By Sergei V. Ketov
The publication is taken into account a scientific presentation of the fashionable quantum box concept of non-linear sigma-models. The content material relies on unique papers. Geometric houses and renormalization of a usual non-linear sigma-model are thought of intimately, and illustrated via 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. purposes of non-linear sigma-models in conformal conception, gauge conception, string conception, and basic relativity are given. The publication addresses graduate scholars and researchers in physics and mathematics.
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Extra resources for Quantum Non-linear Sigma-Models: From Quantum Field Theory to Supersymmetry, Conformal Field Theory, Black Holes and Strings
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57) are enough to perform the subtraction of sub divergences in the basic three-loop integrals Iw and I~,2). 58) , 1 [1 6(41T)3 t 3 1 2] + t2 + ~ In particular, any dependence upon the IR-regulator m 2 disappears after the subtractions, as it should. 58) that we can replace I~) by ~I{ in our calculations of the three-loop RG ,B-functions. This implies that the integral I~) is not independent. In fact, In by the use of its UV-divergence can be reduced to the 'flower'-type ('" integration by parts [92], which simultaneously provides a consistency check of our subtraction approach.
In the DR method the NLSM fields cpa remain massless in d dimensions. Hence, the bare metric g~b and torsion H~bc = a[ah~cJ can be expressed in terms of the renormalized quantities as (cf. 13) where p is the RG scale parameter of dimension of mass. When using the covariant background field method (Sect. 2), the quantum corrections g~;b? and H[~~ll) are all tensors in M, which are polynomial in the curvature, torsion and their covariant derivatives. In particular, all the i-loop corrections have the factor (A 2 ) 1-1.
4. 30). Though being non-minimal in subtracting the sub divergences , this approach allows us to treat all the threeloop UV divergences on an equal footing by reducing them to a few basic divergent integrals [92]. For instance, any three-loop graph of the type al or a2 in Fig. 4 has the UV divergence proportional to without a d-dependent pre-factor. Hence, after subtracting its sub divergences , it only gives rise to a 1/ f3 pole divergence that can be simply ignored in calculating the RG ,a-functions (Sect.