Six Easy Pieces: interplays among dualities in 4d, 3d and 2d

Abstract: In this paper we consider 4d $\mathrm{SU}(N)$ gauge theories with $N+1$ fundamentals, five antifundamentals and a conjugate two index antisymmetric tensor. The model has been shown to be in a mixed phase in the IR, splitting in an interacting non-Abelian Coulomb phase and a free magnetic phase. Through tensor deconfinement, we show that baryonic deformations lead to a non-Abelian free magnetic phase. Along the analysis we obtain a duality with symplectic SQCD that can be further reduced to 3d and 2d. In the 3d case the analysis of the three sphere partition function allows one to obtain dualities between $\mathrm{SU}(N)$ with a two index symmetric tensor and $\mathrm{SO}(N)$ theories. On the other hand, in 2d we recover dualities already known in the literature and propose new ones between special unitary and symplectic gauge theories.

• The paper demonstrates that baryonic deformations in 4d SU(N) gauge theories trigger RG flows that transition mixed phases into free magnetic phases.
• It employs tensor deconfinement to derive integral identities linking partition functions and superconformal indices across different dimensions.
• Dimensional reduction reveals new dualities in 3d and 2d, recovering known (0,2) dualities while extending the framework to novel gauge theory descriptions.

Interplays Among Supersymmetric Dualities in Multiple Dimensions

Introduction and Motivation

The paper "Six Easy Pieces: interplays among dualities in 4d, 3d and 2d" (2512.10495) provides a detailed analysis of supersymmetric dualities in gauge theories across several spacetime dimensions, with a primary focus on constructions involving 4d $\mathrm{SU}(N)$ gauge theories containing $N+1$ fundamentals, five antifundamentals, and a conjugate two-index antisymmetric tensor. The study is situated in the context of tensor deconfinement, building on the observation that many intricate dualities involving matter in tensor representations descend from more basic SQCD (supersymmetric QCD) dualities by the iterative deconfinement of the tensor matter.

A principal outcome is the demonstration that certain baryonic deformations, which are classically irrelevant in the UV but can induce nontrivial RG flows, transform mixed IR phases into free magnetic phases in these models. The subsequent dimensional reduction to 3d and 2d reveals new dualities between special unitary, symplectic, and orthogonal gauge theories and, in two dimensions, recovers and extends known $(0,2)$ dualities.

4d Tensor Deconfined Phases and Baryonic Deformations

Mixed Phases and Tensor Deconfinement

The initial 4d setup yields a mixed phase: in the IR, the system splits into a non-Abelian Coulomb sector plus a free magnetic phase, as established in prior work. Utilizing tensor deconfinement, the authors dissect the antisymmetric tensor of $\mathrm{SU}(N)$ as arising via s-confining $\mathrm{USp}(2m)$ sectors coupled to the original node.

Impact of Baryonic Deformations

A central technical result is that certain baryonic operators (e.g., $\tilde{A}^{n}$, $\tilde{A}^{n-1} \tilde{Q}_1 \tilde{Q}_2$, etc.) trigger flows that alter the IR structure, driving the theory to a free magnetic regime. The investigation distinguishes several classes based on the even or odd nature of $N$ and the structure of the inserted baryonic deformation, and tracks the resulting IR dual descriptions using successive dualities and Higgs mechanisms.

Two alternative deconfinement paths are cataloged:

• Canonical deconfinement: Replacing tensors with a symplectic gauge sector and subsequently dualizing produces $\mathrm{USp}(2M)$ SQCD, which may itself confine, yielding an $\mathrm{USp}(2)$ theory in the IR.
• Non-canonical deconfinement: Nonstandard quivers are constructed, where Higgs flows induced by the deformation produce the same IR dual as the canonical approach.

Crucially, the RG flows induced by these baryonic superpotential deformations respect the expected a-theorem monotonicity and result in genuine IR dualities rather than mixed or runaway behavior.

Dimensional Reduction: 3d and 2d Dualities

Reduction to 3d: Partition Functions and Monopole Constraints

Upon compactification to 3d, the superconformal index is mapped to the $S_b^3$ partition function. The crucial observation is that, after appropriately freezing mass parameters and applying the duplication formula for hyperbolic Gamma functions, the dual pairs obtained between $\mathrm{SU}(N)$ (with a symmetric tensor) and $\mathrm{SO}(K)$ SQCDs do not require parity distinctions for the gauge rank found in 4d.

The reduction produces new integral identities equating the partition functions of unitary gauge theories with symmetric matter and orthogonal gauge theories in a vector representation, subject to linear monopole superpotentials fixing the global charge assignments. This construction is underpinned by explicit tensor deconfinement arguments.

Reduction to 2d: $(0,2)$ Models and Novel Dualities

Reduction to 2d is performed along the prescription mapping the 4d superconformal index to the elliptic genus with a prescribed assignment of $R$-charges. The possible outcome depends on details of the anomaly cancellation and the superpotential:

• $[X, X]$ type cases: The 2d theory lacks charged Fermi multiplets and extra nonperturbative constraints—some instances correspond to previously known dualities, while others reduce further to LG descriptions.
• $[X, V]$ and $[V, X]$ cases: These retain fundamental Fermi multiplets and in certain assignments produce new $\mathrm{SU}(N)/\mathrm{USp}(2M)$ dualities not previously realized in literature.
• $[V, V]$ cases: Even richer multiplet structures emerge, sometimes resulting in two Fermi multiplets and more intricate $J$-term interactions.

Special attention is given to the reduction of models with specific baryonic deformations for $\mathrm{SU}(2n)$, for which a comprehensive catalog of possible consistent reductions is presented. In some scenarios, models with monopole deformations map onto LG phases, generalizing previously established 2d dualities.

Integral Identities, Operator Mapping, and Duality Webs

Beyond the construction of explicit Lagrangians in multiple dimensions, the work systematically demonstrates the equivalence of superconformal indices and partition functions across phases and after dimensional reduction. This provides robust evidence that the dualities established at the Lagrangian level extend to full quantum equivalences, including the chiral ring and protected spectrum. The paper includes detailed operator identifications throughout the sequential duality moves, ensuring the precise mapping of baryonic, mesonic, and monopole operators.

Theoretical and Practical Implications

This analysis confirms that intricate Nambu-Goldstone and IR free structures emerge generically in supersymmetric systems with tensor matter once appropriate baryonic deformations are included. The explicit web of dualities constructed here, extending across dimensions, clarifies how many known—and several new—dualities arise as special cases of a unified tensor deconfinement framework. The strong algebraic machinery involving partition function identities and index manipulations underlines a deep algebraic organizing principle underlying these field theoretical dualities.

Practically, these results present new dual descriptions that can be exploited for nonperturbative computations, e.g., in the context of RG flows, phase structure of QFTs, or categorization of $(0,2)$ conformal field theories. The methodical approach to the reduction also sets a template for extending dualities to further lower dimensions or incorporating higher-rank tensor objects.

Future Directions

Several research avenues are suggested:

• Examining mixed-phase models with extended matter content to search for new $\mathrm{SU}(N)$/$\mathrm{SO}(N)$ (or $\mathrm{USp}(2N)$) dualities in 4d not accessible with minimal flavor assignments.
• Investigating models with multiple coupled gauge nodes or quiver extensions with more intricate tensorial matter, as in the context of bbrane systems.
• Extending the reduction procedure to include nontrivial topology or boundary conditions, connecting to the landscape of dualities in 2d CFTs and surface operators in higher-dimensional theories.
• Integrating the findings with string-theoretic engineering (e.g., brane setups) to provide geometric interpretations of the tensor deconfinement technique and its dimensional reductions.

Conclusion

The work provides a technically thorough and conceptually unified treatment of how baryonic deformations in 4d supersymmetric gauge theories with tensor matter reorganize the infrared phase structure and duality pattern. Through tensor deconfinement, RG flows, and consistent dimensional reductions, the paper establishes new families of dualities in 3d and 2d, extending both the theoretical understanding and the library of tractable IR descriptions for supersymmetric gauge theories. These results reinforce the pivotal role of baryonic deformations and deconfinement in organizing quantum field theory dualities across dimensions.