S-duality Improved Superstring Perturbation Theory

TL;DR

The paper tackles the challenge of obtaining finite-coupling predictions in string theory by leveraging S-duality to interpolate between perturbative weak- and strong-coupling regimes. It defines a renormalized mass function $F(g)$ and constructs a family of interpolants $F_{m,n}(g)$ that reproduce the weak-coupling series $F^W_m(g)$ and the strong-coupling series $F^S_n(g)$, applying this to the mass of the lightest SO(32) spinor state. The leading and first subleading coefficients are computed on both sides: $F^W_2(g)=F^W_3(g)= g^{1/4}(1+K_w g^{2})$ with $K_w\approx0.23$, and $F^S_1(g)= g^{3/4}(1+K_s g^{-1})$ with $K_s\approx0.351$, yielding an interpolant $F_{3,1}(g)= g^{1/4}(1 + 10 K_w g^2 + 10 K_s g^4 + g^5)^{1/10}$ that remains within about 10% of the true function for all $g$. Padé and Kleinert variants provide corroborating accuracy, suggesting the method is a practical route to finite-coupling estimates for non-BPS observables. Overall, the work demonstrates that S-duality–driven interpolation can yield reliable, quantitative insights beyond strictly perturbative regimes in string theory.

Abstract

Strong - weak coupling duality in string theory allows us to compute physical quantities both at the weak coupling end and at the strong coupling end. Furthermore perturbative string theory can be used to compute corrections to the leading order formula at both ends. We explore the possibility of constructing a smooth interpolating formula that agrees with the perturbation expansion at both ends and leads to a fairly accurate determination of the quantity in consideration over the entire range of the coupling constant. We apply this to study the mass of the stable non-BPS state in SO(32) heterotic / type I string theory with encouraging results. In particular our result suggests that after taking into account one loop corrections to the mass in the heterotic and type I string theory, the interpolating function determines the mass within 10% accuracy over the entire range of coupling constant.

Figures (7)

• Figure 1: Graph of $\tan^{-1}F(g)$ vs. $\tan^{-1} g$ for $F=F^S_0$ (thin solid curve), $F^W_0$ (thin dashed curve) and the interpolating functions $F_{0,0}$ (thick dashed curve) and $F_{1,0}$ (thick solid curve). This should be compared with Fig. \ref{['f3']} which describes similar curves after taking into account the first subleading corrections at both ends.
• Figure 2: Graph of $\tan^{-1}F(g)$ vs. $\tan^{-1} g$ for $F=F^S_1$ (thin solid curve), $F=F^W_2(=F^W_3)$ (thin dashed curve) and the interpolating function $F=F_{3,1}$ (the thick solid curve).
• Figure 3: Graph of $F^W_2(g)/F_{3,1}(g)$ (dashed curve) and $F^S_1(g)/F_{3,1}(g)$ (continuous curve) vs. $\tan^{-1}g$.
• Figure 4: Graph of $F_{m,n}(g)/F_{3,1}(g)$ vs. $\tan^{-1}g$ for various $(m,n)$. The labels are as follows: thin dots for $F_{0,0}$, thick dots for $F_{1,0}$, small thin dashes for $F_{2,0}$, small thick dashes for $F_{3,0}$, large thin dashes for $F_{0,1}$, large thick dashes for $F_{1,1}$, continuous thin line for $F_{2,1}$ and continuous thick line for $F_{3,1}$.
• Figure 5: Graph of $P_{m,n}(g)/F_{3,1}(g)$ vs. $\tan^{-1}g$ for various $(m,n)$. The labels are as follows: dots for $P_{1,1}$, dashes for $P_{2,0}$, continuous thick line for $P_{3,1}$ and continuous thin line for unity.