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We call the degenerated point, a node and the surface, a noded Riemann surface. 10). The pinched torus is topologically a sphere with a single 28 CHAPTER 1. 7: (A) Two paths on the sphere. (B) Union of two copies of the sphere with cuts. double point. g. 11). 11). The study of noded Riemann surfaces arised in string theory, where divergences occur due to integration over Mg , which is a non-compact space. One would like to compactify Mg by adding conformal equivalence classes of degenerated surfaces.

117) δBRST ω = 0. 118) The ghost fields are c, ω, which are scalar grassmann fields on M. 122) δBRST E = 0. 123) where E, G are Lagrange multipliers. This leads to the following quantum action Sq = M B1 ∧ dA1 + ∗G ∧ d† B1 + ∗E ∧ d† A + ωd† dω + ∗cd† dc . 124) Here we used that Q commutes with the Hodge star and anti-commutes with d† . The term d† d is just ∆0 . The gaussian ghost terms contribute to the partition function with det(∆0 )2 . The (A1 , B1 , E, G) system is not gaussian, but it can be 56 CHAPTER 2.

Note that the complex number u represents the moduli of the torus, but in the algebraic geometry the limit u → 0 is special. For u = 0 C is still a perfectly regular polynomial, but the corresponding Riemann surface undertakes a radical transformation. 9) This is not a smooth Riemann surface, but it is still a polynomial of the original order in CP2 . We call the degenerated point, a node and the surface, a noded Riemann surface. 10). The pinched torus is topologically a sphere with a single 28 CHAPTER 1.

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Topological Aspects of Quantum Gravity by Weis, M

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