Topological susceptibility of two-dimensional $U(N)$ gauge theories

In this paper, we study the topological susceptibility of two-dimensional $U(N)$ gauge theories. We provide explicit expressions for the partition function and the topological susceptibility at finite lattice spacing and finite volume. We then examine the particularly simple case of the Abelian $U(1)$ theory, the continuum limit, and the infinite volume limit, and we finally discuss the large $N$ limit of our results.

Figure 1

Behavior of the $U(2)$ topological susceptibility as a function of the dimensionless volume $P$ for the value $\beta =1$ of the coupling. Results are shown for three different topologies, corresponding to $g=0$, 1 and 2; the horizontal line denotes the asymptotic $P\to \infty$ value computed by using Eq. (63).

Figure 2

(a) Behavior of the $U(1)$ topological susceptibility as a function of the coupling $\beta$ and of the dimensionless volume $P$. For comparison, points obtained by using numerical lattice simulations are also shown in the case $\beta =2.5$ (error bars are smaller than symbols). (b) Same data as in the upper panel but with quantities normalized by using ${\chi }_t(1,\beta ,1)$.

Figure 3

Behavior of the infinite volume topological susceptibility for $U(N)$ with $N<10$ and $0\le \beta \le 2$, computed using Eq. (63).

Figure 4

(a) Comparison of numerical data obtained by using Eq. (63) (solid lines) and the leading order large $N$ theoretical predictions, which is given by Eqs. (75)–(77) for $\beta <1/2$ (dashed lines) and by Eq. (79) for $\beta >1/2$ (dotted-dashed line). (b) Deviations of numerical data from their expected asymptotic behavior.

Figure 5

Large $N$ scaling of the topological susceptibility at $g=\infty$ for several values of $P$ at coupling $\beta =1$.