Problem 14.5d1

Counting states in heterotic SO(32) string theory | A First Course in String Theory

d) Write a generating function $\displaystyle{f_L(x) = \sum_{r} a(r) x^r}$ for the full set of GSO-truncated states in the left-moving sector (include both NS’+ and R’+ states).

Use the convention where $\displaystyle{a(r)}$ counts the number of states with $\displaystyle{\alpha' M_L^2 = r}$ .

Use $\displaystyle{f_L(x)}$ and an algebraic manipulator to find the total number of states in heterotic string theory at $\displaystyle{\alpha' M_L^2 = 8}$ .

~~~

— This answer is my guess. —

~~~

p.322

$\displaystyle{\alpha' M_L^2 = \alpha' M_R^2}$

$\displaystyle{\frac{1}{2} \alpha' M^2 = \alpha' M_L^2 + \alpha' M_R^2}$

$\displaystyle{\alpha' M^2 = 2 \left( \alpha' M_L^2 + \alpha' M_R^2 \right) = 4 \alpha' M_L^2}$

The left NS’+ sector:

$\displaystyle{ \begin{aligned} \alpha' M_L^2 &= -1 + N^\perp \\ \end{aligned}}$

$\displaystyle{N^\perp:}$

$\displaystyle{\begin{aligned} \left( 1 + \bar \alpha_1 x + (\bar \alpha_1)^2 x^2 + ... \right)^8 \left( 1 + \bar \alpha_2 x + (\bar \alpha_2)^2 x^4 + ... \right)^8 ... \left( 1 + \lambda_{-\frac{1}{2}} x^{-\frac{1}{2}} \right)^{32} \left( 1 + \lambda_{-\frac{3}{2}} x^{-\frac{3}{2}} \right)^{32} ... \\ \end{aligned}}$

$\displaystyle{\begin{aligned} \prod_{r=1}^\infty \frac{1}{(1 - x^r)^8} (1 + x^{r-\frac{1}{2}})^{32} \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} \alpha' M_L^2: \end{aligned}}$

$\displaystyle{\begin{aligned} &f_{L, NS+}(x) \\ &= a_{NS+} (r) x^r \\ &= \frac{1}{x} \prod_{r=1}^\infty \frac{(1 + x^{r-\frac{1}{2}})^{32}}{(1 - x^r)^8} \\ \end{aligned}}$