8.821 String Theory Fall 2008

Transcription

1 MIT OpenCourseWare String Theory Fall 008 For information about citing these materials or our Terms of Use, visit:

2 8.81 Lecture 01: What you need to know about string theory Lecturer: McGreevy Scribe: McGreevy Since I learned that we would get to have this class, I ve been torn between a) starting over with string perturbation theory and b) continuing where we left off. Option a) is favored by some people who didn t take last year s class, and by me when I m feeling like I should do more research. But because of a sneaky and perhaps surprising fact of nature and the history of science there is a way to do option b) which doesn t leave out the people who missed last year s class (and is only not in the interest of the lazy version of me). The fact is this. It is actually rare that the structure of worldsheet perturbation theory is directly used in research in the subject that is called string theory. And further, one of the most important developments in this subject, which is usually called, synecdochically, the AdS/CFT correspondence, can be discussed without actually using this machinery. The most well-developed results involve only classical gravity and quantum field theory. So here s my crazy plan: we will study the AdS/CFT correspondence and its applications and generalizations, without relying on string perturbation theory. Why should we do this? You may have heard that string theory promises to put an end once and for all to that pesky business of physical science. Maybe something like it unifies particle physics and gravity and cooks your breakfast. Frankly, in this capacity, it is at best an idea machine at the moment. But this AdS/CFT correspondence, whereby the string theory under discussion lives not in the space in which our quantum fields are local, but in an auxiliary curved extra-dimensional space (like a souped-up fourier transform space), is where string theory comes the closest to physics. The reason: it offers otherwise-unavailable insight into strongly-coupled field theories (examples of which: QCD in the infrared, high-temperature superconductors, cold atoms at unitarity), and into quantum gravity (questions about which include the black-hole information paradox and the resolution of singularities), and because through this correspondence, gauge theories provide a better description of string theory than the perturbative one. The role of string theory in our discussion will be like its role in the lives of practitioners of the subject: a source of power, a source of inspiration, a source of mystery and a source of vexation. The choice of subjects is motivated mainly by what I want to learn better. After describing how to do calculations using the correspondence, we will focus on physics at finite temperature. For what I think will happen after that, see the syllabus on the course webpage. Suggestions in the spirit described above are very welcome. 1

3 ADMINISTRIVIA please look at course homepage for announcements, syllabus, reading assignments please register I promise to try to go more slowly than last year. coursework: 1. psets. less work than last year. hand them in at lecture or at my office. pset 0 posted, due tomorrow (survey).. scribe notes. there s no textbook. this is a brilliant idea from quantum computing. method of assigning scribe TBD. as you can see, I am writing the scribe notes for the first lecture. 3. end of term project: a brief presentation (or short paper) summarizing a topic of interest. a list of candidate topics will be posted. goal: give some context, say what the crucial point is, say what the implications are. try to save the rest of us from having to read the paper. (benefits: you will learn this subject much better, you will have a chance to practice giving a talk in a friendly environment) Next time, we will start from scratch, and motivate the shocking statement of AdS/CFT duality without reference to string theory. It will be useful, however, for you to have some big picture of the epistemological status of string theory. Today s lecture will contain an unusually high density of statements that I will not explain. I explained many of them last fall; experts please be patient. What you need to know about string theory for this class: 1) It s a quantum theory which at low energy and low curvature 1 reduces to general relativity coupled to some other fields plus calculable higher-derivative corrections. ) It contains D-branes. These have Yang-Mills theories living on them. We will now discuss these statements in just a little more detail. 0.1 how to do string theory (textbook fantasy cartoon version) Pick a background spacetime M, endowed with a metric which in some local coords looks like ds = g µν (x)dx µ dx ν ; there are some other fields to specify, too, but let s ignore them for now. 1 compared to the string scale M s, which we ll introduce below

4 Consider the set of maps X µ : worldsheet Σ (σ,τ) target spacetime M X µ (σ,τ) where σ,τ are local coordinates on the worldsheet. Now try to compute the following kind of path integral I [DX(σ,τ)] exp (is ws [X..]). This is meant to be a proxy for a physical quantity like a scattering amplitude for two strings to go to two strings; the data about the external states are hidden in the measure, hence the. The subscript ws stands for worldsheet ; more on the action below. An analog to keep firmly in mind is the first quantized description of quantum field theory. The Feynman-Kac formula says that a transition amplitude takes the form Here x(τ )=x x(τ 1 )=x 1 is e wl [x(τ)] = x,τ x 1,τ 1 (1) x µ : worldline (τ) target spacetime x µ (τ). can be written as a sum over trajectories interpolating between specified inital and final states. Here x(τ) is a map from the world line of the particle into the target space, which has some coordinates x µ. For example, for a massless charged scalar particle, propagating in a background spacetime with metric g µν and background (abelian) gauge field A µ, the worldline action takes the form S wl [x] = dτg µν (x)ẋ µ ẋ ν + dτ ẋ µ A µ where ẋ τ x. Note that the minimal coupling to the gauge field can be written as dτ ẋ µ A µ = where the second expression is meant to indicate an integral of the one-form A over the image of the worldline in the target space (The expression on the LHS is often called the pullback to the worldline. Those are words.). A few comments about this worldline integral. 1) The spacetime gauge symmetry A A+dΛ requires the worldline not to end, except at a place where we ve explicitly stuck some non-gauge-invariant source (x 1,x in the expression above). ) The path integral in (1) is that of a one-dimensional QFT with scalar fields x µ (τ), and coupling constants determined by g µν,a µ. To understand this last statement, Taylor expand around some point in the target space, which let s call 0. Then for example A wl A µ (x) = A µ (0) + x ρ ρ A µ (0) +... and the coupling to the gauge field becomes dτ (ẋ µ A µ (0) + ẋ µ xρ ρ A µ (0) +...) 3

5 so the Taylor coefficients are literally coupling constants, and an infinite number of them are encoded in the background fields g,a. After all this discussion of the worldline analog, and before we explain more about the, why might it be good to make this replacement of worldlines with worldsheets? Good Thing number 1: many particles from vibrational modes. From far away, meaning when probed at energies E M s, they look like particles. This suggests that there might be some kind of unification, where all the particles (different spin, different quantum numbers) might be made up of one kind of string. To see other potential Good Things about this replacement of particles with strings, let s consider how interactions are included. In field theory, we can include interactions by specifying initial and final particle states (say, localized wavepackets), and summing over all ways of connecting them, e.g. by position-space feynman diagrams. To do this, we need to include all possible interaction vertices, and integrate over the spacetime points where the interactions occur. UV divergences arise when these points collide. To do the same thing in string theory, we see that we can now draw smooth diagrams with no special points (see figure). For closed strings, this is just the sewing together of pants. A similar procedure can be done for open strings (though in fact we must also attach handles there). So, Good Thing Number Two is the fact that one string diagram corresponds to many feynman diagrams. Next lets ask Q: where is the interaction point? A: nowhere. It s in different places in different Lorentz frames. This is a first (correct) indication that strings will have soft UV behavior (Good Thing Number Three). They are floppy at high energies. The crucial point: near any point in the worldsheet (contributing to some amplitude), it is just free propagation. This implies a certain uniqueness (Good Thing Number Four): once the free theory is specified, the interactions are determined. The structure of string perturbation theory is very rigid; it is very diffcult to mess with it (e.g. by including nonlocal interactions) without destroying it. BUT: Only for some choices of (M, ) can we even define I. To see why, let s consider the simplest example where M = IR D 1,1 with g µν = η µν diag( 1,1,1,1...1). Then the worldsheet action is S ws [X] = 1 d σ α X µ α X ν η µν. () 4πα Here α = 1, is a worldsheet index and µ = 0...D 1 indexes the spacetime coords. The quantity 1 has dimensions of length. It is the tension of the string, because it suppresses 4πα configurations with large gradients of X µ ; when it is large, the string is stiff. This is the mass scale Why stop at one-dimensional objects? Why not move up to membranes? It s still true that the interaction points get smoothed out. But the divergences on the worldvolume are worse. d = is a sort of compromise. In fact, some sense can be made of the case of 3-dimensional base space, using the BFSS matrix theory. They turn out to be already d quantized! The spectrum is a continuum, corresponding to the degree of freedom of bubbling off parts of the worldvolume. It s not entirely clear to me what to make of this. 4

6 Crudely rendered perturbative interactions of closed strings. Time goes upwards. Slicing this picture by horizontal planes shows that these are diagrams which contribute to the path integral for to scattering. Tilting the horizontal plane corresponds to boosting your frame of reference and moves the point where you think the splitting and joining happened. alluded to twice so far: 1 M s ; 4πα people will disagree about the factors of and π here. () is the action for free bosons. We can compute anything we want about them. The equations of δs motion 0 = ws δx give the massless d wave equation: ( 0 = τ + ) σ X µ = + X µ where σ ± τ ± σ. This is solved by superpositions of right-movers and left-movers: X µ (σ,τ) = X L µ (σ + ) + X R µ (σ ). To study closed strings, we can treat the spatial direction of the worldsheet as a circle σ σ + π, in which case we can expand in modes: X µ (σ,τ) = α µ n e inσ + + α µn e inσ. n Z For open strings, the boundary conditions will relate the right-moving and left-moving modes: α α. n Z We can quantize this canonically, and find [α µ,α ν ] = nδ n+m η µν, [ α µ,α ν ] = nδ n+m η µν,[α, α ] = 0. n m n m This says that we can think of α n<0 as creation operators and α n>0 as annihilation operators. Let 0 be annihilated by all the α n>0. The zeromodes α 0,α 0 are related to the center-of-mass position and momentum p µ of the string. The worldsheet hamiltonian is µ H ws = α p + α n α n µ a, n>0 where a is a normal-ordering constant, which turns out to be 1. 5

7 Here comes the problem: consider the state BAD α µ=0 n 0 for n > 0. Because of the commutator [α 0 n,α 0 n] = n, we face the choice BAD < 0 or E BAD < 0. Neither is OK, especially since E BAD becomes arbitrarily negative as n gets larger. This kind of problem is familiar from QED or Yang-Mills theory, where the time component of the gauge field presents with very similar symptoms. We need to gauge some symmetry of the theory to decouple the negative-norm states. One way to do this is to gauge the d conformal group. We re going to talk a lot more about conformal symmetry soon, so I ll postpone even the definition. There are fancy motivations for this involving d gravity on the worldsheet that we are not going to talk about now. Of course, like any gauge symmetry, this symmetry is a fake, and one could imagine formulations of the theory wherein it is never introduced. One necessary condition arising from this is that the worldsheet QFT needs to be a conformal field theory (CFT). This is not the CFT of the AdS/CFT correspondence. The generators of the conformal group include, in particular, the hamiltonian, so we must impose 0 = H ws phys as a constraint, like Gauss law in QED. Acting on a state with N oscillator excitations, this equation takes the form 0 = α p + N 1 p + m, which looks like an on-shell condition for a particle in minkowski space. This tells us the targetspace mass of a string state in terms of its worldsheet data. Now we can build physical states 3. The result looks like STRING STATE FIELD MASS α { µ 0 scalar T m = M s α ν } sym tensor g µν m α [µ α ν] 0 AS tensor B µν m 1 1 α µ 1 α 1 µ 0 scalar Φ m α µ αµ n 1 1 α µ n n 0 T µ 1...µ n m = 0 = 0 = 0 = nm s (where [..] is meant to indicate antisymmetrization of indices, {..} is meant to indicate symmetrization of indices). The scalar Φ is called the dilaton. For open strings, we would have, among other states, 3 STRING STATE FIELD MASS α µ 0 vector A µ m 1 Theres one important constraint I didnt mention, called level-matching, which says that the number of rightmoving excitations N and the number of left-moving excitations Ñ should be equal. This follows from demanding that there be no special point on the worldsheet. = 0 6

8 The worldsheet gauge symmetry leads directly to the spacetime gauge symmetries under which e.g. A A + dλ (0) and the two-form B B µν dx µ dx ν B + dλ (1). There s a general theorem that a massless symmetric tensor like g µν which interacts must couple to stress energy, and hence is a graviton. Using the machinery built from the above picture, this mode can be seen to participate in nontrivial scattering amplitudes, which indeed reduce to those computed from GR at low energies. So string theory, however unwieldy this description, is a quantum theory of gravity. 0. the spacetime effective action Almost every string theory result goes through the spacetime effective action: Φ S st [g,b,φ...] d D x [ ] ge R (D) + ( Φ) + H +... where H db i.e.h µνρ [µ B νρ], R (D) is the Ricci scalar made from g µν, and more about the dots below. This action is correct in two a priori different ways. 1) It reproduces scattering amplitudes of the fluctuations δg µν,δb µν,δφ... computed by the worldsheet path integral, at leading order in α E, where E is say the biggest energy of a state involved in the scattering event. That is to say, that the effective action is the leading term in a derivative expansion, where the higher-dimension operators are suppressed by the appropriate number of powers of α ; the coefficients of the corrections are definite and computable. ) To see the second way of thinking about S st, consider a string propagating in a background of g µν,b µν... Its action is S [X] = 1 d β X ν γ αβ β X ν ǫ αβ + Φ(X)R () ws σ [ ] g g +... (3) 4πα µν (X) α X µ + B µν (X) α X µ γ Here, γ αβ is an auxiliary metric on the worldsheet that we can introduce for convenience using the gauge symmetry. This is a d QFT with fields X µ. The bosons X µ are free is the spacetime is flat, and the other fields are trivial. Otherwise, they interact, and the coupling constants are determined by the background fields, like on the worldline above. If the target space is nearly flat, we can treat these interactions perturbatively. The effective, i.e. strength of interaction, is determined by α R (D). In general, these coupling constants will run with scale. A necessary condition for this QFT to be a CFT is that the beta functions for all the coupling constants vanish. It turns out that the leading-order beta functions for g µν,b µν,φ are exactly the equations of motion arising from varying the spacetime effective action S st! 7

9 We can learn some more simple lessons from the worldsheet action (3). 1) The worldsheet metric is innocuous 4. If you tried to take it seriously and give it some dynamics, you would try to add the Einstein-Hilbert term, () ΔS ws = γr. But this term, which is already present, with coefficient determined by the spacetime zeromode Φ 0 of the dilaton field Φ, is topological it is the Euler characteristic of the worldsheet. On a worldsheet Σ with h handles (genus h) and b boundaries, it evaluates to γr () = χ(σ) = h b. Letting g s ln Φ 0, this means that the contribution to the sum over worldsheets of worldsheets with h splittings and joinings is proportional to h g s. Thus g s is the quantity which determines how much strings want to interact, the string coupling. It is the in spacetime. This means that the effective action above is also the leading term in an expansion in powers of g s. What I wrote above is the tree-level effective action. A more correct expression is ( ) ( ) Φ S st [g,b,φ...] d D x [ ] ge R (D) + ( Φ) + H O(α ) 1 + O(g ) s ) Recall that on the worldline with coupling dτa µ ẋ µ = A, gauge invariance under A A + dλ (0) implied that the worldline was not allowed to end randomly, and that the worldline was a monopole source for the gauge field A. The coupling of the string to the AS tensor can similarly be written as a minimal coupling: dσdτ σ X µ τ X ν B µν = B. X(Σ) The gauge invariance B B + dλ (1) (Λ (1) is an arbitrary, well-behaved one-form in the target space) implies that the worldsheet cannot end just anywhere. More on this soon. It also means that fundamental strings are charged under the AS tensor, which is usually called the NSNS B-field, or Kalb-Ramond field. [To be continued] 4 modulo the issue of the conformal anomaly 8

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11 8.81 F008 Lecture 0: String theory summary continued Lecturer: McGreevy Scribe: McGreevy Today: 1. problems with bosonic string. superstrings 3. D-branes Recall: Last time I was showing you how primitive is our description of string theory. Specifically, we are reduced to doing a path sum over worldsheet embeddings. To decouple timelike oscillators, we gauged the worldsheet conformal group. I told you that this leads to a theory of quantum gravity, which reduces to GR plus stuff at low energies. The spacetime effective action which summarizes the scattering amplitudes computed by the world sheet path integrals is d D Φ + H ( ) ( ) S st [g,b,φ...] x [ ] ge R (D) + ( Φ) O(α ) 1 + O(g ) Here H = db, meaning H µνρ = [µ B νρ], and H H = H µνρ H µνρ s To study other backgrounds (which are only weakly curved), we can try to study a worldsheet action in background fields (a nonlinear sigma model): [ ] S ws [X] = 1 d σ g g µν (X) α X µ β X ν γ αβ + B µν (X) α X µ β X ν ǫ αβ () + Φ(X)R γ +... (1) 4πα A few observations about this which we made in the first lecture: conformal invariance constrains form of BG fields to solve spacetime EOM the dilaton term implies that the contribution to the path sum of a worldsheets with more handles are weighted by more factors of g s = e Φ F-strings charged under B-field, and cannot end at some random point in spacetime. The specific objection to the ending of the string is the consistency of the Gauss law for the B-field. This arises from its equation of motion, which in the presence of a fundamental string looks like: 0 = δs st = d H + δ D (string); δb 1

12 which arises by varying the action S[B] = H H + B everywhere string with respect to B. Integrating this (D-)-form over a (D-)-ball containing the string, we get the integrated gauss law: H = Q 1, S D 3 the number of strings inside the (D-3)-sphere (4π is equal to one in this discussion). If the string ended suddenly, the answer here would depend on which choice we made for the interior of the S D 3. problems: 1. bosonic string has a tachyon, a boson field with mass < 0. this is a real instability. conformal anomaly: under a scale transformation on a curved worldsheet, α D 6 δl ws T α R () π 3. perturbation expansion does not converge. worse than QFT. e 1/gs. Problem is solved by string compactification: just study backgrounds where the dimensions you don t like are too small to see. This is an interesting thing that we re not going to talk about. To fix problem 1: superstrings. The RNS description of superstrings is obtained from bosonic strings by changing the worldsheet gauge group to superconformal group no tachyon, still graviton, B, dilaton AND spacetime fermions (which exist) sometimes spacetime susy D c = 10. optimal for susy (will explain later) The supersymmetric ones are the most interesting because we know their stable vacua. There are five flavors of 10d supersymmetric superstrings: IIB, IIA, type I, het E 8, het SO. There duality relations between all of them: they are descriptions of different probes of different states of the same theory. We will focus on the type II theories. They are very closely related to each other. At low energies, they reduce to type II supergravity, with 3 supercharges. A few words about the low-energy spectrum of type II. In addition to the fields which they share in common with the bosonic string, and the spacetime fermions, the type II strings (also type I has these) contain various antisymmetric tensor fields, called Ramond-Ramond (RR) fields. They are like Maxwell fields and like the B-field. They differ from the B-field in that strings are neutral under the associated gauge symmetry. Type IIA has RR potentials of every odd degree (and even-degree field strengths G = dc): C 1,C 3,C 5,C 7,C 9. They are related in pairs by EM duality (like F E = 4 F B of E&M in 4d): e.g. the 1-form potential is the magnetic dual of the 7-form potential: dc 1 = 10 dc 7.

13 Type IIB has even-degree potentials, again related in pairs by EM duality. The middle-degree 5-form field strength is self-dual. (actually all of this is determined by supersymmetry.) The simplest duality relation, namely T-duality, is worth mentioning here if only to convince you that the two type II theories are not rival candidate theories of everything or something silly like that. [T-duality interlude: compactification of string theory on a circle is described on the worldsheet by making some of the fields periodic. Duality of d field theory relates strings on big circles to strings on small circles (in units of α ). The spectrum of strings on a circle is made up of a KK tower of momentum modes, and another tower of winding modes. T-duality interchanges the interpretation of these two towers. The basic idea is just that as the original circle shrinks, the winding modes of the strings start to look more and more like the continuum momentum modes on the big dual circle. In the type II superstring, this is accompanied by an interchange of the two flavors IIA and IIB. To relate the RR potentials: A RR-potential with an index along the circle being dualized loses that index; one without an index along the circle gains an extra index. ] (D-branes) re problem 3: the few insights beyond perturbation come from the following set of observations. The effective action (derived by expanding around particular solutions like flat space) can be used to find other solutions. If these solutions have small curvatures and field gradients (in string units) and if the dilaton is uniformly small, we can trust the leading-order effective action to tell us that it will really be a solution of string theory. In general, we don t know a worldsheet description of most such solutions; the ones where the RR fluxes are nonzero are particularly intractable from the worldsheet point of view. One particularly interesting family of solutions are analogs of the Reissner-Nordstrom black hole solution of 4d Einstein-Maxwell theory, but which carry RR charge. An object which couples minimally to a p + 1-form potential (i.e. which carries the associated monopole moment) has a p + 1-dimensional worldvolume Σ p+1, so that we can have a term in the action of the form S st C p+1 = dσ α 1...dσ α p+1 ǫ α 1..α p+1 α1 X µ 1... αp+1 X µ p+1 C µ1...µ p+1. Σ p+1 In flat space, the stable configurations of such objects are just flat infinite p-dimensional planes in spacetime. These objects have a finite tension, and hence these are not finite-energy excitations above the vacuum; they are different superselection sectors. Nevertheless, they exist (they can preserve half the supersymmetry of the vacuum) and are interesting as we ll see. If the space contains compact factors, we can imagine wrapping such objects on compact submanifolds and getting particle-like excitations. 3

14 The tension of these RR solitons is proportional to g 1 s. They are heavy at weak coupling, (though not as heavy as ordinary GR solitons, which have tension g s ) and are not made of a finite number of fundamental string excitations. This means that the euclidean versions of these objects can contribute the mysterious effects of order e S eucl e 1/gs. At weak coupling, it turns out that these objects have a remarkably simple description. We said earlier that the coupling to the NSNS B-field meant that string worldsheets can t end just anywhere. A Dp-brane is defined to be a (p + 1)-dimensional locus where strings can end. The open strings which end there have tension and hence their light states are localized on the brane. These provide a string theory description of worldvolume degrees of freedom. We saw above that open string spectra contain vector fields; these worldvolume degrees of freedom very generally include a gauge field propagating in p + 1 dimensions. The physics of this vector field is gauge invariant. [It would be more interesting if it weren t. This could be that this is just a result of the fact that we used the gauge principle in constructing the string theory.] So the leading term in the derivative expansion of the effective action for the gauge field is S = d p+1 1 x tr F F + O(α ) g Y M Because this action comes from interactions involving worldsheets with one boundary, it should go like gs h b = gs 1 and therefore gy M g In p+1 dimensions, g has dimensions of length p 3 s. Y M ; these are made up by powers of α. Strings have two ends. Each end can be labelled by which brane it ends on. This means that the string states are N N matrices, where N is the number of D-branes. The ends of the strings are charged particles (quarks and antiquarks). We should try to understand how the ending of the string is consistent with the B-field gauge symmetry. To see this, we use the fact that the end of the string is a charged particle to guess that the coupling of the worldsheet to a background worldvolume gauge field should take the minimal form S A = Σ A where Σ is the boundary of the worldsheet. The variation of the bulk worldsheet action under the B-field gauge transformation B B + dλ is 1 1 δs ws = 4πα dλ = 4πα Λ. Σ We can cancel this if we allow A to vary by A A 4πα Λ. This means that only the combination F 4πα F + B, where F = da is invariant under the B-field gauge symmetry. Hence F must enter the effective action in this combination. To see how this story is modified by the presence of the brane, we need to think about the dependence of the D-brane action on the B-field. Given our previous statement that the D-brane action contains L F F, it must actually be L F F. Σ 4

15 B 8 Dp brane S 7 B 8 string This means that in the presence of a fundamental string ending on a D-brane, the spacetime action for the NSNS B-field is S[B] = d 10 x H H + B + B F + O(B ) everywhere F1 Dp The equation of motion is then 0 = d H + δ 8 (F1) + δ 10 (p+1) (Dp) F ; Integrating this over an 8-ball B 8 that s pierced by the string gives H = n 1 S 7 where S 7 = B 8 is the boundary of the 8-ball and n 1 is the number of strings. If we integrate instead over B 8 which is not pierced by the string, it must instead go through the D-brane, as in the figure. Integrating the EOM for B over B 8 gives 0 = H + F. S 7 B Dp 8 The first term here is the same as before, and is n 1. But this gives a consistent answer because S 7 Dp F is equal to the number of strings ending on the brane, by the gauss law for the worldvolume gauge field. Basically, the worldvolume gauge flux carries the string charge away. Note that IIA has D-branes of these dimensions: D0, D, D4, D6, D8 The D0-brane is just a particle, charged under the RR vector field; The D6 is the associated magnetic charge. These are interchanged under T-duality with the branes of type IIB. while IIB has 5

16 D(-1), D1, D3, D5, D7, D9 The D9 brane fills space and is a big perturbation; they only exist stably in type I. The D(-1)-brane is pointlike in spacetime, and hence is best considered as a contribution to a euclidean path integral. It contributes e 1/gs effects to amplitudes. I forgot to say in lecture that these objects defined by new sectors of open strings can be shown to be charged under the RR tensor fields (and in fact saturate the Dirac quantization condition on the charge), and have tension 1/g s times the right power of α to make up the dimensions. The gravitational back-reaction of a stack of N parallel such D-branes is controlled by G N T g s N λ. When this quantity is small, the description in terms of open strings is good; when this quantity is large, the back-reaction is important and the better description is in terms of the gravitating soliton. More on this soon. 6

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18 8.81 F008 Lecture 03: The decoupling argument; AdS/CFT without string theory, a discovery with hindsight Lecturer: McGreevy September 14, 008 Today: 1. Backreaction and decoupling. A bold assertion 3. Hints, lore, prophesy Recall from last time that: D-branes are subspaces where open strings can end. Nomenclature: Dp-branes have p spatial dimensions. Light open strings are localized at the brane, and represent the fluctuations of the brane. D-branes carry RR charges (which saturate Dirac quantization condition; this is necessary since we have both electric and magnetic charges). This can be seen by computing the amplitude for D-branes to emit RR gauge bosons; it is described by worldsheets as in the figure below. The world volume theory on the brane is a YM theory. Dp Figure 1: The disk amplitude for the emission of a closed string by a D-brane. The disk amplitude for the emission of a graviton gives the tension of the D-brane and is proportional to 1 =. (1) +h+b g s 1 g s

19 The backreaction on the geometry from N coincident D-branes is determined from Einstein equations 1 1 R 8πG N T brane µν Rg µν = µν g s N = g s N = λ, () g s where we used the fact that G N closed string coupling g s. (3) At small λ, the backreaction is negligible and the physics is decribed by closed strings everywhere (these are excitations of empty space) plus open strings (YM +α F 4 + ) localized on the branes (these are excitations of the branes) in flat space. At large λ, the D-branes will gravitationally collapse into an RR soliton (black hole with the same charge). For p > 0, the black hole is not a point particle, but is also extended in p spatial dimensions. The case of p = 3 The 3-brane will fill dimensions (e.g., x µ=0,1,,3 ). We can put the brane at y 1 = = y 6 = 0, i.e, the R 3,1 is at a point in the transverse R 6. Let r = yi be the distance from the brane, then the metric of the brane will take the form and {}}{ R 3,1 6 ds = 1 {}}{ η µν dx µ dx ν + H(r) dyi, (4) H(r) i=1 H(r) = 1 + L r 4, where H(r) 1 for large r and L is like an ADM mass. The RR F 5 satisfies F 5 = N. (6) S 5 at constant r R 6 dy = dr + r dω 5, (5) The solution looks like an RN black hole (Figure ). Far away (large r with H(r) = 1), the solution will asymptote to R 9,1. Near the horizon (r L with H(r) = L /r 4 ), the metric will take the form of an AdS 5 S 5 r dr r/ ds = L η µνdx µ dx ν + L r + L r/ dω 5. (7) }{{}}{{} AdS 5 S 5 The ultra-low energy excitations near the brane can t escape the potential well (throat) and the stuff from infinity can t get in. The absorption cross section of the brane goes like σ ω 3 L 8. (8)

20 R 3,1 r Gravitational potential well Cross-section of S 5 Raduis is constant L Figure : The above conclusion can be reached in another way by noting that for low ω, the wavelength of the excitations will be too large to fit in the throat which has fixed size. Comparison of low energy decoupling at large and small λ (λ = Ng s ) Large λ Small λ Throat states = IIB strings in AdS 5 S 5 related by variation of λ, i.e, dual 4D N = 4, SU(N) YM + + Closed strings in R 9,1 Closed strings in R 9,1 Maldacena: Subtract Closed strings in R 9,1 from both sides. Matching of parameters The parameters of the gauge theory are g YM = g s (we will see that it is really a parameter in N = 4 YM) and the number of colors N. By Gauss s law F 5 = F 5 = N (quantized by Dirac). (9) S 5 S 5 The supergravity equations of motion relate L (size of space) and N (number of branes) as follows: where R G N F 5 F 5... µν = µ... ν, (10) G N g 4 s α, and Fµ... 5 Fν 5... N. (11) 3

21 Dimensional analysis then says R µν L 8. This gives L 4 = g s N = λ ( t Hooft coupling). (1) α In terms of gravitational parameters, this says G N g s (α ) 4, L = N 1/4 1/ G N GN 1 N (in units of AdS raduis, i.e. at fixed λ). (13) A picture of what has happened here which can sometimes be useful is the Picture of the Tents. We draw the distance from the branes as the vertical direction, and keep track only of the size of the longitudinal and transverse directions as a function of this radial coordinate. Flat space looks like this: (Hence, tents.) The picture of the near-horizon limit looks instead like: r R 3,1 r S 5 R ,1 S 3,1 S R R Figure 3: Left: Flat space. Right: AdS 5 S 5. Now it s the Minkowski space IR 3,1 which shrinks at r = 0, and sphere stays finite size. A Bold Assertion Now we back up and try to understand what is being suggested here, without using string theory. We will follow the interesting logic of: Reference: Horowitz-Polchinski, gr-qc/ Assertion: Hidden inside any non-abelian gauge theory is a quantum theory of gravity. What can this possibly mean?? Three hints from the Elders: 1) At the least it must mean that there is some spin-two graviton particle, that is somehow a composite object made of gauge theory degrees of freedom. This statement seems to run afoul of the Weinberg-Witten no-go theorem, which says: Theorem (Weinberg-Witten): A QFT with a Poincaré covariant conserved stress tensor T µν forbids massless particles of spin j > 1 which carry momentum (i.e. with P µ = d D xt 0µ 0). GR gets around this because the total stress tensor (including the gravitational bit) vanishes by δs the metric EOM δg = 0. (Alternatively, the matter stress tensor which doesn t vanish is not µν general-coordinate invariant.) 4

22 Like any good no-go theorem, it is best considered a sign pointing away from wrong directions. The loophole in this case is blindingly obvious in retrospect: the graviton needn t live in the same spacetime as the QFT. ) Hint number two comes from the Holographic Principle [ t Hooft, Susskind]. This is a far-reaching consequence of black hole thermodynamics. The basic fact is that a black hole must be assigned an entropy proportional to the area of its horizon in planck units (much more later). On the other hand, dense matter will collapse into a black hole. The combination of these two observations leads to the following crazy thing: The maximum entropy in a region of space is its area in Planck units. To see this, suppose you have in a volume V (bounded by an area A) a configuration with entropy A S > S BH = 4G (where S N BH is the entropy of the biggest blackhole fittable in V ), but which has less energy. Then by throwing in more stuff (as arbitrarily non-adiabatically as necessary, i.e. you can increase the entropy) you can make a black hole. This would violate the second law of thermodynamics, and you can use it to save the planet from the Humans. This probably means you can t do it, and instead we conclude that the black hole is the most entropic configuration of the theory in this volume. But its entropy goes like the area! This is much smaller than the entropy of a local quantum field theory, even with some UV cutoff, which would have a number of states N s e V ( maximum entropy = ln N s ) Indeed it is smaller (when the linear dimensions are large!) than any system with local degrees of freedom, such as a bunch of spins on a spacetime lattice. We conclude from this that a quantum theory of gravity must have a number of degrees of freedom which scales like that of a QFT in a smaller number of dimensions. This crazy thing is actually true, and AdS/CFT is a precise implementation of it. Actually, we already know some examples like this in low dimensions. One definition of a quantum gravity is a generally-covariant quantum theory. This means that observables (for example the effective action) are independent of the metric: δs eff T µν 0 = =. δg µν We know two ways to accomplish this: 1) Integrate over all metrics. This is how GR works. ) Don t ever introduce a metric. Such a thing is generally called a topological field theory. The best-understood example is Chern-Simons gauge theory in three dimensions, where the variable is a gauge field and the action is S CS = tr A da +... M (where the dots is extra stuff to make the nonabelian case gauge invariant); note that there s no metric anywhere here. With either option (1) or () there are no local observables. But if you put the theory on a space with boundary, there are local observables which live on the boundary. Chern-Simons theory on some manifold M induces a WZW model (a d CFT) on the boundary of M. We will see that the same thing happens for more dynamical quantum gravities. 3) A beautiful hint as to the possible identity of the extra dimensions is this. Wilson taught us that a QFT is best thought of as being sliced up by length (or energy) scale, as a family of trajectories of the renormalization group (RG). A remarkable fact about this is that the RG equations for the 5

23 behavior of the coupling constants as a function of RG scale z is local in scale: z z g = β(g(z)) the RHS depends only on physics at scale z. It is determined by the coupling constant evaluated at the scale z, and we don t need to know its behavior in the deep UV or IR. This fact is not completely independent of locality in spacetime. This opens the possibility that we can associate the extra dimensions raised by the Holographic idea with energy scale. Next we will make simplifying assumptions in an effort to find concrete examples. 6

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25 8.81 F008 Lecture 04 Lecturer: McGreevy September, 008 Today 1. Finish hindsight derivation. What holds up the throat? 3. Initial checks (counting of states) 4. next time: Supersymmetry self-defense Previously, we had three hints for interpreting the bold Assertion that we started with, that hidden inside any non-abelian gauge theory is a quantum theory of gravity. 1. The Weinberg-Witten theorem suggests that the graviton lives on a different space than the Quantum Field Theory.. This comes from the holographic principle. The idea, motivated by black hole thermodynamics, is that the theory of gravity should have a number of degrees of freedom that grows more slowly than the volume (non-extensive) suggesting that the quantum gravity should live is some extra (more) dimensions than the QFT. 3. The structure of the Renormalization Group suggests that we can identify one of these extra dimensions as the RG-scale. In order to make these statements a little more concrete, we will make a number of simplifying assumptions. That means we will not start by writing down the more general case of the correspondence, but we will start with a special case. 1. There are many colors in our non-abelian gauge theory. The motivation for this is best given by the quote: You can hide a lot in a large-n matrix. 1

26 Shenker The idea is that at large N the QFT has many degrees of freedom, thus corresponding to the limit where the extra dimensions will be macroscopic.. We will work in the limit of strong coupling. The motivation for that lies in the fact that we know some things about weakly coupled Field Theories, and they don t seem like Quantum Gravity. This constrains the manner in which we take N to be large, since we would still like the system to be interacting. This means that we don t want vector models. We want the theory to have as much symmetry as possible. However there is a theorem that stands in the way of this. Theorem (Coleman-Mandula) : All bosonic symmetries of a sensible S-matrix (nontrivial, having finite matrix elements) belong to the Poincare group. We are going to use all possible loopholes to circumvent this theorem. The first lies in the word bosonic and the second one in sensible. 3. SuperSymmetry (SUSY) (a) It constrains the form of the interactions. This means that there are fewer candidates for the dual. (b) Supersymmetric theories have more adiabatic invariants, meaning observables that are independent of the coupling. So there are more ways to check the duality. (c) It controls the strong-coupling behavior. The argument is that in non-susy theories, if one takes the strong-coupling limit, it tends not to exist. The examples include QED (where one hits the landau Pole for strong couplings) and the Thirring model, which is exactly solvable, yet doesn t exist above some coupling. (d) Nima says it s simpler ( The most supersymmetric theory in d=4 is the N = 4 SYM and we ll be focusing on this theory for a little while. It has the strange property that, unlike other gauge theories, the beta function of the gauge coupling is exactly zero. This means that the gauge coupling is really a dimensionless parameter, and so, reason number (e) to like SUSY is... (e) SUSY allows a line of fixed points. This means that there is a dimensionless parameter which interpolates between weak and strong coupling. 4. Conformal Invariance: The fact the coupling constant is dimensionless (even quantum-mechanically) says that there is scale invariance. The S-matrix is not finite! Everything is soft gluons. Scale invariance applies to both space and time, so X µ λx µ,where µ = 0,1,,3. As we said, the extra dimension coordinate is to be thought of as an energy scale. Dimensional anal ysis suggests that this will scale under the scale transformation, so z z λ. The most general five-dimensional metric (one extra dimension) with this symmetry and Poincare invariance is of the following form: z dz L z dz ds = ( L ) η µν dx µ dx ν + z L. Let z = L z L = L We can now bring it into the more familiar form (by change of coordinates) ds = ( L ) η µν dx µ dx ν + z L, which is AdS 5. It turns out that this metric also has conformal invariance. So scale and Poincaré implies conformal invariance, at least when there is a gravity dual. This is believed to be true more

27 generally, but there is no proof. Without Poincaré invariance, scale invariance definitely does not imply conformal invariance. We now formulate the guess that a 4-dimensional Conformal field Theory is related to a theory of Gravity on AdS 5 A specific example is N=4 SYM which is related to IIB Supergravity on AdS 5 S 5. Note: On a theory of gravity space-time is a dynamical variable, where you specify asymptotics. This CFT defines a theory of gravity on spaces that are asymptotically AdS 5. Why is AdS 5 a solution? 1. Check on PSet. Use effective field theory (which in this case just means dimensional reduction) to elaborate on my previous statement that the reason why this is a solution is that the flux holds it up. To do this let s consider the relevant part of the supergravity action, which is S = d D x[ GR (D) F g F g ] As an example consider the case where D=10 and g=5 (note that we are working in dimensions where G ( D) N = 1 = 1/5! = = 4π). [Note: The reference for this is Denef et al hep-th/ ] Let s make an Ansatz (called Freund-Rubin) The metric is some space-time (x) plus a q- sphere of radius R, so that ds = g µν dx µ dx ν + R(x) G mn dy m dy n and furthermore for the flux though the q-sphere we have S F q q = N To find the form of this part of the metric as a result of the flux, we integrate over the q-sphere and get the action in D-dimensions, which will contain a term which is S GG m 1 n 1...G q mqnq (F g ) m1...m q (F g ) n1...n q We should then think of R(x) as a moduli field. Now we have to evaluate the following (D) integral S GR = gr q a [R ] q R We will now go to something called the FRAME GAME To make the D-dimensional Einstein-Hilbert term canonical, we can do a Weyl-rescaling of the metric, meaning to define a new metric gµν E = λ(x)g µν. So the new metric is some function of the space-time variables times the old metric. So g E = λ d/ g, R E = λ 1 R. We ll pick λ to absorb the factor of R, so g ERE = λ d/ 1 gr (d) = R q gr (d), λ = R q/d. N We know that the flux is the integral over the q-sphere, so S F = N N q R q, so the flux term of the potential is V R g λ d/ n flux (R) F F = V ol(s q ) gf = g E ( ) S q So V flux (R) N 1 R Rβ. where α > β > 0 α At small R the first term in the effective potential dominates, while for R large, the effective potential approaches zero from below. In between there is some minimum. To find the actual value of this we take 0 = V N α β (R min ) R min. The fact that the potential goes to zero at large R is unavoidable. It is because when R is large, the curvature is small and flux is dilute. So all terms in the potential go away. For special values of α and β, we get that N R 4 L 4 min = 3 R q

28 This endeavor is called flux compactification and it is the way to find vacua of string theory without massless scalars. Furthermore, we can add some effective action for scalars with m R V (R min ), which 1 looks like this : S = d d x g e (R E Λ) R. The cosmological constant is determined as Λ = V eff (R min ) < 0 The fact the the cosmological constant exists, means that the metric is not flat. S The equation of motion is the following: 0 = gµν d R = Λ R µν ( Λ d d ) R µν = g µν ( 1 R Λ) L dr +η µνdx µ dx ν We know the solution for AdS d, which has the metric ds = r Now I will describe the right way to compute curvatures, which is to use what is called tetrad or vielbein or Cartan-Weyl method. There are three steps Rewrite the metric as ds = (e rˆ) + e µˆe νˆη µν,where I define e rˆ L dr and e xˆµ L dxµ r r Cartan 1 demands that they are covariantly constant and is formulated as follows: de a = ω ab e b, where the ω is called the spin connection. For example de rˆ = 0 and e µˆ = L dr dx µ = 1 e rˆ e µˆ ω µˆνˆ = 1 e µˆ = ω rˆµˆ r L L Why do we care about ω? Here s why: Cartan R ab = dω ab + ω ac ω cb = Rµνdx ab µ dx ν From that you can get the Ricci tensor, which appears in the equation of motion, by contracting ρµ some indices R ν µ = R ρν Let s do some examples: R µˆνˆ = ω µˆrˆω rˆνˆ = 1 e µˆ eνˆ R µˆrˆ = dω µˆrˆ = L 1 e rˆ e µˆ R µν L ρσ = L 1 (δ µ ρ δσ ν δ σδ µ ρ ν ) R µr 1 ρr = L δ ν µ d Λ d d(d+1) R µν = g L µν d = Λ = L L Counting of degrees of freedom [Susskind-Witten, hep-th/ ] The holographic principle tells us that the area of the boundary in planck units is the number of degrees of freedom (maximum entropy). 4

29 Area of boundary? = number of dof s of QFT N d 4G N Both sides are equal to infinity, so we need to regulate our counting somehow. The picture we have of AdS is a collection of copies of Minkowski space of varying size and the boundary is the locus where they get really big. The area is defined as A = R gd 3 x = 3 R d 3 x L 3 3 r 3 This is infinite for two reasons: from the integral over x and from the fact that r is going to zero. Let s regulate the Field Theory first. We put the thing on a lattice, introducing a low distance cut-off δ and we put it also in a box of size R. The number of degrees of freedom is the number of cells times N R3, so N d = N. To regulate the integral we write before, we stop not at r = 0 but δ 3 we cut it off at r = δ. A δ = d 3 x L 3 R3L r 3 r=δ = 3 Now we can translate this L (5) G (10) N N L 5 L 3 R 3 A = G (5) δ 3 G (5) N N N G 10 G N = N d = So AdS/CFT is indeed an implementation of the holographic principle. δ3 5

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31 8.81 F008 Lecture 5: SUSY Self-Defense Lecturer: McGreevy September, 008 Today s lecture will teach you enough supersymmetry to defend yourself against a hostile supersymmetric field theory, should you meet one down a dark alleyway. Topics will include 1. SUSY representation theory, including basic ideas regarding the algebra and an explanation of why N = 4 is maximal (a proof due to Nahm). Properties of N = 4 Super Yang-Mills theory, such as the spectrum and where it comes from in string theory. Before plunging in, we might be inclined to wonder: why is supersymmetry so wonderful? In addition to the delightful properties that will be explored in the remainder of this course, there are various reasons arising from pure particle physics: for example, it stabilizes the electroweak hierarchy, changes the trajectory of RG flows such that the gauge couplings unify at very high energies, and halves the exponent in the cosmological constant problem. These are issues that we will not explore further in this lecture. 1 SUSY Representation Theory in d = 4 We begin with a quick review of old-fashioned Poincare symmetry in d = 4: 1.1 Poincare group Recall that the isometry group of Minkowski space is the Poincare group, consisting of translations, rotations, and boosts. The various charges associated with the subgroups of the Poincare group are quite familiar and are listed below Translations Rotations/boosts Charge P µ M µν 1 Subgroup R 3,1 SO(3,1) (SU() SU())