Question about Mode expansion of free compact boson

Question

$(1+1)$-Dim free compact boson, Lagrangian is 
$$mathcal{L}= frac{1}{2}(partial_muphi(sigma,t))^2$$
with $phi(x,t)simphi(x,t)+2pi r$ and periodic boundary condition along $x$, i.e. $phi(sigma,t)= phi(sigma+beta, t)$.

Equation of motion is
$$(partial_t^2 -partial_sigma^2) phi(sigma,t)=0 $$

I saw the following saying but I don't understand:

The mode expansion of $phi(x,t)$ is 
begin{eqnarray}
{  phi(sigma,t) =} nonumber 
&& hat{x} + frac{2 pi}{beta} r w sigma + frac{pi}{beta} hat{p} t
   nonumber 
&& + frac{1}{2}  sum_{n = 1}^{infty} [
   frac{a_n}{sqrt{n}} e^{ - in( sigma + t) frac{2 pi}{beta}}
+ frac{a^{dagger}_n}{sqrt{n}} e^{ in( sigma + t) frac{2 pi}{beta}}
           ]
nonumber 
&& + frac{1}{2}  sum_{n = 1}^{infty} [
   frac{tilde{a}_n}{sqrt{n}} e^{ in( sigma - t) frac{2 pi}{beta}}
+ frac{tilde{a}^{dagger}_n}{sqrt{n}}
e^{ - in( sigma - t) frac{2 pi}{beta}} ]
end{eqnarray}
with $winmathbb{Z}$ is winding number.

My questions:

Why are there second and third terms in compact free boson and no such terms in non-compact free boson?

I know there can exist wind solution for compact case since it's a map from $S$ to $S$. And we can check that $frac{2 pi}{beta} r w sigma$ satisfies the EOM. However why we don't include such a term $propto sigma$ or $propto t$ in non-compact case. And $phi(sigma, t) = alpha sigma$ and $phi(sigma,t)= alpha t$ are certainly solutions of EOM, why we don't include these solutions in mode expansion for non-compact free boson?

How to prove that in compact case, except above modes there are no other solutions which satisfy the EOM and PBC?

Because $phi(sigma, t)= alpha t + alpha' x$ is a solution of EOM and doesn't belong to plane wave. There are also many other solutions like $t^2 +sigma^2$，$ sigma t $ or $sigma^3 + 3 sigma t^2$, etc. Certainly these solutions can't satisfy PBC. But how to prove that in compact case, except plane wave and linear solutions there are no other solutions which can satisfy the EOM and PBC?

Qmechanic · Answer

Those terms include momentum and winding instanton modes $phi_{n,m}$ because the target space (TS) coordinate is in principle many-valued $phi(x,t)simphi(x,t)+2pi r$ with infinitely many branches labelled by an integer. The momentum and winding number tell the monodromy in the $t$- and the $x$-direction on the world sheet (WS), i.e. the change in target space branch.
If we subtract these instanton contributions $phi_{n,m}$, the remainder  $phi-phi_{n,m}$ has TS coordinate within the same target space branch/can be treated as being single-valued on the whole WS, i.e. the boundary conditions (BCs) don't connect different branches, so that the remainder $phi-phi_{n,m}$ is just an ordinary Fourier series, i.e. the oscillator modes.
So the string $phi$ is a sum of an affine instanton part $phi_{n,m}$ and an oscillator part. In the uncompactified case, the instanton part has 1 instanton. (This terminology is used in the same semantic sense that, say, a bicycle without gears has 1 gear.)
There are both BCs in the spatial and temporal directions. E.g. the BCs could be Dirichlet or Periodic BCs.
A Dirichlet BC means that the instanton part may contain terms linear in the corresponding WS coordinate, even in the uncompactified case.

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