ds ds
WCL WC0 = WCt WC−L
→
→
W =Pe
−
R
ds Aµ
d xµ ds
WCt = WCL WC0 WC−1 −L
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2D Integrable Theories The sine-Gordon case A+ A−
= eiβφ T+ + e−iβφ λ T− µ2 1 T− + T+ = −iβ∂− φ T3 + 16 λ
with [ T3 , T± ] = ± T ±
[ T+ , T− ] = 2 T 3
The flat connection condition implies the sine-Gordon eq. 2 iβ µ ∂2φ + sin (βφ) T3 F+− = [∂+ + A+ , ∂− + A− ] = 4 β
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The role of loop spaces
Introduce flat 1-form connection A in loop space F = δA + A ∧ A = 0 and get isospectral evolution W (t) = U W (0) U −1
↔
W =Pe
R
A
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Implementing it For a space-time M of d + 1 dimensions introduce the loop spacea Ωd−1 (M, x0 ) = {γ : S d−1 → M | γ(0) = x0 } Build the loop space connection A out of a tensor Bµ1 ...µd , and a connection Aµ defined on M , as Z A≡ W −1 Bµ1 ...µd W dΣµ1 ...µd−1 δxµd Σ
with Σ being a d − 1 dimensional surface and δxµd variations perpendicular to it. With W same as before. Orlando Alvarez, L.A.F. e J. S´ anchez Guill´ en, hep-th/9710147, Nucl. Phys. B529 (1998) 689-736 a
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Locality in space-time requires δA = 0
and
A∧A=0
which in space-time becomes Fµν = 0
D∧B =0
with Aµ ∈ T and Bµ1 ...µd ∈ P [T , T ] ⊂ T [T , P ] ⊂ P [P , P ] = 0 Conserved currents are J µ = εµ µ1 ...µd W −1 Bµ1 ...µd W
→
∂ µ Jµ = 0
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Where it works • Self-dual Yang-Mills • Bogomolny equations • σ-models in any d • CP 1 and submodel (baby skyrmions) • Skyrme-like models (hopfions) • etc In general “integrability” is accompanied by conformal symmetry. However, we are far from the structures of 2d integrable theories.
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An integrable theory in 4d Consider the following theory in 3 + 1 Minkowski space-time Z 2 S = − d4 x Hµν where Hµν = −2i
(∂µ u∂ν u∗ − ∂ν u∂µ u∗ ) (1+ | u
| 2 )2
= ~n · (∂µ~n ∧ ∂ν ~n)
with ~n ∈ S 2 and u is its stereographic projection ∗ ∗ 2 2 ~n = u + u , −i (u − u ) , | u | −1 / 1+ | u | It is invariant under
• Area preserving diffeomorphisms on S 2 • Conformal symmetry - SO(4, 2)
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Its eq. of motion admit the zero curvature representation 1 ∗ ∗ ∗ Aµ = u T − i∂ u T + (u∂ u − u ∂µ u) T3 ) (−i∂ µ + µ − µ (1+ | u |2 ) 1 (j) (j) ∗ ν ˜µ(j) = K ; K P − K P ≡ H ∂ u B µ µ µν µ 1 −1 (1+ | u |2 ) based on the sl(2) algebra [T3 , T± ] = ± T± ,
[T+ , T− ] = 2 T3
(j)
and where Pm transforms under integer spin j representations of sl(2) (m = −j, −j + 1, . . . , j − 1, j) (j) (j) [T3 , Pm ] = m Pm p (j) (j) [T± , Pm ] = j(j + 1) − m(m ± 1) Pm±1
(j) [Pm
,
(j 0 ) P m0 ]
= 0
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Connection with Yang-Mills Theory Cho-Faddeev-Niemi decomposition ~ µ = Cµ ~n + ∂µ~n ∧ ~n + ρ ∂µ~n + σ ∂µ~n ∧ ~n A Conjecture: At low energies all degrees of freedom are frozen except ~n. The effective Lagrangean of SU (2) YM theory whithout matter is then an extension of the Skyrme-Faddeev modela Z Z 1 2 2 S = m2 d4 x (∂µ~n) + 2 + four derivatives terms d4 x Hµν e a
L.D. Faddeev, A. Niemi, hep-th/9610193, Nature 387:58,1997
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The hidden structures Locality has to be replaced by a weaker condition We need δA + A ∧ A = 0 and not δA = 0
and
A∧A=0
Fundamental objects: not Particles
→
but Fluxes
From Maxwell back to Faraday
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References • Orlando Alvarez, LAF, J. S´anchez Guill´en; hep-th/9710147, Nucl. Phys. B529 (1998) 689-736. • H. Aratyn, LAF, A.H. Zimerman; hep-th/9905079, Physical Review Letters 83 (1999) 1723-1726. • LAF, J. Sanchez-Guillen; hep-th/0010168, Phys. Lett. B. B504 (2001) 195-200. • LAF, A.V. Razumov; hep-th/0012176, Letters in Mathematical Physics 55 (2001) 143-148. • O. Babelon, LAF; hep-th/0210154, JHEP11(2002)020 • LAF; hep-th/0406227, Physics Letters B606 (2005) 417-422. • LAF; hep-th/0601235, JHEP03(2006)075 • A. C. R. do Bonfim, LAF; hep-th/0602234, JHEP03(2006)097
