Our understanding of M-theory is still very incomplete. But possibly, an important clue might be found in a celebrated piece of work that appeared in 1954… Even the interpretation of the name ”M” remains unclear.

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. 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.

An N=4 supersymmetric Yang-Mills theory is determined by its gauge group G. complex coupling  =  /2  +i/g 2. For n=pq, we will consider the case G=SU(n)/Z p so that L G=SU(n)/Z q., Work is in progress on the remaining (simply laced) Lie groups. 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 L G and coupling -1/ . S-duality

In a supersymmetric theory with a massgap, the Witten index Tr(-1) F e -  H gets contributions only from the (normalizable) zero- energy 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 0 BosonsFermions Invariance of the spectrum

Supersymmetric matrix quantum mechanics with 16 supercharges is obtained by dimensional reduction of ten- dimensional supersymmetric Yang-Mills theory. 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. The bosonic variables (in temporal gauge) are nine Lie algebra valued matrices. There is a quartic potential with flat valleys extending to infinity. Matrix quantum mechanics

Yang-Mills theories with gauge groups G and L G are indistinguishable in Minkowski space. We will instead consider the theory on a spatial three-torus T 3 with supersymmetry preserving boundary conditions. This can also be regarded as the six-dimensional (2,0)-theory on a spatial five-torus T 5 =T 2 xT 3. Hopefully, our results can shed some light on the latter theory. x1x1 x3x3 x2x2 1 i/g 2  /2  T3T3 T2T2 Toroidal compactification

The group G adj =SU(n)/Z n has  1 (G adj )  Z n and  3 (G adj )  Z. We will use two important consequences of this (for n=pq): A principal G adj bundle over T 3 is classified by m  M = H 2 (T 3,  1 (G adj )). ”Discrete abelian magnetic ’t Hooft flux”. pm = 0 iff the bundle can be lifted to an SU(n)/C p bundle. There is a group  = Hom(  1 (T 3 ),  1 (G adj )) of ”large” gauge transformations restricted to closed curves in T 3. Physical states are characterized by e  E = Hom( , U(1)). ”Discrete abelian electric ’t Hooft flux.” qe = 0 for states in SU(n)/C p gauge theory. Some algebraic topology

The groups M = H 2 (T 3,  1 (G adj )) and E = Hom( , U(1)) are canonically isomorphic (and non-canonically isomorphic to (  1 (G)) 3 = (Z n ) 3 ). S-duality between the theories with gauge groups G = SU(n)/Z p and L G = SU(n)/Z q, amounts to ”electric-magnetic duality” (m, e)  (e, -m). We will check this by computing spectrum of low-energy states. S-duality again

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(F ij F ij ) 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 T 3. A flat connection is determined by its holonomies U 1,2,3 = Pexp  A along three independent one-cycles of T 3. The holonomies commute as elements of G adj = SU(n)/Z n, but if they are lifted to SU(n) they obey relations like U 1 U 2 U 1 -1 U 2 -1 = m 12  Z n = H 2 (T 2,  1 (G adj )). Here m 12 is the restriction to a two-torus in the 12-plane of m = (m 23, m 31, m 12 )  M = H 2 (T 3,  1 (G adj )). Low energy physics

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(F 0i F 0i ) requires the canonical conjugates F 0i of the holonomies U i 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 (T 3 ),  1 (G adj )). So in this way may construct states with non-trivial values of e  E = Hom( , U(1)).

Quantization of the spinor zero modes   a and their conjugates gives a further 2 8r degeneracy of the low-energy states. 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. Sofar, we have only considered the contribution from the gauge field A. 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(n 1 ) x … x SU(n k ). We rescale the corresponding gauge field components as A i  A i ’ = g -1 A i. The A i ’ 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 low- energy 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.

We consider the G = SU(n)/C p theory, where n = pq. Given m = (m 23, m 31, m 12 )  Z n ) 3  M, we define  u = GCD(m 23, m 31, m 12, n)  q Z. The general form of the holonomies U 1,2,3 is then U 1,2,3 = M 1,2,3  V 1,2,3, where V 1,2,3 are certain SU(n/u) matrices such that V 1 V 2 V 1 -1 V 2 -1 = m 12 etcetera, and M 1,2,3 are arbitrary commuting SU(u) matrices. To construct states with a non-trivial value of e = (e 1, e 2,e 3 )  Z n ) 3  E, we define v = GCD(e 1, e 2,e 3, n)  p Z. and take M 1,2,3 of the form M 1,2,3 = 1 n/v  N 1,2,3, where N 1,2,3 are arbitrary commuting U(uv/n) matrices with (det N 1,2,3 ) n/v = 1. A concrete example

The moduli spaces of the matrices N 1,2,3 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 = (e 1, e 2, e 3 )  (vZ n ) 3. The moduli space of the N 1,2,3 contains subspaces at which the unbroken gauge group is of the form SU(n/v t 1 ) x … x SU(n/v t k ) x U(1) k-1 where t 1 + … + t k = uv/n. Low-energy states with wave functions supported on these loci fall in 6(k-1)-dimensional continua with a further 2 8(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). So we have U 1,2,3 = 1 n/v  N 1,2,3  V 1,2,3

The three conjectures again I.Our results support the hypothesis of S-duality between N = 4 Yang-Mills theories with gauge groups SU(n)/Z p and SU(n)/Z q 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!

The End