nLab multi-trace operator

Idea

A multi trace observable in a gauge theory is a polynomial in single trace operators (see there for more background).

Properties

Under AdS/CFT correspondence

Under the AdS/CFT correspondence, single trace observables in the gauge theory correspond to single particle/string excitations on the gravity-side, while multi-trace observables correspond to multi-particle/string excitations (Liu 98, p. 6 (7 of 39), Andrianapoli-Ferrara 99, p. 13, Chalmers-Schalm 00, Section 7, Aharony-Gubser-Maldacena-Ooguri-Oz 99, p. 75)

The asymptotic boundary conditions for fields $\Phi$ on the AdS-side

$$\Phi(\vec x, r) \overset{r \to 0} \sum_i \alpha_i (\vec x) r^{d -\lambda_i} + \cdots$$

that correspond to multi-trace observables $W(\{\mathcal{O}_i\})$ have coefficients given by the derivative of the multi-trace polynomial by its single-trace variables $\mathcal{O}_i$:

$$\alpha_i = \partial_{\mathcal{O}_i} W (\{\langle\mathcal{O}_i\rangle\})$$

(Witten 01, Section 3)

Examples

Chord diagrams as multi-trace observables in the BMN matrix model

The supersymmetric states of the BMN matrix model are temporally constant complex matrices which are complex metric Lie representations $\mathfrak{g} \otimes V \overset{\rho}{\to} V$ of $\mathfrak{g}=$su(2) (interpreted as fuzzy 2-sphere noncommutative geometries of giant gravitons or equivalently as fuzzy funnels of D0-D2 brane bound states).

A fuzzy 2-sphere-rotation invariant multi-trace observable on these supersymmetric states is hence an expression of the following form:

(from Sati-Schreiber 19c)

Here we are showing the corresponding string diagram/Penrose notation for metric Lie representations, which makes manifest that

1. these multi-trace observables are encoded by Sullivan chord diagrams $D$

2. their value on the supersymmetric states $\mathfrak{su}(2) \otimes V \overset{\rho}{\to}V$ is the evaluation of the corresponding Lie algebra weight system $w_{{}_V}$ on $D$.

Or equivalently, if $\widehat D$ is a horizontal chord diagram whose $\sigma$-permuted closure is $D$ (see here) then the values of the invariant multi-trace observables on the supersymmetric states of the BMN matrix model are the evaluation of $w_{V,\sigma}$ on $\widehat D$, as shown here:

(from Sati-Schreiber 19c)

But since all horizontal weight systems are partitioned Lie algebra weight systems this way, this identifies supersymmetric states of the BMN matrix model as seen by invariant multi-trace observables as horizontal chord diagrams evaluated in Lie algebra weight systems.

References

Discussion of multi-trace operators in super Yang-Mills theory and of their AdS-CFT dual gravity/string theory incarnation:

• Tom Banks, Michael Douglas, Gary Horowitz, Emil Martinec, AdS Dynamics from Conformal Field Theory (arXiv:hep-th/9808016, spire:474214)

• Hong Liu, Scattering in Anti-de Sitter Space and Operator Product Expansion, Phys. Rev. D 60, 106005 (1999) (arXiv:hep-th/9811152)

• Laura Andrianopoli, Sergio Ferrara, On short and long $SU(2,2/4)$ multiplets in the AdS/CFT correspondence, Lett. Math.Phys. 48 (1999) 145-161 (arXiv:hep-th/9812067)

• Gordon Chalmers, Koenraad Schalm, Holographic Normal Ordering and Multi-particle States in the AdS/CFT Correspondence, Phys. Rev. D61:046001, 2000 (arXiv:hep-th/9901144)

• Ofer Aharony, Steven Gubser, Juan Maldacena, Hirosi Ooguri, Yaron Oz, Large $N$ Field Theories, String Theory and Gravity, Phys. Rept. 323:183-386, 2000 (arXiv:hep-th/9905111)

• Massimo Bianchi, Stefano Kovacs, Giancarlo Rossi, Yassen S. Stanev, On the logarithmic behaviour in $\mathcal{N}=4$ SYM theory, JHEP 9908 (1999) 020 (arXiv:hep-th/9906188)

• Witold Skiba, Correlators of Short Multi-Trace Operators in $\mathcal{N}=4$ Supersymmetric Yang-Mills, Phys. Rev. D 60, 105038 (1999) (arXiv:hep-th/9907088)

• Eric D'Hoker, Daniel Freedman, Samir Mathur, A. Matusis, Leonardo Rastelli, Extremal Correlators in the AdS/CFT Correspondence, in: The many faces of the superworld (arXiv:hep-th/9908160)

• Gleb Arutyunov, Sergey Frolov, A. C. Petkou, Perturbative and instanton corrections to the OPE of CPOs in $\mathcal{N}=4$ $SYM_4$, Nucl. Phys. B602:238-260, 2001; Erratum-ibid. B609:540, 2001 (arXiv:hep-th/0010137)

• Ofer Aharony, Micha Berkooz, Eva Silverstein, Multiple-Trace Operators and Non-Local String Theories, JHEP 0108:006, 2001 (arXiv:hep-th/0105309)

• Ofer Aharony, Micha Berkooz, Eva Silverstein, Non-local string theories on $AdS_3$ times $S^3$ and stable non-supersymmetric backgrounds, Phys. Rev. D65:106007, 2002 (arXiv:hep-th/0112178)

• L. Hoffmann, L. Mesref, A. Meziane, W. Rühl, Multi-trace quasi-primary fields of $\mathcal{N}=4$ $SYM_4$ from AdS $n$-point functions, Nucl. Phys. B641 (2002) 188-222 (arXiv:hep-th/0112191)

• Edward Witten, Multi-Trace Operators, Boundary Conditions, And AdS/CFT Correspondence (arXiv:hep-th/0112258)

• Steven Gubser, Indrajit Mitra, Double-trace operators and one-loop vacuum energy in AdS/CFT, Phys. Rev. D67 (2003) 064018 (arXiv:hep-th/0210093)

• Vijay Balasubramanian, Jan de Boer, Bo Feng, Yang-Hui He, Min-xin Huang, Vishnu Jejjala, Asad Naqvi, Multi-Trace Superpotentials vs. Matrix Models, Commun. Math. Phys. 242:361-392, 2003 (arXiv:hep-th/0212082)

• Steven Gubser, Igor Klebanov, A universal result on central charges in the presence of double-trace deformations, Nucl. Phys. B656 (2003) 23-36 (arXiv:hep-th/0212138)

• P. J. Heslop, Paul Howe, Aspects of $\mathcal{N}$=4 SYM, JHEP 0401 (2004) 058 (arXiv:hep-th/0307210)

• Thomas Hartman, Leonardo Rastelli, Double-Trace Deformations, Mixed Boundary Conditions and Functional Determinants in AdS/CFT, JHEP 0801:019, 2008 (arXiv:hep-th/0602106)

Textbook account:

• Sean Hartnoll, Andrew Lucas, Subir Sachdev, Section 1.7.3 of: Holographic quantum matter, MIT Press 2018 (arXiv:1612.07324, publisher)

  (with an eye towards AdS/CFT in condensed matter physics)