Gauge fields in Polyakov's treatment of renormalization for nonlinear sigma model

I am deriving the results of renormalization for nonlinear sigma model using Polyakov approach. I am closely following chapter 2 of Polyakov's book--- ``Gauge fields and strings''.

The action for the nonlinear sigma model (NLSM) is \begin{equation} S= \frac{1}{2e_0^2}\int d^2x (\partial_\mu \mathbf{n})^2. \end{equation}

Polyakov breaks the $N$-dimensional unit vector $\mathbf{n}$ up into slow ($e_a,\mathbf{n}_0$) and fast ($\varphi_a$) variables as \begin{equation} \mathbf{n}(x) = \sqrt{1-\varphi^2}\mathbf{n}_0(x)+\sum_{a=1}^{N-1}\varphi_ae_a(x) \end{equation} where $\varphi^2 = \sum_{a=1}^{N-1}(\varphi_a)^2$. The vectors $e_a$ and $\mathbf{n}_0$ are orthogonal unit vectors.

He then introduces the gauge fields $A_\mu^{ab}$ and $B_\mu^a$ by \begin{eqnarray} \partial_\mu \mathbf{n}_0 &=& \sum_{a}^{}B_\mu^a e_a\\ \partial_\mu e_a &=& \sum_{b}^{} A_\mu^{ab} e_b - B_\mu^a\mathbf{n}_0 \end{eqnarray} where $a,b=1,2,\dots,N-1$ denote the transverse directions; $\mathbf{n}_0 \cdot e_a=0$ and $e_a \cdot e_b=0$.

Using this parametrization, the action of NLSM becomes \begin{equation} S= \frac{1}{2e_0^2}\int \left\{ \left( \partial_\mu \sqrt{1-{\varphi}^2} -B_\mu^a\varphi^a\right)^2 + \left( \partial_\mu \varphi^a- A_\mu^{ab}\varphi^b +B_\mu^a\sqrt{1-{\varphi}^2}\right)^2\right\}d^2x \end{equation}

The second order correction is given as \begin{equation} S^{(II)}= \frac{1}{2e_0^2}\int \left\{ \left( \partial_\mu \varphi^a -A_\mu^{ab}\varphi^b\right)^2 + B_\mu^a B_\mu^b\left(\varphi^a \varphi^b-{\varphi}^2 \delta^{ab}\right)\right\}d^2x+\frac{1}{2e_0^2}\int \left(B_\mu^a \right)^2 d^2x \end{equation} At this level, he clearly ignores terms like \begin{equation} B_\mu^a \partial_\mu \varphi^a\quad \text{and}\quad B_\mu^a A_\mu^{ab}\varphi^b. \end{equation} On which basis, these terms can be ignored? This is my first question. Secondly, how to perform integration over ${\varphi}$?

Moreover...

• How the terms in action $S^{(II)}$ change under the continuous rotation of transverse coordinate system?


• How can we prove that the Lagrangian can only depend on derivatives of gauge fields?



• What is importance of gauge fields?

Hints of these questions are given Assa Auerbach book---``Interacting Electrons and Quantum Magnetism '' [Chapter 13; section 13.3 Poor Man's renormalization]. But it is not clear to me. I would appreciate very much if some one help me in understanding the Mathematics and Physics related to my questions.