Relevant Deformations, Brane Brick Models and Triality

TL;DR

This work extends the study of relevant deformations in 2d $(0,2)$ gauge theories realized on D1-branes at toric CY$_4$ singularities beyond pure mass terms, leveraging the SE$_7$ base volume monotonicity to infer UV$ o$IR directionality. By tying gauge theory fields to GLSM/divisor data via forward/brick-match mappings, the authors extract scaling dimensions from divisor volumes and systematically identify the sets of terms needed to realize target deformations. The paper demonstrates non-mass deformations mapping to mass deformations under triality, resolves puzzles about non-holomorphic couplings through precise triality rules, and provides evidence that birational transformations preserve the Hilbert series when refined only by $U(1)_R$, extending known invariants to a broader deformation class. Collectively, these results deepen the geometric understanding of IR dynamics in brane brick models and highlight robust invariants under a wide class of deformations with potential implications for dualities and moduli-space engineering in toric CY$_4$ setups.

Abstract

We extend the study of relevant deformations connecting 2d (0,2) gauge theories on D1-branes probing toric Calabi-Yau 4-folds beyond pure mass deformations. The underlying geometry provides powerful insights when field-theoretic tools are still lacking. We observe that the volume of the Sasaki-Einstein base of the Calabi-Yau 4-fold grows towards the IR, signaling the relevance of deformations. We exploit the map between gauge theory fields and GLSM fields to compute scaling dimensions directly from divisor volumes, allowing for a sharper determination of whether terms in the Lagrangian are relevant or irrelevant. Moreover, this map provides a systematic way to determine the precise set of terms needed to realize a given deformation. We also explore the interplay between general relevant deformations and triality, studying cases where non-mass deformations are mapped to mass deformations in a dual theory, and resolving puzzles that seem to require non-holomorphic couplings in one of the dual phases. Finally, we present evidence that when the Hilbert series of the mesonic moduli space is refined only under the U(1) R-symmetry, it becomes invariant even under non-mass relevant deformations of the brane brick models corresponding to toric Calabi-Yau 4-folds related by a birational transformation, extending previous results to a broader class of deformations.

Figures (19)

• Figure 1: Mesons resulting from the composition of an incoming chiral field with an outgoing chiral or Fermi field at node $i$.
• Figure 6: Toric diagrams for $P_{+-}(\textrm{SPP}/\mathbb{Z}_2)$ and $P^2_{+-}(\textrm{PdP}_3)$. We have labeled points anticipating the corresponding brick matchings.
• Figure 7: Quiver diagram for a toric phase of $P_{+-}(\textrm{SPP}/\mathbb{Z}_2)$.
• Figure 8: Toric diagrams for $\textrm{SPP} \times \mathbb{C}$ and $D_3$. We have labeled points anticipating the corresponding brick matchings.
• Figure 9: Quiver diagram for $\textrm{SPP} \times \mathbb{C}$ .