[Paper Review] Aspects of Massive ABJM Models with Inhomogeneous Mass Parameters

Kyung Kiu Kim Yoonbai Kim, O-Kab Kwon+1 more|arXiv (Cornell University)|Oct 11, 2019

Black Holes and Theoretical Physics44 references4 citations

TL;DR

This paper constructs ${\cal N}=1$ and ${\cal N}=2$ inhomogeneously mass-deformed ABJM (ImABJM) models using arbitrary spatially dependent mass-functions, generalizing the previously known ${\cal N}=3$ case. It derives two distinct vacuum solutions—diagonal and GRVV-type—depending on the integral of the periodic mass-function over its period, with explicit examples provided and conformal primary operators analyzed in the classical limit.

ABSTRACT

Recently, ${\cal N} =3$ mass-deformed ABJM model with arbitrary mass-function depending on a spatial coordinate was constructed. In addition to the ${\cal N} = 3$ case, we construct lower supersymmetric ${\cal N} =1$ and ${\cal N} =2$ inhomogeneously mass-deformed ABJM (ImABJM) models, which require three and two arbitrary mass-functions, respectively. We also construct general vacuum solutions of the ${\cal N} = 3$ ImABJM model for any periodic mass-function. There are two classes of vacua, which are diagonal type and GRVV type according to reference value of mass-functions. We provide explicit examples of the vacuum solutions and discuss related operators.

Motivation & Objective

• To extend the ${\cal N}=3$ inhomogeneously mass-deformed ABJM model to lower supersymmetries (${{\cal N}=1}$, ${{\cal N}=2}$) with arbitrary spatial mass-functions.
• To derive general vacuum solutions for the ${\cal N}=3$ ImABJM model under periodic mass-functions.
• To classify vacuum configurations into two types—diagonal and GRVV-type—based on the integral of the mass-function over its period.
• To analyze the corresponding conformal primary operators (CPOs) of dimension $\Delta=1,2$ in the classical limit and their gravity duals.

Proposed method

• Derive supersymmetry conditions for space-dependent mass matrices and ${{\cal N}=6}$ parameters, reducing to ${{\cal N}=1}$ and ${{\cal N}=2}}$ models via specific constraints on mass-functions.
• Construct vacuum solutions by solving the BPS equations $\delta\psi_A = 0$ for fermion variations, reducing the energy to a boundary term.
• Classify vacuum solutions into two types: diagonal (when $\int_0^\tau m(x)\,dx = 0$) and GRVV-type (when $\int_0^\tau m(x)\,dx \neq 0$).
• Use explicit ansätze for scalar fields $Y^A_0$ as $P(x)\tilde{Y}_D^A$ or $P(x)\tilde{Y}_{\text{GRVV}}^A$, with $P(x)$ and $Q(x)$ solving ODEs derived from BPS conditions.
• Construct solutions for various mass-function profiles: sinusoidal, logarithmic, and delta-function arrays, ensuring regularity via control parameters like $C_1$.
• Analyze the classical limit of vacuum expectation values (vevs) of CPOs with $\Delta=1,2$, linking them to gravity duals via holographic methods.

Experimental results

Research questions

• RQ1 How can ${\cal N}=1$ and ${\cal N}=2$ supersymmetric ImABJM models be constructed with arbitrary spatially dependent mass-functions?
• RQ2 What are the general vacuum solutions of the ${\cal N}=3$ ImABJM model when the mass-function is periodic?
• RQ3 What determines the two distinct vacuum types—diagonal and GRVV-type—in the ${\cal N}=3$ ImABJM model?
• RQ4 How do the conformal dimensions $\Delta=1$ and $\Delta=2$ of chiral primary operators relate to the classical vevs and their gravity duals in the weakly coupled regime?
• RQ5 What are the physical implications of vacuum solutions with delta-function mass-functions, and how do junction conditions affect field configurations?

Key findings

• The ${\cal N}=1$ and ${\cal N}=2$ ImABJM models require two and three arbitrary mass-functions, respectively, extending the ${\cal N}=3$ model with one mass-function.
• Vacuum solutions for the ${\cal N}=3$ ImABJM model fall into two classes: diagonal type when $\int_0^\tau m(x)\,dx = 0$, and GRVV-type when the integral is non-zero.
• For the diagonal type, scalar fields $Y_0^A$ are diagonal matrices, and solutions are regular when $Q(x)=0$ and $P(x)$ is derived from $m(x) = \frac{q\sin(qx)}{2(C_1 + \sin^2(qx/2))}$.
• For the GRVV-type, $Y_0^A$ are proportional to GRVV matrices, and solutions remain regular even for singular mass-functions, e.g., $m(x) = q\frac{((\sin^2(qx) - C_1)^3 + \sin(2qx))}{\sin^2(qx) - C_1}$ with $C_1 > 1$.
• Solutions with delta-function mass-functions satisfy junction conditions: $P(x_{i}+\epsilon)/P(x_{i}-\epsilon) = e^{q_i}$ and $Q(x_{i}+\epsilon)/Q(x_{i}-\epsilon) = e^{-q_i}$, enabling piecewise constant or power-law profiles.
• Classical vevs of CPOs with $\Delta=1,2$ are derived from the vacuum configurations, with gravity duals linked via holographic methods, particularly in the weak-coupling limit.