Gauge theory effective actions from open strings
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In an introductory chapter, a summary of the construction of string theories is given, for both the bosonic string and the RNS superstring. Relevant mathematical technology is introduced, including super-Riemann surfaces. Conformal field theory is discussed and BRST quantization of the string is explained. (Super) Schottky groups for the construction of higher-genus Riemann surfaces are introduced. As an example of the use of Schottky groups and super-Riemann surfaces, the one-loop gluon two point function is calculated from string theory. The incorporation of background gauge fields into string theory via nontrivial monodromies (twists) is discussed. The two loop Prym period matrix determinant is computed in the Schottky parametrization. The string theory model with N parallel separated D3-branes is introduced, and the formulae for the the vacuum amplitude are written down. A manifestly symmetric parametrization of two loop Schottky space is introduced. The relationship between worldsheet moduli and Feynman graph Schwinger times is given. The 0 ! 0 limit of the amplitude is written down explicitly. The lagrangian for the corresponding gauge theory is found, making use of a generalization of Gervais-Neveu gauge which accounts for scalar VEVs. Propagators in the given gauge field background are written down. All of the 1PI two-loop Feynman diagrams are written down, including diagrams with vertices with an odd number of scalars. Illustrative example Feynman graphs are computed explicitly in position space. These results are compared with the preceding string theory results and exact agreement is obtained for the 1PI diagrams. An example application is given: the computation of the function of scalar QED at two loops with the same methods, leading to the same result as found in the literature.
AuthorsPlayle, Samuel Rhys
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