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

CHAPTER 2. BOSONIC STRINGS 232.6.2 Energy-momentum tensor and HamiltonianWe have come to a point where we can no longer ignore the energy-momentum tensorT αβ , defined asT αβ = − 2 1 δL√ σ, (2.58)T −h δhαβ or if you prefer, as the variation <strong>of</strong> the Lagrangian (density) with respect to the (worldsheet) metric. In this equation, L σ represents the Lagrangian density as it appears inthe Polyakov action defined in Eq. 2.5. Using the Polyakov action andone finds that∫δSδh αβ = −1 2 T= − 1 2 T √ ∫−h{( √ ) ∂ −hd 2 σ∂h αβd 2 σδ √ −h = − 1 2√−hhαβ δh αβ ,h γδ η µν ∂ γ X µ ∂ δ X ν + √ }−hη µν ∂ α X µ ∂ β X ν δh αβ{− 1 }2 h αβh γδ ∂ γ X µ ∂ δ X µ + ∂ α X µ ∂ β X µ δh αβ , (2.59)implying thatT αβ = ∂ α X · ∂ β X − 1 2 h αβh γδ ∂ γ X · ∂ δ X. (2.60)Demanding that the field equations for the world sheet metric,δLδh αβ= 0, are satisfied,we need to impose thatT αβ = 0. (2.61)These equations are called the constraint equations, as they lay extra constraints on thesolutios <strong>of</strong> the EOM for X µ . In the particular case <strong>of</strong> bosonic string theory, they also gounder the name <strong>of</strong> Virasoro constraints, for reasons that will become clear in a second.We have seen that one <strong>of</strong> the local symmetries <strong>of</strong> the world sheet is Weyl invariance. Thissymmetry has a side effect, namely it induces the tracelessness <strong>of</strong> the energy-momentumtensor, i.e. h αβ T αβ = 0. To show this, consider a general action S (h αβ , φ i ), where φ irepresents some fields satisfying the field equations δL = 0. Supposing that under anδφ iinfinitesimal Weyl rescaling with parameter Λ these transform ash αβ −→ h αβ + 2Λh αβ ; φ i −→ φ i + d i Λφ i , (2.62)where d i represents the conformal weight (see Chapter 4) <strong>of</strong> φ i , and assuming the actionto be scale invariant, we find that∫ {0 = δS = d 2 σ 2 δL h αβ + ∑ }δLd i φ i Λ. (2.63)δh αβ δφi iDue to the field equations <strong>of</strong> φ i , only the first term is non-trivial. Using Eq. 2.58, wefind that∫0 = −T d 2 σ √ )−h(T αβ h αβ , (2.64)