Integrability and the conformal field theory of the Higgs branch

Abstract

In the context of the AdS 3 /CFT 2 correspondence, we investigate the Higgs branch CFT 2 . Witten showed that states localised near the small instanton singularity can be described in terms of vector multiplet variables. This theory has a planar, weakcoupling limit, in which anomalous dimensions of single-trace composite operators can be calculated. At one loop, the calculation reduces to finding the spectrum of a spin-chain with nearest-neighbour interactions. This CFT 2 spin-chain matches precisely the one that was previously found as the weak-coupling limit of the integrable system describing the AdS 3 side of the duality. We compute the one-loop dilatation operator in a non-trivial compact subsector and show that it corresponds to an integrable spin-chain Hamiltonian. This provides the first direct evidence of integrability on the CFT 2 side of the correspondence.

The AdS3× S3× S3× S1worldsheetSmatrix

“…Recently, some signs of integrability on the CFT side have also been identified [100] in the CFT dual to strings on AdS 3 × S 3 × T 4 . Much less is known about the CFT 2 dual of strings on AdS 3 × S 3 × S 3 × S 1 [51,[101][102][103].…”

Closed strings and moduli in AdS3/CFT2

“…Nonetheless, as the recent findings for the WZW theory at k = 1 suggest [36][37][38], the exact relationship to the Sym N orbifold most likely needs to be revisited, and the H (0,0) sector contains a wealth of non-protected information with which to test any such conjectures. It seems plausible that the more complete holographic dual will have to incorporate aspects of an effective Higgs branch CFT [7], where integrability has been found [65].…”

Integrable open spin-chains inAdS3/CFT2correspondences

“…Let us now consider two left-moving magnon excitations |ϕ r p ϕ s q of momenta p and q. The non-trivial action of the fermionic generators of the su(1|1) 2 c algebra is 16) and similarly for primed left-moving magnon excitations. The action on two right-moving magnon excitations |φ r pφ s q , or a left-and a right-mover, |ϕ r pφ s q or |φ r p ϕ s q , is obtained by interchanging L ↔ R indices and using barred notation for the action on the right-movers.…”

“…We will explain this argument in detail in subsequent sections. 16 This composition is well-defined, since |0 B is a state in a one-dimensional vector space ( ∼ = C) and Wp ⊗ C ∼ = Wp. 17 A boundary state is created when the boundary absorbs magnon excitations via the so-called boundary bootstrap procedure (see, for example, Section 3 in [68] …”

“…The dual field theories have been the subject of recent interest [14][15][16][17], but much still remains to be understood. On the string theory side, the IIB supergravity backgrounds AdS 3 × S 3 × M 4 , with M 4 = S 3 × S 1 or T 4 , are known [18,19].…”