Large N volume independence

Volume independence of non-Abelian gauge theories is expected to provide both analytical insight into their non-perturbative aspects and also constitute a cost-effective numerical approach for extracting properties of infinite volume theories from lattice simulations using only one or a few lattice sites. Two viable schemes in which one can construct a fully reduced theory are known. These involve the addition of either center stabilizing deformations to pure YM theory, or light adjoint fermions endowed with periodic boundary conditions. When combined with large N orientifold equivalence, which relates QCD(adj) ( i.e., Yang-Mills with adjoint fermions) to QCD(AS) (Yang-Mills with antisymmetric representation fermions), the validity of large N volume independence in QCD(adj) has practical import for the understanding of real QCD at N=3.

Non-perturbative large-N equivalences

The fact that the Z_N center symmetry does not break in the continuum limit of massless QCD(adj) when compactified on R³ × S¹, no matter how small the R¹S¹ radius, is unambiguous. During the workshop, concerns were raised that in lattice formulations, this does not necessarily imply that the one-site reduced model for lattice regularized QCD(adj) will have unbroken center symmetry. A practical question which is currently being studied is the determination of the center symmetry realization in specific lattice regularized theories. Are there lattice artifacts that may lead to center symmetry broken phases? How fine a discretization is needed to have unbroken center symmetry (upon which large N volume independence relies)? In N  = 1 SYM theory [which is N_f=1 QCD(adj)], our analytic understanding of the presence of unbroken center symmetry crucially relies on supersymmetry, which is broken in conventional lattice formulations. In the continuum, the preservation of center symmetry is also related to the presence of discrete chiral symmetry. For N_f=2 QCD(adj) theory, taking the strict chiral limit on the lattice may not be necessary, and a few groups are exploring this issue.

Instead of relying on dynamical fermions to obtain valid fully reduced models, one can also add explicit center stabilizing terms to the action. However, for a d-dimensional compactification, the number of required stabilizing terms grows as N^d, making simulations of the reduced model rather costly. It is not currently know if this construction is preferable over conventional lattice formulations requiring sufficiently large volumes.

Exploration of strongly-coupled models of new physics

During the workshop, exploratory lattice studies to determine the conformal window of QCD-like gauge theories were discussed. These studies are particularly important if a strongly coupled Higgs-sector is found at the LHC. Walking technicolor is known to present a plausible solution to the electro-weak symmetry breaking problem. However, the determination of the class of QCD-like theories where this scenario can work requires non-perturbative study. QCD-like theories with 2-index symmetric representation and adjoint representation fields are likely candidates, and initial lattice results for such theories were discussed. Although not fully conclusive, largely due to the fact that the massless limit is numerically difficult to reach, there is some useful lattice data indicating conformal behavior in a few examples. We need new analytical and numerical input to fully understand this class of generalized QCD-like theories as a function of N_c and N_f.

New analytic methods to study QCD-like gauge theories on R³ × S¹

This part of the workshop highlighted two areas of progress. One is that phase transitions associated with changes in realization of center symmetry - analogs of confinement-deconfinement transitions - can be accessible using weak coupling techniques when a QCD-like theory has at least one compactified direction. With double-trace deformations of the type mentioned above, the confinement-deconfinement transition can be pushed to small radius and can be examined both analytically and numerically. This is the first time that the transitions of this type can be analyzed analytically, in a regime where one has full control over the (continuum) theory.

It was also pointed that S¹ compactified small-radius center stabilized theories with only discrete global symmetries should be continuously connected to the large radius regime, whereas theories with continuous global symmetries should have a single chiral transition which is not associated with any change in center symmetry. Verifying these expectations with numerical studies is highly desirable.

An index theorem for topological excitations on R³ × S¹, new topological excitations such as magnetic bions and magnetic quintets responsible for confinement in various vector-like and chiral theories, and duality in gauge theories, were discussed in detail. Since the regime where these excitations are visible is a semi-classical window in which analytical methods are reliable, it is also desirable to check predictions of semi-classical analysis using lattice methods which can probe portions of the phase diagram at larger radius.

Strongly coupled chiral gauge theories

One question of importance is whether non-perturbative techniques on R³ × S¹ can be usefully applied to pin down the conformal window of asymptotically free theories, or other scenarios motivated by walking technicolor. We also expect the R³ × S¹ index theorem and fuller understanding of non-perturbative excitations to be useful in the context of supersymmetric theories, particularly for resolving some puzzles concerning dynamical supersymmetry breaking which rely on assumptions about confinement.