# Derivation of non-local conserved charges

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.

• This is the celebrated nonlocal Yangian construction. It is reviewed in, e.g. Curtright & Zachos 1992, i.e.. I assume you can easily check conservation of the charge you wrote down? Commented May 17, 2018 at 1:38
• @CosmasZachos I have copied them, and deleted my answer, will try to make a simple answer, so you may delete these since they are irrelevant to the question, thanks Commented May 17, 2018 at 14:36
• Commented May 17, 2018 at 18:31

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 jμ, 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.
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.