MIT OpenCourseWarehttp://ocw.mit.edu 8.821 String Theory Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics String Theory (8.821) – Prof. J. McGreevy – Fall 2008 Problem Set 4 Fields in AdS 1. One more symmetry. Define xA ≡ (z, xµ), so that the Poincar´e patch metric takes the form dz2 + d~x2 dxAdxAds2 = = . 2 2z zShow that this metric is preserved by the following inversion symmetry: AxA x → xBxB i.e. z z2+z~x2 , xµz2x+µ ~x2 . This is a conformal transformation which is not → →continuously connected to the identity. [Bonus question: what does this do to the region of the global AdS space covered by the Poincar´e patch coordinates?] 2. Schr¨odinger description of AdS instabilities.1 a) Consider the scalar wave equation in poincar´e AdS, in momentum space: 0 = (� − m 2)φ, with φ(x, z) = eik·xφ(z) By redefining the independent variable, rewrite the scalar wave equation as a Sch¨odinger equation: (−∂2 + V (z))Ψ(z) = ω2Ψ(z) z with ω2 playing the role o f energy. (Hint: Rewrite Ψ(z) = A(z)φ(z), and choose the function A to set to zero the term in the wave equation multiplying Ψ′(z).) 1I learned about this trick from Sean Hartnoll. If you get stuck or want to see a recent application of this technique, see e.g. appendix A of 0810.1563. 1� � b) A property of a field configuration which generally determines whether it is allowed to participate in the physics (i.e. whether it is fixed by boundary con-ditions, or whether it can happen on its own) is whether it is normalizable. In Euclidean space, this generally just means that the euclidean a ction evaluated on the configuration is finite, since then it contributes to the path integral with a finite Boltzmann factor e−S[φ]. In Minkowski sig nat ure, a field configuration should be considered normalizible if it has finite energy; for a scalar field, this energy is E[φ] ≡ TABξA n B Σ where the integral is over some fixed-time slice Σ, n is a unit normal vector to Σ, ξ is a time-like killing vector , and TAB is the stress tensor for t he bulk field, 2 δS TAB ≡√g δgAB . Given the redefinition you found in part a), what is the relationship between normalizable wave functions in the usual QM sense (i.e. kΨk2 < ∞) and nor-malizable solutions of the AdS wave equation (i.e. those with finite energy)? c) Set ~k = 0. Show that when m2 passes through the value −|m2 | from above, BFthis Schr¨odinger equation develops a single normalizable negative-energy state.2 (Recall that the BF bound is m2 ≥ mBF 2 = −4DL22 .) This corresponds by the map above to a normalizable mode with imaginary frequency – i.e. a linear instability. 3. Dimensions of vector operators. By studying the boundary behavior o f solutions to the bulk Proca equations, derived from the action �� 2 � Sbulk[A] = −κdp+2 x√g 1 FµνFµν + mAµAµ4 2AdS compute the scaling dimension Δ of the current coupled to a bulk vector field Aµ of arbitary mass: dp+1 JµSbdy = x√γAµ. ∂AdS For purposes of this problem, define Δ to be the power of z of the subleading power-law solution of the bulk equation near the boundary at z = 0. 2If you want to learn more about potentials of this form, see H. Hammer and B. Swingle, Annals Phys. 321 (2006) 306-317 [arXiv:quant-ph/0503074v1]. 2� � � 4. Saturating the unitarity bound.3 For this problem we will work in Euclidean AdS. Feel free to redo the problem for the Lorenzian case (using the notion of normalizability described in problem two). a) Consider the usual bulk action for a (free) scalar field in AdSD + 1: dD2φ2Susual[φ] = −21 xdz√g � (∂φ)2 + m � Show that for m2L2 > −D4 2 + 1, Susual[z Δ− ] = ∞. That is, the integr al over the radial direction in this action diverges near z = 0 when evaluated on a solution of the form φ ∼ zΔ− , where Δ is the smaller root −of Δ(Δ −D) = L2m2; such a solution is non-normalizable. On the other hand, show that zΔ+ gives a finite value, and hence a solution with these asymptotics is normalizable. With this action, show that the smallest conformal dimension Δ = Δ+ of a scalar operator in the dual theory is Δ ≥D .2 b) Show that the following modified action dD + 1 2)φSKW[φ] ≡ − 21 x√g � φ(−� + m � = Susual[φ] − ∂AdS √γφ∂nφ. produces the same bulk equations, but has the following property: in the win-dow of mass values (we could call this the KW window) D2 D2 −4 < L2 m 2 < −4 + 1, both r oots Δ give normalizable solutions with respect to the action SKW (i.e. ± SKW[zΔ± ] < ∞). What is the lowest conformal dimension you can obtain now? (Hint: you can now consider Δ+ to be the dimension of the source, and Δ to be the dimension −of the operator.) 3This result is from Klebanov-Witten, hep-th/9905104.