# Physics 250: Bonus Problem Assignment

The bonus problem is being offered to the most advanced students in the class, as a possible alternative to all the regular homework assignments.

The problem involves a tachyonic (but otherwise perturbatively consistent, i.e. modular invariant etc.) heterotic string theory in ten (uncompactified, flat) spacetime dimensions, and with the Yang-Mills gauge group given by (just one copy of) E_8. This theory was first found in the following paper (it appears there as the final model listed in Table I):

H. Kawai, D.C. Lewellen and S.-H. H. Tye, " Classification of Closed Fermionic String Models," Phys. Rev. D34 (1986) 3794.

Here is the list of tasks to be performed in this bonus assignment. The questions/tasks that appear between stars (like *this*) are purely voluntary, and represent a challenge for the truly inquisitive student; they may not even have a well-understood answer!

(1) Consider the free string spectrum of the model. Discuss the massless states, the tachyonic state(s), and all states at the first massive level above the massless level where physical states survive the GSO projection (i.e., write the states in terms of oscillators acting on the ground state). In particular, discuss in detail the GSO projection in various sectors of the theory. Discuss transformation properties of these states under E_8 as well as the space-time Lorentz group. Write down the vertex operators associated with all the massless states.

(2) In the fermionic representation of the model, write down the explicit prescription for summing over spin structures in the one-loop string amplitude, i.e., on a torus. What is the generalization of this prescription to higher-order amplitudes, i.e., to a genus g Riemann surface?

(3) Find a reasonably compact expression for the one-loop vacuum amplitude in the theory. How much physical information can you extract from this expression?

(4) Find the E_8 current algebra on the worldsheet in terms of the 32 worldsheet fermions of [Kawai, Lewellen and Tye], and verify the operator product expansion of E_8.
(4a) What is the level of the E_8 current algebra?
(4b) What is the central charge of the E_8 current algebra? (verify)

(5) Write the spectrum in terms of the E_8 current algebra and a minimum number n of extra fermions. What is the value of n? *Is there a good physical reason for precisely this value of n?*

(6) Write down the relevant worldsheet operator corresponding to the tachyon. *What is the outcome of the relevant deformation of the worldsheet theory by this operator?*

(7) *Can you find a bosonic realization of this heterotic model, in terms of a lattice compactification?*

(8) **What is the outcome of the tachyon condensation in this model?**

horava@socrates.berkeley.edu