Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 306 / 4

Varying the gauge-fixed Polyakov action gives the bulk wave equation and the boundary termIt vanishes in each target direction either through the Neumann boundary conditionor through the Dirichlet boundary conditionwhich fixes the endpoint positions.

Along the common D-brane worldvolume, the Neumann solution isIn each transverse direction, the endpoint conditions $X^I(0,\tau )=x_1^I$ and $X^I(\pi ,\tau )=x_2^I$ giveThe linear term is the classical stretch between the two parallel D-branes.

The zero-mode Virasoro constraint iswhere the string level operator isSince the string tension isthe mass formula is

For a string beginning and ending on the same brane, the $N=1$ states are massless. Oscillators polarized tangentially give a gauge field on the $(p+1)$-dimensional worldvolume, while transverse polarizations give scalar fields describing fluctuations of the brane position. Strings joining two separated branes have the additional stretching mass $T|x_2-x_1|$ and are charged under the two endpoint gauge groups. When the two branes coincide, these off-diagonal states also become massless and Coincident-D-brane gauge enhancement enlarges $U(1)\times U(1)$ to $U(2)$. The bosonic open string also retains its $N=0$ tachyon.