# Hong Liu Lectures on holography [YouTube](https://www.youtube.com/watch?v=EUnGZoBa3nc&list=PLUl4u3cNGP633VWvZh23bP6dG80gW34SU) [notes](https://ocw.mit.edu/courses/physics/8-821-string-theory-and-holographic-duality-fall-2014/lecture-notes/) ## 1. Hints for holography ### 1.1 Prelude: Gravity v.s. other interactions - we don't know how to quantise gravity. Holography is one realisation of quantum gravity: QG = field theory on fixed spacetime - idea of emergence - 1967 Sakharov has idea of emergence of gravity - 1950's hydrodynamics is not fundamental but emerges at some effective description of the molecules -> maybe gravity is also not fundamental and emerges somehow - from field theory perspective, natural to ask whether massless spin-2 particles can arise as bound states of low spins - if yes -> gravity emerges - e.g. strong interaction: gluons + quarks combine to give spin-2 particles (but massive) - -> can we make these massive ones massless somehow? No: [[0151 Weinberg-Witten theorem]] - **Theorem 1**: a theory that allows the construction of a Lorentz-covariant, conserved current $J^\mu$ cannot contain massless particles of spin gt; 1/2$ with non-vanishing value of $\int J^0 d^3x$ (i.e. charge) - **Theorem 2**: a theory that allows a Lorentz-covariant, conserved $T^{\mu\nu}$ cannot contain massless particles of spin $j>1$ - **Remarks**: - the theorems apply to both elementary and composite particles - the theorem is compatible with QED - photons: spin 1 but no charge - electrons: spin 1/2 - YM SU(2): - with $A^3_{\mu}, \,A^{\pm}_\mu=\frac{1}{\sqrt{2}}(A_\mu^1\pm iA_\mu^2)$, $A^{\pm}_\mu$ massless spin-1 charged under U(1) subalgebra generated by $\sigma^3/2$ - BUT: there does not exist a conserved Lorentz-covariant current for this U(1) - gauge-invariant and conserved current is not Lorentz-covariant - Lorentz-covariant and gauge invariant current is not conserved - these theorems do not forbid graviton from GR - In GR, there is no *conserved* Lorentz-covariant $T^{\mu\nu}$ - $\nabla_\mu T^{\mu\nu}=0$ <=> *covariantly conserved* (but not conserved) - *conserved* means ordinary derivatives - theorem 2 => none of renormalisable QFTs in Minkowski can have emergent gravity - hidden assumption: the particles live in the same spacetime in the original theory! - **Proof**: Suppose there exist massless particles of spin-j. - one particle state: $|k^\mu,\sigma\rangle$, $\sigma=\pm j$ = helicity - consider rotation $\hat{R}(\theta,\hat k)|k,\sigma\rangle=e^{i\sigma \theta}|k,\sigma\rangle$ around its spatial direction - now conserved current $J^\mu$ => $Q=\int d^3 x J^0$ - $T^{\mu\nu}$=> $\hat P^\mu=\int T^{0\mu}$, $\hat P^\mu|k,\sigma\rangle=k^\mu|k,\sigma\rangle$ - if charged under $\hat Q$: $\hat Q|k,\sigma\rangle=q|k,\sigma\rangle$ - want to show: 1) if $q\ne0$, $j\le1/2$; 2) $j\le1$. - i) Lorentz symmetry: $\left\langle k, \sigma\left|J^{\mu}\right| k^\prime, \sigma\right\rangle$ -> $\frac{qk^\mu}{k^0}\frac{1}{(2\pi)^3}$ and $\left\langle k, \sigma\left|T^{\mu\nu}\right| k^\prime, \sigma\right\rangle$ -> $\frac{k^\mu k^\nu}{k^0}\frac{1}{(2\pi)^3}$ as $k\rightarrow k^\prime$ - intuition: when $\mu=0$, $|k,\sigma\rangle$ is eigenstate of $J^0$ with eigenvalue $q$; when $\mu\ne0$ is must carry a $\mu$ index and must reduce to $q$ when $\mu=0$ - ii) for massless particles, $k^2=0={k^\prime}^2$ => $k.k^\prime<0$, $k+k^\prime$ timelike. -> choose a frame s.t. $\vec k+\vec k^\prime=0$. so $k^\mu=(E,0,0,E),{k^\prime}^\mu=(E,0,0,-E)$ - iii) under rotation of $\theta$ around 3-direction, $\hat R(\theta)|k,j\rangle=e^{\mathrm{i}j\theta}|k,j\rangle$, $\hat R(\theta)|k^\prime j\rangle=e^{-\mathrm{i}j\theta}|k^\prime,j\rangle$. Consider $\langle k^\prime,j|\hat R^{-1}(\theta)J^\mu \hat R(\theta)|k,j\rangle$ => $e^{2 i j{\theta}}\langle k^\prime, j| J^{k}|k, j\rangle ={\Lambda^\mu}_\nu\langle k^\prime, j| J^{k}|k, j\rangle$ ==(4)==. - can do it for $T^{\mu\nu}$ similarly => ==(5)== - now $\Lambda$ can only have eigenvalue $e^{\pm i\theta}, 1$ - => $j\le1/2$ from (4) and $j\le1$ from (5). QED ### 1.2 Black hole thermodynamics - when Schwarzschild radius much larger than Compton, expect quantum effect to be negligible, but this is not true: - BH can have macroscopic quantum effects! - [[0017 Planck length]] is the minimal scale one can probe - different regimes - in particular, semiclassical: $\hbar$ finite, expand in $G_N$ - classical BH - Rindler horizon etc ### 1.3 Holographic principle - in non-gravitational system, number of d.o.f. is proportional to volume, but can easily violate entropy bound - so QG leads to a huge reduction of the number of d.o.f. - i.e. coupling any QFT to gravity leads to formation of BH before too many d.o.f. are stuffed into any given region - **holographic principle**: in QG, a region of boundary area A can be fully described by no more than A/4 d.o.f. (so the number of Hilbert space would be $e^{A/4}$) ### 1.4 Large N matrix theories - QCD: $SU(3)$ gauge theory + quarks - $L = -\frac{1}{g_{YM}}\left[-\frac{1}{4}TrFF- i \Psi(\cancel{D}-m)\Psi\right]$ - 't Hooft (1974): take number of colours as a parameter, i.e. $A_\mu$ are now N by N matrices, and do expansion in 1/N. - **Surprise**: $1/N$ expansion <-> String Theory - key: fields are matrices - **Illustration example**. Consider scalar theory - $\mathcal{L}=-\frac{1}{g^{2}} \operatorname{Tr}\left[\frac{1}{2} \partial_{\mu} \Phi \partial^{\mu} \Phi+\frac{1}{4} \Phi^{4}\right]$ - $\left\langle\Phi_{b}^{a}(x) \Phi_{d}^{c}(y)\right\rangle=g^{2} \delta_{d}^{a} \delta_{b}^{c} G(x-y)$ can represent using **double lines** - 4-vertex: $\frac{1}{g^{2}} \delta_{h}^{a} \delta_{b}^{c} \delta_{d}^{e} \delta_{g}^{f}$ - vacuum diagrams - ![[Rsc0010_ribbons_vac.png|400]] ![[Rsc0010_ribbons_vac_cd.png|400]] - number of closed single-line loops = power of N (because each loop is taking a trace) - **Observations** - (b) and (d) can be drawn on a torus without crossing lines ![[Rsc0010_torus_bd.png|400]] - power of N = number of faces on torus - **Fact**: any orientable 2d surface is classified topologically by an integer h, called genus. Equal to the number of holes. - **Claim 1**. For any non-planar diagram, there exists an integer $h$, such that the diagram can be straightened out on a genus-$h$ surface, but not on a surface with a smaller genus. - **Claim 2**. For any diagram, the power N coming from contracting propagators is given by the number of faces on such a genus-h surface. - => a vacuum diagram has $A\sim(g^2)^{E-V}N^F$ - E: no. of propagators; V: vertices; F: faces - seems no sensible N -> infinity limit - but 't Hooft: possible to have finite limit if $g \to 0$. - define $\lambda\equiv g^2N$ ('t Hooft coupling) - then $A\equiv (g^2N)^{E-V}N^{F+V-E}=\lambda^{L-1}N^{2-2h}$ - $L=E-(V-1)$ = number of loops - number of loops = \# undetermined momentum; each propagator carries a momentum; each vertex fixes a momentum conservation, although the overall momentum conservation is guaranteed (so $V-1$ independent constraints) - $\chi=F+V-E=2-2h$ - this now has a well-defined large-$N$ limit as $h\ge0$. - to leading order in large $N$: planar diagrams (genus 0) has $N^2(c_0+c_1\lambda+...)=N^2f_0(\lambda)$。 - turns out to be convergent: the number of *planar* diagrams grows very slowly with the power of $\lambda$ - in general, **all vacuum diagrams**: $\log Z=\sum_{h=0}^\infty N^{2-2h}f_h(\lambda)=N^2f_0(\lambda)+f_1+\frac{1}{N^2}f_2(\lambda)+\dots$ - **a heuristic way** to understand this expansion - $Z=\int D\Phi e^{iS[\Psi]}$ - $\mathcal{L}=-\frac{N}{\lambda}Tr(\dots)$: this is order $N^2$ (trace is a sum of $N$ things) - so (leap of faith) if we do saddle point approximation, we get powers of $1/N^2$ in the expansion - **Claim**. For *any* Lagrangian of matrix valued fields of the form $\mathcal{L}=\frac{N}{\lambda}Tr(\dots)$, we have $\log Z=\sum_{h=0}^\infty N^{2-2h}f_h(\lambda)=N^2f_0(\lambda)+f_1+\frac{1}{N^2}f_2(\lambda)+\dots$ - **Summary**. In the 't Hooft limit, $1/N$ expansion = expansion in the topology of Feynman diagrams. - In **gauge theories**, $\mathcal{L}=\mathcal{L}(A_\mu,\Phi,\dots)$ - allowed **local operators**: $Tr(F_{\mu\nu}F^{\mu\nu})$, $Tr(\Phi^k)$ <- single trace: denote $O_n$ - $Tr(F_{\mu\nu}F^{\mu\nu})Tr(\Phi^2)$ <- multi-trace" denote $O_1O_2O_3...$ - **general observables**: correlation functions of gauge invariant operators - **Trick to obtain $N$ dependence** of correlation functions - $Z\left[J_{1}, \ldots ,J_{n}\right]=\int D{A_\mu}D\Phi \exp \left[i S_{0}+i N\int J_i(x) O_i(x)\right]$ - => $\left\langle O(x_1) \cdots O_{n}(x_{n})\right\rangle_C=\frac{1}{i^n}\frac{\delta^n\log Z}{\delta J_1(x_1)\dots\delta J_n(x_n)}\frac{1}{N^n}$ - now notice that the whole exponent has $N \text{Tr}(\dots)$ just like before => $\log Z(J_1,\dots, J_n)=\sum_{h=0}^\infty N^{2-2h}f_h(\lambda,J_i)$ - => ==$\left\langle O(x_1) \cdots O_{n}(x_{n})\right\rangle_C=\sum_{h=0}^\infty N^{2-n-2h}f_h(\lambda,J_i)\sim N^{2-n}(1+\mathcal{O}(\frac{1}{N^2})+\dots)$== - **Physical implications**. 1. $O_1\dots O_n(x)|0\rangle$ = $n$-particle state - (a) $\langle O_iO_j\rangle\sim \delta_{ij}$ by choice of basis - (b) $\langle O_i(x) :O_1O_2:(y)\rangle\sim 1/N\rightarrow 0$. i.e. no mixing between single particle and multi-particle states - (c) $\langle :O_1O_2:(x) :O_1O_2:(y)\rangle$ $= \langle O_1(x)O_2(y)\rangle\langle O_1(x)O_2(y)\rangle+\langle O_1O_2O_1O_2\rangle_C$ second terms goes to 0 at large N, so this is like two particles propagating on their own - n.b. may not be physical particles states, just saying they are like them. in QCD they can be short-lived glueball states 2. glueball fluctuations are suppressed - suppose $\langle O\rangle\ne0\sim N$ - variance = $\langle O^2\rangle-\langle O\rangle^2=\langle O^2\rangle_C\sim N^0$ - so $\sqrt{\text{variance}}/\langle O\rangle\sim 1/N \rightarrow 0$ - disconnected correlation always factorise -> like a classical theory - $\langle O_1O_2\rangle=\langle O_1\rangle\langle O_2\rangle+\langle O_1O_2\rangle_C$ first term always dominate 3. If we interpret $\langle O_1(x_1)\dots O_n(x_n)\rangle_C$ as scattering amplitudes of $n$ glueballs (we call each single particle state a glueball, then the scatterings are classical (i.e. tree-level) - (a) think of a 3-pt vertex as a 3-pt. correlator so $\sim 1/N\equiv \tilde{g}$ => can convince oneself that tree-level $n$-particle amplitudes $\sim \tilde{g}^{n-2}\sim N^{2-n}$ - (b) can also include higher order vertices (4, 5, ...) - (c) to leading order in N, in any correlation function, there are only one-particle intermediate states. - try inserting complete basis (which are called intermediate states), then will find that contribution from multi-particle states vanish for large N - contribution from multi-particle states form loops: e.g. $\langle O_1O_2O_3\rangle$ contains possibility of 1 -> i,j (2 particles) -> 2,3, where the splitting and joining would form a loop - **Summary**. at large $N$, we have a *classical* theory of glueballs, with interactions among them given by $\tilde{g}\sim1/N$. - gauge theory at large N but finite $\hbar$ = glueball theory with effective $\tilde{\hbar}\rightarrow0$ - perturbation in $1/N$ <-> semi-classical expansion in $\tilde{\hbar}\sim1/N$ - **Caveat**. We are talking about loops of glueballs (gauge-invariant composite objects) which are suppressed. The gauge-dependent fields such as original *gluons* can have loops in their amplitudes. ### 1.5 Large N expansion as a String theory - QFT: 2nd quantized approach. - String theory: a generalisation of the 1st quantised approach: theory of particles (interaction needs to be added by hand, e.g. a particle splitting into 2 particles; adding interaction = specifying what paths to include in the PI for the particles) - **Vacuum energy**. $Z=\sum_{h}\sum_{\text{all surf. with genus } h}e^{-S_{NG}}$, where we use Euclidean Nambu-Goto action now. - whenever you have a discrete sum, can add a weight by hand: $Z=\sum_{h}e^{\lambda\chi}\sum_{\text{all surf. with genus } h}e^{-S_{NG}}$, where $\lambda$ is like a chemical potential for topology. - turns out that this weighting is what we get if we quantise strings properly, some other powers of $\chi$ will give different theories - write ==$g_S=e^\lambda$== - e.g. genus 0: order $g_S^{-2}$; genus 1: order 1 etc - **Remarkable fact**. Summing over topology automatically includes interactions of the string. In fact this fully specifies string interactions. - effectively giving a factor of $g_S$ for each splitting or joining (adding a genus gives $g_S^2$, and adding a genus adds two vertices) - n.b. can slice the diagrams in different ways and can in principle get a string splitting into 3, but it is consistent to have only 1-to-2 splitting. - **With external strings**. Since now the worldsheets have $n$ boundaries, $\chi=2-2h-n$, the amplitude is given by ==$A_n=\sum_{h=0}^\infty g_S^{n-2+2h}F_n^{(h)}$== - identical structure with large-$N$ theories! - with $g_S\leftrightarrow 1/N$, external strings $\leftrightarrow$ glueballs (single-trace operators), sum over string worldsheets of genus $h$ $\leftrightarrow$ sum over Feynman diagrams with genus $h$ - **BUT**, can you really identify the two ($f_n^{(h)}$ and $F_n^{(h)}$)? - $f_n^{(h)}$ = $\sum_\text{Feynman diagrams of genus h}G=\sum_\text{all triangulation of surface with genus h}G$ = discrete form of summing over all surfaces - so they are really the same if $G=e^{-S_{NG}}$ - what string theory it corresponds to will depend on the nature of Feynman diagrams: given a QFT at large $N$, need to figure out that string theory it maps to. - **Comment**. The identification is difficult: 1. $G$ is expressed as products of field theory propagators integrated over spacetime. they look nothing like $e^{-S_{string}}$ - $S_{string}$ is a map from worldsheet to target space, which needs to be chosen. need to choose (a) the target space, (b) what $S_{String}$ looks like, (c) additional internal d.o.f. on the worldsheet e.g. fermions in Superstring 2. String theory: continuum. Feynman diagrams: at best discrete version. - expect geometric picture for $G$ to emerge only at strong 't Hooft coupling (at very strong coupling, Feynman diagrams with infinite number of vertices dominate: more like continuum) - recall $A\equiv (g^2N)^{E-V}N^{F+V-E}=\lambda^{L-1}N^{2-2h}$ - with fixed number of lines at each vertex, number of loops is proportional to the number of vertices: think about squares on the floor - alternative - $\mathcal{L}=\frac{N}{\lambda}Tr(\frac{1}{2}(\partial\Phi)^2+\frac{1}{4}\Phi^4)$ and $\lambda$ goes to the $\Phi^4$ after rescaling -> i.e. propagator $g^2$, vertex $1/g^2$ becomes propagator $1$, vertex $g^2$ - so no need to count number of propagators; just count vertices 3. For simple systems like a matrix integral ($dM$) or matrix quantum mechanics ($DM(t)$: paths), can guess the corresponding string theory. both are simple. - see [[Klebanov1991]]@[arXiv](https://arxiv.org/abs/hep-th/9108019) sec. 2 - **Generalisations**. - so far matrix valued fields, adjoint rep of the $U(N)$ gauge group. Can also include fields in the fundamental rep. (quarks). These have single index, so are represented by single lines. => Feynman diagrams can be classified by 2d surfaces with boundaries (for vacuum diagrams) - <-> a String theory with both closed and open strings - so far used $U(N)$ as gauge group. Can consider $SO(N)$ or $Sp(N)$ so that fields are $\Phi_{ab}$ -> no up and down indices -> lines are not directed -> non-orientable surfaces - <-> non-orientable string theories - Take large $N$ generalisation of QCD in 3+1 d Mink, can we say anything about its String theory description? - natural guess: a String theory in $M_{3+1}$, $S_{NG}=\frac{1}{2\pi\alpha^\prime}\int dA$. But it does not work: - such a string theory is inconsistent except for $D=26$, or 10 if add fermions - take a string in 10d with $M_{3+1}\times N$ with $N$ compact - would not work, because string theory always contain gravity, and if put in compact space, it always leads to massless spin-2 particle in the $M_{3+1}$ part -> not okay for QCD in 3+1 [[0151 Weinberg-Witten theorem]] - -> what to do now? - (a) more exotic action - (b) other target space - Hints for looking for a string theory in *4+1* non-compact spacetime - holographic principle: String theory contains gravity, and theories with gravity have d.o.f. proportional to area -> an equivalent non-gravitational description should be one dimension lower. (this is hindsight: no one realised it before the discovery) - consistency of string theory itself (sometimes adding d.o.f. in order to have the correct symmetry leads to one higher dimension) - Now suppose a $4+1$ manifold, Y, describes QCD - by Poincare invariance of 3+1, the metric must be of the form $d s^{2}=a^{2}(z)\left(d z^{2}+ \eta_{\mu\nu} d X^{\mu} d X^\nu\right)$ - if the field theory is CFT -> can show that this metric must be AdS (easily) - **History** - late 60's and early 70's, String theory developed to understand strong interactions - 1971 Asymptotic freedom. Eliminated the hope of using string theory to describe strong interactions: QCD very different from string theory - 1974 - 't Hooft: large N like a string theory - Schwarz etc: string theory as a gravity - Hawking radiation - quarks observed (charmonium) - Wilson: lattice QCD. quarks can be confined through the strings. -> revived string theory - 1993, 1994, 't Hooft: holographic principle - 1995 Polchinski: D-branes - 1997 June, Polyakov: realised that it should be 4+1 not 3+1 - 1997 November, Maldacena: construction using D-branes - 1998 February, Witten: made the connection to holographic principle ## 2. Deriving AdS/CFT 1. spectrum of closed and open strings - closed string -> gravity - open string -> gauge theory 2. D-branes - special role of connecting gravity and gauge theory ### 2.1 More about string theory #### 2.1.1 General setup - setup - consider a string with $ds^2=g_{\mu\nu}(X)dX^\mu dX^\nu$ - worldsheet parameterised by $X^\mu(\sigma^a)$ with $\sigma^a=(\tau,\sigma), a=0,1$ - induced metric $ds_{ind}^2=h_{ab}d\sigma^a d\sigma^b$, $h_{ab}=g_{\mu\nu}\partial_a X^\mu \partial_b X^\nu$ - action $S_{NG}=\frac{1}{2\pi\alpha^\prime}\int d^2\sigma \sqrt{-\det h}$ - inconvenient so rewrite as (**Polyakov** form): - $S_P[\gamma.X]=-\frac{1}{4\pi\alpha^\prime}\int d^2\sigma\sqrt{-\gamma}\gamma^{ab}(\sigma^a)\partial_a X^\mu\partial_bX^\nu g_{\mu\nu}(X^\mu)$ - $\gamma^{ab}$ is like Lagrangian multiplier - **sigma model** - Polyakov action has the form of a scalar field on a 2-dim spacetime, but now both $\gamma, X^\mu$ dynamics - -> can be considered as 2d gravity coupled to $D$ scalar fields - path integral over both: $\int D\gamma_{ab}DX^\mu e^{iS_P[\gamma,X]}$ - n.b. one of the scalar fields has the wrong sign for KE term - will be resolved by integration over $\gamma$ - symmetries of the action - (a) Poincare symmetry of $X^\mu$ - (b) 2d coordinate transformation (reparameterisation) - (c) Weyl scaling of $\gamma_{ab}$ - a bit surprising - since NG action has no $\gamma$, this is needed if Polyakov is equivalent to NG - **Euler action** - can use (a,b,c) to (almost) determine the action - another action consistent with 3 symmetries: **Euler action** $S_{Euler}=\frac{\lambda}{4\pi}\int d^2\sigma\sqrt{-\gamma}R$ - but this is a total derivative -> actually topological: depends on genus - in Euclidean path integral $e^{-S_{Euler}}=e^{-\lambda \chi}$ - categorise the symmetries - (a) is global -> conserved charges - (b,c) gauge symmetry #### 2.1.2 Light cone quantisation - why **canonical quantisation** - PI is simpler but does not give string spectrum - basic picture - closed string: each oscillation mode gives rise to a particle - open string: also get gauge fields - infinite number of types of particles - quantization **procedure** - write down EOM - fix gauge - find complete set of classical solutions - promote classical worldsheet fields to quantum operators satisfying canonical commutation relations - then classical solutions become solutions to operator equations - parameters in classical solutions (for SHO: amplitude of oscillations, a and a*) -> creation and annihilation operators - use ladders to find the spectrum - gauge freedom - use 3 gauge freedoms to fix 3 d.o.f. of $\gamma_{ab}$ to something fixed value - this is can be done locally. as spectrum can be obtained using local stuff, no need to worry about global issues - but will worry about global issues when doing PI - lightcone gauge - first solve for true d.o.f. then quantise (rather than quantise and then impose constraints as operator constraints) - $d s^{2}=-d t^{2}+d s^{2}=-2 d \sigma^{+} d \sigma^{-}$ - this metric is preserved by **residual gauge** freedom: $\tilde{\sigma}^{+}=f\left(\sigma^{+}\right)$, $\tilde{\sigma}^{-}=g\left(\sigma^{-}\right)$ followed by a Weyl scaling (this combination is the residual gauge freedom) - lightcone gauge: $\tau=X^+/v^+$ for constant $v^+$, i.e. identify worldsheet time with lightcone time - consequence - $\partial_\tau X^+=v^+$, $\partial_\sigma X^+=0$ - $X^-$ can be fully solved in terms of $X^is - => only independent d.o.f. are $X^i$ ($X^+$ known by fixing the gauge) - -> great: $X^i$ are just free scalar fields - comments - Lorentz symmetry is not manifest in lightcone gauge - but the remaining rotation group for $X^i$, $SO(D-2)$ is manifest - **mode expansion** - $X^{\mu}(\sigma, \tau)=x^{\mu}+v^{\mu} \tau+X_{R}^{\mu}(\tau-\sigma)+X_{L}^{\mu}(\tau+\sigma)$ - for closed string: $X^{\mu}(\sigma, \tau)=x^{\mu}+v^{\mu} \tau+i \sqrt{\frac{\alpha^{\prime}}{2}} \sum_{n \neq 0} \frac{1}{n}\left(\alpha_{n}^{\mu} e^{-i n(\tau+\sigma)}+\tilde{\alpha}_{n}^{\mu} e^{-i n(\tau-\sigma)}\right)$ - open string: - $X_{R}^{\mu}=X_{L}^{\mu}$ => $\alpha_{n}^{\mu}=\tilde{\alpha}_{n}^{\mu}$ - => $X^{\mu}(\sigma, \tau)=x^{\mu}+v^{\mu} \tau+i \sqrt{2 \alpha^{\prime}} \sum_{n \neq 0} \frac{1}{n} \alpha_{n}^{\mu} e^{-i n \tau} \cos n \sigma$ - remarks - CoM motion given by $x^{\mu}+v^{\mu}\tau$ - $\alpha^\mu_n$ and $\tilde\alpha^\mu_n$ are oscillations - closed: left and right moving - open: only standing wave - in lightcone gauge, $X^-$ can be solved by inserting the mode expansion, equating coefficients of different modes - => find $v^-, \alpha_n^-, \tilde\alpha_n^-$ - Virasoro constraints then imply (looking at zero modes) => - closed - $2 v^{+} v^{-}=v_{i}^{2}+\alpha^{\prime} \sum_{n \neq 0}\left(\alpha_{-n}^{i} \alpha_{n}^{i}+\tilde{\alpha}_{-n}^{i} \tilde{\alpha}_{n}^{i}\right)$ ==(19)== - and $\sum_{n \neq 0} \alpha_{-n}^{i} \alpha_{n}^{i}=\sum_{n \neq 0} \tilde{\alpha}_{-n}^{i} \tilde{\alpha}_{n}^{i}$ - overall left-moving must be same as right-moving - a result of having periodic boundary condition -> no special point - open - $2 v^{+} v^{-}=v_{i}^{2}+2 \alpha^{\prime} \sum_{n \neq 0} \alpha_{-n}^{i} \alpha_{n}^{i}$ ==(18)== - looking at n modes will get additional equations - global symmetry => conserved currents on the WS - translation symmetry => conserved current = spacetime momentum density of the string $\Pi^\mu_\tau$ - now (18) => $M^{2}=\frac{1}{2 \alpha^{\prime}} \sum_{n \neq 0} \alpha_{-n}^{i} \alpha_{n}^{i}$. i.e. mass of the open string is decided by the oscillations on the string - for closed string, (19) => $M^{2}=\frac{1}{\alpha^{\prime}} \sum_{n \neq 0}\left(\alpha_{-n}^{i} \alpha_{n}^{i}+\tilde{\alpha}_{-n}^{i} \tilde{\alpha}_{n}^{i}\right)=\frac{2}{\alpha^{\prime}} \sum_{n \neq 0} \alpha_{-n}^{i} \alpha_{n}^{i}$ - these are **mass-shell** conditions - quantization (finally!) - independent d.o.f.: $X^i$ (free scalar fields), $x^-$, $p^+$ (ordinary QM for the CoM of the string) - canonical momentum $\Pi^{i}=\frac{1}{2 \pi \alpha^{\prime}} \partial_{\tau} X^{i}$ - agrees with the conserved current, but physically this is different: this is the conjugate of the scalar fields - impose commutations as usual - $\left[X^{i}(\sigma, \tau), \Pi^{j}\left(\sigma^{\prime}, \tau\right)\right]=i \delta^{i j} \delta\left(\sigma-\sigma^{\prime}\right)$ - find $\left[x^{i}, p^{j}\right]=i \delta^{i j}, \quad\left[\alpha_{n}^{i}, \alpha_{m}^{j}\right]=\left[\tilde{\alpha}_{n}^{i}, \tilde{\alpha}_{m}^{j}\right]=n \delta^{i j} \delta_{n+m, 0}$ (all others vanish) - -> need to rescale $\frac{1}{\sqrt{n}} \alpha_{n}^{i}=a_{n}^{i}$ etc - spectrum - just use ladder operators on $|0,p\rangle$ (one vacuum for each momentum eigenstate) - but need to satisfy the level matching condition - for closed string, two sets of creations; open string, one set - quantum mass-shell condition - $\alphas become operators, and do not commute - deal with it just like for SHM - for closed strings: $M^{2}=\frac{2}{\alpha^{\prime}} \sum_{i=2}^{D-1} \sum_{n \neq 0} n\left(N_{n}^{i}+\tilde{N}_{n}^{i}\right)+a_{0}=\frac{4}{\alpha^{\prime}} \sum_{i=2}^{D-1} \sum_{n \neq 0} n N_{n}^{i}+a_{c}$ where $a_{c}=\frac{2(D-2)}{\alpha^{\prime}} \sum_{n=1}^{\infty} n=\frac{2(D-2)}{\alpha^{\prime}} \zeta(-1)=-\frac{D-2}{6 \alpha^{\prime}}$ - for open strings: $M^{2}=\frac{1}{\alpha^{\prime}} \sum_{i=2}^{D-1} \sum_{n \neq 0} n N_{n}^{i}+a_{o}$ where $a_{o}=\frac{(D-2)}{2 \alpha^{\prime}} \sum_{n=1}^{\infty} n=\frac{(D-2)}{2 \alpha^{\prime}} \zeta(-1)=-\frac{D-2}{24 \alpha^{\prime}}$ - **spectrum for open strings** - (open) for $|0,p^\mu\rangle$, $M^2$ negative for $D>2$ -> ==ground states are tachyons== - $\alpha^i_{-1}|0,p^\mu\rangle$ transforms like a vector under $SO(D-2)$ - mass: $M^{2}=\frac{1}{\alpha^{\prime}}\left(1-\frac{D-2}{24}\right)=\frac{26-D}{24\alpha^\prime}$ - but this vector has only $D-2$ components -> must be massless -> ==$D=26$== (==first excited state can be thought of as a photon==) - $D\ne26$: Lorentz symmetry lost (as this particles do not all into representations of Lorentz group) -> quantisation is inconsistent - **more rigorous derivation**: let $a_0$ undetermined, and check that Lorentz algebra is only satisfied at $D=26$ for the quantum generators - **non-flat spacetimes**: same conclusion - turns out higher excitations all positive for $D=26$, with spacing given by $1/\alpha^\prime$ - e.g. at mass $1/\alpha^\prime$, we have $\alpha_{-1}^{i} \alpha_{-1}^{j}|0, p\rangle$ as a tensor and $\alpha_{-2}^{i}|0, p\rangle$ as a vector -> but only together they form a tensor representation - ==higher excited states give infinite number of massive particles== - **spectrum for closed strings** - ground state as before - single alpha state not allowed: left and right moving need to match; so first excited state given by $\alpha^j_{-1}\alpha^i_{-1}|0,p^\mu\rangle$, with $M^2=\frac{26-D}{6\alpha^\prime}$ - again need $D=26$ to form rep. of Lorentz - turns out to be a reducible rep. -> reduce to - a massless scalar (trace part), $\phi$ - a massless spin-2 particle (symmetric traceless part), $h_{\mu\nu}$ - and a massless (1,1) rep. (antisym traceless part), $B_{\mu\nu}$ - remarks - general principle (Lorentz covariance, unitarity) => at low energies, any massless vector field -> photon; any massless spin 2 -> Einstein graviton. - this is confirmed by explicit string theory calculation of scattering amplitudes: at $E\ll1/\alpha^\prime$, massless exchange dominate, $\mathcal{A}_4$ reduces to that of Einstein gravity with matter fields $B_{\mu\nu}$ and $\phi$ - at tree level, effective action $S_{e f f}=-\frac{1}{2 \kappa_{0}^{2}} \int d^{26} x \sqrt{-G} e^{-2 \Phi}\left(R-4 G_{\mu \nu} \partial^{\mu} \Phi \partial^{\nu} \Phi+\frac{1}{12} H_{\mu \nu \lambda} H^{\mu \nu \lambda}\right)$ w/ $H_{\mu \nu \lambda}=\partial_{\mu} B_{\nu \lambda}+\partial_{\nu} B_{\lambda \mu}+\partial_{\lambda} B_{\mu \nu}$ - loops are divergent -> non-renormalisable - the string action, however, is UV finite - to match, $\mathcal{A}_4\sim G_N$ for Einstein and $\mathcal{A}_4\sim g_S^2$ => $G_N\sim g_S^2$ - $S\sim \frac{1}{g_S^2}\int e^{-2\phi}R$ => integration over $\phi$ effectively changes coupling constant - $g_S=e^{\langle\phi\rangle}$, the expectation value of $\phi$ - now we see that ==the only free parameter is String theory is actually fixed== - superstring (D=10) - add fermions on the worldsheet: $S=-\frac{1}{4 \pi \alpha^{\prime}} \int d^{2} \sigma\left(\partial_{a} X^{\mu} \partial^{a} X_{\mu}+i \bar{\Psi} \gamma^{a} \partial_{a} \Psi\right)$ - these fermions do not have geometric interpretation, so thought of as internal d.o.f. - two quantisation schemes exist that lead to no tachyons: Type IIA and Type IIB (periodic and anti-periodic boundary conditions) - these worldsheet fermions can now generate spacetime fermions - low energy: Type IIA and IIB supergravity #### 2.1.3 D-branes - **definition** - D-branes: defects where open strings can end. require them to stay at the branes by Dirichlet boundary conditions - **Dp branes**: p+1 Neumann BCs -> Dp branes are p+1 dimensional surfaces - if Neumann BC in all directions -> can end anywhere -> space-filling brane - bosonic: D25 brane - superstring: D9 brane - note - $X^0$ cannot have Dirichlet BC - smallest p: D0 brane: a point moving in time - other BCs - more than one brane - 4 types of BC: both end on $\vec a$, both end on $\vec b$, a to b, and b to a - branes of different dimensions - if one end is on a Dp brane, other end is on Dq brane, impose p+1 Neumann on this end, and q+1 Neumann on the other end - symmetry - breaks Lorentz and Poincare to Poincare (1,p) $\times$ SO(D-1-p): translation on the brane and rotation transverse to the brane - **open string spectrum on a Dp-brane** - write $\mu=(\alpha,a), \alpha=0,...,p; a = p+1,...,D-1$ - $x^a=b^a$ (Dirichlet BC), $p^a=0$ - quantisation the same as before. oscillations happen still in all directions -> zero-point energy the same & spectrum the same - zero-point energy etc would change if in between different dimensional Dp branes - now for the spectrum, will have $|0,p^\alpha\rangle$ rather than $|0,p^\mu\rangle$ - since strings now *live* on the branes, states should form rep.s of Poincare (1,p) times SO(D-1-p) - massless states: - write $\alpha = +, -, i$ - first excited states are - $\alpha_{-1}^{i}\left|0, p^{\alpha}\right\rangle$: massless particle - n.b. at the particular state: although it is one dimensional, it is just a *particle*. to see the one-dimensional structure of it, need to probe higher excited states of it - $\alpha_{-1}^{a}\left|0, p^{\alpha}\right\rangle$: scalar fields: because $a$ is not a index on the brane - i.e. a vector field $A_\alpha$ and D-1-p scalar fields $\phi^a$ - these scalar fields describe the motion and fluctuation of the brane in transverse directions - comments - $\phi^a$ excitations modify these Dirichlet BC (as they move the brane location) - by general principle, the low energy effective action must be that of a Maxwell field. but here can actually work out the precise prefactor in the action $S=-T_{p} \int d^{p+1} x\left(1+\frac{1}{4} F^{\alpha \beta} F_{\alpha \beta}+\frac{1}{2} \partial^{\alpha} \Phi^{a} \partial_{\alpha} \Phi^{a}+\ldots\right)$ - $T_p$ = mass of Dp brane per unit volume (tension) - when $A_\alpha=0$, $S=-\int d t M_{D p}\left(1+\frac{1}{2} \dot{\Phi}^{a}+. .\right)$, which is just like some relativistic classical object -> these $\phi^a$ do describe the motion of the brane - can actually go to higher orders to check that it really behaves like a relativistic object - tension of a D-brane - mass of a D-brane = vacuum energy of open strings living on it $M_{Dp}=T_pV_p=E_{vac}$ - $E_{vac}$ = sum of vac diagrams of open strings = sum of all 2d surfaces with at least one boundaries (but no external open string) (in Euclidean PI) - now $g_S^{-\chi}$ where $\chi=2-2h-b$ (because now can have boundaries) - at weak coupling, the single boundary and no genus diagram dominate -> ==$T_p\sim 1/g_S$== - alternatively, think about the Newton's force between two branes, it is proportional to $G_N T_p^2$ by Newton's law, but also $\sim g_S$ because the leading diagram of *closed* string exchange is just the boundary of a cylinder with no genus -> $T_p\sim1/g_S$ - **channel duality** - a tree diagram of closed string connecting two branes = a loop diagram of an open string -> hints at that certain closed string process can be completely described by open strings - basis of AdS/CFT - closed strings contain gravitons, open string contain gauge fields - why modes describing motions of D-branes appear as massless modes? - underlying Mink. is translation invariant -> no potential term for $\phi$ fields -> no mass term - i.e. $\phi^a$ are Goldstone bosons for breaking translational symmetries (choosing the position of D brane SSB translation symmetry) - strength of open string interactions - consider a 2d sheet with a inner boundary: the inner boundary adds a factor of $g_S$ because of Euler number, but can also be thought of as open string splitting and joining, giving a factor of $g_o^2$ => ==$g_S\sim g_o^2$== - multiple D-branes on top of each other - with two branes, states become $|\psi,IJ\rangle$, $I,J=1,2$ -> 4 copies of each spectrum; or each open string excitation becomes 2 by 2 matrix - n brane -> $n\times n$ matrices - remarks - strings interact by joining their ends -> matrix products in $I,J$ indices - $(\phi^a)^1{}_2$ and $(\phi^a)^2{}_1$ are complex conjugates - under phase $e^{i\theta_I}$ at one end and opposite phase at other end, $(\phi^a)^I{}_I$ is invariant; but $(\phi^a)^I{}_J$ and $(\phi^a)^J{}_I$ transform with opposite phases - for coincident branes, since branes are indistinguishable, we can reshuffle their indices -> $U(n)$ symmetry when branes are coincidental - $\left|\Psi^{\prime} ; I J\right\rangle=U_{I K} U_{J L}^{\dagger}|\Psi ; K L\rangle$; $\Psi \rightarrow \Psi^{\prime}=U \Psi U^{\dagger}$ - i.e. each open string excitation transforms under the adjoint rep of this U(n). on the string WS, this U(n) is a global symmetry; but in spacetime, i.e. in the worldvolume of D branes, this is a gauge symmetry - $A_\alpha{}^I{}_J$ must be the corresponding gauge bosons: the only way a field can transform like this at low energies must be YM. - low energy effective action: $S=-\frac{1}{g_{Y M}^{2}} \int d^{p+1} x \operatorname{Tr}\left(\frac{1}{4} F^{\alpha \beta} F_{\alpha \beta}-\frac{1}{2} D^{\alpha} \Phi^{a} D_{\alpha} \Phi^{a}+\left[\Phi^{a}, \Phi^{b}\right]^{2}+\ldots\right)$ with $g_{YM}\sim g_o\sim g_S^{1/2}$ -> (dim. ana.) ==$g_{Y M}^{2}=d_{p} g_{s} \alpha^{\prime \frac{p-3}{2}}$== - separating the coincidental branes - consider two branes separated by a distance $d$ - costs an energy $M=T_s d$, i.e. elastic energy of string - now $A_\alpha, \phi^a$ no longer massless (but those starting and ending on the same brane still massless) - -> $U(2)\rightarrow U(1)\times U(1)$ - i.e. ==separation of branes <-> Higgs mechanism== - more generally, separate into $k$ stacks of brane - **D branes in superstring** - in bosonic string, always have open string tachyon -> unstable - in super string, D branes of certain dimensions do not have tachyons -> stable. three comment: - (1) stable D branes always have conserved charge - (2) worldvolume theory always supersymmetric - in addition to $A_\alpha,\phi^a$, also have massless fermions (SYM) - more on (1) - *bosonic* part of massless closed superstring spectrum: $h_{\mu\nu},B_{\mu\nu},\phi$ plus $C_\mu^{(1)},C_{\mu\nu\rho}^{(3)}$ (in IIA) or $\chi,C_{\mu\nu}^{(2)}, C_{\mu\nu\lambda\rho}^{(4)}$ (in IIB) (RR fields) - n.b. these are *spacetime* bosons, whether they arise from boson or fermion excitations of the string is a separate issue (these extra ones actually arise from WS fermions) - these anti-symmetric potential are generalisations of Maxwell fields - can define conserved charges associated with these gauge symmetries - for one form $A$, its source is a point particle; for $p+1$-form, it's a p-dimensional object - can do Hodge dual for these generalised forms to get analogies of magnetic fields. can couple a (D-n-3)-dim objeect to the dual n-forms $\tilde C$. in terms of original $C$, this is a magnetic object (generalisation of magnetic monopole) - RR fields can give rise to extended fields that couple to them -> turns out to be D branes. because they couple to those gauge fields, they are stable objects (at least the minimally charged ones) - IIA: - $C^{(1)}$: D0 (electric) and D6 (magnetic); - $C^{(3)}$: D2 (electric) and D4 (magnetic); - IIB: - $C^{(2)}$: D-string (electric) and D5 (magnetic); - $C^{(4)}$: D3 (self-dual -> electric and magnetic); - important: 4 dimensional theory - scalar fields couple to D-instantons (makes sense in Euclidean) - other branes would not be stable objects (tachyons): cannot find conserved charge for them to couple - on these branes, these branes are SYM - D3: $\mathcal{N}=4$ SYM ### 2.2 D-branes as spacetime geometries - **spacetime around branes** - consider charged particle at origin of Minkowski -> get RN BH - same story for higher dimensional objects ([[HorowitzStrominger1991]]) - D3-brane in IIB supergravity in $D= 10$ - region of validity: $g_S\ll1$, energy $\ll1/\alpha^\prime$, curvature $\ll1/\alpha^\prime$ (strings are like particles; decouple massive modes) (can achieve by taking $\alpha^\prime\rightarrow0,g_S\rightarrow 0$) - D3 charged under $C_+^{(4)}$ - e-charge, $q_3=\int_{S^5}\star F^{(5)}$ - m-charge, $g_3=\int_{S^5} F^{(5)}$ - they are equal <- self-dual - why $S^5$: D3 is 4d -> transverse direction is 6d -> D3 is like a point charge in 6d -> sphere surrounding it is 5d - Dirac quantization -> $g_3=q_3=\sqrt{2\pi}N$ for N of them - tension of D3: $T_3=\frac{q_3}{16\pi G_N}=\frac{N}{(2\pi)^3g_S\alpha^{\prime2}}$ - N.b. only special objects have this tension: BPS - symmetry: Poincare (1,3) x SO(6) - full solution for D3 brane - $d s^{2}=f(r)\left(-d t^{2}+d \vec{x}^{2}\right)+h(r)\left(d r^{2}+r^{2} d \Omega_{e}^{2}\right)$ - $f(r)=\frac{1}{h(r)}=H^{-1 / 2}(r), H(r)=1+\frac{R^{4}}{r^4}, \quad R^{4}=N \frac{4}{\pi^{2}} G_{N} T_{3}=N 4 \pi g_{s} \alpha^{\prime}$ - in order for SUGRA to be valid $\alpha^\prime R^{-2}\ll 1$ -> $g_S N\gg1$ & $g_S\ll1$ - near $r=0$, an infinite throat, with constant radius for $S^5$ -> $S^5$ decouples -> get $AdS_5\times S^5$ - now we have two descriptions of D3-branes - (A) D-branes in flat Mink_10 where open strings live - (B) spacetime metric & $F_5$ flux on $S^5$. only closed strings. target space now $AdS_5\times S^5$ - by consistency A = B - both descriptions can in principle be valid for all $\alpha^\prime$ and $g_S$ - but taking a low energy limit -> AdS/CFT - low energy limit - fix some energy E, take $\alpha^\prime\rightarrow0$ (equivalent to taking E-> 0 while fixing $\alpha^\prime$); i.e. $\alpha^\prime E^2\rightarrow0$ (no restriction on $g_S$) - now for (A): - open string -> $\mathcal{N}=4$ SYM (==$g_{YM}^2=4\pi g_S$==) - closed string -> graviton, dilaton, etc (but only massless modes) - coupling between massless closed and open strings, or closed string themselves, $G_N\sim g_S^2\alpha^{\prime4}$; in low energy limit $G_N E^8\rightarrow0$ - i.e. we get ==$\mathcal{N}=4$ SYM + free gravitons== - for (B) - define E w.r.t. $t$, time at infinity -> even in low E limit, gravitational effects can be large due to gravitational redshift: $E_\tau=H^{1/4}E$ - at large $r$, $E^2\alpha^\prime\rightarrow0$ => all massive closed strings decouple - small radius, $E_\tau^2\frac{r^2}{R^2}\alpha^\prime\rightarrow0$ => $E_{\tau}^{2} \frac{r^{2}}{\sqrt{4 \pi g_{s} N}} \rightarrow 0$ - miraculously $\alpha^\prime$ cancel. so $E_\tau$ can be anything as long as $r$ is small enough; i.e. anything is allowed in deep throat, including massive string modes - i.e. we have ==free graviton at $r=\infty$ + full string theory (closed only: no branes) in $AdS_5\times S^5$, which decouple== - comment: low energy limits help to decouple massive modes, but once this is done, can extend to higher energies (and infinity radius) ### 2.3 AdS/CFT duality #### 2.3.1 AdS spacetime #### 2.3.2 String theory in AdS5 x S5 - parameters - only $\alpha^\prime/R^2, g_S$ - Newton $G_N=(2\pi)^7g_S^2\alpha^{\prime4}$, so alternatively: $G_N/R^8$ and $\alpha^\prime/R^2$ - **classical gravity limit**: $g_S\rightarrow0$ (or $G_N/R^8\rightarrow0$) and $\alpha^\prime/R^2\rightarrow0$ => IIB SUGRA - **classical string limit**: $\alpha^\prime/R^2$ arbitrary, but $g_S\rightarrow0$ - compactness - convenient to express 10-d fields in terms of harmonics on $S^5$ - gravity is essentially 5d as higher harmonics develop mass due to curvature #### 2.3.3 N=4 SYM (3+1d) - field content - $A_\mu,\phi^i, i=1,...,6$ - fermions $\chi^A_\alpha, A=1,...,4$, $\alpha=$ spinor index - all in adjoint rep of U(N), each N x N matrix - U(1) decouples - U(N) = SU(N) x U(1) - no matter how many D-branes, the CoM motion always decouples - can also check from the low energy effective action - properties - $\mathcal{N}=4$: conserved charge given by 4 Weyl spinors - actually the maximally allowed no. in 4d for renormalisable theories - $g_{YM}$ dimensionless classically and *remains* so at quantum - i.e. $\beta$ function is zero - not just scale invariant, actually conformal - refined statement - (A) D-brane in Mink_10 at low energy limit -> $\mathcal{N}=4$ SYM with SU(N) + decoupled U(1) + decoupled gravity - (B) in the throat, string in $AdS_5\times S^5$ + decoupled gravity + decoupled CoM (can choose position of throat) - => $\mathcal{N}=4$ SYM with SU(N) on $R^{1,3}$ = IIB String in AdS${}_5\times S^5$ in Poincare - N on the LHS is related to flux on RHS (and thus Newton constant; see later) - remarks - $R^{1,3}$ is the boundary of AdS${}_5$ - RHS is a 5d gravity theory (can dimensionally reduce the compact space) - => realisation of holographic principle - non-trivial prediction: global AdS and its boundary - at finite $G_N$, we only require asymptotic geometry to be fixed, bulk will fluctuate ## 3 Duality toolbox ### 3.1 General aspects #### 3.1.1 IR/UV connection - we saw that smaller $r$ <-> lower energy (recall throat picture or redshift equation) - -> $r$ can be considered as the energy scale of CFT - relating bulk (local) proper quantities to YM: - $E_{YM}=\frac{R}{z}E_{loc}$ & $d_{YM}=\frac{z}{R}d_{loc}$ - for bulk process at different $z$, same $E_{loc}$ and $d_{loc}$ but and $d_{YM}\sim z$ - -> as $z\rightarrow0$ (imprecisely "IR" from bulk point of view), $E_{YM}\rightarrow\infty$, $d_{YM}\rightarrow0$, i.e. UV process in YM. vice versa. - pictorially, need a larger region on CFT to describe a more interior bulk process (entanglement wedge) - remarks - $z=\epsilon$ cutoff <-> UV cutoff on CFT - for CFT in $R^{1,3}$, there exists arbitrarily low-energy excitations <-> $z \rightarrow\infty$ (deep interior) region; but if the bulk ends at some finite proper distance (no "more interior" points exist) <-> boundary theory in gapped - e.g. global AdS <-> gauge theory on $S^3\times R$ which is gapped due to compact direction #### 3.1.2 Matching of symmetries - what symmetries - conformal SO(4,2) <-> isometry of AdS${}_5$ - global SO(6) (internal symmetry of 6 scalar fields) <-> isometry of $S^5$ - SUSY 4 + 4 (conformal sym does not commute with the 4 SUSY generators so extra 4) Weyl spinors (two complex components) so 32 real charges <-> same number of large *local* SUSY in this string theory - pattern: all symmetries on global on CFT, but local in bulk - remarks - isometry is subgroup of diffeo. (local sym) - why only isometry (not full diffeo.)? - this is the subgroup that leaves the asymptotic metric invariant - large gauge transformation can be considered as the global part of diffeo. - actually more general: large gauge <-> global symmetry on CFT #### 3.1.3 Matching of parameters - $g_{YM}^2=4\pi g_S$ - $\lambda\equiv g_{YM}^2 N=R^4/\alpha^{\prime2}$ - ${\pi^{4}}/{2 N^{2}}=G_N/R^8$ - $SU(N)$ <-> flux $N$ - Newton in 5d: $\frac{1}{G_{5}}=\frac{V_{5}}{G_{N}} \Longrightarrow \frac{G_{5}}{R^{3}}=\frac{\pi}{2 N^{2}}$ - semiclassical gravity limit: $\hbar =1, \hbar G_N\rightarrow0, \alpha^\prime\rightarrow0$ i.e. QFT in curved spacetime - <-> in YM, large N (expansion in topologies) and large $\lambda$ (diagrams with more vertices dominate) - $\alpha^\prime/R^2$ expansion <-> expansion in $1/\sqrt{\lambda}$ in YM - $G_N$ expansion <-> expansion in $1/N^2$ in YM - classical string limit: $G_N\to 0$ (large $N$) but arbitrary $\alpha^\prime$ ($\lambda$) #### 3.1.4 Matching of the spectrum - Hilbert space: rep. of $SO(d,2)$ for both sides - conformal local operators <-> bulk fields - e.g. scalar operator <-> scalar field; vec operator $J_\mu$ <-> vector field $A_M$; tensor operator $T_{\mu\nu}$ <-> tensor fields $h_{MN}$ - $\mathcal{L}_{\mathcal{N}=4}$ <-> dilaton $\Phi$ - SO(6) current $J^a_\mu$ <-> bulk gauge field $A_M^a$ (dimensionally reducing $S^5$ gives gauge fields corresponding to generators of $S^5$) - stress tensor in YM <-> metric perturn. #### 3.1.5 Mass-dimension relation - consider gravity action $S=\frac{1}{2 \kappa^{2}} \int d^{d+1} x \sqrt{-g}\left(\mathcal{R}-2 \Lambda+\mathcal{L}_{\text {matter}}\right), \quad 2 \kappa^{2}=16 \pi G_{d+1}$ with $\mathcal{L}_{\text {matter}}=-\frac{1}{2}(\partial \Phi)^{2}-\frac{1}{2} m^{2} \Phi^{2}-V(\Phi)+\cdots$ - convenient to canonically normalise: $\Phi \rightarrow \kappa \Phi \quad g_{M N}=g_{M N}^{(0)}+\kappa h_{M N}$ - then $\kappa \sim G_{d+1}^{1 / 2} \sim N^{-1}$ in units of curvature, $\Phi, h_{M N} \sim O(1)$ - non-linear terms are $O(\kappa)$ or higher -> neglect -> free scalar field - now solve the free scalar field like in Xi String 2 notes - -> $\Phi(x, z)=A(x) z^{d-\Delta}+B(x) z^{\Delta}, \quad z \rightarrow 0$ - comments - **BF bound**: in AdS, constant modes ($\vec k =0$) not alllowed -> forced to have some kinetic energy, which can compensate some negative $m^2$ - **normalisability and BC** - use KG inner product - can check which modes are normalisable - $z^\Delta$ always normalisable - $z^{d-\Delta}$ normalisable if $0 \leqslant v<1$ - BC - $\nu>1$: easy: need to throw away the non-renormalisable one -> $A=0$ - $0 \leqslant v<1$: $A=0$ (standard quantisation) or $B=0$ (alternative quantisation) - have to choose one in order for energy conservation - so now "normalisable" refers to the one specified by quantisation - normalisable modes - used to build up Hilbert space - -> mapped to states on the CFT - boundary value of bulk fields <-> expectation of operator (see later) - non-normalisable modes - NOT part of Hilbert space - if present, should be viewed as defining the background - this changes the boundary theory, rather than the state: $\int d^{d} x \phi_{0}(x) \mathcal{O}(x) \Longleftrightarrow \phi_{0}(x)=\lim _{z \rightarrow 0} z^{\Delta-d} \Phi(z, x)$ - equation above implies that $\Delta$ is the scaling dimension of $O$ - boundary scaling $x^{\mu} \rightarrow x^{\prime \mu}=\lambda x^{\mu}$ defines scaling dimension $\mathcal{O}(x) \rightarrow \mathcal{O}^{\prime}\left(x^{\prime}\right)=\lambda^{-\Delta} \mathcal{O}(x)$ - corresponding bulk scaling $x^{\mu} \rightarrow x^{\prime \mu}=\lambda x^{\mu}, z \rightarrow z^{\prime}=\lambda z$ changes $\Phi$ to $\Phi^\prime$ - facts: $\Phi$ is scalar & boundary action is conformally invariant: $\Phi(z, x)=\Phi^{\prime}\left(z^{\prime}, x^{\prime}\right), \quad \int d^{d} x \phi_{0}(x) \mathcal{O}(x)=\int d^{d} x^{\prime} \phi_{0}^{\prime}\left(x^{\prime}\right) \mathcal{O}^{\prime}\left(x^{\prime}\right)$ - so $\int d^{d} x^{\prime} \phi_{0}^{\prime}\left(x^{\prime}\right) \mathcal{O}^{\prime}\left(x^{\prime}\right)=\lambda^{\Delta} \int d^{d} x \phi_{0}(x) \mathcal{O}^{\prime}\left(x^{\prime}\right)=\int d^{d} x \phi_{0}(x) \mathcal{O}(x)$ $\Longrightarrow \mathcal{O}^{\prime}\left(x^{\prime}\right)=\lambda^{-\Delta} \mathcal{O}(x)$ - **mass-dimension relation** in stardard quantisation - $\Delta=\frac{d}{2}+\sqrt{\frac{d^{2}}{4}+m^{2} R^{2}}$ - $m=0$ <-> $\Delta=d$ marginal - in CFT (in particular UV), no change <-> asymptotic field $\phi(z)z^{d-\Delta}$ constant - $m^{2}<0, \Delta<d$ relevant - less important in CFT UV <-> $\phi(z)z^{d-\Delta}\rightarrow0$ asymptotically #### Euclidean correlation functions - $Z_{C F T}[\phi(x)]=\left\langle e^{\int d^{d} x \phi(x) \mathcal{O}(x)}\right\rangle_{E}$ - expect $Z_{C F T}[\phi(x)]=Z_{b u l k}\left[\left.\Phi\right|_{\partial A d S}=\phi(x)\right]$ - remarks - $\phi(x)$ in $Z_{CFT}$ should be infinitesimal <- we are viewing it as generating function, not really changing the theory - both sides actually divergent - CFT: usual UV div. - bulk: volume divergence - -> need to renormalise: add CT - conditions on CT: - must be local <- in CFT, short distance divergence must come from single points in the UV limit - covariant - n.b. the *finite* piece in the CT actually have no ambiguity by above criteria - general n-pt functions: just do n derivatives of Z - **1-pt function** - just take one derivative - ==$\langle\mathcal{O}(x)\rangle_{\phi} \sim \lim _{z \rightarrow 0} z^{d-\Delta} \Pi_{c}^{(R)}(z, x)$== - $\frac{\delta S_{E}^{(R)}\left[\Phi_{c}\right]}{\delta \Phi_{c}(z, x)} \sim \Pi_{c}(z, x)$ is "canonical momentum", treating $z$ as time, evaluated at classical solution (thus the subscript) - this equation is familiar from Hamilton-Jacobi theory: variation of action with respect to the end point gives momentum at that point - in stardard quantisation, $A(x)$ <-> $\phi(x)$, $\langle\mathcal{O}(x)\rangle_{\phi}=2 \nu B(x)$ - **2-pt function** - include quadratic terms in action, do expansion of solution, add CT etc. - $G_{E}(k)=\frac{\delta^{2} S}{\delta \phi(k) \delta \phi(-k)}=\frac{\delta}{\delta \phi(k)}\langle\mathcal{O}(k)\rangle_{\phi}=\frac{\langle\mathcal{O}(k)\rangle_{\phi}}{\phi(k)}=\frac{2 \nu B(k)}{A(k)}$ - to find the relation between B and A: impose boundary condition, and require solution be smooth at interior - **higher point function** - treat higher order interactions as perturbations because coupling small for small $G_N$ - solve perturbatively in $\phi$: $\Phi_{c}=\Phi_{1}+\Phi_{2}+\cdots$ where $\Phi_1$ is linear in $\phi$ etc - then substitute solution back to action $S\left[\Phi_{c}\right]=S_{2}[\phi]+S_{3}[\phi]+\cdots$ where $S_2$ is quadratic which gives 2-pt function, etc - HARD -> use Feynman diagrams just like in QFT - one major difference: sources lie at the boundary - -> two types of propagators: bulk-to-bulk & bulk-to-boundary - bulk-to-boundary: need $K\left(z, x ; x^{\prime}\right) \rightarrow z^{d-\Delta} \delta^{(d)}\left(x-x^{\prime}\right)$ because it should approach a source: - i.e. writing $\Phi(z, x)=\int d^{d} x^{\prime} K\left(z, x ; x^{\prime}\right) \phi\left(x^{\prime}\right)$, will give $\Phi(x, z) \stackrel{z \rightarrow 0}{\longrightarrow} z^{d-\Delta} \phi\left(x\right)$ (what we want) - in other words, the bulk-to-bdy propagator has the non-renormalisable BC, while bulk-to-bulk only have normalisable modes - remarks - procedure so far is tree-level: evaluating on classical solutions. to go to loops, expand further $Z_{C F T}=\int_{\left.\Phi\right|_{\partial A d S}=\phi} D \Phi e^{S_{E}[\Phi]}=e^{S_{E}\left[\Phi_{c}\right]} \int D \chi e^{S_{E}\left[\Phi_{c}+\chi\right]-S_{E}\left[\Phi_{c}\right]}$, i.e. include fluctuations around saddle - can also calculate $n$-pt correlation in the bulk then take to infinity, but normally just use b-to-bdy propagators directly #### 3.1.7 Wilson loops - $W_r[C]=\operatorname{Tr} \mathcal{P} \exp \left[i \int_{C} A_{\mu} d x^{\mu}\right]$ - $r$: choice of rep., usually fundamental - $\mathcal{P}$: path ordering. (matrices do not commute, here choose to order by path) - physical meaning: phase factor after transporting an *external* particle in a r-rep. along the loop - **external**: infinite mass -> localised path - call the particle a quark from now on - rectangular loop, $T\gg L$, $\langle W(C)\rangle \simeq e^{-i E T}$, where $E$ is potential energy between quark and anti-quark - what is a particle in fundamental rep. in SYM? - consider N+1 brane, and separate into N and 1. then SU(N+1) -> SU(N) x U(1) - the string connecting then carries an index -> fundamental rep of $SU(N)$ - mass = $|\vec r|/2\pi \alpha^\prime$ - in the limit of low energy $\alpha^\prime\rightarrow0$ and small separation $\vec r\rightarrow0$ with $|\vec r|/\alpha^\prime$ finite => particles remain in SYM - what is it on bulk side? - N D3 branes disappear into r=0, but the single D3 brane remains at finite distance - can check that mass = $r/2\pi \alpha^\prime$ still in AdS_5 - this is a consistency check: low-energy physics should not change - ==to get external particle, need mass infinite -> this D3-brane lies on the boundary of AdS5, i.e. string ending on boundary== - because strings pull the brane, the open strings should now couple to the scalar field on the brane (which describes motion of brane) -> modified Wilson loop from bulk point of view - the Wilson loop duality - $\langle W(C)\rangle=Z_{\text {string}}[\partial \Sigma=C]$ - $Z_{\text {string}}[\partial \Sigma=C]=\int_{\partial \Sigma=C} D X e^{i S_{\text {string}}}$ - a limit where calculation is easy: - $g_S\rightarrow0$, so neglect different topologies - $\alpha^\prime\rightarrow0$, evaluate PI using saddle point - so $\langle W(C)\rangle=Z_{\text {string}}[\partial \Sigma=C]=e^{i S_{c l}[\partial \Sigma=C]}$ - Example 1: particle at fixed position - YM result: just $e^{-iMT}$ - bulk result: string not moving. get $S_{NG}=-\frac{1}{2\pi\alpha^\prime}T\Lambda$, where $\Lambda$ is cutoff radius - interestingly, bulk predicts that $M=\frac{\sqrt{\lambda}}{2\pi}\frac{1}{\epsilon}$, where $\epsilon=R^2/\Lambda$ - curiously, this self-energy is sqrt. recall a electron has self energy $e^2\sim\alpha$. so this is a prediction of strong coupling - Example 2: static potential between quark and anti-quark - YM: $\langle W(C)\rangle=e^{-i E_{t o t} T}$ with $E_{t o t}=2 M+V(L)$. BUT we do not know this potential energy in strong coupling! - need to find classical string location by extremising action (subtlety: need to do cutoff) - result: potential $V(L)=-\frac{\sqrt{\lambda}}{L} \frac{4 \pi^{2}}{\Gamma^{4}(1 / 4)}$ - negative: expected for a potential - $1/L$ just by dimension - $\sqrt{\lambda}$: again, strong coupling result; we know $\sim \lambda$ at weak coupling - so in a plot against $\lambda$: linear at small values, sqrt at large values ### 3.2 Generalisations #### 3.2.1 Finite temperature - finite temperature CFT on $R^{1,3}$, the only scale is $T$, physics at all temperatures are the same (related by a scaling a units) - CFT on sphere: now have a dimensionless number (combining size of sphere - chosen to be R for simplicity - and temperature): $RT$ - richer physics - now thermal gas allowed: Euclidean bulk metric well-defined (in flat case, Euclidean metric singular) - big black hole always have larger entropy than smaller one (for same temperature) -> big ones dominate -> good, because big ones have positive specific heat, and small ones have negative specific heat (CFT should always have positive specific heat) - trick: subtracting the pure AdS action from the solution of interest is equivalent to doing some renormalisation (i.e. CT term) - even simpler shortcut: assume entropy given by area of BH and use thermo relations - since physics only depend on $RT$ - large $R$ fixed T <-> large $T$ fixed R - so large temperature can be also viewed as flat limit (i.e. ==theory on $R^{1,3}$ is the high temperature limit==) - **Hawking-Page transition from CFT point of view** - consider a system of 2N SHO - $\mathcal{L}=\frac{1}{2} T r \dot{A}^{2}+\frac{1}{2} T r \dot{B}^{2}-\frac{1}{2} \operatorname{Tr} A^{2}-\frac{1}{2} \operatorname{Tr} B^{2}$ - by simple statistics - $D(E) \sim O\left(N^{0}\right) \quad$ for $E \sim O\left(N^{0}\right)$ - $D(E) \sim e^{O\left(N^{2}\right)} \quad$ for $E \sim O\left(N^{2}\right)$ - partition function $Z=\int d E e^{-\beta E} D(E)$ - for $\beta\sim O(N^0)$, naively only get contribution from $E \sim O\left(N^{0}\right)$, but for $E \sim O\left(N^{2}\right)$, the contribution is $\int d E e^{-\# \beta N^{2}} e^{\# N^{2}}$ - so when $\beta$ sufficiently small, $O(N^2)$ states dominate and $Z \sim e^{O\left(N^{2}\right)}$ - so a phase transition $F \sim O\left(N^{0}\right)$ to $F \sim O\left(N^{2}\right)$ when temperature is increased - conclusion - Thermal AdS $\Longleftrightarrow$ states with $E \sim O\left(N^{0}\right)$ - Big BH $\Longleftrightarrow$ states with $E \sim O\left(N^{2}\right)$ ### 3.3 HEE - [[0145 Generalised area]] #### 3.3.1 Entanglement entropy - properties - subadditivity - $|S(A)-S(B)| \leq S(A B) \leq S(A)+S(B)$ - assume A and B have no overlap - strong subadditivity - $S(A C)+S(B C) \geq S(A B C)+S(C)$ and $S(A C)+S(B C) \geq S(A)+S(B)$ - $S(A C)+S(B C) \geq S(A)+S(B)$ - also in this way of writing, no intersections #### 3.3.2 EE in many-body systems - for a general local Hamiltonian, the ground state has $S_{A}=\# \frac{A r e a(\partial A)}{\epsilon^{d-2}}+\cdots$ - each two lattice points separated by $\epsilon$, then this is just the number of lattice points (or d.o.f.) at the boundary - this leading term characterises the *short-distance* behaviour - the dots characterise long range - can characterise **topological order** - Xiao-Gang Wen: for gapped systems, can have non-trivial long-range correlations; but cannot be seen using standard observables like correlations functions of local operators; however, can actually just see it by computing EE of the wavefunction itself - $S_{A}=\# \frac{L(\partial A)}{\epsilon}-\gamma$. the constant is non-zero for topologically ordered systems - characterise no. of d.o.f. of a QFT - long known: 1+1 CFT: central charge is used to count d.o.f. - turns out $S_{A}=\frac{c}{3} \log \frac{l}{\epsilon}$ #### 3.3.3 Holographic - [[0007 RT surface]] - can show strong subadditivity using RT surfaces - can also find the 1+1 CFT formula in the last subsection - RT is always perpendicular to boundary -> can use to show that it always gives area law