nLab multi-trace operator



Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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


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 Φ\Phi on the AdS-side

Φ(x,r)r0 iα i(x)r dλ i+ \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({𝒪 i})W(\{\mathcal{O}_i\}) have coefficients given by the derivative of the multi-trace polynomial by its single-trace variables 𝒪 i\mathcal{O}_i:

α i= 𝒪 iW({𝒪 i}) \alpha_i = \partial_{\mathcal{O}_i} W (\{\langle\mathcal{O}_i\rangle\})

(Witten 01, Section 3)


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 𝔤VρV\mathfrak{g} \otimes V \overset{\rho}{\to} V of 𝔤=\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 DD

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

Or equivalently, if D^\widehat D is a horizontal chord diagram whose σ\sigma-permuted closure is DD (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,σw_{V,\sigma} on D^\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.


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

Textbook account:

Last revised on February 5, 2020 at 10:44:50. See the history of this page for a list of all contributions to it.