PROBLEM 7.4

Generalize the analysis of Exercise 7.6 to the heterotic string. In particular,

verify that the Wilson lines, together with the B and G ¬elds, have the right

number of parameters to describe the moduli space M0 16+n,n in Eq. (7.121).

PROBLEM 7.5

In addition to the SO(32) and E8 — E8 heterotic string theories, there is

a third tachyon-free ten-dimensional heterotic string theory that has an

SO(16) — SO(16) gauge group. This theory is not supersymmetric. In-

vent a plausible set of GSO projection rules for the fermionic formulation of

this theory that gives an SO(16) — SO(16) gauge group and does not give

any gravitinos. Find the complete massless spectrum.

PROBLEM 7.6

The SO(16) — SO(16) heterotic string theory, constructed in the previous

problem, is a chiral theory. Using the rules described in Chapter 5, construct

the anomaly 12-form. Show that anomaly cancellation is possible by showing

that this 12-form factorizes into the product of a four-form and an eight-

form.

PROBLEM 7.7

The ten-dimensional SO(32) and E8 — E8 string theories have the same

number of states at the massless level. Construct the spectrum at the ¬rst

excited level explicitly in each case using the formulation with 32 left-moving

fermions. What is the number of left-moving states at the ¬rst excited level

Homework Problems 293

in each case? Show that the numbers are the same and that they agree with

the result obtained in Exercise 7.10.

PROBLEM 7.8

(i) Consider a two-dimensional lattice generated by the basis vectors

e2 = (1, ’1)

e1 = (1, 1) and

with a standard Euclidean scalar product. Construct the dual lattice

Λ . Is Λ: unimodular, integral, even or self-dual? How about Λ ?

(ii) Find a pair of basis vectors that generate a two-dimensional even

self-dual Lorentzian lattice.

PROBLEM 7.9

Consider the Euclideanized world-sheet theory for a string coordinate X

1 ¯

‚X ‚Xd2 z.

S[X] =

π M

Suppose that X is circular, so that X ∼ X + 2πR and that the world sheet

M is a torus so that z ∼ z + 1 ∼ z + „ . De¬ne winding numbers W1 and

W2 by

X(z + 1, z + 1) = X(z, z ) + 2πRW1 ,

¯ ¯

X(z + „, z + „ ) = X(z, z ) + 2πRW2 .

¯¯ ¯

(i) Find the classical solution Xcl with these winding numbers.

(ii) Evaluate the action Scl (W1 , W2 ) = S[Xcl ].

(iii) Recast the classical partition function

e’Scl (W1 ,W2 )

Zcl =

W1 ,W2

by performing a Poisson resummation. Is the result consistent with

T-duality?

PROBLEM 7.10

Consider a Euclidean lattice generated by basis vectors ei , i = 1, . . . , 8,

294 The heterotic string

whose inner products ei · ej are described by the following metric:

«

2 ’1 0 0 0 0 0 0

¬ ’1 2 0·

’1 0 0 0 0

¬ ·

¬ 0 ’1 0·

2 ’1 0 0 0

¬ ·

¬ ·

’1 2 ’1 0

¬0 0 0 0·

·.

¬

0 ’1 2 ’1 0 ’1 ·

¬0 0

¬ ·

¬0 0 ’1 2 ’1 0 ·

0 0

¬ ·

0 0

0 ’1 2

0 0 0

0 ’1 0

0 0 0 0 2

This is the Cartan matrix for the Lie group E8 .

(i) Find a set of basis vectors that gives this metric.

(ii) Prove that the lattice is even and self-dual. It is the E8 lattice.

PROBLEM 7.11

As stated in Section 7.4, in 16 dimensions there are only two Euclidean even

self-dual lattices. One of them, the E8 — E8 lattice, is given by combining

two of the E8 lattices in the previous problem. Construct the other even

self-dual lattice in 16 dimensions and show that it is the Spin(32)/ 2 weight

lattice.

PROBLEM 7.12

Show that the spectrum of the bosonic string compacti¬ed on a two-torus

parametrized using the two complex coordinates „ and ρ de¬ned in Exer-

cise 7.8 is invariant under the set of duality transformations SL(2, )„ —

SL(2, )ρ generated by

’„ +1 ρ’ρ+1

„

.

’ ’1/„ ρ ’ ’1/ρ

„

Moreover, show that the spectrum is invariant under the following inter-

changes of coordinates:

U : („, ρ) ’ (ρ, „ ) V : (σ, „ ) ’ (’¯ , ’¯).

and σ„

These results imply that the moduli space is given by two copies of the

moduli space of a single torus dividing out by the symmetries U and V .

PROBLEM 7.13

Consider the bosonic string compacti¬ed on a square T 3