Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 4 / i

Table of contents
- Solution i

Solution

In Euclidean worldsheet signature, the string nonlinear sigma model action can be written

S = \frac{1}{4 π α ^{'}} \int_{Σ} d^{2} σ h [h^{ab} G_{μν} (X) \partial_{a} X^{μ} \partial_{b} X^{ν} + i ϵ^{ab} B_{μν} (X) \partial_{a} X^{μ} \partial_{b} X^{ν} + α^{'} Φ (X) R^{(2)} (h)] .

The factor of

i

in the B-field term is absent in Lorentzian signature. Here

G

is the target metric,

B

the Kalb–Ramond field, and

Φ

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