Spherical Coulomb Branch Algebras

• Spherical Coulomb branch algebras are commutative and quantized rings derived from 3d N=4 gauge theories using the BFN convolution construction on the affine Grassmannian.
• They connect geometric representation theory with difference-operator algebras, bridging frameworks such as Cherednik algebras, Yangians, and DAHA.
• Explicit constructions involve monopole operators, minuscule generators, and abelianization techniques, leading to diverse topological and algebraic extensions.

Spherical Coulomb branch algebras are the commutative and quantized algebras attached to Coulomb branches of $3$-dimensional $\mathcal N=4$ gauge theories in the Braverman–Finkelberg–Nakajima program, together with related Weyl-invariant or idempotent corners that appear in Cherednik, Yangian, DAHA, and physical star-product realizations. In the cotangent case $\mathbf M=\mathbf N\oplus \mathbf N^*$, the spherical Coulomb branch algebra is $A:=H_*^{G_O}(R)$, its quantization is $A_\hbar:=H_*^{G_O\rtimes \mathbb C^\times}(R)$, and the Coulomb branch variety is $\mathcal M_C:=\operatorname{Spec} A$ (Braverman et al., 2016). For Jordan quiver gauge theories, the quantized Coulomb branch is isomorphic to spherical graded or cyclotomic rational Cherednik algebras, so the subject sits simultaneously in geometric representation theory, difference-operator algebra, and the algebra of monopole operators (Kodera et al., 2016).

1. Foundational constructions

The first systematic mathematical proposal for Coulomb branches uses the gauged nonlinear sigma-model on the $2$-sphere $S^2\cong \mathbb{CP}^1$. In that formulation, a $3$d $\mathcal N=4$ theory is specified by a compact Lie group $\mathcal N=4$0 and a quaternionic representation $\mathcal N=4$1, and the “spherical” coordinate ring is proposed from vanishing cycle cohomology for a holomorphic Chern–Simons functional on the space of fields. The basic equations are

$\mathcal N=4$2

and the holomorphic Chern–Simons functional is

$\mathcal N=4$3

For $\mathcal N=4$4, the resulting graded dimensions agree exactly with the monopole formula

$\mathcal N=4$5

with

$\mathcal N=4$6

This “spherical” terminology refers here to the $\mathcal N=4$7-based construction and to the $\mathcal N=4$8-invariant viewpoint that appears in the sequel (Nakajima, 2015).

The BFN construction replaces the $\mathcal N=4$9-model by a convolution algebra on the affine Grassmannian. For $\mathbf M=\mathbf N\oplus \mathbf N^*$0, $\mathbf M=\mathbf N\oplus \mathbf N^*$1, $\mathbf M=\mathbf N\oplus \mathbf N^*$2, and cotangent matter $\mathbf M=\mathbf N\oplus \mathbf N^*$3, one defines

$\mathbf M=\mathbf N\oplus \mathbf N^*$4

The convolution product on equivariant Borel–Moore homology yields

$\mathbf M=\mathbf N\oplus \mathbf N^*$5

with $\mathbf M=\mathbf N\oplus \mathbf N^*$6 commutative, $\mathbf M=\mathbf N\oplus \mathbf N^*$7 a flat deformation, and $\mathbf M=\mathbf N\oplus \mathbf N^*$8 the Coulomb branch (Braverman et al., 2016). The same framework supplies the Cartan subalgebra $\mathbf M=\mathbf N\oplus \mathbf N^*$9 and the integrable-system map

$A:=H_*^{G_O}(R)$0

whose generic fiber is $A:=H_*^{G_O}(R)$1. In the cotangent case, the Poisson bracket on $A:=H_*^{G_O}(R)$2 is the semiclassical limit of the commutator and is symplectic on the smooth locus (Braverman et al., 2016).

2. The meaning of “spherical”

In the cited literature, the adjective “spherical” has several related meanings. In the $A:=H_*^{G_O}(R)$3-based approach it refers to the gauged sigma-model on the $A:=H_*^{G_O}(R)$4-sphere and, in the sequel’s convolution picture, to the $A:=H_*^{G_O}(R)$5-invariant setting (Nakajima, 2015). In the BFN homological definition it refers to the $A:=H_*^{G_O}(R)$6-equivariant convolution algebra, in contrast with the Iwahori version

$A:=H_*^{G_O}(R)$7

for which the spherical algebra is obtained as the corner $A:=H_*^{G_O}(R)$8 cut out by the idempotent attached to the trivial representation of the finite Weyl group (Hilburn et al., 2020).

On the algebraic side, “spherical” often means an idempotent subalgebra $A:=H_*^{G_O}(R)$9. For the Jordan quiver this is the spherical graded Cherednik algebra $A_\hbar:=H_*^{G_O\rtimes \mathbb C^\times}(R)$0 in the unframed case and the spherical cyclotomic rational Cherednik algebra $A_\hbar:=H_*^{G_O\rtimes \mathbb C^\times}(R)$1 in the framed case (Kodera et al., 2016). For star-shaped quivers, Webster’s spherical subalgebra is

$A_\hbar:=H_*^{G_O\rtimes \mathbb C^\times}(R)$2

where the BGG–Demazure operators are $A_\hbar:=H_*^{G_O\rtimes \mathbb C^\times}(R)$3 (Dimofte et al., 2018).

A closely related use appears in physical star-product constructions. There the gauge-invariant operator algebra is obtained by Weyl averaging the abelianized monopole-difference operators, and the gauge-invariant, Weyl-averaged sector is identified with the spherical Coulomb branch algebra $A_\hbar:=H_*^{G_O\rtimes \mathbb C^\times}(R)$4 (Dedushenko et al., 2018). Sphere quantization sharpens this by introducing a spherical vector $A_\hbar:=H_*^{G_O\rtimes \mathbb C^\times}(R)$5, a positive twisted trace, and commuting left and right actions of the Coulomb-branch algebra on a Hilbert space completion (Gaiotto, 2023). A common misconception is that “spherical” names a single uniform construction; the sources instead use it for a family of equivalent or parallel operations: $A_\hbar:=H_*^{G_O\rtimes \mathbb C^\times}(R)$6-origin, $A_\hbar:=H_*^{G_O\rtimes \mathbb C^\times}(R)$7-equivariance, Weyl averaging, and idempotent projection.

3. Jordan quiver gauge theories and Cherednik algebras

The most explicit identification is due to Kodera–Nakajima for the $A_\hbar:=H_*^{G_O\rtimes \mathbb C^\times}(R)$8d $A_\hbar:=H_*^{G_O\rtimes \mathbb C^\times}(R)$9 Jordan quiver gauge theory with gauge group $\mathcal M_C:=\operatorname{Spec} A$0 and matter representation

$\mathcal M_C:=\operatorname{Spec} A$1

The quantized Coulomb branch $\mathcal M_C:=\operatorname{Spec} A$2 is a flat $\mathcal M_C:=\operatorname{Spec} A$3-graded deformation over $\mathcal M_C:=\operatorname{Spec} A$4 of the commutative Coulomb branch, and the main theorem states: $\mathcal M_C:=\operatorname{Spec} A$5 with the same parameters $\mathcal M_C:=\operatorname{Spec} A$6 on both sides and cyclotomic parameters related by

$\mathcal M_C:=\operatorname{Spec} A$7

The overall shift $\mathcal M_C:=\operatorname{Spec} A$8 is irrelevant, and setting $\mathcal M_C:=\operatorname{Spec} A$9 loses no generality (Kodera et al., 2016).

The BFN difference-operator model is central in this identification. Let $2$0 be the ring of $2$1-difference operators on $2$2, generated by $2$3 with

$2$4

After localizing by inverting $2$5, one defines

$2$6

$2$7

The quantized Coulomb branch is the subalgebra of the localized difference-operator ring generated by the operators $2$8, $2$9, and symmetric polynomials in $S^2\cong \mathbb{CP}^1$0 (Kodera et al., 2016).

On the Cherednik side, the embeddings are explicit. There is an embedding

$S^2\cong \mathbb{CP}^1$1

such that

$S^2\cong \mathbb{CP}^1$2

and

$S^2\cong \mathbb{CP}^1$3

This realizes the isomorphism through the basic Coulomb generators $S^2\cong \mathbb{CP}^1$4, $S^2\cong \mathbb{CP}^1$5, and symmetric functions in the $S^2\cong \mathbb{CP}^1$6 (Kodera et al., 2016).

The same paper also isolates small-rank examples. For $S^2\cong \mathbb{CP}^1$7, $S^2\cong \mathbb{CP}^1$8,

$S^2\cong \mathbb{CP}^1$9

and similarly $3$0. For $3$1, $3$2, one checks $3$3 with $3$4, identifying the Coulomb branch with the spherical rational Cherednik algebra in that case (Kodera et al., 2016).

4. Generators, abelianization, and Yangian structures

For general quiver gauge theories, the spherical quantized Coulomb branch admits both intrinsic convolution definitions and explicit generating sets. In Weekes’ formulation, for a simple quiver with gauge group $3$5, the spherical quantized Coulomb branch is

$3$6

the Iwahori algebra is

$3$7

and

$3$8

where $3$9 is the spherical idempotent. Moreover, $\mathcal N=4$0 is a matrix algebra over $\mathcal N=4$1 of rank $\mathcal N=4$2 (Weekes, 2019).

The same work proves that $\mathcal N=4$3 is generated by dressed minuscule monopole operators $\mathcal N=4$4, $\mathcal N=4$5 together with $\mathcal N=4$6, and that there is a surjective homomorphism

$\mathcal N=4$7

In finite ADE type this extends to a surjection of graded $\mathcal N=4$8-algebras

$\mathcal N=4$9

This places spherical Coulomb branch algebras within the same presentation-theoretic range as shifted Yangians, and in finite ADE with $\mathcal N=4$00 dominant the surjection becomes an isomorphism (Weekes, 2019).

The Jordan quiver case admits a parallel Yangian statement. For the framed theory, if $\mathcal N=4$01 denotes the algebra obtained by deforming a subalgebra of the affine Yangian of $\mathcal N=4$02, then there is a surjective homomorphism

$\mathcal N=4$03

sending

$\mathcal N=4$04

Accordingly, the quantized Coulomb branch is a deformation of a subquotient of $\mathcal N=4$05 (Kodera et al., 2016).

Abelianization techniques give another explicit incarnation. For star-shaped quivers, the quantized abelianized algebra $\mathcal N=4$06 contains rescaled monopoles

$\mathcal N=4$07

and BGG–Demazure operators $\mathcal N=4$08. The spherical Coulomb branch algebra is then

$\mathcal N=4$09

For star-shaped quivers there is strong evidence that

$\mathcal N=4$10

so the spherical subalgebra coincides with the full quantized Coulomb-branch algebra in that setting (Dimofte et al., 2018).

5. Modules, traces, and physical realizations

The BFN construction supports a Springer-theoretic representation theory. For a representation $\mathcal N=4$11 of a reductive group $\mathcal N=4$12, one sets

$\mathcal N=4$13

and for $\mathcal N=4$14 defines the BFN Springer fibre

$\mathcal N=4$15

If $\mathcal N=4$16 admits a dualizing sheaf, then

$\mathcal N=4$17

carries a natural action of the quantized spherical Coulomb branch algebra $\mathcal N=4$18. Under mild hypotheses these modules are weight modules, and if $\mathcal N=4$19 is $\mathcal N=4$20-semistable then $\mathcal N=4$21 lies in category $\mathcal N=4$22 (Hilburn et al., 2020). This extends the analogy between Springer fibres and representation spaces from the nilpotent cone to Coulomb branches.

A complementary realization comes from the $\mathcal N=4$23-dimensional topological sector on $\mathcal N=4$24. In that framework the Coulomb-branch operator algebra is generated by vectormultiplet scalars and monopole operators represented by explicit difference operators acting on hemisphere wavefunctions. The gauge-invariant, Weyl-averaged sector is the spherical Coulomb branch algebra $\mathcal N=4$25, and the trace map

$\mathcal N=4$26

computes protected correlation functions (Dedushenko et al., 2018). For good and ugly theories, this trace guarantees the truncation property of the star product. The physical construction thus gives a direct star-product realization of the spherical BFN algebra.

Sphere quantization sharpens the analytic structure. The spherical Coulomb branch algebra is presented as the protected noncommutative algebra of Coulomb-branch local operators quantized by $\mathcal N=4$27 correlation functions; it comes with a positive twisted trace $\mathcal N=4$28, an anti-linear involution $\mathcal N=4$29, and a Hermitian pairing

$\mathcal N=4$30

The resulting Hilbert space carries commuting actions of $\mathcal N=4$31 and $\mathcal N=4$32 and a distinguished spherical vector $\mathcal N=4$33, while Coulomb-branch flavor symmetries lead to Harish–Chandra bimodules $\mathcal N=4$34 and direct-integral decompositions

$\mathcal N=4$35

This gives an analytic form of spherical representation theory for Coulomb-branch algebras (Gaiotto, 2023).

For noncotangent theories with half-hypermultiplets in general pseudoreal representations, the hemisphere construction uses Dirichlet boundary conditions for gauge fields and a careful treatment of half-hypermultiplet boundary data. Under anomaly-free assumptions, the hemisphere partition function $\mathcal N=4$36 is a cyclic vector for a spherical module, and the authors expect

$\mathcal N=4$37

after appropriate completion at $\mathcal N=4$38 and localization of equivariant parameters, although a full proof is not given (Dedushenko et al., 30 Dec 2025). This suggests that spherical Coulomb branch algebra methods extend beyond cotangent matter, but only with explicit anomaly constraints.

6. DAHA, skein, imprimitive, and K-theoretic extensions

Recent work places spherical Coulomb branch algebras in a broader web of Hecke-type structures. For coadjoint matter, the affine-flag or Iwahori Coulomb branch

$\mathcal N=4$39

is isomorphic to the trigonometric DAHA $\mathcal N=4$40, and the Grassmannian spherical Coulomb branch satisfies

$\mathcal N=4$41

where $\mathcal N=4$42 is the spherical tDAHA and the shift $\mathcal N=4$43 is explicit (Chan et al., 27 Jan 2026). In the same setting, the action on $\mathcal N=4$44 is computed on stable envelopes by

$\mathcal N=4$45

This identifies the spherical Coulomb branch with a concrete Demazure–Lusztig action.

For the imprimitive reflection groups $\mathcal N=4$46, the same $\mathcal N=4$47d $\mathcal N=4$48 theory that yields $\mathcal N=4$49 acquires a $\mathcal N=4$50 realization after adding vortex-type line defects and imposing $\mathcal N=4$51-cyclic masses. The quantized Ext-algebras of local operators then identify with graded pieces of the partially spherical and spherical rational Cherednik algebras: $\mathcal N=4$52 Direct sums over degrees recover $\mathcal N=4$53 and $\mathcal N=4$54, and these spherical algebras are shown to be principal Galois orders (LePage et al., 2019).

In multiplicative form, spherical quantized K-theoretic Coulomb branches also meet low-dimensional topology. For compact oriented surfaces of genus at most one with boundary, one associates a quantized K-theoretic Coulomb branch $\mathcal N=4$55. For the three-holed sphere, four-holed sphere, and one-holed torus, this algebra is related to the Kauffman bracket skein algebra, with parameter matching

$\mathcal N=4$56

In the four-holed sphere and one-holed torus cases the identification is made through explicit spherical DAHA difference-reflection operators and BFN minuscule monopole operators (Allegretti et al., 2024).

A $\mathcal N=4$57d analogue appears for $\mathcal N=4$58 gauge theory with four fundamentals and one anti-symmetric hypermultiplet. There the algebra of BPS loop operators in the $\mathcal N=4$59-background is a deformation quantization of the Coulomb branch, and for $\mathcal N=4$60 the quantized Coulomb branch agrees with the polynomial representation of the spherical DAHA of $\mathcal N=4$61-type. For general $\mathcal N=4$62, the minimal ’t Hooft loop agrees with the Koornwinder operator, giving evidence that the quantized Coulomb branch is isomorphic to the spherical DAHA of $\mathcal N=4$63-type (Yoshida, 19 Mar 2025).

These extensions show that spherical Coulomb branch algebras do not form a single isolated class. They recur as spherical Cherednik algebras, spherical tDAHA, shifted-Yangian quotients, Ext-algebras of line defects, Weyl-invariant star-product algebras, and quantized K-theoretic algebras related to skein theory. A plausible implication is that “spherical Coulomb branch algebra” is best understood as a structural role—$\mathcal N=4$64-equivariant, Weyl-invariant, or idempotent-projected—rather than as a single presentation.