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In this appendix, we shall define and study complex line bundles over an arbitrary compact Riemann surface, provide their topological classification in terms divisors, and give the Riemann–Roch theorem. We shall prove various dimension formulas, including for the dimension of the moduli space of complex or conformal structures on a Riemann surface. We then discuss sections of line bundles from a more physics-oriented point of view in terms of spaces of vector fields, differential forms, and spinor fields.
In this chapter, the geometric description of generic branes in Yang–Mills matrix model is elaborated, and structures familiar from gravity are identified. The dynamics resulting from the classical model is interpreted as pre-gravity.
An introduction to field Lagrangians for scalars, vectors, and the Einstein–Hilbert Lagrangian for gravity provides a venue to think about coupling together different field theories. The natural expression of that coupling comes from an action, and we show how the “Euler–Lagrange” field equations enforce the universal coupling of all physical theories to gravity. As an example, the combined field equations of electricity & magnetism and gravity are solved in the spherically symmetric case to give the Reissner–Nordstrøm spacetime associated with the exterior of charged, massive, spherically symmetric central bodies.
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