# Zero-point energy? (D-branes C.Johnson's book)

In Johnson's Book about D-branes (11.2) gives the zero point energy. This is given using (2.80) $$\begin{equation} +\frac12\omega: \text{for boson and}\;\; -\frac12\omega: \text{for fermion}. \end{equation}$$

where $$\begin{equation} \omega = \frac1{24} - \frac18(2\theta - 1)^2,\;\; \begin{cases} \theta = 0& \text{(for integer modes)},\\ \theta = 1/2& \text{(for half-integer modes)}. \end{cases} \end{equation}$$ Then $$\begin{equation} \omega = \begin{cases} -\displaystyle\frac1{12}& r\in\mathbb Z,\\ +\displaystyle\frac1{24}& r\in\mathbb Z+1/2. \end{cases} \end{equation}$$ We obtain $$\begin{equation} +\frac1{24}\text{: for periodic fermion},\;\; -\frac1{48}\text{: for anti-periodic fermion}. \end{equation}$$

Using them, in (11.2) the zero point energy is calculated as $$\begin{equation} (8-\nu)\Big(-\frac1{24} - \frac1{48}\Big) + \nu\Big(\frac1{48} + \frac1{24}\Big)\qquad —-(11.2) \end{equation}$$

PS. Then in the first parenthesis of (11.2), The first term $$-1/24$$ represents the periodic boson and the second term is the anti-periodic fermion. In the second parenthesis the first term $$1/48$$ is anti-periodic boson and the second term is periodic fermion.

Please tell me my thoughts are incorrect.

• I seem to remember this is well explained in Polchinski’s first volume. – Oбжорoв Dec 2 '20 at 10:10
• I know how to obtain the zero point energy. What I want to know is what each term in (11.2) means. $\nu$is the sum of numbers of dimensions where Dirichlet and Neumann boundary condition is imposed. – KoKo_physmath Dec 3 '20 at 6:02
• Ok - not exactly sure I understand your question. What is equation (11.2) and what is $\nu$? – Oбжорoв Dec 6 '20 at 13:02
• I added the equation number (11.2). The author distinguished the boundary conditions on the both sides of the string by DD, NN, ND or DN. These are Dirichlet or Neumann boundary conditions. $\nu$ means the number of ND plus DN directions. – KoKo_physmath Dec 6 '20 at 13:09