Complete spectrum of long operators in Script N = 4 SYM at one loop

Abstract

We construct the complete spectral curve for an arbitrary local operator, including fermions and covariant derivatives, of one-loop N = 4 gauge theory in the thermodynamic limit. This curve perfectly reproduces the Frolov-Tseytlin limit of the full spectral curve of classical strings on AdS 5 × S 5 derived in hep-th/0502226. To complete the comparison we introduce stacks, novel bound states of roots of different flavors which arise in the thermodynamic limit of the corresponding Bethe ansatz equations. We furthermore show the equivalence of various types of Bethe equations for the underlying su(2, 2|4) superalgebra, in particular of the type "Beauty" and "Beast".Non-abelian gauge theories discovered more than 40 years ago still do not have a satisfactory quantitative description at the intermediate and strong coupling regimes, in spite of their great importance for fundamental physics and substantial efforts of many theoretical and computational physicists and mathematicians. The most important of them, QCD and the standard model, are well studied perturbatively and to some extent we know the qualitative picture of strong coupling phenomena, such as confinement or instanton-induced processes. Nowadays lattice calculations provide some reasonable quantitative data confirming the qualitative picture, but they cannot replace systematic analytical methods, still absent.A new boost to the study of these questions was given by supersymmetry: Supersymmetric Yang-Mills (SYM) theories appeared to have special BPS sectors where certain physical quantities, protected from renormalization by supersymmetry, can be computed exactly. Outstanding examples are the Seiberg-Witten low-energy effective action in N = 2 SYM and the Dijkgraaf-Vafa effective potential in N = 1 SYM. This shed a good deal of light on the role of confinement and of the Higgs mechanism in strongly coupled supersymmetric gauge theories. Still, the BPS sector is only a tiny fraction of the theory, the rest being as difficult to access as in non-supersymmetric theories.New hope came from an issue which was not expected to play any role in interacting four-dimensional theories: Quantum integrability, discovered in a pioneering work of Hans Bethe [1] in relation to the Heisenberg XXX chain, usually applicable exclusively to exact solutions in (1+1)-dimensional theories, made its breakthrough into fourdimensional large-N c gauge theories. Here, the simplifications of large-N c lead to an essentially two-dimensional description of parts of the theory bypassing a no-go theorem for four dimensions. The first traces of integrability were observed and proved by Lipatov for reggeized gluons in QCD [2]. In [3] the equivalence of the reggeon Hamiltonian and the Hamiltonian of the Heisenberg XXX 0 spin chain was shown. This result was independently derived by Faddeev and Korchemsky in [4] where also the Bethe and Baxter equations for this model are analyzed.The observations of integrability look especially interesting and hopeful in the maximally supersymmetric gauge the...

Double scaling and finite size corrections in spin chain

“…The appropriate thermodynamical limit in terms of non-hyperelliptic curve, corresponding to very long operators in SYM was constructed in particular sectors in [4,7,20,37] and for the full psu(2, 2|4) SYM chain in [11]. Finding the finite size corrections to these solutions might be extremely useful for getting some clues for the quantization of the string sigma model on the other side of AdS/CFT correspondence.…”

Integrability, spin-chains and the AdS3/CFT2 correspondence

“…The procedure for fermionic duality on Bethe equations used in this section closely follows the general prescription discussed in [72][73][74][75]. The case at hand is very similar to the osp(6|4) spin-chain studied in [49,57].…”

A large spin limit of strings on in a non-compact sector

“…This suggests another interesting window to explore the AdS/CFT correspondence. Our analysis may be extended to other non-compact sectors along the line of [13][14][15][16][17].…”

“…In particular, in that (Frolov-Tseytlin) limit [4,9], the scaling dimension/energy universally exhibits the BMN scaling [11]. A systematic way to show the matching between the two sides is to compare the algebraic curves and the differentials thereon which are associated with the integrability [8,10,[12][13][14][15][16][17].…”