Abstract:
We construct a new and efficient cluster algorithm for updating strongly coupled U(N) lattice gauge theories with staggered fermions in the chiral limit. The algorithm uses the constrained monomer-dimer representation of the theory and should also be of interest to researchers working on other models with similar constraints. Using the new algorithm we address questions related to the chiral limit of strongly coupled U(N) gauge theories beyond the mean field approximation. We show that the infinite volume chiral condensate is non-zero in three and four dimensions. However, on a square lattice of size $L$ we find $\sum_x < \bar\psi\psi(x) \bar\psi\psi(0) > \sim L^{2-\eta}$ for large $L$ where $\eta = 0.420(3)/N + 0.078(4)/N^2$. These results differ from an earlier conclusion obtained using a different algorithm. Here we argue that the earlier calculations were misleading due to uncontrolled autocorrelation times encountered by the previous algorithm.

Abstract:
We consider lattice gauge theories at strong coupling with gauge group $U(N_C)$, or $SU(N_C)$ restricted to the meson sector, and coupled to $N_F$ flavors of fundamental representation staggered fermions. We study the formation of a chiral condensate by means of resummation of a hopping expansion. Different classes of graphs become dominant as the parameter $(N_F/N_C)$ is varied. By performing graph resummation we obtain an equation for determining the condensate as a function of $(N_F/N_C)$ and mass $m$. For values of $(N_F/N_C)$ below a critical value one reproduces the well-known result of the existence of a non-vanishing condensate solution in the $m=0$ limit. Above the critical $(N_F/N_C)$ value, however, no such solution exists, its abrupt disappearance indicating a first order transition to a chirally symmetric phase with composite (colorless) excitation spectrum.

Abstract:
We analyse - within the hamiltonian formalism with staggered fermions - the patterns of chiral symmetry breaking for the strongly coupled Schwinger and $U({\cal N}_c)$-color `t Hooft models with one and two flavor of fermions. Using the correspondence between these strongly coupled gauge models and antiferromagnetic spin chains, we provide a rather intuitive picture of their ground states, elucidate their patterns of chiral symmetry breaking, and compute the pertinent chiral condensates. Our analysis evidences an intriguing relationship between the values of the lattice chiral condensates of the `t Hooft and Schwinger models with one flavor of fermions.

Abstract:
We show that at sufficiently large chemical potential SU(N) lattice gauge theories in the strong coupling limit with staggered fermions are in a chirally symmetric phase. The proof employs a polymer cluster expansion which exploits the anisotropy between timelike and spacelike directions in the presence of a quark chemical potential $\mu$. The expansion is shown to converge in the infinite volume limit at any temperature for sufficiently large $\mu$. All expectations of chirally non-invariant local fermion operators vanish identically, or, equivalently, their correlations cluster exponentially, within the expansion. The expansion itself may serve as a computational tool at large $\mu$ and strong coupling.

Abstract:
This note is presenting the generating functions which count the BPS operators in the chiral ring of a N=2 quiver gauge theory that lives on N D3 branes probing an ALE singularity. The difficulty in this computation arises from the fact that this quiver gauge theory has a moduli space of vacua that splits into many branches -- the Higgs, the Coulomb and mixed branches. As a result there can be operators which explore those different branches and the counting gets complicated by having to deal with such operators while avoiding over or under counting. The solution to this problem turns out to be very elegant and is presented in this note. Some surprises with "surgery" of generating functions arises.

Abstract:
We present a formulation of chiral gauge theories, which admits more general spectra of Dirac operators and reveals considerably more possibilities for the structure of the chiral projections. Our two forms of correlation functions both also apply in the presence of zero modes and for any value of the index. The decomposition of the total set of pairs of bases into equivalence classes is carefully analyzed. Transformation properties are derived.

Abstract:
We present a general formulation of chiral gauge theories, which admits Dirac operators with more general spectra, reveals considerably more possibilities for the structure of the chiral projections, and nevertheless allows appropriate realizations. In our analyses we use two forms of the correlation functions which both also apply in the presence of zero modes and for any value of the index. To account properly for the conditions on the bases the concept of equivalence classes of pairs of them is introduced. The behaviors under gauge transformations and under CP transformations are unambiguously derived.

Abstract:
Recently, a new constraint on the structure of a wide class of strongly coupled field theories has been proposed. It takes the form of an inequality limiting the number of degrees of freedom in the infrared description of a theory to be no larger than the number of underlying, ultraviolet degrees of freedom. Here we apply this inequality to chiral gauge theories. For some models we find that it is always satisfied, while for others we find that the assumption of the validity of the inequality implies a strong additional restriction on the spectrum of massless composite particles.

Abstract:
The possibility of Lorentz symmetry breaking (LSB) has attracted considerable attention in recent years. Spontaneous LSB, in particular, offers the attractive prospect of the graviton as a Nambu-Golstone boson. Here we consider the question of spontaneous LSB in lattice gauge theories via formation of fermion condensates in the strong coupling and large N limits. We employ naive massless fermions in a fermionic hopping expansion in the presence of sources coupled to various condensate operators of interest. The expansion is summed in the large N limit in two equivalent ways: (i) direct summation of all leading N graphs; and (ii) construction of the corresponding large N effective action for composite operators. When sources are turned off a variety of fermionic condensates is found to persist. These include the chiral symmetry breaking condensates, thus recovering previous results, but also some LSB condensates, in particular, axial vector and rank-2 tensor condensates. Furthermore, in the presence of internal (global) symmetry groups, formation of condensates "locking" internal and external (Lorentz subgroup) symmetries is found to also become possible. Some implications and open questions are briefly discussed.