Semiclassical Strings on AdS_5 x S^5/Z_M and Operators in Orbifold Field Theories

Abstract

We show agreements, at one-loop level of field theory, between energies of semiclassical string states on AdS_5 x S^5/Z_M and anomalous dimensions of operators in N=0,1,2 orbifold field theories originating from N=4 SYM. On field theory side, one-loop anomalous dimension matrices can be regarded as Hamiltonians of spin chains with twisted boundary conditions. These are solvable by Bethe ansatz. On string side, twisted sectors emerge and we obtain some string configurations in twisted sectors. In SU(2) subsectors, we compare anomalous dimensions with string energies and see agreements. We also see agreements between sigma models of both sides in SU(2) and SU(3) subsectors.

Figures (4)

• Figure 1: Interaction between the $L$-th site and the first ($(L+1)$-th) site. Each $\phi$ is $X$ or $Y^{\dagger}$ and $i$, $i+1$ and $i+2$ represent the gauge groups.
• Figure 2: The interactions which generate the phase shift at the end (the beginning) of the chain. $A$ and $B$ are fields consisting of operators. $A$ is $(i, \overline{i+p})$-type and $B$ is $(i, \overline{i+q})$-type operator. In the $SU(2)_L$ case, $A=X,\ B=Y$ and $p=1, q=-1$.
• Figure 3: Diagrams which contribute to $Z$-factor.
• Figure 4: Diagram from yukawa coupling $\text{tr} X^\dagger\chi\psi$. A diagram (A) consist of $NM\times NM$ matrix valued fields \ref{['scalar']} in orbifold theory. $\chi$ is gaugino and $\psi$ is ${\cal N}=1$ superpartner of $X$. This diagram (A) includes only one diagram (B) which is concerned with a component field $X_{i,i+1}$, which is an $N\times N$ matrix. Suffix of field $X_{i,i+1}$, $\chi_{i,i}$, $\psi_{i,i+1}$ represent gauge group 'name'. A diagram (C) is a corresponding diagram of ${\cal N}=4$$U(N)$ SYM.