Luiz A. Ferreira- Solitons and Integrable Field Theories

8/3/2019 Luiz A. Ferreira- Solitons and Integrable Field Theories

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Solitons and Integrable Field Theories

Luiz A. Ferreira

Instituto de Fsica de Sao Carlos, USP, [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 possibilityLook for operators that evolve as ( isospectral evolution )

W (t) = U W (0) U1

Eigenvalues are conserved charges

W (t) | = | d

d t= 0


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A standard resultA : Flat connection on space-time ( F = 0 )

d W

d s + Ad x

d s W = 0 W = P e

dsA dxd s

WCL WC0 = WCt WCL WCt = WCL WC0 W1CL


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2D Integrable Theories

The sine-Gordon case

A+ = ei T+ + e

i T

A = i T3 +2

16

T +

1

T+

with

[ T3 , T ] = T [ T+ , T ] = 2 T3

The flat connection condition implies the sine-Gordon eq.

F+ = [+ + A+ , + A] =i

4

2 +

2

sin()

T3


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The role of loop spaces

Introduce flat 1-form connection A in loop space

F = A + A A = 0

and get isospectral evolution

W(t) = UW(0) U1 W = PeA


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Implementing it

For a space-time M of d + 1 dimensions introduce the loop spacea

d1 (M, x0) = { : Sd1 M | (0) = x0}

Build the loop space connection A out of a tensor B1...d , and a

connection A defined on M, as

A

W1 B1...d W d1...d1 xd

with being a d 1 dimensional surface and xd variationsperpendicular to it. With W same as before.

a Orlando Alvarez, L.A.F. e J. Sanchez Guillen,

hep-th/9710147, Nucl. Phys. B529 (1998) 689-736


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Locality in space-time requires

A = 0 and A A = 0

which in space-time becomes

F = 0 D B = 0

with A T and B1...d P

[ T , T ] T

[ T , P ] P

[ P , P ] = 0

Conserved currents are

J = 1...d W1 B1...d W J = 0


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Where it works

Self-dual Yang-Mills Bogomolny equations

-models in any d

CP1 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

S =

d4x H2

where

H

= 2i(uu

uu)

(1+ | u |2)2= n (

n

n)

with n S2 and u is its stereographic projection

n = u + u, i (u u) , | u |2 1 / 1+ | u |2It is invariant under

Area preserving diffeomorphisms on S2

Conformal symmetry - SO(4, 2)


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Its eq. of motion admit the zero curvature representation

A =1

(1+ | u |2)(iu T+ iu

T + (uu uu) T3)

B(j) =1

(1+ | u |2)

K P(j)1 K

P

(j)1

; K H u

based on the sl(2) algebra

[T3

, T] = T , [T+

, T] = 2 T3

and where P(j)m transforms under integer spin j representations of

sl(2) (m = j, j + 1, . . . , j 1, j)

[T3 , P(j)m ] = m P(j)m

[T , P(j)m ] =

j(j + 1) m(m 1) P

(j)m1

[P(j)m , P(j)m ] = 0


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Connection with Yang-Mills Theory

Cho-Faddeev-Niemi decomposition

A = C n + n n + n + n n

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

S = m

2 d

4

x (n)

2

+

1

e2

d

4

x H

2

+ four derivatives terms

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

A + A A = 0

and not

A = 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. Sanchez Guillen; hep-th/9710147, Nucl.

Phys. B529 (1998) 689-736.

H. Aratyn, LAF, A.H. Zimerman; hep-th/9905079, Physical Re-

view 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

Luiz A. Ferreira- Solitons and Integrable Field Theories

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