Topological effects in continuum two-dimensional $U(N)$ gauge theories

We study the $\theta$ dependence of the continuum limit of 2D $U(N)$ gauge theories defined on compact manifolds, with special emphasis on spherical ($g=0$) and toroidal ($g=1$) topologies. We find that the coupling between $U(1)$ and $SU(N)$ degrees of freedom survives the continuum limit, leading to observable deviations of the continuum topological susceptibility from the $U(1)$ behavior, especially for $g=0$, in which case deviations remain even in the large $N$ limit.

Figure 1

Large $N$ behavior of the topological susceptibility for $g=2$: deviations of $R^{(2)}(N,X)$ from $R^{(2)}(1,X)$ are shown for $N$ up to 6.

Figure 2

Large $N$ behavior of the topological susceptibility for $g=1$: deviations of $R^{(1)}(N,X)$ from $R^{(1)}(1,X)$ are shown for $N$ up to 8.

Figure 3

Large $N$ behavior of the topological susceptibility for $g=0$. The vertical line at $X={\pi }^2$ denotes the position of the Douglas-Kazakov transition. For $10\le X\le 11$ the $N\to \infty$ extrapolation of $R^{(0)}(N,X)$ is also shown, which is obtained from the results of Monte-Carlo simulations performed at $N\ge 30$, see the text for more details.

Figure 4

Large $N$ behavior of the topological susceptibility for $g=0$ and several $X$ values.

Figure 5

Large $N$ behavior of $\partial R^{(0)}(N,X)/\partial X$. For the $N$ values shown in this figure the peak seems to approach the Douglas-Kazakov transition, indicated by the dashed vertical line.

Figure 6

Large $N$ behavior of $\partial R^{(0)}(N,X)/\partial X$. The dashed vertical line denotes the location of the Douglas-Kazakov transition and it is clear that the peak moves into the large-area phase. Continuous lines are quadratic fit and they are drawn just to guide the eye.