DBI Analysis of Open String Bound States on Non-compact D-branes

CHAPTER 4. CONFORMAL INVARIANCE 57dealt with, the anti-symmetric Kalb-Ramond field B µν and the dilaton field Φ. These aresometimes referred to as the universal factor. So we should also find a way to incorporatethese last two fields in our action since they are equally part <strong>of</strong> the background. Ourprevious recipe can be applied without too much difficulty to the B µν field, 6S B = − 14πα ′ ∫d 2 σǫ αβ B µν (X) ∂ α X µ ∂ β X ν , (4.16)in which ǫ αβ is the usual anti-symmetric tensor density with components ǫ 01 = −ǫ 10 = 1and ǫ 00 = ǫ 11 = 0, but the dilaton coupling is considerably more subtle. The right wayto incorporate it turns out to beS Φ = − 14π∫d 2 σ √ −hΦ(X) R (2) , (4.17)in which R (2) again represents the world sheet Ricci scalar. Notice that this action isone order higher in α ′ than Eqs. 4.15 and 4.16. The full updated Polyakov action thusbecomesS σ = S g + S B + S Φ = − 14πα ′ ∫− 14π∫( √−hh )d 2 σαβ g µν + ǫ αβ B µν ∂ α X µ ∂ β X νd 2 σ √ −hΦ(X)R (2) . (4.18)Unfortunately, some bad news awaits us (yet again). This updated form still preservesparameterization invariance, and classicaly, both S g and S B conserve Weyl invariance,but S Φ does not. 7 This is bad news, as we needed Weyl invariance to be able to gaugefix our metric, in turn to be able to quantize our theory. Moreover, when consideringEq. 4.18 as describing a quantum field theory, Weyl invariance gets broken (for nontrivialmetrics) in general due to quantum effects. Recall that a consequence <strong>of</strong> Weylinvariance was the tracelessness <strong>of</strong> the enery-momentum tensor. Hence, breakdown <strong>of</strong>Weyl invariance will be reflected in the trace <strong>of</strong> T αβ . More generally speaking, breakdown<strong>of</strong> scale invariance in quantum field theories is usually associated with the non-vanishing<strong>of</strong> the β-function. Calculations <strong>of</strong> the less enjoyable variety based on an α ′ powerexpansion <strong>of</strong> Eq. 4.18 reveal thatT α α = − 12α ′βg µνh αβ ∂ α X µ ∂ β X ν − 12α ′βB µνǫ αβ ∂ α X µ ∂ β X ν − 1 2 βΦ R (2) , (4.19)6 Some authors include a complex factor i in this action.7 To be exact, it does not except when Φ is constant. In that case, S Φ ∼ χ M = − 14πRd 2 σ √ −hR (2) ,which is the Euler characteristic <strong>of</strong> the manifold. This is a topological invariant, and thus unaffected byWeyl rescalings.