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

CHAPTER 4. CONFORMAL INVARIANCE 56where D (z ′ ) zstands for the Schwarzian derivative, 3D ( z ′) z = ∂ z (z ′ )∂z 3 (z ′ ) − 3 (2 ∂2z z ′) 2(∂ z z ′ ) 2 . (4.13)The important thing is that we are now faced with a conformal anomaly c, <strong>of</strong>ten alsocalled central charge, a term that signals the breakdown <strong>of</strong> conformal invariance forgeneral conformal transformations in the quantum theory if c ≠ 0. The consequencehere<strong>of</strong> is that the trace <strong>of</strong> the energy-momentum tensor no longer vanishes, but nowbecomesT α α = − c12 R(2) , (4.14)with R (2) representing the Ricci scalar <strong>of</strong> the world sheet. This implies that we willrun into problems when considering non-flat world sheet space-times, or at least severerestrictions as to the possible consistent configurations.On a final note for interested and/or intruiged parties, all one could ever wish toknow about conformal field theory, and then some, can be found in [4].4.2 <strong>String</strong> Theory vs. QFT vs. General RelativityRefreshing the Polyakov action we used in order to study the bosonic string (Eq. 2.5),S σ = − 1 2 T ∫d 2 σ √ −hh αβ η µν ∂ α X µ ∂ β X ν ,we see that a possible generalization to curved backgrounds naturally imposes itself,namelyS g = − 1 ∫4πα ′ d 2 σ √ −hh αβ g µν (X)∂ α X µ ∂ β X ν , (4.15)where we have simply replaced the Minkwoskian metric with an arbitrary, non-trivial,field-dependent metric. One might argue that this is actually using results before havingobtained them, as the g µν field came about by constructing the spectrum <strong>of</strong> the quantizedclosed string (or if you prefer: is generated by the strings themselves), but it can beshown without great difficulty 4 by considering perturbations around η µν that this actiondoes indeed comply with the way the graviton arises in string theory.That seems easy enough, but we have seen in the previous chapters that the bosonicstring theory, as well as the (NS,NS) sector <strong>of</strong> both Type II superstring theories containnot one, but three massless bosonic fields: 5 the space-time metric g µν which we just3 Note that it seems like everyone uses their own little symbol to represent this function. We follow[14].4 You might ask yourself: “So why don’t you do it here?” The answer is that this short demonstrationdeals with vertex operators, and hence string interactions, which the author <strong>of</strong> this thesis has purposfullyleft out <strong>of</strong> his writings.5 The Type II theories also contain massless fields in the other sectors as we saw <strong>of</strong> course, but thefields originating from the (NS,NS) sector are the only ones they have in common with each other, andwith the bosonic theory.