Fukaya categories of Coulomb branches as unique deformations

Abstract: The symplectic geometry of Coulomb branches is complicated and it is particularly difficult to determine their Fukaya categories. Relative Fukaya categories present an approach to circumvent these difficulties by first computing the Fukaya category of the complement of a divisor and then solving a deformation problem. In this paper, we apply this approach to the specific case of horizontal Hilbert schemes by removing their matter divisor and narrowing down the set of possible deformations through an additional $\mathbb{Z}²$-grading. We utilize an existing description of the Fukaya category after removal of the matter divisor, in particular we use a specific generating Lagrangian and the identification between its endomorphism algebra and the NilHecke algebra. The core of this paper consists of solving the deformation problem, after which we recover the result of Aganagic et al.

• The paper proves that the relative Fukaya category of planar horizontal Hilbert schemes is equivalent to the NilHecke algebra, using deformation rigidity instead of explicit holomorphic disk counts.
• A Z²-grading and Hochschild cohomology calculation show that the undeformed algebra has exactly one nontrivial deformation direction, parameterized by a scalar.
• The result recovers the wrapped Fukaya category–KLRW algebra correspondence in this setting while highlighting open questions about extending deformation uniqueness to broader Coulomb branches.

The paper "Fukaya categories of Coulomb branches as unique deformations" by Jasper van de Kreeke (2607.00146) provides an alternative proof of the identification between the wrapped Fukaya category of a multiplicative Coulomb branch and a Khovanov–Lauda–Rouquier–Webb (KLRW) algebra, in the special case of planar horizontal Hilbert schemes. Rather than performing the explicit disk-counting computations of Aganagic–Danilenko–Li–Shende–Zhou [ADLSZ], the author recovers their result by showing that the relative Fukaya category is forced to be the NilHecke algebra because it is the unique non-trivial $\mathbb{Z}^2$-graded deformation of a common base algebra.

Background and strategy

Coulomb branches of 3d $\mathcal{N}=4$ gauge theories were given a rigorous mathematical definition as spectra of K-theoretic algebras by Braverman–Finkelberg–Nakajima. Aganagic et al. computed the Fukaya categories of multiplicative Coulomb branches of framed quiver settings via an explicit generating system of Lagrangians $T_\theta$ indexed by strand diagrams, establishing an equivalence

$$TwKLRW(\Gamma, d) \simeq relFuk(\mathcal{M}^\times(\Gamma,d), W).$$

This paper restricts to planar horizontal Hilbert schemes $Y = Hilb_k(\mathbb{C}^* \times \mathbb{C})$, for which the relevant KLRW algebras are the classical NilHecke algebras $NH_k$. The proof strategy follows Seidel's relative Fukaya technique: remove the matter divisor (the big diagonal where two $y_i$-coordinates coincide), compute the Fukaya category of the complement using the known description from ADLSZ, and then solve the resulting deformation problem. The key observation is that both the relative Fukaya category $relFuk(Y_W)_L$ and the algebra $NH_k$ are deformations of the same base object—the $\hbar = 0$ specialization $\mathcal{N}=4$0—and that an additional $\mathcal{N}=4$1-grading leaves only one possible non-trivial deformation over $\mathcal{N}=4$2, forcing the two to be equivalent.

Graded deformation theory framework

A substantial portion of the paper develops a graded version of $\mathcal{N}=4$3-deformation theory. The author defines completed $\mathcal{N}=4$4-graded deformation bases—complete local Noetherian $\mathcal{N}=4$5-algebras whose quotients $\mathcal{N}=4$6 carry compatible $\mathcal{N}=4$7-gradings—and notes that such bases need not admit direct sum decompositions into homogeneous components (e.g., $\mathcal{N}=4$8 with $\mathcal{N}=4$9 is not itself $T_\theta$0-graded as a vector space). On this foundation, graded Maurer-Cartan elements and graded gauge equivalences are defined for $T_\theta$1-algebras and $T_\theta$2-categories, with the Hochschild DGLA controlling deformations.

Two technical results carry significant weight. First, a graded push-pull lemma establishes that any graded Maurer-Cartan element is homogeneously gauge equivalent to its image under the minimal model inclusion-composition $T_\theta$3. Second, the deformation theory of quiver algebras with reduction systems (following Chouhy–Solotar and Barmeier–Wang) is extended to the graded setting: when the quiver and reduction relations are $T_\theta$4-homogeneous, the projective bimodule resolution, the induced $T_\theta$5-structure on $T_\theta$6, and the quasi-isomorphism to the Hochschild DGLA are all $T_\theta$7-homogeneous. This allows infinitesimal deformations to be read off directly as deformed reduction rules that remain reduction-unique.

The paper also includes supporting lemmas on perfect modules over deformed algebras, showing that flat deformations lift generators, kernels, images, and projective resolutions—establishing that $T_\theta$8 is a loose object-cloning deformation of $T_\theta$9.

Geometry of the horizontal Hilbert scheme

The horizontal Hilbert scheme $$TwKLRW(\Gamma, d) \simeq relFuk(\mathcal{M}^\times(\Gamma,d), W).$$0 is realized as the symmetrized variety of points $$TwKLRW(\Gamma, d) \simeq relFuk(\mathcal{M}^\times(\Gamma,d), W).$$1 with $$TwKLRW(\Gamma, d) \simeq relFuk(\mathcal{M}^\times(\Gamma,d), W).$$2, together with slope data $$TwKLRW(\Gamma, d) \simeq relFuk(\mathcal{M}^\times(\Gamma,d), W).$$3 satisfying $$TwKLRW(\Gamma, d) \simeq relFuk(\mathcal{M}^\times(\Gamma,d), W).$$4. Three Liouville structures are constructed:

• $$TwKLRW(\Gamma, d) \simeq relFuk(\mathcal{M}^\times(\Gamma,d), W).$$5: the full Hilbert scheme with pullback Liouville form from an affine embedding.
• $$TwKLRW(\Gamma, d) \simeq relFuk(\mathcal{M}^\times(\Gamma,d), W).$$6: $$TwKLRW(\Gamma, d) \simeq relFuk(\mathcal{M}^\times(\Gamma,d), W).$$7 cut along the locus $$TwKLRW(\Gamma, d) \simeq relFuk(\mathcal{M}^\times(\Gamma,d), W).$$8 for the potential $$TwKLRW(\Gamma, d) \simeq relFuk(\mathcal{M}^\times(\Gamma,d), W).$$9, forming a Liouville sector via chimney attachment.
• $Y = Hilb_k(\mathbb{C}^* \times \mathbb{C})$0: obtained by removing a neighborhood of the big diagonal, truncating ends in the $Y = Hilb_k(\mathbb{C}^* \times \mathbb{C})$1- and $Y = Hilb_k(\mathbb{C}^* \times \mathbb{C})$2-directions, smoothing corners, quotienting by $Y = Hilb_k(\mathbb{C}^* \times \mathbb{C})$3, and applying the same potential cut.

A simplified model expresses $Y = Hilb_k(\mathbb{C}^* \times \mathbb{C})$4 as the symmetric product of $Y = Hilb_k(\mathbb{C}^* \times \mathbb{C})$5 base rectangles (with two stops each) and $Y = Hilb_k(\mathbb{C}^* \times \mathbb{C})$6 fiber cylinders (one stop each), which facilitates explicit Floer-theoretic computation. The standard Lagrangian $Y = Hilb_k(\mathbb{C}^* \times \mathbb{C})$7 is a product of radial segments in the annuli and vertical segments in the rectangles; its wrappings $Y = Hilb_k(\mathbb{C}^* \times \mathbb{C})$8 wind positively around the fiber cylinders. The key input from ADLSZ is the identification

$Y = Hilb_k(\mathbb{C}^* \times \mathbb{C})$9

where generators $NH_k$0 correspond to self-intersections in the fiber direction and $NH_k$1 to crossings of adjacent strands.

The $NH_k$2-grading and relative Fukaya category

The central innovation is the introduction of two independent winding-number gradings via grading functions

$NH_k$3

For contractible Lagrangians avoiding the zero loci of these functions, degrees of intersection points and holomorphic disks are defined by integrals of $NH_k$4 along boundary paths. Non-negativity of disk degrees follows from the argument principle applied to $NH_k$5. Applying the construction twice yields a $NH_k$6-graded deformation $NH_k$7 over $NH_k$8; setting $NH_k$9 produces the desired deformation over $y_i$0.

On the algebraic side, $y_i$1 carries the $y_i$2-grading $y_i$3, $y_i$4, and all defining relations ($y_i$5, dot-passing relations with coefficient $y_i$6, braid relations) are homogeneous. Graphically, the $y_i$7-degree of $y_i$8 equals 1 because $y_i$9 makes a half-turn around zero, while the $relFuk(Y_W)_L$0-degree of $relFuk(Y_W)_L$1 equals $relFuk(Y_W)_L$2 because $relFuk(Y_W)_L$3 winds negatively around the origin. The existing identification $relFuk(Y_W)_L$4 is verified to respect this $relFuk(Y_W)_L$5-grading.

Hochschild cohomology computation

Presenting $relFuk(Y_W)_L$6 as a quiver algebra with a reduction-unique reduction system, the author computes the degree-$relFuk(Y_W)_L$7 component of Hochschild cohomology. The space $relFuk(Y_W)_L$8 has basis elements determined by values on the ambiguity paths $relFuk(Y_W)_L$9, $NH_k$0, and $NH_k$1 ($NH_k$2), with all other ambiguities mapping to zero. A lengthy but systematic evaluation of $NH_k$3 on 1-ambiguities forces $NH_k$4 for all off-diagonal indices and $NH_k$5 for a single scalar $NH_k$6. The result is:

$NH_k$7

where $NH_k$8, $NH_k$9, and $\hbar = 0$0 vanishes on all other degree-$\hbar = 0$1 ambiguities. Moreover, $\hbar = 0$2 for $\hbar = 0$3. Consequently,

$\hbar = 0$4

and distinct values of $\hbar = 0$5 yield pairwise non-gauge-equivalent Maurer-Cartan elements. All deformations and functors are automatically locally algebraic (defined over $\hbar = 0$6 rather than requiring formal power series).

Classification and main theorem

Every non-trivial $\hbar = 0$7-graded deformation is classified up to gauge equivalence by a scalar $\hbar = 0$8, extracted concretely as the coefficient satisfying $\hbar = 0$9. Explicit representatives $\mathcal{N}=4$00 are given by deformed reduction rules $\mathcal{N}=4$01 on the ambiguities $\mathcal{N}=4$02. Using the graded push-pull lemma and a substitution automorphism of the deformation base, any non-trivial deformation is shown to be $\mathcal{N}=4$03-isomorphic (though not necessarily $\mathcal{N}=4$04-linearly isomorphic) to the standard deformation $\mathcal{N}=4$05.

Since ADLSZ established that $\mathcal{N}=4$06 has nonzero scalar $\mathcal{N}=4$07—i.e., it is a non-trivial deformation—the uniqueness result immediately yields the main theorem:

Theorem. There is an isomorphism of $\mathcal{N}=4$08-categories

$\mathcal{N}=4$09

Specializing at $\mathcal{N}=4$10 recovers the ADLSZ result $\mathcal{N}=4$11 without performing their final disk-counting assembly. The functor $\mathcal{N}=4$12 may involve a rescaling $\mathcal{N}=4$13, so it is not canonical over the deformation base.

Limitations and open questions

Several caveats qualify the scope of the result. First, the argument relies on the prior computation of $\mathcal{N}=4$14 from ADLSZ section 7, so the paper does not eliminate dependence on earlier work entirely—it circumvents only the disk-counting of ADLSZ section 8. Second, the non-triviality of the deformation $\mathcal{N}=4$15 (nonzero scalar $\mathcal{N}=4$16) is invoked from ADLSZ rather than proven intrinsically here; the appendix offers a partial geometric approach via three explicitly constructed Lagrangians with intersection products $\mathcal{N}=4$17, but this is presented as indicative rather than complete. Third, several technical points are flagged as incomplete in the source: the definition of Liouville manifolds-with-boundary is deferred, order conventions in the Barmeier–Wang resolution require correction, and the claim that every completed graded deformation base arises as a completion of a graded noetherian algebra is left unresolved. Fourth, the generalization beyond planar horizontal Hilbert schemes—to arbitrary framed quivers or non-planar Coulomb branches—is not addressed; whether the corresponding KLRW algebras also admit unique $\mathcal{N}=4$18-graded deformations remains open. Finally, the appendix observes empirically that certain families of twisted complexes appear uncurvable in $\mathcal{N}=4$19, but no general criterion is established.

Conclusion

This paper demonstrates that for horizontal Hilbert schemes, the equivalence between Fukaya categories of Coulomb branches and NilHecke algebras can be obtained through rigidity of graded deformations rather than explicit holomorphic disk enumeration. The essential mechanism—an additional $\mathcal{N}=4$20-grading collapsing the deformation space to a one-parameter family with a unique non-trivial class—provides a template that may extend to other Coulomb branch settings where generating Lagrangians and divisor complements are understood, though confirming this requires resolving the open questions identified above.