Using the character expansion method, we generalize several wellknown integrals over the unitary group to the case where general complex matrices appear in the integrand. These integrals are of interest in the theory of random matrices and may also find applications in lattice gauge theory.

Quantum spin and quantum link models provide an unconventional regularization of field theory in which classical fields arise via dimensional reduction of discrete variables. This D-theory regularization leads to the same continuum theories as the conventional approach. We show this by deriving the low-energy effective Lagrangians of D-theory models using coherent state path integral techniques. We illustrate our method for the

We extend Zirnbauer's color-flavor transformation in the fermionic sector to the case of the special unitary group. The transformation allows a certain integral over SU(N c ) color matrices to be transformed into an integral over flavor matrices which parameterize the coset space U(2N f )/U(N f )×U(N f ). Integrals of the type considered appear, for example, in the partition function of lattice gauge theory.

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The phase diagrams of cuprate superconductors and of QCD at non-zero baryon chemical potential are qualitatively similar. The Néel phase of the cuprates corresponds to the chirally broken phase of QCD, and the high-temperature superconducting phase corresponds to the color superconducting phase. In the SO(5) theory for the cuprates the SO(3)s spin rotational symmetry and the U (1)em gauge symmetry of electromagnetism are dynamically unified. This suggests that the SU (2)L ⊗ SU (2)R ⊗ U (1)B chiral symmetry of QCD and the SU (3)c color gauge symmetry may get unified to SO(10). Dynamical enhancement of symmetry from SO(2)s ⊗ Z Z(2) to SO(3)s is known to occur in anisotropic antiferromagnets. In these systems the staggered magnetization flops from an easy 3-axis into the 12-plane at a critical value of the external magnetic field. Similarly, the phase transitions in the SO(5) and SO(10) models are flop transitions of a "superspin". Despite this fact, a renormalization group flow analysis in 4 − ǫ dimensions indicates that a point with full SO(5) or SO(10) symmetry exists neither in the cuprates nor in QCD.Understanding QCD at non-zero baryon chemical potential µ is very important in the context of both heavy ion and neutron star physics. While asymptotically large values of µ are accessible in perturbative QCD calculations, such values are not realized in actual physical systems. Investigations of the phenomenologically relevant regime at intermediate µ require the use of nonperturbative methods. Unfortunately, first-principles lattice calculations in this regime are presently prevented by the notorious complex action problem. Conjectures for the QCD phase diagram at non-zero µ are thus based on model calculations. These calculations reveal interesting phenomena such as color superconductivity [1] but one cannot expect the results to be quantitatively correct.While it is very important to develop quantitative methods to understand QCD at nonzero µ, here we ask if further qualitative insight can be gained through analogies with related condensed matter systems. In particular, the phase diagram of high-temperature cuprate superconductors is qualitatively similar to the one conjectured for two flavor QCD. The ordi-nary hadronic phase of QCD at small µ in which the chiral SO(4) = SU (2) L ⊗ SU (2) R symmetry is spontaneously broken down to SO(3) = SU (2) L=R corresponds to the antiferromagnetic Néel phase of the undoped cuprates in which the SO(3) s spin rotational symmetry is broken down to SO(2) s due to the spontaneous generation of a staggered magnetization. The hightemperature superconducting phase of the doped cuprates with spontaneous U (1) em breaking corresponds to the color superconducting phase of two flavor QCD in which the SU (3) c gauge symmetry is expected to break down to SU (2) c . Finally, the quark-gluon plasma corresponds to the high-temperature metallic phase of the cuprates.QCD in the color superconducting phase is a genuine high-temperature superconductor. The mechanism that leads to quark Cooper ...