Twisted Traces in Quantum Coulomb Branches

• The paper introduces a framework integrating diagrammatic relations and flavor twist data into quantum cylindrical KLRW algebras to yield twisted trace evaluations.
• It demonstrates that change-of-flavor bimodules and noncommutative resolutions capture wall-crossing phenomena and grading shifts critical for link invariants.
• The work unifies representation theory, knot homology, and categorification by leveraging twisted traces to model quantum Coulomb branches in minuscule ADE types.

Twisted traces for quantum Coulomb branches represent a class of trace functionals and diagrammatic or algebraic structures that encode, via intrinsic twist data, the symmetries, wall-crossing behaviors, and categorified invariants of noncommutative or quantized Coulomb branch algebras. In the context of quiver gauge theories and their knot homology invariants, these traces are realized through explicit modifications of diagrammatic algebras (notably quantum cylindrical KLRW algebras), are deeply intertwined with flavor parameter twists and bimodule correspondences, and play a critical role in the manifestation and categorification of quantum link invariants for minuscule ADE representations.

1. Noncommutative Resolutions, Twisting, and Diagrammatic Algebras

The quantized Coulomb branch $M$ of a generic quiver gauge theory can be described as the derived spectrum of the endomorphism algebra $R$ of a tilting bundle. In the cylindrical KLRW algebraic approach, $R$ is presented by cylindrical diagrams whose strands are labeled by flavor or longitude data. The central innovation of the twisted setting is to allow the intersection number between a corporeal strand and its ghost counterpart to be prescribed by an integer-valued function $d$, rather than forced to vanish as in the untwisted case (see Definition 6.17). This modification is both in the local diagrammatic relations (such as relation (3’) in Section 6.2.2) and in the construction of change-of-flavor bimodules (capturing shifts in parameters across $\mathbb{R}/\mathbb{Z}$).

This framework naturally encodes a twisted trace, since the value of a diagrammatic invariant (such as a closed diagram or link) may depend nontrivially on the twist $d$—the trace is not invariant under cyclic permutation but is instead equivariant with respect to a twisting automorphism arising from the flavor/cocharge data.

2. Quantum Cylindrical KLRW Algebras and Coulomb Branch Quantization

Within the quantum (deformed) cylindrical KLRW algebra $R$, the relations depend on both deformation parameters (such as $h$) and twisting data ($\beta$, $d$). The algebra $R$ is generated as the quotient by “lifted” cylindrical diagrams modulo quantum-deformed relations (see eqs. (8.3a–8.3p)). The isomorphism between the classical/untwisted Coulomb branch and the idempotent truncation of $R$ (Theorem 6.15, 8.11) is extended in the twisted case by tracking the effect of the twist $d$ in cylindrical diagrams, particularly how certain ghost-corporeal crossings introduce extra sign or grading factors. These local modifications ensure that the trace (i.e., the algebraic evaluation of closed diagrams) acquires a twist, and the resulting “twisted trace” is intrinsic to the algebraic model of the quantum Coulomb branch.

3. Change-of-Flavor Bimodules and the Twisted Trace in Tangle Invariants

Twisted traces also arise intrinsically from the construction and composition of change-of-flavor (or charge) bimodules, as detailed in Section 8.4. These bimodules, denoted $T^{\varphi+\nu}_\varphi$, are constructed via cylindrical diagrams with local relations twisted by the shift $d_e$. This Morita equivalence encodes the passage from one flavor (longitude) configuration to another and ensures that annular diagrammatic invariants are computed with the appropriate twisted trace.

This is especially salient in the tangle (and link) categorification framework: functors assigned to annular tangles (Theorem 7.10) and constructed via cup/cap bimodules involve variants of the twisted trace, as the diagrammatic relations carry explicit dependence on the twisting data. For closed annular links, the evaluation is strictly a twisted trace—integrating the link diagram over the noncommutative resolution with the twist incorporated in the local relations, thus capturing framing anomalies and grading shifts critical in knot homology.

4. Derived Equivalences, Tilting Bundles, and Diagrammatic Traces

In the setting of coherent sheaves and derived categories, twisting data plays a pivotal role in the algebraic structures underlying noncommutative resolutions. Explicitly, the endomorphism algebra of a tilting bundle $Q$ on the (BFN) resolution of $M$ is isomorphic to the quantum cylindrical KLRW algebra $R$. The trace on $R$ (as a central character or cyclic quotient) corresponds to a twisted center on the derived category of coherent sheaves. The twist matches any change-of-flavor (or wall-crossing) parameters, and tilting generators allow for the explicit computation of derived equivalences (see Theorem 8.14). In the realization of noncommutative Springer resolutions or Slodowy slices (Appendix A), the general framework of twisted traces specializes to recover known cases such as rational Cherednik algebras and noncommutative Springers.

5. Implications for Representation Theory and Knot Homology

The twisted-trace framework supports the categorification of quantum link invariants, especially those arising as Reshetikhin-Turaev invariants for minuscule representations in ADE types. In practice, the invariants computed—either as homological invariants agreeing with categorifications of Khovanov-Rozansky type-A/DADE knot homology, or as link invariants in the annular tangle calculus—depend crucially on the twisted trace arising from the local diagrammatic relations and global flavor-twist data. In particular, the non-commutative and wall-crossing aspects feed into the Morita/derived equivalence properties of the relevant categories, which are inextricably linked to the trace structures on diagrammatic and module categories.

6. Summary and Outlook

Twisted traces in quantum Coulomb branches, as realized in the framework of quantum cylindrical KLRW algebras and their tilting module categories, are not ad hoc functionals but arise systematically from the underlying diagrammatics, flavor-twist data, and wall-crossing/morita contexts. The twist manifests in both local (diagrammatic relations, annular tangle invariants) and global (change-of-flavor bimodules, derived equivalences, categorifications of link invariants) structures. This approach unifies representation-theoretic, topological, and geometric aspects of quantum Coulomb branches and their categorified invariants, providing a flexible and robust template for further developments in knot homology, categorified quantum invariants, and noncommutative symplectic resolutions (Webster, 2022).