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On Melonic Supertensor Models

Chi-Ming Chang, Sean Colin-Ellerin, Mukund Rangamani

TL;DR

The paper investigates a class of $ N=2$ supersymmetric tensor models with melonic large-$N$ dominance and dynamical bosons, asking whether SUSY can survive melonic IR physics. It finds a definite tension: the IR fixed point is conformal and non-supersymmetric once a UV regulator is applied to control bosonic divergences, with SUSY breaking occurring explicitly along the RG flow and no goldstino in the spectrum. Finite-$N$ analysis shows SUSY is preserved there, but the large-$N$ limit introduces $ O(N)^{q-1}$-generated light modes that complicate the IR structure. The spectrum of singlet operators is computed via a conformal (SL$(2,R)$) framework, revealing emergent time reparametrization, affine $U(1)_R$, and local scaling symmetries, while the four-point functions are obtained by resumming ladder diagrams and analyzed through conformal eigenfunctions. Overall, the work demonstrates that melonic dominance and supersymmetry are not naturally compatible in this class of quantum-mechanical models, with implications for holography and the structure of near-AdS$_2$ dynamics.

Abstract

We investigate a class of supersymmetric quantum mechanical theories (with two supercharges) having tensor-valued degrees of freedom which are dominated by melon diagrams in the large $N$ limit. One motivation was to examine the interplay between supersymmetry and melonic dominance and potential implications for building toy models of holography. We find a definite tension between supersymmetry (with dynamical bosons) and melonic dominance in this class of systems. More specifically, our theories attain a low energy non-supersymmetric conformal fixed point. The origin of supersymmetry breaking lies in the need to regularize bosonic and fermionic degrees of freedom independently. We investigate various aspects of the low energy spectrum and also comment on related examples with different numbers of supercharges. Along the way we also derive some technical results for $SL(2,{\mathbb R})$ wavefunctions for fermionic excitations.

On Melonic Supertensor Models

TL;DR

The paper investigates a class of supersymmetric tensor models with melonic large- dominance and dynamical bosons, asking whether SUSY can survive melonic IR physics. It finds a definite tension: the IR fixed point is conformal and non-supersymmetric once a UV regulator is applied to control bosonic divergences, with SUSY breaking occurring explicitly along the RG flow and no goldstino in the spectrum. Finite- analysis shows SUSY is preserved there, but the large- limit introduces -generated light modes that complicate the IR structure. The spectrum of singlet operators is computed via a conformal (SL) framework, revealing emergent time reparametrization, affine , and local scaling symmetries, while the four-point functions are obtained by resumming ladder diagrams and analyzed through conformal eigenfunctions. Overall, the work demonstrates that melonic dominance and supersymmetry are not naturally compatible in this class of quantum-mechanical models, with implications for holography and the structure of near-AdS dynamics.

Abstract

We investigate a class of supersymmetric quantum mechanical theories (with two supercharges) having tensor-valued degrees of freedom which are dominated by melon diagrams in the large limit. One motivation was to examine the interplay between supersymmetry and melonic dominance and potential implications for building toy models of holography. We find a definite tension between supersymmetry (with dynamical bosons) and melonic dominance in this class of systems. More specifically, our theories attain a low energy non-supersymmetric conformal fixed point. The origin of supersymmetry breaking lies in the need to regularize bosonic and fermionic degrees of freedom independently. We investigate various aspects of the low energy spectrum and also comment on related examples with different numbers of supercharges. Along the way we also derive some technical results for wavefunctions for fermionic excitations.

Paper Structure

This paper contains 37 sections, 149 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Outline of the paper:
  3. Bosonic tensor model revisited
  4. The $\mathcal{N}=2$ supersymmetric tensor model
  5. The model
  6. Hilbert space at finite $N$
  7. Melonic dominance and low energy conformal symmetry
  8. The conformal fixed point and IR symmetries
  9. The fixed point solution:
  10. Local symmetries in the IR:
  11. The RG flow and supersymmetry breaking
  12. Comments on supersymmetry breaking:
  13. Four-point functions and operator spectrum
  14. Resumming ladder supergraphs
  15. Conformal eigenfunctions
  16. ...and 22 more sections

Figures (6)

  • Figure 1: The leading order large $N$ contribution to the boson propagator which leads to the Schwinger-Dyson equation \ref{['eqn:SDeqn']}.
  • Figure 2: Comparison of numerical (red) and analytic (blue) solutions of the regularized bosonic Schwinger-Dyson equations \ref{['eqn:massiveSDeqn']} for two different values of $\beta J$ as indicated. The numerical simulation is carried out with the imaginary time circle discretized by a lattice with $200$ points (see footnote \ref{['fn:resolution']}).
  • Figure 3: Supergraph representation of the super-Schwinger-Dyson equation.
  • Figure 4: Comparison of numerical (red) and analytic (blue) solutions of the supersymmetric Schwinger-Dyson equations for $q=6$ and $b_\phi \, J^{2\Delta_\phi+1}=1$ (for different choices of the dimensionless coupling $\beta J$ as indicated). The numerical calculation is still with a discretized temporal grid with 200 points (see footnote \ref{['fn:resolution']} for comments on increasing the resolution). For $G^{FF}$ we have specifically chosen a larger value of $\beta J$ to separate out the curves for ease of visualization.
  • Figure 5: The leading $1/N$ correction to the four-point function of superfield $\Phi^{A_q}$.
  • ...and 1 more figures