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Week 1
1) Show that S=k_B (Area)/4l_p^2 solves dE=TdS with E=M, T the Hawking temperature of a 4d black hole, and A the horizon area of the black hole. Extra credit: show that a similar formula holds in D spacetime dimensions, S=k_B(Area)/4l_p^{D-2}, where the horizon area is that of a D-2 dimensional sphere which surrounds the black hole. 2) Show that a 5d scalar, with 5d mass M, leads to an infinite tower of 4d scalars indexed by n. Find their masses m_n, in terms of M and the radius R_5 of the 5th dimension, when the compact 5th dimension is spacelike. Find also m_n if the extra dimension is timelike and note that this case is no good. (Due April 11) top |
| Week 2 1) Show that S=-m\int ds=-m\int d\tau \sqrt{-dX^\mu/d\tau dX_\mu /d\tau} becomes, in the non-relativistic limit, the usual action S\approx \int dt (T-V), with T=(1/2)m(d X^i/dt)^2 and V=m, with t=X^0 the time component of X^\mu and X^i the spatial components of X^\mu. 2) A "primary" operator \Phi (z,zbar) transforms under conformal mapping (z,zbar)->(f(z),fbar(zbar)) to (\partial f/\partial z)^h(\partial fbar /\partial zbar)^{hbar} \Phi (f(z),fbar(zbar)). Find how \partial _z \Phi transforms and show that it is almost primary, of dimensions (h+1, hbar), but that, for non-zero h, there is an extra term which makes it not primary. (Due April 18) top |
| Week 3 1) Use the OPE between T(z) and a primary field \Phi (w,wbar) and the expression for the generator L_n, in terms of a contour integral of z^{n+1}T(z), to show that the commutator of L_n with \Phi (w,wbar) is [L_n,\Phi]=(h(n+1)w^n+w^{n+1}\partial _w)\Phi (w,wbar). 2) Show that Mobius transformations of the complex plane, via z\rightarrow (az+b)/(cz+d), with (a,b,c,d) arbitrary complex numbers satisfying ad-bc=1, satisfy the defining conditions of a group: a) the composition of any two symmetry operations on z, e.g. first by (a_1, b_1,c_1,d_1) and then by (a_2,b_2,c_2,d_2), is itself a symmetry operation on z of the above type for some (a_3,b_3,c_3,d_3) which also satisfy (ad-bc)=1. Show that this is a non-Abelian group, since changing the order of the above two symmetry operations gives a different result. b) the symmetry operations are associative c) every symmetry operation has an inverse operation, also with (ad-bc)=1, so that the composition of the symmetry and inverse symmetry leave z unchanged. You can save ink in showing the above if you write the symmetry transformations as matrices, with the composition of two operations on z given by matrix multiplication. This symmetry group is called SL(2,C), that of complex 2x2 matrices with unit determinant. (Due April 25) top |
| Week 4 Consider fields b(z) and c(z) with the two point functions b(z)c(w)=c(z)b(w)=1/(z-w) and stress tensor T(z)=(1-\lambda) (\partial b(z)) c(z) - \lambda b(z)\partial c(z). b(z) is a field with h=\lambda and c(z) has h=1-\lambda and these fields anticommute. \lambda is an arbitrary parameter. Find the two point function expectation value of T(z)T(w) and thus find the central charge c as a function of \lambda. (Do not confuse the central charge c(\lambda) with the field c(z)!). We will soon see that the Faddeev-Popov ghosts needed for properly treating the functional integral over the world-sheet metric h_{ab} are of exactly the above form, with \lambda =2. In this case the central charge thus obtained from the above should equal -26. Extra credit: Find the expectation value of T(z)J(w) for J=-bc the ghost number current. top |
| Week 5 Consider the open string state at oscillator level N=2, with \epsilon _{\mu \nu} \alpha _{-1}^\mu \alpha _{-1}^\nu acting on the momentum eigenstate with momentum p^\mu. Find all of the conditions on the polarization tensor \epsilon _{\mu \nu} and the momentum p^\mu for this to be a physical state, i.e. primary with h=1. (See web lecture notes (week 5, p. 17) if this is too hard to read.) (Due May 9) top |
| Week 6 Consider a primary state of L_0 eigenvalue h. Act on this state with (L_{-2}+bL_{-1}L_{-1}) to obtain a descendent state. For what choice of of the constant b (as a function of h) is this descendent annihilated by L_1? Taking this value for b, what is the value of the central charge (as a function of h) for which this descendent also annihilated by L_2. Verify that for h = -1 (in which case the descendent is a physical state, with h=1) the special value of the central charge is 26. Use only the Virasoro algebra and the definition of a primary state for this problem. This problem shows that there are many null states for c=26, corresponding to an enormous spacetime gauge symmetry. (Due May 16) top |
| Week 7 top |
| Week 8 In lecture, (b_0c_0+c_0b_0)=1 was used to argue that the ghost vacuum expectation value, without any ghost zero mode insertions, vanishes. Show that the above identity does not imply the vanishing of the ghost vacuum expectation value with a c_0 (and barc_0) zero mode insertion. But show that there are two similar identities which could be used to show that this expectation value, with only the c_0 (and bar c_0) zero mode soaked up, also vanishes. Finally, verify that we can set the ghost vacuum expectation value, with all three of the ghost (and bar ghosts) zero modes inserted, to unity; i.e. there is no identity of the above type which would require it to vanish. top |
| Week 9 Verify that (ST)^3 maps \tau back to \tau. Find all values of \tau in the fundamental domain which are mapped to themselves by ST. (Due June 6) top |
| Week 10 1) Consider the SU(2) currents J_i(z) at the self-dual radius given in lecture (J_3 is proportional to \partial X and J_\pm = J_1\pm iJ_2 =exp(\pm 2iX_L/\sqrt{\alpha '})). Define the corresponding conserved charges to be Q_i. By considering the current-current OPEs, verify that [Q_i, J_j(z)]=i\epsilon ^{ijk}J_k(z). 2) Find the solution X_{cl} (z,zbar), on the torus z\sim z+2\pi \sim z+2\pi \tau, which satisfies the X equation of motion \partial \bar \partial X=0, with the boundary condition that X winds w times around the circle of radius R under shifting z by 2\pi, and m times around the circle of radius R under shifting z by 2\pi \tau. Verify that the classical action for this X_{cl} is S_{cl}(m,w)=\pi R^2| m-w\tau |^2/ \alpha ' \tau _2. 3) Verify that \sum e^{-S_{cl}(m,w)} is modular invariant, where the sum is over all integers m and w. (Due June 13. PLEASE LEAVE IN MY MAILBOX) top |