Softly broken MQCD and the theta angle

Abstract

We consider a family of non-supersymmetric MQCD five-brane configurations introduced by Witten, and discuss the dependence of the curves on the microscopic theta angle and its relation with CP. We find evidence for a non-trivial spectral flow of the curves (vacua) and for the level-crossing of adjacent curves at a particular value of the theta angle, with spontaneous breaking of CP symmetry, providing an MQCD analogue of the phase transitions in theta proposed by 't Hooft. November 1997The modelling of gauge dynamics via brane configurations of weakly coupled string theory [1,2,3,4], or M-theory [5,6,7,8,9], provides a geometrical interpretation of various field theoretical strong coupling phenomena. In some cases, the geometrical viewpoint can be used to discover new exact solutions [5] of N = 2 theories. In the context of N = 1 theories, it provides a semiclassical approach to such thorny problems as confinement and chiral symmetry breaking [6,7]. These constructions are similar in spirit to previous work on geometric engineering of gauge theories (see for example [10]). Here the role of a nontrivial string compactification is played by some complicated brane configuration sitting in flat space at small string coupling, and carefully adjusted to provide a weakly coupled supersymmetric Yang-Mills theory at intermediate scales. Since the gauge theory lives on the branes world-volume, one can relate many classical and semiclassical features of gauge theories to some properties of brane dynamics. The real power of the method arises when taking the strong coupling limit of the brane configuration. If a strong coupling dual is available, one can describe many non-perturbative, infrared properties of the gauge theory, just by reading the tree-level data of the dual brane configuration.In four-dimensional models one uses the duality between type IIA strings and Mtheory [5]. Here one maps the type IIA brane configuration to a single smooth five-brane, whose world-volume is appropriately embedded in the flat eleven-dimensional background, as a product M 4 × Σ, with M 4 the four-dimensional Minkowski space, and Σ a holomorphic curve with respect to a given complex structure of the background. In general, the detailed physics of the resulting M-theory model differs from the Yang-Mills theory we are interested in; however, to the extent that some observables are protected by supersymmetry, we can calculate them in the deformed strong coupling model. This is the case, for example, of all holomorphic quantities determined by the Seiberg-Witten curve in N = 2 models [5], including BPS spectra [11]. For N = 1 models the agreement is only qualitative in principle, but some observables can be accurately matched, like superpotentials and gaugino condensates [6,8].In the absence of some unbroken supersymmetry on the world-volume, we cannot use holomorphy to accurately match observables, and a working assumption must be made that no phase transitions occur in the way to strong coupling. Still, qualitative features based o...

Precision“measurement” of the potential in MQCD

“…Using the relation of the rotation parameter α to the gaugino mass given in [25,26] we get in our notations the plausible result m λ = tan α, and the pure YM theory is given by α = π/2 and µ = 1 . In this case the energy takes the form…”

“…Using the rotation suggested in [25,26] we keep p the same as for N = 1 and choose q = ζ(sin α, i sin α, cos α, −i cos α)…”

Branes and theta dependence

“…The effective low energy four dimensional gauge theory in x 0 , ..., x 3 described by this system is SU (N c ) Yang-Mills with softly broken supersymmetry [12,5]. The real parameter ǫ describes the soft breaking of supersymmetry: 3 for ǫ = 0 we have a holomorphic curve and a realization of N = 1 supersymmetric Yang-Mills [13,12] while for ǫ = 0 we get a soft breaking of supersymmetry by gaugino mass.…”

Kähler potential and higher derivative terms from M-theory fivebrane

“…Similar webs of intersecting branes of Type IIA string theory describing N = 1 gauge theories in four dimensions [7,8] can be realized by a single fivebrane of M theory wrapping a Riemann surface. The fivebrane configurations corresponding to these N = 1 supersymmetric gauge theories encode the information about the N = 1 moduli spaces of vacua [9][10][11][12][13][14][15][16][17][18][19][20] So far the fivebrane construction has been successfully applied to the computation of holomorphic (or rather BPS) quantities of the four dimensional supersymmetric gauge theory. Of particular importance are the non-holomorphic quantities such as higher derivative terms in N = 2 theories and the Kähler potential of N = 1 supersymmetric gauge theories.…”