Large N duality beyond the genus expansion

TL;DR

The paper analyzes non-perturbative aspects of the large-N duality between Chern-Simons theory and topological strings, revealing a rich phase structure in the complex plane of the 't Hooft parameter governed by large-N instantons and a generalized Stokes phenomenon. By studying CS on L(2,1) and its two-cut matrix-model realization, the authors show that the dominant saddle, and thus the relevant target geometry, can change with t, necessitating instanton corrections to the genus expansion. They classify real, imaginary, and complex g_s regimes, deriving phase boundaries and interpreting transitions as deformations of Stokes phenomena, with theta-function oscillations appearing in the interior saddle case. Numerical tests on the exact CS partition function corroborate the predicted instanton-corrected asymptotics, highlighting the importance of non-perturbative effects and background independence in this duality context.

Abstract

We study non-perturbative aspects of the large N duality between Chern-Simons theory and topological strings, and we find a rich structure of large N phase transitions in the complex plane of the 't Hooft parameter. These transitions are due to large N instanton effects, and they can be regarded as a deformation of the Stokes phenomenon. Moreover, we show that, for generic values of the 't Hooft coupling, instanton effects are not exponentially suppressed at large N and they correct the genus expansion. This phenomenon was first discovered in the context of matrix models, and we interpret it as a generalization of the oscillatory asymptotics along anti-Stokes lines. In the string dual, the instanton effects can be interpreted as corrections to the saddle string geometry due to discretized neighboring geometries. As a mathematical application, we obtain the 1/N asymptotics of the partition function of Chern-Simons theory on L(2,1), and we test it numerically to high precision in order to exhibit the importance of instanton effects.

Figures (13)

• Figure 1: In this figure we plot $\log |Z(N, N_2, g_s)|$ as a function of $N_2=0, \cdots, N$, for $N=12$ and for $g_s=\pm\pi/24$ respectively on the left and on the right. The dominant configurations are $(N_1^*,N_2^*)=(N,0)$ and $(0,N)$, respectively, in agreement with the symmetry (\ref{['signchange']}).
• Figure 2: In this figure, the crosses represent the right hand side of (\ref{['iasn']}), for $N=24$ and different values of $t$, while the continuous line represents the value of $A(t)/t$ obtained from (\ref{['ainst']}).
• Figure 3: In this figure we plot $\log |Z(N, N_2, g_s)|$ as a function of $N_2=0, \cdots, N$, on the left for $N=12$ and $g_s=\pi {\rm i} /48$, on the right for $N=13$ and $g_s=\pi {\rm i}/52$. The largest value is obtained when $N_2={N/2}$ for $N$ even and when $N=(N\pm 1)/2$ for $N$ odd.
• Figure 4: On the left column we plot the sequence ${\rm Re}\, f_{e}(\ell,z)$ as well as its Richardson transforms ${\rm Re}\, f^{(k)}_{e}(\ell,z)$ for $k=1,2,3$, and for $\ell=2\cdots 14$. On the right column we plot the sequence ${\rm Re}\, f_{o}(\ell,z)$ as well as its Richardson transforms ${\rm Re}\, f^{(k)}_{o}(\ell,z)$ for $k=1,2,3$ for $\ell=2\cdots 14$. The plots on the top are for $z=1/4$, and the plots of the bottom are for $z=1/8$. The dashed lines in the plots show the expected value $-{\rm Re}\, F_0( z)/(\pi z)^2$.
• Figure 5: The sequence ${\rm Re}\, \Theta_{eo}(z)$ for $z=1/4,1/8$, together with its Richardson transforms ${\rm Re}\, \Theta^{(k)}_{eo}(z)$ for $k=1,2,3$. The dashed lines in both plots show the expected value ${\rm Re}(\log\vartheta_e(z)-\log\vartheta_o(z))$.