Chapter 8.8/8.87 Holographic Duality Fall 04 Chapter : Deriving AdS/CFT MIT OpenCourseWare Lecture Notes Hong Liu, Fall 04 Lecture 0 In this chapter, we will focus on:. The spectrum of closed and open strings (for gravity and gauge theories.. The physics of D-branes nonperturbative objects in string theory. 3. D-branes as classical gravity (general relativity solutions..: PERTURBATIVE (BOSONIC STRING THEORY..: GENERAL SET UP Consider a string moving in a spacetime M with the metric (µ, ν = 0,,..., d : ds = G µν (XdX µ dx ν, ( with the worldsheet Σ parametrizations (a = 0, : X µ (σ, τ = X µ (σ a. ( The induced metric on Σ is written as: h ab = G µν a X µ b X ν, ds ind = h ab (σ, τ dσ a dσ b. (3 The string action is defined to be proportional to the area of Σ, written in the following Nambu-Goto form: S NG [X µ ] = da = d σ h ; [α ' ] = L α ' = ls, T = πα ' Σ πα ' Σ πα ', (4 with l s is the string length scale (from dimensional analysis and T is the string tension. To understand this action, at least at classical level, let s take a look a nearly static string configuration (small fluctuations extends in X -direction in a flat spacetime R,d and choose the worldsheet parametrizations σ 0 = X 0 and σ = X (the remaining coordinates are X i, with i =, 3,..., d : + ( 0 X i 0 X i X h i ab = G µν ax µ b X ν = (5 0 X i X i + ( X i h = ( 0 X i + ( X i + ( 0 X i X i ( 0 X j X j. (6 This gives the Nambu-Goto Lagrangian at lowest orders: S NG = d σl NG, L NG = T + T ( 0 X i ( X i Σ + O ( X i 4. (7 From the 0th order, it s straight-forward to see why T can be identified as the string tension (the energy per unit length of the string. The next order shows that the string fluctuations (waves propagate with the speed of light.
Chapter 8.8/8.87 Holographic Duality Fall 04 Since the nonpolynomial nature of S NG is inconvenient for calculations, it s much easier to work with the Polyakov s action, which is equivalent to the Nambu-Goto s action at classical level: S γ ab P [γ ab, X µ ] = d σ γγ ab a X µ b X ν G µν ; = γ ab (σ, τ. (8 4πα ' Σ To see this, note that worldsheet stress-energy tensor is defined as: δs P T ab = 4π. (9 γδγ ab Since δs P = 0 for variations around the classical solution (on-shell, the equation of motion for γ ab is T ab = 0. Using: δ γ = γγ ab δγ ab, (0 then the stress-energy tensor can be found: ( ( ( d σ δs P = δ γγ ab h ab + γδγ ab h ab = γ γ ab γ cd h cd h ab δγ ab ( Σ 4πα 4πα ( ( T ab = γ ab γ cd h cd h ab = α α Gµν γ abγ cd cx µ d X ν ax µ b X ν = 0. ( This means γ ab = Bh ab, with B = B(σ, τ can be arbitrary. Integrate out the worldsheet intrinsic metric field γ ab : S P [γ ab = Bh ab, X µ ] = d σ(b h(bh ab (h d ab = σ h = S NG [X µ ]. (3 4πα Σ πα Σ In quantum field theories and effective field theories, if different theories are related by field redefinitions that preserve symmetries and have the same -particle states (Representation Independence Theorem, then the results of observables in spacetime of these theories should agree (although the off-shell process might be different but the on-shell calculations are the same, order by order. Hence it is very tempting to say that S NG and S P produce the same physics (not only classically but also quantumly, however, the argument is only done on actions of polynomial form. Therefore, one shouldn t think S NG and S P describe the same physics at quantum level; S NG simply provoke S P, and the later is much more simple for quantization and computational purposes. Equation (5 has the form of a D field theory of d scalar fields X µ couples with gravity the D metric γ ab (note that, X 0 field has opposite sign in its kinetic term. Historically, (5 is called a nonlinear σ-model (for S P, the scalar fields X µ which takes on values in a nonlinear target manifold Σ. The string path integral quantization will be based on (after taking care of gauge redundancy and adding insertions in...: is P [γ ab,x µ ] Dγ ab DX µ e..., (4 and this method of quantization has computational advantages as well as conceptual simplicity (especially for a general worldsheet Σ with nontrivial topology, but for the sake of understanding the physical spectrum of strings (the worldsheet describes the noninteracting asymptotic state is topologically trivial, then canonical quantization is more convenient. The Polyakov Lagrangian: S P = d σl P. (5 Σ Work with flat spacetime G µν = η µν (M = R,d, then the Polyakov action from the D worldsheet perspective has set of global symmetry (spacetime Poincare symmetry and sets of local gauge symmetries (worldsheet diffeomorphism symmetry for reparametrization of the worldsheet and Weyl symmetry for rescaling of the intrinsic metric:. Global Poincare transformation (translation and Lorentz rotation:. Local diffeomorphism transformation (σ a σ 'a : µ X µ (σ, τ X µ (σ, τ + a ; X µ (σ, τ Λ ν µ X ν (σ, τ. (6 σ 'a σ 'b X µ (σ, τ X 'µ (σ ', τ ' = X µ (σ, τ, γ ab (σ, τ γ 'ab (σ ', τ ' = γ cd (σ, τ. (7 σ c σ d
Chapter 8.8/8.87 Holographic Duality Fall 04 3. Local Weyl transformation: γ ab (σ, τ e ω(σ,τ γ ab (σ, τ. (8 Note that, conformal symmetry = Diff Weyl γ ab symmetry. These symmetries (Poincare and Diff Weyl can be used as the guiding principles to (almost uniquely determined the string action in equation (5. Indeed, for example, in a topological invariant of D oriented closed surfaces, the D Einstein-Hilbert action also fits the bill: S χ [γ ab ] = λ d σ γr = λχ(σ, χ(σ = g. (9 4π Σ Because of the topology nature, S χ plays the role similar to the vertex weight in quantum field theory. The closed λ string coupling is defined to be g c = e. For more generalization, a topological invariant of D oriented (both closed and open surfaces that comes from Einstein-Hilbert action and boundary extrinsic curvature action (K = ±t a n b \ at b is the extrinsic curvature of Σ with t n and the sign ± depends on the type of boundary timelike: +, spacelike: should be included: ( S χ[γ ab ] = λ d σ γr + dsk = λχ(σ ; χ(σ = g b. (0 4π Σ π Σ It s simple to prove this action is diffeomorphism invariant. To see that S χ is indeed Weyl invariant γ ab e ω γ ab, note that: Hence: Γ a bc Γ a bc + a bωδ + cωδ a ω ω d ωγ ad γ bc, R e ω (R \ a a ω, t e t, n a e n a, ds e ω c b ds. ( Using Stokes theorem: a K e ω (K t t an b b ω = e ω (K + n a aω. ( ( S χ λ d σ γ(r \ a a ω + ds(k + n a aω (3 4π Σ π Σ = S χ + λ ( d σ γ\ a a ω + dsn a aω = S χ (4 π Σ Σ Mathematically, the action S χ only depend on the topological properties of the worldsheet Σ, with χ(σ is known as the Euler-characteristic of Σ (g is the number of genus and b is the number of open boundary. D gravity is dynamically trivial as S χ gives no dynamics: indeed, in D, from a purely symmetrical fact: This means the worldsheet Ricci scalar: R abcd = R cdab = R bacd = R abdc R abcd = Rε ab ε cd. (5 R = R, R ab = γ cd R abcd = Rγ ab, (6 and the equation of motion for Einstein-Hilbert action (the worldsheet stress-energy tensor T ab = 0 on-shell is satisfied naturally. Because of the topology nature, S χ in string theory plays the role similar to the vertex weight in quantum field theory:. Add a handle to a closed string worldsheet (creates an extra emission and reabsorption of a closed string will increase the number of genus by and change the string path weight. In terms of Feynman diagrams this corresponds to the weight of vertices, which can be read-off as the closed string coupling g c = e λ.. Add a hole to an open string worldsheet (creates an extra emission and reabsorption of an open string will increase the number of boundary by and change the string path weight. In terms of Feynman diagrams this corresponds to the weight of vertices, which can be read-off as the open string coupling g λ/ o = e. In oriented string theories, because of a different between Euclidean and Lorentzian signature, the worldsheet Σ can be treated as a Riemann surface (orientable complex manifold for calculations with imaginary time via analytic continuation (Wick s rotation. For unoriented string theories (eg. type I superstring with orientifold planes, one needs extra works to understand the perturbative regime (eg. worldsheets with cross-caps. 3
Chapter 8.8/8.87 Holographic Duality Fall 04..: LIGHT-CONE QUANTIZATION Each physical oscillation mode of a string corresponds to a particle in spacetime. For massless mode, closed string gives a spin particle (graviton and open string gives a spin particle (gauge particle, like photon or gluon. The gauge symmetries (Diff Weyl indicate the redundancy in degrees of freedom, and this must be fixed during quantization. For a general topology of the worldsheet Σ, path integral works best for calculation (eg. S-matrix, but for understanding the particle spectrum of the theory by quantizing on a trivial topology of single string propagation (long cylinder for closed string and long sheet for open string, then canonical quantization (old covariant and light-cone quantization is simpler and faster. It can be shown that these methods of quantization yields the same results for the particle contents []. The canonical quantization procedure:. Write down the classical equation of motion.. Fix the gauge symmetries. 3. Find the complete set of classical solution. 4. Promote classical fields (on worldsheet to quantum operators, satisfying canonical quantization condition. The classical solutions become solutions to operator equation, and the parameters in classical solutions become creation and annihilation operators. 5. Read-of the spectrum by acting creation operators on the vacuum of the (D worldsheet theory. For single string propagation, closed string (worldsheet topology S R can gets spatial parametrization σ [0, π] with: X µ (σ, τ = X µ (σ, τ, γ ab (σ, τ = γ ab (σ + π, τ, (7 while open string (worldsheet topology L R can gets spatial parametrization σ [0, π] with boundary conditions at σ = [0, π] which can be found by enforcing the nonlocal contribution to δs P when varying δx µ goes away: δs P = dτ γδx µ σ X µ = 0 (8 πα ' With this, one can impose at the ends either Dirichlet condition (δx µ = 0 or Neumann condition (γ σa a X µ = 0. Or in other words, n a a X µ = 0 with n Σ (this indeed gives a glimpse of a nonperturbative object in string theory the D-brane, where strings can end. Let s just look at the Neumann condition (in all directions for the moment. Note that, the boundary condition (at the ends of open string it is not for the sake of simplicity, but for consistency only for appropriate boundary conditions, can equations of motion be consistently imposed. Indeed, when D-branes are included, the boundary contributions to the stringy dynamics is very important. The classical equation of motion from (5:. For γ ab :. For X µ : T ab = a X µ b X µ γ ab γ cd c X µ d X ν = 0 (9 a ( γγ ab b X µ = 0 (30 4
Chapter 8.8/8.87 Holographic Duality Fall 04 By diffeormophism, the metric can be put in the conformally flat form γ ab = e ω(σ,τ η ab, and Weyl rescaling can be used to get γ ab = η ab (note that, after this setting one still have a gauge redundancy, called the conformal symmetry. Indeed, by using the similar trick on a general worldsheet Σ, the whole surface can be cover with a single flat D sheet with appropriate holes, parametrizations and identifications (which give rise to the moduli and curvature. This trick is extremely useful for studying string scattering via the path integral quantization, and has a natural generalization to a superworldsheet in superstring. For example, genus g = Riemann surfaces can be built out of a complex plane parametrize by z by cutting-off the regions: and then glueing the annuli: by the transition function: z ( E w, z, E «, (3 E w D 0 = {z 0 C ( E w < z 0 < w }, D = {z 0 C ( E < z 0 < } (3 E w E w The complex modulus can be read-off from w. z 0 = f, (z = wz. (33 To get genus g = surfaces, firstly cut-off more disks: z a ( E u, z b ( E u, (34 and specify the annular regions D a and D b, then glue these together by a transition map f a,b (z. The transition functions f 0, (z and f a,b (z should be in total depend on 3 complex parameters (up to conformal transformation, which is the 3 complex moduli of this Riemann surfaces. Higher genus (superriemann surfaces, or any other general (superworldsheet Σ, follow a similar procedure. With γ ab = η ab, equation (6 can be rewritten: Equation (3 can be further simplified: τ X µ σ X µ = ( τ σ X µ = ( τ σ ( τ + σ X µ = 0. (35 T ττ = T σσ = ( τ X µ τ X µ + σ X µ σ X µ = 0. (36 T τσ = T στ = τ X µ σ X µ = 0. (37 For open string with Neumann condition, σ X µ (σ = 0, π; τ = 0. Note that equation (3 can be derived from the action obtained by directly setting the flat worldsheet metric for the action given in equation (3 (which gives a bunch of free scalar fields. However, quantization of strings is different from merely quantization of S P [γ ab = η ab, X µ ], as one needs to impose the conditions that the stress tensor following from that is 0, which are equation (3 and (33 the Virasoro constraints (nonlinear constraint equations. µ Equation (3 can be immediately solved (x, v µ can be arbitrary constants: µ X µ (σ, τ = x + v µ τ + X µ (τ σ + X µ (τ + σ (38 For closed strings, X µ and X µ R L are independent periodic functions of period π. For open strings, from the Neumann condition, then at the ends σ X R = σ X L as X µ = X µ R L and is periodic in π. From here, one could quantize S P [γ ab = η ab, X µ ] first and then impose the Virasoro constraints at quantum level later, or firstly solve these constraints explicitly at classical level and then find unconstrained degrees of freedom hence only quantize these. Light-cone gauge quantization follows the nd approach. R L 5
Chapter 8.8/8.87 Holographic Duality Fall 04 After fixing the worldsheet metric, one still have residual gauge freedom (conformal symmetry. Let s introduce: τ ± σ σ ± =, ds = dσ + dσ, (39 hence this symmetry can be viewed as the preservation of γ ab = η ab (up to a Weyl rescaling as: Since: σ + σ + = f(σ +, σ σ = g(σ, ds + f gdσ + dσ. (40 f(τ + σ + g(τ σ τ = τ, (4 which has the same form as the classical solution of X µ, then one can fix the gauge completely by choosing appropriate f and g so that: X + X 0 ± X τ = v +, X ± =. (4 This is known as the light-cone gauge, as the worldsheet time is fixed by the spacetime light-cone coordinate. With X µ = (X +, X, X i (the transverse directions i =, 3,..., d : dx µ dx µ = dx + dx + dx i dx i. (43 In light-cone gauge, the Virasoro constraints become: v + τ X = ( τ X i + ( σ X i, (44 v + σ X = τ X i σ X i, (45 therefore, X can be fully solved in terms of X i (σ, τ. Therefore, the independence degrees of freedom are X i. Since X 0 has a wrong sign for its kinetic terms, no X 0 in these degrees of freedom actually partly solve a problem of unitarity at quantum level. Reference [] Polchinski, J. (03. String Theory, v.. Lexington, KY: Cambridge University Press. 6
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