Energy-momentum Tensors in Matrix Theory and in Noncommutative Gauge Theories

Noncommutative gauge theories can be realized by considering branes in string theory with a strong NS-NS two-form field [1]–[11]. In our previous paper [12], we derived the energy-momentum tensors of these theories by computing disk amplitudes with one closed string and an arbitrary number of open strings and by taking the Seiberg-Witten limit [10] of these amplitudes. We found that the energymomentum tensors involve the open Wilson lines [13]–[18]. However, they do not reduce to the ones in the commutative theories in the limit θ → 0, where θ is the noncommutative parameter. This is because the Seiberg-Witten limit does not commute with the commutative limit. We also found that the energy-momentum tensors are conserved in interesting ways. In particular, in theories derived from bosonic string, the energy-momentum tensors are kinematically conserved, namely, the conservation law holds identically for any field configuration irrespective of the equations of motion. It turns out that we can use the same method to derive the energy-momentum tensor of Matrix Theory [33]. If we perform the dimensional reduction of the noncommutative theories along their noncommutative directions, the Seiberg-Witten limit reduces to the DKPS limit [35] (or the Sen-Seiberg limit [36, 37] in the context of Matrix Theory [33, 38]). Therefore, the energy-momentum tensor of Matrix Theory can be derived as a special case of the computation in [12] where there are no noncommutative directions. In [39, 40], the energy-momentum tensor of Matrix Theory was deduced from computations of one-loop amplitudes in Matrix Theory and from their comparison with graviton exchange amplitudes. The energy-momentum tensor we derive from the disk amplitudes perfectly agrees with that obtained in [39, 40] including the structure of the higher moments.