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By Sergei V. Ketov
The publication is taken into account a scientific presentation of the fashionable quantum box conception of non-linear sigma-models. The content material is predicated on unique papers. Geometric houses and renormalization of a well-known non-linear sigma-model are thought of intimately, and illustrated by means of specific multi-loop calculations in perturbation concept. a few non-perturbative effects are derived for the conformally invariant non-linear sigma-models. Supersymmetric extensions are given for many buildings, with emphasis on their relation to complicated geometry. functions of non-linear sigma-models in conformal conception, gauge thought, string thought, and basic relativity are given. The publication addresses graduate scholars and researchers in physics and mathematics.
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Additional info for Quantum Non-linear Sigma-Models: From Quantum Field Theory to Supersymmetry, Conformal Field Theory, Black Holes and Strings
Example text
73) is obviously consistent with the non-renormalization of the WZNW term, as well as the existence of the exact fixed point at 'f72 = 1. 74) yields HVB qo - 8, qr VB = -10/3, q~VB = -5/3 . 48). The three-loop RG (3-functions of the general NLSM with non-vanishing torsion are very complicated, while they depend upon the renormalization 40 2. Classical Structure and Renormalization prescription used in actual calculations [92]. The generic structure of the three-loop contribution is /3~;) '" RR2 + (\7 R)2 + R 2\7 H + R(\7 R)H + (\72 R) (\7 H) + RRH2 + R(\7 H)2 + (\7 R)(\7 H)H + RH(\72 H) + (\72 R)H2 + RH2 (\7 H) + (\7 R)H3 + (\7 H)3 + (\72 H) (\7 H)H + (\73 H)H2 + RH4 + (\7 H)2 H2 + H4 (\7 H) .
The three-loop Feynman graph whose UV-divergence is given by I~) 34 2. Classical Structure and Renormalization (1) 1(2) E = ·3 1 f (2) ddpddqdd s (27r)3d Fig. 1. The three-loop graphs whose UV-divergences are given by J;;) and JIf), respectively q. 53) . All the 'f/- and p-type graphs in Fig. 4 do not lead to new types of UV divergences. Integrating by parts in them always results in either a tadpole or a cancellation of one of the propagators, thus effectively reducing the calculation oftheir UV-divergences to the other types in Fig.
26) of the NLSM with torsion, whose metric and the torsion potential are given by 13 h (B) = -cabeBe ab 1 + B2 . 19) with respect to Ba first, we arrive at the constraint 13 The (up and down) positions of the flat SU(2) indices ofthe B-field are irrelevant, while it matters for the 'world' indices of the NLSM metric, torsion and curvature tensors. 46 2. 14). 19) now yields the dual NLSM with vanishing torsion. 16) for the corresponding vielbein. Their solution in the SU(2) case reads where the trace goes over the SU(2) group matrices.