Confining Phase of N=1 Sp(Nc)Sp(N_c) Gauge Theories via M Theory Fivebrane

The moduli space of vacua for the confining phase of N=1 $Sp(N_c)$ supersymmetric gauge theories in four dimensions is studied by M theory fivebrane. We construct M theory fivebrane configuration corresponding to the perturbation of superpotential in which the power of adjoint field is related to the number of NS'5 branes in type IIA brane configuration. We interpret the dyon vacuum expectation values in field theory results as the brane geometry and the comparison with meson vevs will turn out that the low energy effective superpotential with enhanced gauge group SU(2) is exact.Comment: 14 pages, late

Geometry, D-Branes and N=1 Duality in Four Dimensions II

We study N=1 dualities in four dimensional supersymmetric gauge theories in terms of wrapping D 6-branes around 3-cycles of Calabi-Yau threefolds in type IIA string theory. We generalize the recent work of geometrical realization for the models which have the superpotential corresponding to an $A_k$ type singularity, to various models presented by Brodie and Strassler, consisting of $D_{k+2}$ superpotential of the form $W=Tr X^{k+1} + Tr XY^2$. We discuss a large number of representations for the field $Y$, but with $X$ always in the adjoint (symmetric) [antisymmetric] representation for $SU (SO) [Sp]$ gauge groups.Comment: 15 pages, late

c=5/2 Free Fermion Model of WB_{2} Algebra

We investigate the explicit construction of the $WB_{2}$ algebra, which is closed and associative for all values of the central charge $c$, using the Jacobi identity and show the agreement with the results studied previously. Then we illustrate a realization of $c=\frac{5}{2}$ free fermion model, which is $m \rightarrow \infty$ limit of unitary minimal series, $c ( WB_{2} )=\frac{5}{2} (1-\frac{12}{ (m+3)(m+4) })$ based on the cosets $( \hat{B_{2}} \oplus \hat{B_{2}}, \hat{B_{2} })$ at level $(1,m).$ We confirm by explicit computations that the bosonic currents in the $WB_{2}$ algebra are indeed given by the Casimir operators of $\hat{B_{2}}$ .Comment: 16 page

Higher Spin Currents in the Orthogonal Coset Theory

In the coset model $(D_N^{(1)} \oplus D_N^{(1)},D_N^{(1)})$ at levels $(k_1,k_2)$, the higher spin $4$ current that contains the quartic WZW currents contracted with completely symmetric $SO(2N)$ invariant $d$ tensor of rank $4$ is obtained. The three-point functions with two scalars are obtained for any finite $N$ and $k_2$ with $k_1=1$. They are determined also in the large $N$ 't Hooft limit. When one of the levels is the dual Coxeter number of $SO(2N)$, $k_1=2N-2$, the higher spin $\frac{7}{2}$ current, which contains the septic adjoint fermions contracted with the above $d$ tensor and the triple product of structure constants, is obtained from the operator product expansion (OPE) between the spin $\frac{3}{2}$ current living in the ${\cal N}=1$ superconformal algebra and the above higher spin $4$ current. The OPEs between the higher spin $\frac{7}{2}, 4$ currents are described. For $k_1=k_2=2N-2$ where both levels are equal to the dual Coxeter number of $SO(2N)$, the higher spin $3$ current of $U(1)$ charge $\frac{4}{3}$, which contains the six product of spin $\frac{1}{2}$ (two) adjoint fermions contracted with the product of $d$ tensor and two structure constants, is obtained. The corresponding ${\cal N}=2$ higher spin multiplet is determined by calculating the remaining higher spin $\frac{7}{2}, \frac{7}{2}, 4$ currents with the help of two spin $\frac{3}{2}$ currents in the ${\cal N}=2$ superconformal algebra. The other ${\cal N}=2$ higher spin multiplet, whose $U(1)$ charge is opposite to the one of above ${\cal N}=2$ higher spin multiplet, is obtained. The OPE between these two ${\cal N}=2$ higher spin mutiplets is also discussed.Comment: 96 page