String Theory and M-Theory

682 Gauge theory/string theory dualities
string field theory. (Its construction is a long story that we won’t pursue
here.) The 1/J corrections are obtained by keeping track of the leading
1/R 2 corrections to the Penrose limit. This is straightforward, in principle,
but rather complicated in practice.
The M-theory plane-wave duality
Let us now mention the corresponding results for M-theory. The AdS4 × S 7
and the AdS7 × S 4 solutions have identical Penrose limits. This background
turns out to be given by
where
ds 2 = 2dx + dx − + g++(x I )(dx + ) 2 +
9
I=1
dx I dx I , (12.171)
g++(x I ) = −µ 2 ((r3/3) 2 + (r6/6) 2 ). (12.172)
The coordinate r3 is the radial coordinate for three of the x I s and r6 is the
radial coordinate for the other six of them. The transverse symmetry in
this case is SO(3) × SO(6). The M theory four-form field strength takes the
form
F4 ∼ µ dx + ∧ dx 1 ∧ dx 2 ∧ dx 3 . (12.173)
The dual gauge theory in this case is a version of Matrix theory. It is a
massive deformation of the original Matrix-theory proposal for a dual description
of M-theory in flat 11-dimensional space-time, which was discussed
in Section 12.2.
EXERCISES
EXERCISE 12.11
Starting from the light-cone gauge action in Eq. (12.160), generalize the
analysis given in Chapter 2 to derive the mode expansions of the bosonic
fields and the quantization conditions. Also, derive the corresponding formulas
for the fermions.