Hermiticity versus self-adjointness of various operators.

Constant Magnetic Field on a Torus
----------------------------------

In this subsection we impose a torus boundary condition over a rectangular region of size $L_x \times L_y$. This will lead to a quantization condition for the magnetic flux. Since the magnetic field is constant, it obviously is periodic. On the other hand, the vector potential of the infinite volume theory $A_x(x,y) = 0$, $A_y(x,y) = B x$ obeys the conditions $$\begin{aligned}
&&A_x(x + L_x,y) = A_x(x,y), \nonumber \\
&&A_y(x + L_x,y) = A_y(x,y) + B L_x = A_y(x,y) + {\partial}_y (B L_x y), \nonumber \\
&&A_x(x,y + L_y) = A_x(x,y), \nonumber \\
&&A_y(x,y + L_y) = A_y(x,y).\end{aligned}$$ As a gauge-dependent quantity, the vector potential is periodic only up to gauge transformations, i.e. $$A_i(x + L_x,y) = A_i(x,y) - {\partial}_i \varphi_x(y), \
A_i(x,y + L_y) = A_i(x,y) - {\partial}_i \varphi_y(x).$$ The gauge transformations $\varphi_x(y)$ and $\varphi_y(x)$ are transition functions in a fiber bundle which specify the boundary condition. In our case the transition functions are given by $$\varphi_x(y) = \frac{\theta_x}{e} - B L_x y, \ 
\varphi_y(x) = \frac{\theta_y}{e}.$$

Besides the field strength, gauge theories on a periodic volume possess additional gauge invariant quantities — the so-called Polyakov 