Nonrelativistic Corners of N=4 Supersymmetric Yang-Mills Theory

Abstract

We show that N = 4 super Yang-Mills (SYM) theory on R×S 3 with gauge group SU(N ) is described in a near-BPS limit by a simple lower-dimensional non-relativistic field theory with SU(1, 1) × U(1) invariant interactions. In this limit, a single complex adjoint scalar field survives, and part of its interaction is obtained by exactly integrating out the gauge boson of the SYM theory. Taking into account normal ordering, the interactions match the one-loop dilatation operator of the SU(1, 1) sector, establishing the consistency of the limit at the quantum level. We discover a tantalizing field-theoretic structure, corresponding to a (1 + 1)-dimensional complex chiral boson on a circle coupled to a non-dynamical gauge field, both in the adjoint representation of SU(N ). Our findings equivalently apply to other BPS bounds and point towards the existence of new non-relativistic field theories that correspond to non-relativistic corners of N = 4 super Yang-Mills theory, paving the way for a new approach to understand its strongly coupled finite-N dynamics.

Symmetry structure of the interactions in near-BPS corners of $$ \mathcal{N} $$ = 4 super-Yang-Mills

“…This is a typical feature of non-relativistic theories that goes also in hand with the U(1) global symmetry responsible for the conservation of particle number. The same phenomenon was observed in near-BPS limits with SU(1, 1) subgroup for the dynamical modes of the scalar fields [11,12].…”

“…We start in section 3.2 with the SU(1, 2) sector. Since the field content surviving the decoupling limit is entirely given by the gauge fields, it is an instructive example that shows one of the main novelties with respect to the SU(1, 1) limits [11,12]. Then we add scalars and fermions to the theory by considering the generalization to the SU(1, 2|2) case in section 3.3.…”

“…A novel approach to obtain an effective description of the degrees of freedom surviving the near-BPS limits in terms of the Hamiltonian of a lower-dimensional field theory was found in [11], and then pursued in [12] for all the cases containing SU(1, 1) as a subset of the spin group. The technique is the following: we put classical N = 4 SYM on R × S 3 and we take a decoupling limit by zooming in towards a BPS bound of the CFT.…”

Null reductions of the M5-brane

“…In contrast, the field theories we obtain do not have higher derivatives but involve Lagrange multiplier constraints that reduce the dynamics to motion on a moduli space of anti-selfdual gauge fields [9,10], in line with the DLCQ description of the M5-brane [11,12]. Other classes of theories without Lorentz invariance but related to String/M-Theory have recently received attention in works such as [13][14][15][16][17][18].…”

Nonrelativistic spinning strings

“…During last one decade, a series of work [1]- [7] have been put forward which essentially argues about a more tractable limit of the AdS 5 /CFT 4 correspondence [8]- [9] where both the descriptions are under control and hence the corresponding spectrum could be subjected to a precise test. Here, ∆ is the energy of a state in N = 4 SYM on R×S 3 and J is a linear sum over Cartan charges.…”