Question: Using fsolve with a dispersion relation

I've been given the relationship:

R*sqrt(sigma)-1-(1)/(4)*(l-(2*kappa*R^2)/(R*sqrt(sigma)))/(1+coth(R*sqrt(sigma)))-kappa*R^2/6+m/(2*R), for the particular case where m=l=0, and where

R=exp[(kappa*(R^2))/6]. On a sketch of R vs kappa, apparently sigma >0 on the lower branch, but there should be some sigma < 0 on the upper branch.

To attempt to solve this whole problem, I use the following commands:

restart;with(plots):

l:=0;

m:=0;

kappa:=6*log(R)/(R^2); (Re-arranging for kappa)

eq01:=R*sqrt(sigma)-1-(1)/(4)*(l-(2*kappa*R^2)/(R*sqrt(sigma)))/(1+coth(R*sqrt(sigma)))-kappa*R^2/6+m/(2*R)=0;

eq02:=R-1.5=0;  (here any value of R can be inputted)

fsolve({eq01,eq02},{sigma,R},complex);

For this particular problem, I have obtained a negative sigma. Am I attempting it the right way? Is there another way of attempting it?