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Our understanding of M-theory is still very incomplete.
Even the interpretation of the name ”M” remains unclear.
But possibly, an important clue might be found in a celebrated
piece of work that appeared in 1954…
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I will report on some work in progress in collaboration with
Niclas Wyllard at Chalmers University in Göteborg, Sweden
on N=4 supersymmetric Yang-Mills theory in 3+1 dimensions .
Hopefully, some of this is new, although it appears that similar
arguments have been given in the context of matrix models.
We are investigating the relationship between three
(rather uncontroversial) conjectures:
I.
S-duality.
II.
Invariance of the low-energy spectrum of a
supersymmetric theory under smooth deformations.
III. The normalizable zero-energy states of supersymmetric
matrix quantum mechanics.
S-duality
An N=4 supersymmetric Yang-Mills theory is determined by its
•
gauge group G.
•
complex coupling =/2+i/g2.
S-duality states that this description is slightly redundant:.
We get an equivalent theory if we replace G and  by
• gauge group G and coupling +1
• Gauge group LG and coupling -1/.
For n=pq, we will consider the case
G=SU(n)/Zp so that LG=SU(n)/Zq.,
Work is in progress on the remaining (simply laced) Lie groups.
Invariance of the spectrum
In a supersymmetric theory with a massgap, the Witten index
Tr(-1)Fe-H gets contributions only from the (normalizable) zeroenergy states and is invariant under smooth deformations.
We conjecture that this remains true also for theories without a
massgap. A weaker and a stronger form of the conjecture is
• The number of normalizable zero-energy states (counted
with signs) is invariant.
• Also the quantum numbers of continua of states extending
down to zero energy is invariant.
Energy
Bosons Fermions
0
Matrix quantum mechanics
Supersymmetric matrix quantum mechanics with 16
supercharges is obtained by dimensional reduction of tendimensional supersymmetric Yang-Mills theory.
The bosonic variables (in temporal gauge) are nine Lie
algebra valued matrices. There is a quartic potential with flat
valleys extending to infinity.
A determination of the spectrum from first principles is a subtle
and still unsolved problem. However,
• For G=SU(m), the duality between type IIA string theory and
M-theory on a circle predicts one normalizable state.
• For other G, the predictions are not yet as clear, but we hope
that our methods can improve this situation.
• Theories with less supersymmetry should have no
normalizable states.
Toroidal compactification
Yang-Mills theories with gauge groups G and LG are
indistinguishable in Minkowski space.
We will instead consider the theory on a spatial three-torus T3
with supersymmetry preserving boundary conditions.
This can also be regarded as the six-dimensional (2,0)-theory
on a spatial five-torus T5=T2xT3.
Hopefully, our results can shed some light on the latter theory.
x3
i/g2
x2
x1
T3
/2
T2
1
Some algebraic topology
The group Gadj=SU(n)/Zn has 1(Gadj)Zn and 3(Gadj) Z.
We will use two important consequences of this (for n=pq):
A principal Gadj bundle over T3 is classified by
m  M = H2(T3, 1(Gadj)) .
”Discrete abelian magnetic ’t Hooft flux”.
pm = 0 iff the bundle can be lifted to an SU(n)/Cp bundle.
There is a group  = Hom(1(T3), 1(Gadj)) of ”large” gauge
transformations restricted to closed curves in T3 .
Physical states are characterized by
e  E = Hom(, U(1)) .
”Discrete abelian electric ’t Hooft flux.”
qe = 0 for states in SU(n)/Cp gauge theory.
S-duality again
The groups
M = H2(T3, 1(Gadj)) and E = Hom(, U(1))
are canonically isomorphic
(and non-canonically isomorphic to (1(G))3 = (Zn)3 ).
S-duality between the theories with gauge groups
G = SU(n)/Zp and LG = SU(n)/Zq,
amounts to ”electric-magnetic duality” (m, e)  (e, -m).
We will check this by computing spectrum of low-energy
states.
Low energy physics
To investigate the spectrum of low-energy states, we will
consider the conditions for different contributions to the energy
to (almost) vanish.
The magnetic contribution Tr(FijFij) requires the spatial
components of the field strength to vanish. So the wave
function is supported on the moduli space M of flat
connections over T3.
A flat connection is determined by its holonomies
U1,2,3 = Pexp  A along three independent one-cycles of T3.
The holonomies commute as elements of Gadj = SU(n)/Zn,
but if they are lifted to SU(n) they obey relations like
U1 U2 U1-1 U2-1 = m12  Zn = H2 (T2, 1 (Gadj)).
Here m12 is the restriction to a two-torus in the 12-plane of
m = (m23, m31, m12)  M = H2 (T3, 1 (Gadj)) .
At a generic point in M, the gauge group is broken to U(1)r for
some r.
Only abelian degrees of freedom need then be considered at
low energies.
The electric contribution Tr(F0iF0i) requires the canonical
conjugates F0i of the holonomies Ui to vanish.
So the wave function of a low-energy state is locally constant
on the moduli space M of flat connections.
But the wave function might take different values on different
components of M.
And these components might be permuted by large gauge
transformations in  = Hom(1(T3), 1(Gadj)) .
So in this way may construct states with non-trivial values of
e  E = Hom(, U(1)) .
Sofar, we have only considered the contribution from the
gauge field A.
The scalar field contribution to the energy is proportional to
Tr(AA), where A are the canonical conjugates of the scalar
fields A in the 6 of the SU(4)R R-symmetry .
So in the U(1)r theory, we get a 6r-dimensional continuum of
”plane-wave” states |A> of arbitrarily low energies.
Quantization of the spinor zero modes a and their conjugates
gives a further 28r degeneracy of the low-energy states.
Finally, we have to project onto the part of the spectrum that is
invariant under the Weyl group.
This description of the low-energy physics breaks down at
singular loci of the moduli space M, where the unbroken
subgroup contains non-abelian factors SU(n1) x … x SU(nk).
We rescale the corresponding gauge field components as
Ai  Ai’ = g-1 Ai. The Ai’ are canonically normalized and
periodic with period ~g-1. So in the weak-coupling limit g  0,
they become non-periodic like the scalar fields A. The lowenergy theory is then given by supersymmetric matrix
quantum mechanics with 16 supercharges.
This theory has a single normalizable ground state and a
continuum of low-energy states. The wave function of the
normalizable state is localized in a region of size g around the
singular locus of M. The continuum states match onto the
spectrum of continuum states supported on all of M.
A concrete example
We consider the G = SU(n)/Cp theory, where n = pq.
Given m = (m23, m31, m12)Zn)3  M, we define
u = GCD(m23, m31, m12, n)  q Z.
The general form of the holonomies U1,2,3 is then
U1,2,3 = M1,2,3  V1,2,3 ,
where V1,2,3 are certain SU(n/u) matrices such that
V1V2V1-1V2-1 = m12 etcetera,
and M1,2,3 are arbitrary commuting SU(u) matrices.
To construct states with a non-trivial value of
e = (e1, e2,e3) Zn)3  E , we define
v = GCD(e1, e2,e3, n)  p Z.
and take M1,2,3 of the form M1,2,3 = 1n/v  N1,2,3 ,
where N1,2,3 are arbitrary commuting U(uv/n) matrices with
(det N1,2,3)n/v = 1.
So we have U1,2,3 = 1n/v  N1,2,3  V1,2,3
The moduli spaces of the matrices N1,2,3 have n/v components.
These are permuted by large gauge transformations.
By taking linear combinations of locally constant wave
functions supported on these components, we can construct
states with arbitrary e = (e1, e2, e3)  (vZn)3 .
The moduli space of the N1,2,3 contains subspaces at which
the unbroken gauge group is of the form
SU(n/v t1) x … x SU(n/v tk) x U(1)k-1 where t1+ … + tk = uv/n.
Low-energy states with wave functions supported on these
loci fall in 6(k-1)-dimensional continua with a further 28(k-1) fold
degeneracy.
Magnetic and electric quantum numbers appear rather
differently in these considerations, but the final result is indeed
invariant under S-duality acting as (m, e)  (e, -m).
The three conjectures again
I.
Our results support the hypothesis of S-duality between
N = 4 Yang-Mills theories with gauge groups SU(n)/Zp
and SU(n)/Zq theories for n = pq.
II. We have used the supposed invariance of the spectrum
of (un)-normalizable low-energy states under a smooth
deformation from weak coupling (where we explicitly
constructed the states) to strong coupling.
III. Localization of the wave function to submanifolds of the
moduli space of flat connections is crucial. This is due
to the presence of normalizable states in matrix
quantum mechanics with 16 supercharges.
But it remains to use these results to learn more about the
structure of (2, 0 ) theory in 5+1 dimensions!
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The End