N=4 SYM on R x S^3 and Theories with 16 Supercharges

TL;DR

The work presents a unified harmonic-expansion approach to ${\cal N}=4$ SYM on ${\mathbb R}\times S^3$ and its 16-supercharge truncations, producing a common KK framework that yields the plane wave matrix model, ${\cal N}=4$ SYM on ${\mathbb R}\times S^2$, and ${\cal N}=4$ SYM on ${\mathbb R}\times S^3/Z_k$. It shows that 1-loop corrections organize into an integrable ${\rm SO}(6)$ spin chain and demonstrates the stability of a time-dependent BPS solution—which corresponds to an ${\rm AdS}$ giant graviton—under quantum fluctuations in both the original and truncated theories. The results establish a concrete link between KK-truncated gauge theories and their holographic dual bubbling geometries, illustrating universal features of the gauge/gravity correspondence across a family of theories with 16 supercharges. This framework lays groundwork for exploring nontrivial vacua, thermodynamics, and time-dependent dynamics in truncated sectors while preserving key integrable structures.

Abstract

We study N=4 SYM on R x S^3 and theories with 16 supercharges arising as its consistent truncations. These theories include the plane wave matrix model, N=4 SYM on R x S^2 and N=4 SYM on R x S^3/Z_k, and their gravity duals were studied by Lin and Maldacena. We make a harmonic expansion of the original N=4 SYM on R x S^3 and obtain each of the truncated theories by keeping a part of the Kaluza-Klein modes. This enables us to analyze all the theories in a unified way. We explicitly construct some nontrivial vacua of N=4 SYM on R x S^2. We perform 1-loop analysis of the original and truncated theories. In particular, we examine states regarded as the integrable SO(6) spin chain and a time-dependent BPS solution, which is considered to correspond to the AdS giant graviton in the original theory.

Figures (12)

• Figure 1: The KK tower of $\mathcal{N}=4$ super Yang-Mills on $R \times S^3$. The first number, the second number and the third number in the parentheses represent $J$, $\tilde{J}$ and the dimension of the representation of $SU(4)$, respectively. The solid and dotted arrows represent the superconformal transformation in the free theory for the creation operator caused by $\eta_{m+}$ and $\eta_{m-}^*$, respectively.
• Figure 2: Diagrams for the one-loop self-energy of $X_{AB}$. The curly line represents the propagator of $A_i$. The wavy line represents the propagator of $A_0$. The solid line represents the propagator of $X_{AB}$. The dashed line represents the propagator of $\psi^A$.
• Figure 3: Region of the regularized summations over $J_1$ and $J_2$
• Figure 4: Diagram determining $\Phi_i$. The curly line represents the propagator of $A_i$. The dotted line represents the propagator of the ghost.
• Figure 5: BPS solution