Derivation of non-local conserved charges

Question

Consider a 2D sigma model with a symmetry group $G$ and whose generators obey $$[T_A, T_B] = f^C_{AB} T_C$$ and whose conserved currents are Lie algebra-valued, i.e. $$j_{\mu} = j_{\mu}^A T_A$$ and $$\partial_{\mu} j^{\mu A} = 0.$$

From this it is straightforward to construct a conserved quantity $$Q^A = \int_{-\infty}^{\infty} j^{0A} (x,t) dx$$ where $$\dot{Q}^A = 0.$$

In integrable systems, we can generate an infinite amount of additional, non-local conserved charges if the current satisfies the flatness condition $$ \partial_{\mu} j_{\nu} - \partial_{\nu} j_{\mu} + [j_{\mu}, j_{\nu} ] = 0 ~, $$ where indices are raised/lowered with a Lorentzian metric. The first of these non-local charges is given as

$$ Q^{A~(1)} = f^A_{BC} \int_{-\infty}^{\infty} dx \int_x^{\infty} dy \; j^{0B}(x,t) j^{0C}(y,t) - 2 \int_{-\infty}^{\infty} dx \; j_1^A(x,t) ~. $$

I don't understand how this result is obtained from the flatness condition, and I can't find a derivation anywhere (the result is just stated, e.g. in section 2 of http://arxiv.org/abs/hep-th/0308089 and equations (11-12) of http://arxiv.org/abs/hep-th/0404003).

Any help would be appreciated.

Answer 1

Your sources are particularly unhelpful, bibliographically speaking. The original recursive construction of these nonlocal currents is, of course, E Brézin, C Itzykson, J Zinn-Justin, J-B Zuber, Phys Lett B 82 (1979) 442-444, (https://doi.org/10.1016/0370-2693(79)90263-6 ) . But, by now, the Yangian structure of them has been streamlined much better -- see below.

To start with, reassure yourself that, indeed, $\dot{Q}^{(1)}$ vanishes, since it is the charge of the first non-local current, $$ j^{(1)}_\mu(x)=\tilde {j}_\mu(x) +\tfrac{1}{2}[~j_\mu (x), \int_{-\infty}^{x} \!\!dy ~~ j_0(y)], $$ where I have reverted to Lie-Algebra-valued currents j_μ, as in your flatness condition, $$ (\partial_\mu + j_{\mu} ) ~\tilde{j}_\mu =0~. $$ By virtue of this flatness condition and conservation, in 2 dimensions, you then see that $$ \partial^\mu \int_{-\infty}^{x} dy ~~j_0(y) =\tilde{j}^\mu . $$

It is then straightforward to see, given the conservation of $j_\mu$, that $$ \partial^\mu j_\mu^{(1)}= \partial\cdot \tilde{j}+ \tfrac{1}{2}[j_\mu (x), \partial^\mu\int_{-\infty}^{x} dy ~~j_0(y)]= \tfrac{1}{2}[~j_\mu (x),\tilde{j}^\mu ] - \tfrac{1}{2}[j_\mu (x), \tilde{j}_\mu] =0~, $$ leading to the time invariance of the corresponding charge.

It is much more efficient to think of all these nonlocal currents as just one conserved master current which, however, depends on an arbitrary spectral parameter θ; so a generating function. The coefficients of each power of θ will then be separately conserved, and the 0th order will be the starting local current, the term linear in θ the above nonlocal current, and so on ad infinitum, for increasingly nonlocal currents. See T Curtright & C Zachos, Nucl Phys B 402 (1993) 604-612 (https://doi.org/10.1016/0550-3213(93)90120-E) .

First define (with Polyakov), $$ C_\mu (x,\theta)= \tfrac{1}{2} (1-\cosh \theta)~ j_\mu -\tfrac{1}{2}\sinh\theta ~\tilde{j}_\mu \\ K_\mu (x,\theta)=\cosh \theta ~j_\mu+\sinh \theta ~\tilde{j}_\mu \\ \chi (x,\theta)= \operatorname {P} e^{-\int_{-\infty}^x dy C_1 (y,\theta)}, $$ the last a path ordered exponential. These are seen to satisfy $$ (\partial^\mu+C^{\mu}) \tilde {C} _\mu= 0\\ \partial^\mu K_\mu +[C^\mu , K_\mu ]=0\\ \partial_\mu \chi=-C_\mu \chi . $$

Given these, you may prove conservation of the master Yangian current, $$ {\cal J}_\mu(x,\theta)=\chi^{-1} K_\mu ~\chi= j_\mu+\theta ~j^{(1)}_\mu+… $$ So the above conservation proof of the first nonlocal current is but the warmup for the conservation proof of all nonlocal currents.