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

CHAPTER 2. BOSONIC STRINGS 14moment we consider space-time to be flat. Note that the minus sign preceding thedeterminant is only needed if a negative time-component is present, as is the case here,but if we were to e.g. consider a Euclidean space-time, we should disregard it. Furthernote that the expressionη αβ = η µν ∂ α X µ ∂ β X ν (2.4)defines a metric on the world sheet. This expression is <strong>of</strong>ten called the pullback η αβ <strong>of</strong>the tensor η µν , in this case the space-time metric, on the world sheet.Eq. 2.1 however is a classical action that does not lend itself very well to quantizationbecause <strong>of</strong> the presence <strong>of</strong> a square root. 1 Note that we have not yet made use <strong>of</strong> ametric explicitely defined on the world sheet (we only used the pullback from the externalspace-time). Doing so allows us to write a new action, called the Polyakov action orstring sigma model action. Writing this world sheet metric as h αβ , this action takes thefollowing form:S σ = − 1 2 T ∫d 2 σ √ −hh αβ η µν ∂ α X µ ∂ β X ν . (2.5)Here, h = det h αβ . The √ −h factor is actually needed to make the integral measure(d 2 σ √ −h) reparameterization invariant, a symmetry that will prove itself very usefullater on. The Nambu-Goto and Polyakov actions are equivalent at the classical level,which can be demonstrated by computing the equations <strong>of</strong> motion for h αβ , and usingthem to eliminate h αβ from the Polyakov action, which gives back the Nambu-Gotoaction, a point we will come back to later.Before going on, we will agree to the convention that indices (µ, ν, . . .) refer tospace-time, whilst indices (α, β, . . .) refer to the world sheet.2.2 SymmetriesNow that we have our action, we are interested in the symmetries <strong>of</strong> this action, especiallythe local ones, as we will need these to fix our metric (i.e. choose a gauge), a stepwe need to undertake if we wish to be able to quantize our theory. Keep in mind thatwe are still considering Minkowski space-time, which comes down to considering thefirst term in a perturbative expansion around a flat background. Going to the case <strong>of</strong>a curved background can be achieved by simply replacing η µν by g µν . 2 The symmetries<strong>of</strong> the Polyakov action are:Global symmetries• Poincaré transformations: the usual translations, rotations and boosts in spacetime,which are represented byδX µ = a µ νX ν + b µ ,δh αβ = 0,(2.6)1 In the case <strong>of</strong> a pointlike particle, it is also inadequate to describe massless particles, given thatT 0 = m.2 Note that this is true for the classical theory, but not for the quantum theory.