String Theory and M-Theory

12.2 Matrix theory 637
in the type IIA theory. However, we know that, from the M-theory viewpoint,
D4-branes are really M5-branes that wrap the extra spatial dimension
that M-theory provides. Thus, the D4-brane system is better viewed as a
set of M5-branes wrapping a T 5 . The desired SL(5, ) duality group is
then recognized to be the modular group of this torus. This six-dimensional
world-volume theory is believed to be a well-defined quantum field theory.
The reason its discovery was made relatively recently is that it is strongly
coupled in the UV, and therefore it does not have a simple Lagrangian description.
The situation for p > 4 is even more challenging and has not been worked
out in detail. However, it should be clear already that Matrix theory is
capable of capturing a great deal of subtle physics. In fact, its validity can
be deduced from the gauge theory/string theory dualities considered in the
next section.
EXERCISES
EXERCISE 12.3
Show that the fermionic propagator can be expressed in terms of the bosonic
propagator as indicated in Eq. (12.65).
SOLUTION
Comparing Eqs (12.63), (12.65) and (12.66), we need to show that
(∂τ − vτγ1 − bγ2)(∂τ + vτγ1 + bγ2) = ∂ 2 τ − r 2 + vγ1,
where we have used r 2 = b 2 + (vτ) 2 and mF = vτγ1 + bγ2. This follows from
some simple gamma matrix algebra and the derivative acting on the τ term.
Thus, one obtains the desired relation between the bosonic and fermionic
propagators. ✷
EXERCISE 12.4
Show that the only terms in the Matrix theory effective action up to three
loops are the terms appearing in Table (12.76).
SOLUTION
The solution follows from dimensional analysis. Since the action is dimen-