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By Kiritsis E.
In those lecture notes, an advent to superstring concept is gifted. Classical strings, covariant and light-cone quantization, supersymmetric strings, anomaly cancelation, compactification, T-duality, supersymmetry breaking, and thresholdcorrections to low-energy couplings are mentioned. a quick creation to non-perturbative duality symmetries is usually incorporated.
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There are certainly several, constructed out of products of derivatives of X. We will consider another interesting class, the “vertex” operators Va (z)=: eiaX(z) :. The OPE with the stress-tensor is T (z)Va (w, w) ¯ = − 21 : ∂X(z)∂X(z) : ∞ in an : X n (w, w) ¯ : . n! 7) For all terms in the expansion there can be either one or two contractions. We obtain T (z)Va (w) = − 21 [ia∂ XX ]2 eiaX(w) − 21 2ia : ∂X(z)∂ XX eiaX(w) : + . . a2 /2 iaX(w) ia∂X(z) iaX(w) e + e + ... = (z − w)2 z−w = a2 /2 1 Va (w) + ∂Va (w) + .
9) z z z w − w w = Figure 5: Rearrangement of contours relevant for commutators. where Φi are primary fields. Viewed as a function of z, F N is meromorphic with poles when z → zi . 8). A meromorphic function on the sphere is uniquely specified by its poles and residues. Thus, we obtain N N ∂zi hi Φi (zi , z¯i ) . 10) + F N (z, zi , z¯i ) = 2 z − zi i=1 i=1 (z − zi ) This Ward identity expresses correlation functions of primary fields with an insertion of the stress-tensor in terms of the correlator of the primary fields themselves.
1 ∂z3 z n+1 = 61 (n+1)n(n−1)w n−2 . We integrate z=w the last term by parts and combine it with the second term. This gives (n−m)w n+m+1 T (w). Performing the w integration leads to the Virasoro algebra [Ln , Lm ] = (n − m)Ln+m + 51 c 3 (n − n)δn+m,0 . 8) The analogous calculation for T¯ (¯ z ) yields ¯n, L ¯ m ] = (n − m)L ¯ n+m + c¯ (n3 − n)δn+m,0 . 9) Since T T¯ has no singularities in its OPE, ¯ m ] = 0. 10) Every conformally invariant theory realizes the conformal algebra, and its spectrum forms representations of it.