Lecture Notes on String Theory Kevin Zhou [email protected] I don't actually know any string theory; these incomplete notes are just a collection of the prereq- uisites one needs to know before starting to learn string theory, beyond the usual ones of quantum field theory and general relativity. Topics covered include the quantization of constrained systems, 2D conformal field theory, and the quantization of the bosonic string. The primary sources were: • David Tong's String Theory lecture notes. These clear lecture notes follow the first volume of Polchinski, but with a much more approachable style, and hence are accessible after a good course on quantum field theory. • Timo Weigand's String Theory lecture notes. As with the quantum field theory notes, these notes are slightly more comprehensive and precise, at the cost of being slightly drier; also contains a short introduction to the superstring. • Zwiebach, A First Course in String Theory. A very basic introduction, covering bosonic string theory in the first part, and a wide variety of developments, such as D-branes and AdS/CFT in the second. The book is clearly written and accessible even without any field theory background, and in fact might be useful as indirect preparation for field theory. It has the benefit of explicitly showing many steps of logic skipped in most books. The downside is that the entire 300 pages of the first part barely covers the first 30 pages of Polchinski. • Henneaux and Teitelboim, Quantization of Gauge Systems. The most thorough book by far on the subject; if you're wondering how constrained quantization, Grassmann variables, or BRST symmetry really work, this is the place to go. Naturally, more formal than any of the other books on this list. The most relevant chapters for these notes are 1, 4, 6, and 13. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Preliminaries 3 1.1 Miscellanea.........................................3 1.2 Classical Strings......................................6 1.3 String Motion........................................ 11 1.4 Light Cone Gauge..................................... 15 1.5 Light Cone Fields...................................... 20 1.6 The Veneziano Amplitude................................. 23 2 Constrained Systems 26 2.1 Constrained Hamiltonian Systems............................ 26 2.2 Dirac Brackets and Gauge Fixing............................. 29 2.3 General Covariance..................................... 32 2.4 Constrained Quantization................................. 34 2.5 Classical Point Particle................................... 37 3 The Bosonic String 42 3.1 The Polyakov Action.................................... 42 3.2 Old Covariant Quantization................................ 47 3.3 Computing Spectra..................................... 52 4 Conformal Field Theory 57 4.1 Conformal Transformations................................ 57 4.2 Elementary Aspects.................................... 60 4.3 Free Scalar Field...................................... 65 4.4 Central Charges...................................... 68 4.5 The Virasoro Algebra................................... 72 4.6 The State-Operator Correspondence........................... 75 5 String Interactions 80 5.1 Motivation......................................... 80 5.2 Vertex Operators...................................... 84 5.3 Bosonic Open Strings................................... 86 6 String Compactification 90 6.1 Kaluza{Klein Theory.................................... 90 6.2 Worldsheet Theory..................................... 91 3 1. Preliminaries 1 Preliminaries 1.1 Miscellanea As a first pass, we'll follow Zwiebach's text. In this first section, we establish some conventions. • There are three fundamental dimensions: mass, length, and time. Charge is not a independent 2 dimension; this is clearest in the cgs system, where the Coulomb force law is F = q1q2=r . That is, one can define charges purely in terms of the forces they produce. In the SI system, all occurrences of the unit of charge in measurable quantities are canceled out by 0 or µ0. • String theory is said to have no adjustable parameters. This means that it has no dimensionless parameters; there is a single dimensionful parameter, the string length `s. When string theory was considered as a theory of hadrons, `s was thought to be on the nuclear scale, but now it is viewed as much smaller. • We use the (− + ++) metric convention, but define the interval ds2 to measure proper time, 2 µ ν −ds = ηµνdx dx : • We will perform the quantization of the relativistic string in light cone coordinates, where 1 x± = p (x0 ± x1): 2 The other coordinates stay the same, so in xµ, the µ index runs over (+; −; 2; 3). The light cone coordinate axes are tilted so that they lie on the light cone, e.g. a photon moving to the right has x− = 0. • The metric in light cone coordinates is 0 −1 1 2 + − 2 2 3 2 B−1 C −ds = −2dx dx + (dx ) + (dx ) ; ηµν = B C : @ 1 A 1 For example, we have − + µ − + + − 2 2 3 3 a+ = −a ; a− = −a ; a bµ = −a b − a b + a b + a b : • None of the coordinates are true time coordinates, but we will conventionally think of x+ as the light cone time coordinate. This isn't completely unnatural, since dx±/dτ > 0 for almost all particles. • We think of dx−=dx+ as a \light cone velocity". For a particle with speed v, the light cone velocity is (1 − v)=(1 + v). In particular, it is zero for a massless particle moving to the right, infinite for a massless particle moving to the left, and one when v = 0. • Next we must define the light cone energy. Energy is the conjugate variable to time, and µ + − 2 3 pµx = p+x + p−x + p2x + p3x − which motivates us to define the light cone energy as −p+ = p . Defining energy as a conjugate variable is useful as, e.g. the Schrodinger equation i @ =@x0 = p0 becomes i @ =@x+ = p− in lightcone coordinates. 4 1. Preliminaries Next, we consider the possibility of extra spatial dimensions. We maintain only one time dimension, since it is difficult to construct a consistent theory with more than one. • The extra dimensions can be topologically nontrivial; for example, for a single dimension we may identify x ∼ x + 2πR. The interval 0 ≤ x < 2πR is a fundamental domain for this identification, as every point is identified with exactly one point in the fundamental domain. We then construct the space by identifying points on the boundary, getting a circle. • Sometimes identifications have fixed points. The resulting space is not a manifold, since it is singular at the fixed point, but an orbifold. For example, identifying x ∼ −x gives the half-line 1 x ≥ 0, called the R =Z2 orbifold. • As another example, consider the identification z ∼ e2πi=N z in the complex plane, giving the C=ZN orbifold. It is a cone which is singular at its vertex; one fundamental domain is 0 ≤ θ < 2π=N. • Physics on spaces with generic singularities is typically complicated and possibly even inconsis- tent. Orbifolds are interesting because they have \tractable" singularities, so we may quantize strings on them. • For point particles, a small compactified dimension creates new energy levels, but they are very high if the dimension is small. By contrast, for a string, new low-lying states can appear if the dimension is much smaller than the string length, corresponding to the string wrapping around it. This is a consequence of T duality, as we'll see. • Compactified dimensions lead to subtleties with gauge theory. It can be the case that two configurations with the same fields are not related by a gauge transformation; then we must consider the states physically distinct. It also also be the case that some configuration of fields cannot be realized by a potential, even if we allow the potential be defined on patches and related between patches with gauge transformations; such states are forbidden. Note. A quick review of the nonrelativistic string. The Lagrangian is 1 @y 2 1 @y 2 L = µ − T : 2 @t 2 @x When we attempt to extremize the action, we run into a boundary term Z x=a @y δS ⊃ −T dt δy : @x x=0 To remove this unwanted boundary term, we need to apply boundary conditions. One acceptable boundary condition is Neumann boundary conditions (free ends), where @y = 0: @x x=0;a Alternatively, we could use Dirichlet boundary conditions (fixed ends), @y = 0 @t x=0;a 5 1. Preliminaries which ensures that y is constant on the boundaries, and hence δy = 0 there. A third alternative is that the string is closed, so that there are no boundaries at all. In the initial days of string theory, open strings were given Neumann boundary conditions, but it was later realized that they could have Dirichlet boundary conditions if they attached to an extended object called a Dp-brane, where D stands for Dirichlet and p is the number of spatial dimensions. (One can think of Neumann boundary conditions as the special case where the D-brane fills all space.) Remarkably, it turns out that D-branes are physical objects in their own right, and arise naturally from string theory without being introduced by hand. Finally, it will be useful later to consider the conjugate momenta, @L @y @L @y Pt = = µ ; Px = = −T : @y_ @t @y0 @x Here Pt is simply the usual momentum density in the y-direction. The equation of motion is @Pt @Px + = 0: @t @x In this notation, Neumann and Dirichlet boundary conditions set Px = 0 and Pt = 0 respectively. Note. Gravity and electromagnetism in higher dimensions. We will use 2πd=2 πd=2 vol(Sd−1) = ; vol(Bd) = : Γ(d=2) Γ(1 + d=2) In d spatial dimensions, Gauss's law continues to hold, so the field of a point charge is 1 q E(r) = vol(Sd−1) rd−1 in cgs units. As for gravitation, we have Gm g(r) = r2 in D = 4 spacetime dimensions. More generally, the gravitational field is determined by r2V (D) = 4πG(D)ρ, g = −∇V where G(D) is the gravitational constant in D dimensions.