Useful relations among the generators in the defining and adjoint  representations of SU(N)

Haber, Howard E.

doi:10.21468/scipostphyslectnotes.21

Cited by 27 publications

(15 citation statements)

References 6 publications

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“…The difficulty originates from the complexity of structure constants of the SU(N ) gauge group. To explain this, we will introduce following notations for the SU(N ) color group [62]. We denote as T A the generators of SU(N ) gauge group, where A = 1, .…”

Section: Jhep02(2022)191mentioning

confidence: 99%

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Spin Matrix Theory in near $$ \frac{1}{8} $$-BPS corners of $$ \mathcal{N} $$ = 4 super-Yang-Mills

Baiguera

Harmark

Lei

2022

J. High Energ. Phys.

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We consider limits of $$ \mathcal{N} $$ N = 4 super-Yang-Mills (SYM) theory that approach BPS bounds. These limits result in non-relativistic theories that describe the effective dynamics near the BPS bounds and upon quantization are known as Spin Matrix Theories. The near-BPS theories can be obtained by reducing $$ \mathcal{N} $$ N = 4 SYM on a three-sphere and integrating out the fields that become non-dynamical in the limits. In the previous works [1–3] we have considered various SU(1,1) and SU(1,2) types of subsectors in this limit. In the current work, we will construct the remaining Spin Matrix Theories defined near the $$ \frac{1}{8} $$ 1 8 -BPS subsectors, which include the PSU(1,1|2) and SU(2|3) cases. We derive the Hamiltonians by applying the spherical reduction algorithm and show that they match with the spin chain result, coming from the loop corrections to the dilatation operator. In the PSU(1,1|2) case, we prove the positivity of the spectrum by constructing cubic supercharges using the enhanced PSU(1|1)2 symmetry and show that they close to the interacting Hamiltonian. We finally analyse the symmetry structure of the sectors in view of an interpretation of the interactions in terms of fundamental blocks.

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Section: Jhep02(2022)191mentioning

confidence: 99%

“…The d ABC are simply vanishing in the SU(2) gauge group, while in the SU(3) case, eq. (5.38) can be simplified to [62,64]…”

Section: Jhep02(2022)191mentioning

confidence: 99%

Spin Matrix Theory in near $$ \frac{1}{8} $$-BPS corners of $$ \mathcal{N} $$ = 4 super-Yang-Mills

Baiguera

Harmark

Lei

2022

J. High Energ. Phys.

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“…with N i being the dimension of the fundamental representation of each copy of the SU(N i ) group, see [48] whose normalization for the d abc tensor we adopt. From the one-loop logarithmic divergences of the effective action (3.9) we extract the counterterms…”

Section: Example: Symmetric Multi-adjoint Scalarmentioning

confidence: 99%

Worldline description of a bi-adjoint scalar and the zeroth copy

Bastianelli

Comberiati

Cruz

2021

J. High Energ. Phys.

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Bi-adjoint scalars are helpful in studying properties of color/kinematics duality and the double copy, which relates scattering amplitudes of gauge and gravity theories. Here we study bi-adjoint scalars from a worldline perspective. We show how a global G × $$ \overset{\sim }{G} $$ G ~ symmetry group may be realized by worldline degrees of freedom. The worldline action gives rise to vertex operators, which are compared to similar ones describing the coupling to gauge fields and gravity, thus exposing the color/kinematics interplay in this framework. The action is quantized by path integrals to find a worldline representation of the one-loop QFT effective action of the bi-adjoint scalar cubic theory. As simple applications, we recover the one-loop beta function of the theory in six dimensions, verifying its vanishing, and compute the self-energy correction to the propagator. The model is easily extendable to that of a particle carrying an arbitrary representation of direct products of global symmetry groups, including the multi-adjoint particle, whose one-loop beta function we reproduce as well.

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Section: Fermionic F-termmentioning

confidence: 99%

Spin Matrix Theory in near $\frac{1}{8}$-BPS corners of $\mathcal{N} = 4$ super-Yang-Mills

Baiguera,

Harmark,

Lei

2021

Preprint

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We consider limits of N = 4 super-Yang-Mills (SYM) theory that approach BPS bounds. These limits result in non-relativistic theories that describe the effective dynamics near the BPS bounds and upon quantization are known as Spin Matrix Theories. The near-BPS theories can be obtained by reducing N = 4 SYM on a three-sphere and integrating out the fields that become non-dynamical in the limits. In the previous works [1-3] we have considered various SU(1,1) and SU(1,2) types of subsectors in this limit. In the current work, we will construct the remaining Spin Matrix Theories defined near the 1 8 -BPS subsectors, which include the PSU(1,1|2) and SU(2|3) cases. We derive the Hamiltonians by applying the spherical reduction algorithm and show that they match with the spin chain result, coming from the loop corrections to the dilatation operator. In the PSU(1,1|2) case, we prove the positivity of the spectrum by constructing cubic supercharges using the enhanced PSU(1|1) 2 symmetry and show that they close to the interacting Hamiltonian. We finally analyse the symmetry structure of the sectors in view of an interpretation of the interactions in terms of fundamental blocks.

show abstract

Useful relations among the generators in the defining and adjoint representations of SU(N)

Cited by 27 publications

References 6 publications

Spin Matrix Theory in near $$ \frac{1}{8} $$-BPS corners of $$ \mathcal{N} $$ = 4 super-Yang-Mills

Spin Matrix Theory in near $$ \frac{1}{8} $$-BPS corners of $$ \mathcal{N} $$ = 4 super-Yang-Mills

Worldline description of a bi-adjoint scalar and the zeroth copy

Spin Matrix Theory in near $\frac{1}{8}$-BPS corners of $\mathcal{N} = 4$ super-Yang-Mills

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