3D World Volume Supersymmetry

• 3D world volume supersymmetry is a framework describing supersymmetric dynamics on branes and membranes, incorporating both linear and nonlinear realizations.
• It involves algebraic structures like exceptional Lie superalgebras and dualities that relate vector and tensor multiplet formulations through constrained superfield techniques.
• The approach underpins effective actions for D2-branes, supermembranes, and topological models, offering insights into partial symmetry breaking and κ-symmetry in curved backgrounds.

Three-dimensional (3D) world volume supersymmetry encompasses the algebraic structures, field-theoretic realizations, and effective actions describing supersymmetric dynamics on brane, membrane, and field-theoretic worldvolumes in three spacetime dimensions. This domain covers nonlinear and linear supersymmetry, partial symmetry breaking, Goldstone multiplets, dualities, and the geometric and algebraic constraints governing supersymmetry realization. Worldvolume supersymmetry is central to effective field theories for D2-branes, supermembranes, topological models, and domain wall dynamics in 3D, integrating constrained superfield techniques, exceptional Lie superalgebras, κ-symmetry, and self-duality.

1. Algebraic Structures of World Volume Supersymmetry

3D world volume supersymmetry is anchored in several algebraic frameworks:

• Linear and Nonlinear Realizations: The partial breaking of extended supersymmetry, such as ${\cal N}=2 \rightarrow {\cal N}=1}$, yields models where one supersymmetry is linearly realized and the remainder emerges nonlinearly. This framework is exemplified in D2-brane effective actions employing nilpotent ${\cal N}=2$ superfields—where constraints like $\mathbb{W}^2=0$ or $\mathbb{X}^2=0$ produce Goldstone multiplet incarnations (Maxwell or Tensor) for the broken symmetry (Hu et al., 2022).
• Superalgebra Dualities: Duality transformations invert dimensionless parameters, connecting vector and tensor Goldstone formulations via 3D vector–tensor duality (the coupling ratio $\lambda$ maps to its inverse under duality).
• Exceptional Lie Superalgebras: The rigid ${\cal N}=4$ worldvolume supersymmetry—especially on AdS$_3$ backgrounds—is captured by the exceptional superalgebra $D(2,1;\alpha)$, with bosonic subalgebra $$\mathfrak{sl}(2)\oplus\mathfrak{sl}(2)\oplus\mathfrak{su}(2)$$ and intricate fermion shifts and central extensions (Andrianopoli et al., 2021).
• F-brane Virasoro–κ Algebra: In F-theory, the worldvolume is governed by a generalized (super) Virasoro algebra, with worldvolume coordinates and currents extended by κ-symmetry partners. The full algebra includes first-class constraints for all generators and incorporates self-duality to eliminate coordinate doubling (Linch et al., 2017).

2. Supersymmetric Actions and Constrained Superfields

The construction of effective supersymmetric actions on 3D worldvolumes relies on constrained superfields and Goldstone multiplets:

• Born–Infeld Extensions: For D2-branes, the Born–Infeld (BI) action receives a supersymmetric completion dictated by nilpotent constraints on ${\cal N}=2$ superfields. For the Maxwell–Goldstone multiplet, the action reads

$S_{\rm MG} = \tau \int d^3x\,d^2\theta\,X$

where $X$ solves $X=W^\alpha W_\alpha + \frac{1}{2}(D^2 X)X$ and $X^2 = 0$. The explicit component expansion yields bosonic BI terms and nonpolynomial fermionic completions governed by the 3D Cecotti–Ferrara function (Hu et al., 2022).

• Tensor–Goldstone Dual: The chiral multiplet perspective produces a tensor–Goldstone action,

$$S_{\rm TG} = -\tilde{\tau} \int d^3x\,d^2\theta\,\tilde{X}$$

with analogous constraints and dual couplings. The Maxwell and Tensor formulations are related by integrating out auxiliary superfields and inverting the coupling ratio.

• Deformations by Chern–Simons Terms: A gauge-invariant Chern–Simons–like mass term,

$$S_m = \frac{m}{2} \int d^3x\,d^2\theta\,\Gamma^\alpha W_\alpha$$

can deform the Maxwell–Goldstone action, preserving the nonlinearly realized supersymmetry and modifying the manifestly realized one (Hu et al., 2022).

3. Membrane Worldvolume Supersymmetry

Supersymmetry on the 3D worldvolume of membranes (notably the BST supermembrane) reflects residual target-space structures after static gauge and κ-symmetry gauge fixing:

• $N=1$ Supersymmetric Extensions: The bosonic membrane derivative expansion in static gauge admits an $N=1$ 3D supersymmetric analog using $\hat{D}$ copies of 3D scalar multiplets, with superspace actions and quartic derivative completions matching the structure of four-derivative bosonic terms (Tseytlin et al., 4 Dec 2025).
• $N=8$ Supersymmetry in $D=11$ Supermembranes: The $N=8$ worldvolume supersymmetry realized after gauge fixing requires the fermion to transform as an $SO(8)$ spinor (not a vector), affecting closure of the full superalgebra. The explicit matching of $N=1$ and $N=8$ actions holds precisely in $D=4,5$ but not in $D=11$, due to an extra anti-self-dual $SO(8)$ tensor term absent in pure $N=1$ invariants.
• Quantum 1-loop Amplitudes: The difference in field-theoretic backgrounds manifests in the one-loop 2-to-2 worldvolume scattering amplitudes, which agree in $D=4,5$ but differ for $D=11$ supermembranes (Tseytlin et al., 4 Dec 2025).

4. Supersymmetry on Curved Worldvolumes and Rigid Backgrounds

Superspace techniques generalize supersymmetric models to curved worldvolumes (notably AdS$_3$ embeddings):

• AdS$_3$ Superspace Models: Supersymmetry generators in AdS$_3$ backgrounds are realized as "square roots" of isometry transformations, with the supercharges squaring to rotations or boosts rather than translations. The algebra $\{Q_i,Q_j\} = 2(\sigma^{AB})_{ij}L_{AB}$ demonstrates mass splitting between bosonic and fermionic fields, absent in flat-space (McKeon, 2013).
• Twisted Rigid Supersymmetry ($D(2,1;\alpha)$): Rigid $N=4$ supersymmetry on AdS$_3$ employs hypermultiplets and supercoset constructions; twists involving $SL(2)$ groups produce Rozansky–Witten-like topological sectors and, alternately, massive Dirac structures analogously to AVZ unconventional supersymmetry (Andrianopoli et al., 2021).
• Dualities and Effective Theories on Domain Walls: IR supersymmetry enhancement may occur on BPS domain walls; for instance, 2D $(2,0)$ chiral supersymmetry emerges on walls in several dual 3D models, though precise enhancement depends on Morse index differences and field content (1904.02722).

5. Geometric and Algebraic Constraints: Virasoro–κ Structure and Self-Duality

F-brane superspace formalism generalizes the worldvolume coordinate system and algebraic constraints:

• Worldvolume Superspace: Each generator (including $\kappa$-symmetry) is associated with its own worldvolume coordinate $$(\sigma^{\alpha\beta}, \theta^\alpha, \kappa_\alpha)$$. Constraints, e.g., $P_m + B_{mn}\partial_\sigma X^n = 0$, enforce self-duality and eliminate coordinate doubling (Linch et al., 2017).
• Extended Affine Superalgebra: The algebra incorporates graded commutators among supersymmetry currents $D_\alpha$, Virasoro generators $P_{\alpha\beta}$, $\kappa$-partners $d_\alpha$, and first-class bilinears $S_{AB}$. Closure is guaranteed by specific Jacobi identities and self-duality constraints on background field strengths.

Structure/Formalism	Core Feature	Reference
Nilpotent superfields	Partial SUSY breaking, nonlinearity	(Hu et al., 2022)
$D(2,1;\alpha)$ superalgebra	Rigid AdS$_3$ $N=4$ SUSY, exceptional twist	(Andrianopoli et al., 2021)
F-brane Virasoro–κ algebra	Extended supergeometry, self-duality	(Linch et al., 2017)
BST supermembrane gauge fix	Residual $N=8$ SUSY, $SO(8)$ spinor	(Tseytlin et al., 4 Dec 2025)
Domain wall enhancements	IR supersymmetry on effective theories	(1904.02722)

6. Dualities, Effective Theories, and Physical Interpretation

Dualities between vector and tensor multiplet descriptions, as well as topological twisting, underpin much of the richness in 3D worldvolume supersymmetry:

• Vector–Tensor Duality: Integration over unconstrained superfields or Lagrange multipliers in parent actions yields dual formulations, with the coupling ratio inverted.
• Rozansky–Witten Twist: The first twist in $D(2,1;\alpha)$ models yields a topological sector with a Chern–Simons odd-connection and fermionic gauge-fixing, recapitulating Rozansky–Witten theory in three dimensions.
• Massive Dirac Twists: Identifying one $SL(2)$ with Lorentz symmetry recasts hyperini into spin-3/2 and spin-1/2 fields, yielding massive Dirac equations and connections to 3D unconventional supersymmetry (Andrianopoli et al., 2021).
• Domain Walls and IR Enhancement: Effective 2D theories living on 3D domain walls sometimes enjoy supersymmetry enhancement ($N=(2,0)$), though absent in theories where Morse index conditions preclude it. Matching of central charges, BPS equations, and wall sigma models across dualities is observed (1904.02722).

7. Open Directions and Mathematical Consequences

Worldvolume supersymmetry in 3D presents several directions for further inquiry:

• The extension of component superfield solutions, explicit metrics, and Wess–Zumino terms in F-brane superspace remains incomplete, with only the algebraic backbone established (Linch et al., 2017).
• Embedding techniques in AdS$_3$ and AdS$_4$ suggest distinct structures for boson-fermion mass splitting and symmetry algebra closure, not always providing a true "square root" of background isometries (McKeon, 2013).
• Consistency conditions and Bianchi identities in extended Virasoro–κ superspace await full geometric realization.
• Nonlinear and partially broken supersymmetry frameworks continue to be central in describing effective D-brane and membrane worldvolume actions, with implications for duality relations and nonperturbative dynamics (Hu et al., 2022, Tseytlin et al., 4 Dec 2025).

A plausible implication is that the precise realization of worldvolume supersymmetry—linear, nonlinear, rigid, topological, or partially broken—directly dictates the spectrum, duality properties, and effective dynamics on brane and membrane worldvolumes, providing deep insights into the geometric and quantum field content of M-theory, superstring models, and topological field theories in three dimensions.