1d Seiberg Duality and Triality in SQM

• 1d Seiberg Duality is an IR equivalence in N=2 quantum mechanics featuring U(Nc) gauge theories with chiral, anti-chiral, and fermi multiplets, realized via gauge rank mutations.
• The matching of flavoured Witten indices and cohomological computations on Grassmannian manifolds confirms consistent Higgs-phase dynamics and wall-crossing transitions.
• An exact order-three triality emerges under specific conditions, linking repeated dual mutations to dual Grassmannian descriptions and confinement phenomena in quiver quantum mechanics.

One-dimensional (1d) Seiberg duality refers to a class of infrared (IR) dualities for $\mathcal{N}=2$ supersymmetric gauged quantum mechanics, typically with unitary gauge group $U(N_c)$ and various matter representations. These dualities are generalizations of higher-dimensional Seiberg dualities and manifest as equivalences between distinct 1d gauge theories—often called “mutations”—invariant under the operation of exchanging chirality and gauge rank data. Their primary signatures include matching of supersymmetric ground states, flavoured Witten indices, and physical observables across dual theories, often accompanied by nontrivial wall-crossing phenomena and, in several cases, an enhancement to exact triality symmetry (Closset et al., 2 Dec 2025).

1. Definitions and Algebraic Structure

1d $\mathcal{N}=2$ supersymmetric quantum mechanics relevant to Seiberg duality involve

• Gauge group $U(N_c)$
• Fayet–Iliopoulos (FI) parameter $\zeta\neq0$
• Bare 1d Chern–Simons (CS) (Wilson-line) charge $Q_c\in\mathbb{Z}$, so the effective CS level is $q_c=Q_c-\frac{1}{2}(n_1-n_2-n_3)$
• Matter content:
  • $n_1$ fundamental chiral multiplets $\Phi_i^\alpha$
  • $n_2$ antifundamental chiral multiplets $U(N_c)$0
  • $U(N_c)$1 fundamental fermi multiplets $U(N_c)$2
  • Optionally, in the $U(N_c)$3-SQCD variant, a gauge-neutral fermi $U(N_c)$4 with $U(N_c)$5, $U(N_c)$6

The theory is denoted as

$U(N_c)$7

with the sign of $U(N_c)$8 distinguishing two distinct duality actions:

• Right mutation ($U(N_c)$9):

$\mathcal{N}=2$0

with $\mathcal{N}=2$1 and flavors permuting as $\mathcal{N}=2$2.

• Left mutation ($\mathcal{N}=2$3):

$\mathcal{N}=2$4

with $\mathcal{N}=2$5.

These mutations are involutive and inverse to each other. When $\mathcal{N}=2$6 and $\mathcal{N}=2$7, repeated mutation leads to an order-three triality (Closset et al., 2 Dec 2025).

2. Flavoured Witten Index and Index Matching

The flavoured Witten index is a protected quantity capturing the supersymmetric ground state structure, defined as

$\mathcal{N}=2$8

where it localizes to a Jeffrey–Kirwan (JK) residue on the $\mathcal{N}=2$9 Cartan torus,

$U(N_c)$0

with

$U(N_c)$1

Explicit JK-residue computations confirm that the indices of dual pairs agree, up to contact-term factors and sign conventions dictated by mutation rules:

$U(N_c)$2

$U(N_c)$3

This rigorous matching validates the IR equivalence of 1d Seiberg-dual pairs (Closset et al., 2 Dec 2025).

3. Higgs-Branch Analysis and Dual Grassmannian Manifolds

In the deep Higgs phase ($U(N_c)$4), the gauge symmetry is fully broken, yielding a nonlinear sigma model on Grassmannian target spaces:

$U(N_c)$5

The associated vector bundles $U(N_c)$6 encode CS charge, matter representations:

$U(N_c)$7

$U(N_c)$8

Supersymmetric ground states correspond to sheaf cohomology groups $U(N_c)$9, computable via Borel–Weil–Bott theory. The Grassmannian duality

$\zeta\neq0$0

ensures that under parameter shifts, the respective bundles map to one another, providing a geometric proof of both right and left mutations (Closset et al., 2 Dec 2025).

4. Coulomb-Branch Dynamics and Wall-Crossing Phenomena

For abelian $\zeta\neq0$1 gauge group, the Coulomb-branch analysis involves real mass deformations and the quantum dynamics of $\zeta\neq0$2. Integrating out matter generates a real superpotential $\zeta\neq0$3, resulting in an effective potential $\zeta\neq0$4. Supersymmetric vacua are localized at the critical points of $\zeta\neq0$5; these match precisely the Higgs-branch ground state structure and the cohomology result for $\zeta\neq0$6.

Wall-crossing occurs as $\zeta\neq0$7 crosses zero, causing the index to jump by contributions from poles at $\zeta\neq0$8 and $\zeta\neq0$9. The condition for trivial wall-crossing (i.e., index continuity) for $Q_c\in\mathbb{Z}$0 is:

$Q_c\in\mathbb{Z}$1

or equivalently,

$Q_c\in\mathbb{Z}$2

Generalizing, the triviality window is

$Q_c\in\mathbb{Z}$3

Within this regime, both Higgs phases are smoothly connected, leading to enhanced IR duality and triality (Closset et al., 2 Dec 2025).

5. Enhancement to Triality and Confinement Limits

When $Q_c\in\mathbb{Z}$4 and $Q_c\in\mathbb{Z}$5, repeated right and left mutations traverse three distinct dual gauge sectors:

$Q_c\in\mathbb{Z}$6

This cyclic structure is an exact order-three triality, matching the dimensional reduction of the $Q_c\in\mathbb{Z}$7 $Q_c\in\mathbb{Z}$8 Gadde–Gukov–Putrov triality and anomaly cancellation ($Q_c\in\mathbb{Z}$9). If $q_c=Q_c-\frac{1}{2}(n_1-n_2-n_3)$0 or $q_c=Q_c-\frac{1}{2}(n_1-n_2-n_3)$1, the gauge sector completely confines, leaving only free mesons $q_c=Q_c-\frac{1}{2}(n_1-n_2-n_3)$2 in the IR (Closset et al., 2 Dec 2025).

6. 1d Seiberg Duality in Quiver Quantum Mechanics and Wall-Crossing

Quiver quantum mechanics provides a natural setting for 1d Seiberg duality, as evidenced in bound states of D4–D2–D0 branes on singular spaces such as the resolved conifold and orbifold $q_c=Q_c-\frac{1}{2}(n_1-n_2-n_3)$3. The FI parameter dependence of the Witten index translates directly into Kähler moduli dependence for BPS indices. Wall-crossing is realized as cascades of sequential Seiberg dualities within the quiver, with precise rules for node dualization, gauge rank mutation, FI parameter shifts, and meson introduction.

For the conifold quiver, duality cascades drive transitions between dual frames, with wall-crossed partition functions assembled via fugacity-redefinitions and product formulas consistent with the semi-primitive wall-crossing paradigm.

For $q_c=Q_c-\frac{1}{2}(n_1-n_2-n_3)$4, partition functions are invariant across all chambers and coincide with characters of the affine $q_c=Q_c-\frac{1}{2}(n_1-n_2-n_3)$5 algebra, indicating trivial wall-crossing. This structure realizes a universal pattern of Seiberg duality enacted by quiver mutations in $q_c=Q_c-\frac{1}{2}(n_1-n_2-n_3)$6 quantum mechanics (Nishinaka, 2013).

7. Physical Implications and Research Directions

The study of 1d Seiberg duality has illuminated several fundamental aspects of supersymmetric gauge theory in low dimensions, including the categorization of IR-equivalent models via index theory, geometric interpretation through Grassmannian duality, and the universal emergence of triality both in gauge and solitonic sectors. In geometric engineering contexts, quiver mutations realize wall-crossing phenomena and encode BPS spectra.

A plausible implication is the utility of 1d duality as a laboratory for testing duality principles and wall-crossing formulas in higher dimensions, as well as exploring connections to enumerative geometry and topological field theory via cohomological computations and partition functions. The triality structure, especially in correspondence with $q_c=Q_c-\frac{1}{2}(n_1-n_2-n_3)$7 anomaly cancellation and reduction, further suggests deep intertwined symmetry hierarchies in quantum field theory.