A Short Course in Differential Geometry and Topology by A. T. Fomenko

By A. T. Fomenko

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5 to determine exactly when this gauge group enhancement occurs. The vertex operator for one of our new massless states will be as follows. For the left-moving part we want simply 8X" to give a vector index. For the rightmoving part we require another operator with conformal weight one. pR = -2. a = -2. The charge of such a state, a, with respect to the U(1)24 gauge group is simply given by the coordinates of a. This follows from the conformal field theory of free bosons and we refer to (41) for details.

The terms are identified as follows. g,, is a Riemannian metric on X and I3,j are the components of a real 2-form, B, on X. tk is the "dilaton" and is a real number (which might depend on x) and R(2) is the scalar curvature of E. This two-dimensional theory is known as the "non-linear a-model". In order to obtain a valid string theory, we require that the resulting twodimensional theory is conformally invariant with a specific value of the "central charge". ) Conformal invariance puts constraints on the various parameters above [42, 43].

Proposition 1 now tells us something interesting. Despite the fact that the conformal field theory approach to the type IIA string insisted that it could never have any gauge group other than U(1)24, the dual picture in the heterotic string dictates otherwise. There must be some points in the moduli space of a type IL4 string compactified on a K3 surface where the conformal field theory misses part of the story and we really do get an enhanced gauge group. Since we know exactly where the enhanced gauge groups appear in the moduli space of the heterotic string and we know exactly how to map this to the moduli space of K3 surfaces, we should be able to see exactly when the conformal field theory goes awry.

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