nLab multi-trace operator

Redirected from "multi-trace operators".
Contents

Context

Quantum Field Theory

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

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

1. Idea

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

2. 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 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)

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 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.

4. References

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.