Long-range Bethe ansätze for gauge theory and strings

Abstract

We generalize various existing higher-loop Bethe ansätze for simple sectors of the integrable long-range dynamic spin chain describing planar N = 4 Super Yang-Mills Theory to the full psu(2, 2|4) symmetry and, asymptotically, to arbitrary loop order. We perform a large number of tests of our conjectured equations, such as internal consistency, comparison to direct three-loop diagonalization and expected thermodynamic behavior. In the special case of the su(1|2) subsector, corresponding to a long-range t-J model, we are able to derive, up to three loops, the Smatrix and the associated nested Bethe ansatz from the gauge theory dilatation operator. We conjecture novel all-order S-matrices for the su(1|2) and su(1, 1|2) subsectors, and show that they satisfy the Yang-Baxter equation. Throughout the paper, we muse about the idea that quantum string theory on AdS 5 × S 5 is also described by a psu(2, 2|4) spin chain. We propose asymptotic all-order Bethe equations for this putative "string chain", which differ in a systematic fashion from the gauge theory equations.Recently a powerful new tool for the study of planar non-abelian gauge theories and strings on curved space-times, as well as the conjectured dualities linking the two, has become available. Integrability has made its appearance in N = 4 Super Yang-Mills theory and in IIB string theory on the AdS 5 ×S 5 background. It is beginning to shed entirely new light on the AdS/CFT duality. Proving or disproving part of the gauge/string correspondence suddenly seems to be within reach. The central new tool is a technique widely known as the Bethe ansatz. It dates back to the year 1931 when Hans Bethe solved the Heisenberg spin chain in his pioneering work [1]. Its impact on condensed matter theory and mathematical physics cannot be underestimated.The first, crucial observation in the context of the gauge/string duality was made by Minahan and Zarembo [2]. They noticed that the conformal quantum operators in the scalar field sector of N = 4 gauge theory are, at the planar one-loop level, in one-to-one correspondence with the translationally invariant eigenstates of an integrable so(6) magnetic quantum spin chain. The spin chain Hamiltonian corresponds to the gauge theoretic planar one-loop dilatation operator, whose "energy" eigenvalues yield the scaling weights of the conformal operators. This observation turned out to be a first hint at a very deep structure. The result generalizes to all local operators of the planar one-loop N = 4 theory [3]. What is more, evidence was found that integrability extends beyond the one-loop approximation [4].First indications that planar gauge theories may contain hidden integrable structures were discovered in a QCD context in seminal work by Lipatov [5]. References to further interesting work on integrability in QCD may be found in [6]. New aspects of the more recent developments [2][3][4] when comparing to these important earlier insights are that (i) the integrability links space-time to internal symmetries, (ii) the studied spin ...

Universality of three gaugino anomalous dimensions in Script N = 4 SYM

“…This kind of phenomena has been first discussed in [4]. Here, we have given a simple explicit example.…”

“…A nested Bethe Ansatz valid in this sector is described in [25], according to the methods of [4,16]. Up to three loops, the Bethe Ansatz equations read…”

“…However, it seems that the psu´¾ ¾ µ symmetry still has to be fully exploited. This is clear in the deep discussion of [4] about degeneracies in the spectum of anomalous dimensions. The Bethe Ansatz equations have remarkable structural properties related to supersymmetry.…”

“…In particular, one can consider the maximally supersymmetric AE super Yang-Mills theory which is UV finite and superconformally invariant at the quantum level. There, asymptotic all-loop Bethe Ansatz equations are available for the full set of psu´¾ ¾ µ operators [4].…”

“…There, all conformal operators fall in a single supermultiplet [19,20,21] and the anomalous dimension of different channels are related by supersymmetry and can be expressed in terms of a single universal function ÙÒ Ú . The methods of [4] suggest instead that one can expect hidden relations between operators with different twists.…”