The matter part of the <massless closed-string vertex operator> is
$$
V_\epsilon=\epsilon_{\mu\nu}:\!\partial X^\mu\bar\partial X^\nu e^{ik\cdot X}\!:,
\qquad k^2=0,
\qquad k^\mu\epsilon_{\mu\nu}=\epsilon_{\mu\nu}k^\nu=0.
$$
Its symmetric trace-free polarization is the graviton, while its antisymmetric polarization is the <Kalb–Ramond field>, or B-field. The <mass-shell condition> $k^2=0$ and transversality make $V_\epsilon$ a <primary operator> of <conformal weights> $(1,1)$, as required for an <integrated string vertex operator>.

The graviton polarization has the linearized <gauge redundancy>
$$
\epsilon_{\mu\nu}\sim\epsilon_{\mu\nu}+k_\mu\xi_\nu+k_\nu\xi_\mu,
$$
while the antisymmetric polarization obeys
$$
\epsilon_{\mu\nu}\sim\epsilon_{\mu\nu}+k_\mu\Lambda_\nu-k_\nu\Lambda_\mu.
$$
In either case the change in the integrated vertex is a worldsheet <total derivative>, hence vanishes on a closed worldsheet; in covariant language it is <BRST cohomology>[BRST-exact]. This <string-state gauge redundancy> is the vertex-operator form of linearized target-space diffeomorphism or two-form gauge invariance.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 306 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-306/3/c/solution