Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Integrability and the Conformal Field Theory of the Higgs branch

Olof Ohlsson Sax, Alessandro Sfondrini, Bogdan Stefanski

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, weak-coupling 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.

Integrability and the Conformal Field Theory of the Higgs branch

Abstract

In the context of the AdS/CFT correspondence, we investigate the Higgs branch CFT. Witten showed that states localised near the small instanton singularity can be described in terms of vector multiplet variables. This theory has a planar, weak-coupling 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 spin-chain matches precisely the one that was previously found as the weak-coupling limit of the integrable system describing the AdS 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 side of the correspondence.

Paper Structure

This paper contains 16 sections, 79 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The D1-D5 system
  3. Field content in the UV
  4. Two-dimensional action by dimensional reduction
  5. Effective action at the origin of the Higgs branch
  6. Superconformal representations of the fields
  7. The u(1,1|2) superalgebra
  8. psu(1,1|2) + psu(1,1|2) decomposition of the fields
  9. Spin-chain states
  10. One-loop dilatation operator in the so(4) sector
  11. "One-loop" Feynman diagrams
  12. Computation of one-loop dilatation operator
  13. Conclusions
  14. Gamma matrices
  15. G-functions
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Example of counting of factors of $N_c$ and $N_f$ for three diagrams. The double lines represent adjoint fields and the single line give particles transforming in the fundamental representation of the gauge group. Each closed loop gives a factor of $N_c$ or $N_f$ depending on which representation the fields in the loop transform under. Additionally, each propagator for a field in the vector multiplet give a factor $1/N_f$. The first two diagrams are proportional to $N_c/N_f$ and hence count as one-loop diagrams. The dotted propagator in the third diagram corresponds to a fermion in the adjoint hypermultiplet. This diagram is proportional to $(N_c/N_f)^2$ and is therefore suppressed in the weakly-coupled planar limit.
  • Figure 2: The flavour structure of the fermion box diagram for the two fermion chiralities. The dashed red lines represent the $\mathrm{su}(2)_{\hbox{\tiny L}}$ flavour, while the $\mathrm{su}(2)_{\hbox{\tiny R}}$ flavour is represented by solid blue lines. A vertical line connecting the operator at the bottom with an external line corresponds to a Kronecker $\delta$ while a line connecting two sites of the operator or two external lines corresponds to a Levi-Civita tensor $\epsilon$.