The role of Coulomb branches in 2D gauge theory

I give a simple construction of certain Coulomb branches $C_{3,4}(G;E)$ of gauge theory in 3 and 4 dimensions defined by Nakajima et al. for a compact Lie group $G$ and a polarisable quaternionic representation $E$. The manifolds $C(G; 0)$ are abelian group schemes (over the bases of regular adjoint $G_c$-orbits, respectively conjugacy classes), and $C(G;E)$ is glued together from two copies of $C(G;0)$ shifted by a rational Lagrangian section $\varepsilon_V$, the Euler class of the index bundle of a polarisation $V$ of $E$. Extending the interpretation of $C_3(G;0)$ as"classifying space"for topological 2D gauge theories, I characterise functions on $C_3(G;E)$ as operators on the equivariant quantum cohomologies of $M\times V$, for all compact symplectic $G$-manifolds $M$. The non-commutative version has an analogous description in terms of the $\Gamma$-function of $V$, appearing to play the role of Fourier transformed J-function of the gauged linear Sigma-model $V/G$.


Introduction
Associated to a compact connected Lie group G and a quaternionic representation E, there are expected to be Coulomb branches C 3,4 (G; E) of N = 4 SUSY gauge theory in dimensions 3 and 4, with matter fields in the representation E. They ought to be components of the moduli space of vacua, representing solutions of the monopole equations with singularities. Following early physics leads [SW, CH] and more recent calculations [CHMZ], a precise definition for these spaces was proposed in the series of papers [N, BFN] by Nakajima and collaborators in the case when E is polarisable (isomorphic to V ⊕ V ∨ for some complex representation V ). Abelian groups were handled independently by Bullimore, Dimofte and Gaitto [BDG] from a physics perspective, while the case of the zero representation had been developed in [BFM], although only later recognised as such [T1, T2].
The C 3,4 are expected to be hyperkähler (insofar as this makes sense for singular spaces), with C 3 carrying an SU(2) hyperkähler rotation. They are constructed in [BFN] as algebraic Poisson spaces, with C × -action in the case of C 3 . We shall rediscover them as such in a simpler construction, which illuminates their relevance to 2-dimensional gauge theory: the C 3,4 for polarised E are built from their more basic versions for the zero representation E = 0. Specifically, they are affinisations of a space constructed by partial identification of two copies of C (G; 0). The identification is implemented by a Lagrangian shift along the fibers of the (Toda) integrable system structure of the C 3,4 , and its effect is to impose growth conditions, selecting a subring of regular functions. The non-commutative versions quantise this Lagrangian shift of the C 3 into conjugation by the Γ-class of the representation (respectively, a specialisation of its Jackson-Γ p version for C 4 ).
The reconstruction results, Theorems 1, 2 and 4, are more elementary than their 2D gauge theory interpretation, but it is the latter which seems to give them meaning. In compromise, I have attempted to isolate the gauge theory comments (for which a rigorous treatment has not yet been published) into paragraphs whose omission does not harm the remaining mathematics. I have also separated the non-commutative version of the story into the final section: its meshing with quantum cohomology theory is still incomplete.
A pedestrian angle on this paper's results is the abelianisation underlying the calculations -a reduction to the Cartan subgroup H and Weyl group W . This is seen in the description of the Euler Lagrangians (4.2) which are used to build the "material" Coulomb branches from C (G; 0), and is closely related to the abelianised index formula in [TW], which ends up governing the Gauged Linear Sigma model (GLSM). Oversimplifying a bit, the interesting difference between G and its abelian reduction is already contained in C (G; 0), the effect of adding a polarised representation being captured by a calculus reminiscent of toric geometry. Abelianization also has an explicit manifestation, similar to the Weyl character formula, in an isomorphism C 3,4 (G; E) ∼ = C 3,4 H; E ⊖ (g/h) ⊕2 ) /W, whenever the formal difference on the right is a genuine representation of H; a quick argument has been included in the appendix, as it appeared not to be well known.
A qualification is in order: the simple characterization applies to the variants of C 3,4 enhanced by the (complex) mass parameters [BDG], or by the more general flavor symmetries [BFN,3(vii)]. The original spaces are subsequently recovered by setting the mass parameters to zero; however, at least one parameter, effecting a compactification of V , must be initially turned on. The moral explanation is easily expressed in physics language, and in a way that can be made mathematically precise. What my construction does is characterize the 3-dimensional topological gauge theories underlying the C 3,4 by means of their 2D topological boundary theories -a characterization accurate enough, at least, to determine their expected Coulomb branches. For pure gauge theory (E = 0), I explained in [T2, §6] in what sense the (A-models of) flag varieties of G supply a complete family of boundary theories, the Coulomb branch being akin to a direct integral of those: more precisely, it has a Lagrangian foliation by the mirrors of flag varieties. With E-matter added, a new boundary theory, the GLSM of V by G (again in the A-version) must be introduced as a factor, carrying the action of the matter fields. Since V is not compact, this model must be regularized by the inclusion of mass parameters. There is a mathematically sound version of this statement: the GLSM is a 2D TQFT over the ring of rational functions in the complex mass parameters, and has singularities at zero mass.
The same perspective points to a difficulty in extending these constructions when E cannot be polarised. There is no a priori reason why a 3D TQFT should be characterized by its topological boundary theories; Chern-Simons theory (for general levels) is a notorious counter-example [KS]. A Ginvariant Lagrangian V ⊂ E seems to provide (in addition to the flag varieties) a generating boundary condition for the 3D gauge theory with matter -specifically, it is a domain wall between G-gauge theory with and without matter. No substitute is apparent in general. Clearly this deserves further thought. One obstacle is that 3D gauge theory gives only a partially defined TQFT, so its mathematical structure is incompletely settled, and the list of desiderata for a presumptive reconstruction is not known with clarity. as a classifying space for topological 2-dimensional gauge theories was developed in [T1, T2], where the space was denoted BFM(G ∨ ). As we now recall, this virtue of C 3 (G; 0) must be read in the sense of semiclassical symplectic calculus, and not as a spectral theoremà la Gelfand-Naimark. It gives the "mirror description" of the gauged A-models in 2 dimensions.
(2.2) Relation with quantum cohomology. A partial summary of the classifying property of C 3 (G; 0) is that its regular functions (sometimes called the ring of chiral operators) act on the equivariant quantum cohomologies QH * G (M ) of compact G-Hamiltonian symplectic manifolds M , in a manner making the E 2 structure 1 on QH * G (M ) compatible with the E 3 structure defined by the Poisson tensor on C 3 . This lays out QH * G (M ) as a sheaf over C 3 , which turns out to have Lagrangian support (Remark 2.3 below). This construction generalises Seidel's theorem [S] on the action of π 1 G on QH * (M ), as well as the shift operators on QH * and their equivariant extensions [OP]. In fact, these latter ingredients are the "leading order" description of the story of [T2] in the case of torus actions. A similar narrative applies to C 4 (G; 0) and equivariant quantum K-theory (minding, however, the orbifold nature of C 4 for general G, see §3) even though the general framework for K-theoretic mirror symmetry is incompletely understood.
2.3 Remark. The shortest argument for the Lagrangian property of QH * G (M ) passes to the noncommutative Coulomb branches of §7, over which the versions of QH * G (M ) equivariant under loop rotation (which are related to cyclic homology of the Fukaya category) are naturally modules. The Lagrangian property is now a consequence of the integrability of characteristics [G] supplemented by finiteness of QH * G (M ) over H * (BG).
(2.4) Coulomb branches with matter. The universal property of the C (G; 0) leaves the spaces C (G; E) in search of a rôle. Their new characterisation addresses this riddle. Namely, the Seidel shift operators act on QH * (M ) only when M is compact; for more general spaces, the most we expect is an action on the symplectic cohomology, when the latter is defined [R]. Equivariant symplectic cohomology SH * G (X) is sometimes a localisation of QH * G (X), in which case the space C 3 (G; 0) will capture a dense open part of QH * G (X), with portions lost at infinity. Notably, this is the case when X = M × V , with compact M and a linear G-space V . The lost part of QH * G (M × V ) can be captured in a second chart of C (G; 0), shifted from the original by the effect of the functor M → M × V .
This shift is implemented as follows. The tensor product defines a symmetric monoidal structure on 2-dimensional TQFTs with G-gauge symmetry. This structure is mirrored in the classifying space C 3 (G; 0) into a multiplication along an abelian group structure over Spec H G * (point). (The latter is isomorphic to the space g reg C /G C of regular adjoint orbits, and the projection exhibits C 3 (G; 0) as a fiberwise group-completion of the classical Toda integrable system; see §3.3.) The operation QH * G (M ) SH * G (M × V ) is implemented by multiplication by a certain rational Lagrangian section ε V of this group scheme, whose structure sheaf is SH * G (V ). The Lagrangian ε V should be regarded as the gauged B-model mirror of V : see Remark 4.3.
The precise statement of the main results requires preparation and is postponed to §4; the remainder of this section develops two key examples.
(2.5) Example I: G = U 1 , with the standard representation L. We have with C × dual to U 1 : the coordinates τ and z on the two factors generate H 2 (BU 1 ) and π 1 U 1 . The canonical symplectic form dτ ∧dz/z also admits an intrinsic topological definition, in terms of a natural circle action on BU 1 × ΩU 1 (cf. §3.1 and §7.2 below). One usually defines the toric mirror of the space L as the function (super-potential) ψ(z) = z on the space C × . The differential dψ defines the Lagrangian ε L := {τ = z} ⊂ T ∨ C × . View ε L instead as the rational section τ → z = τ of the projection T ∨ C × → C to the τ -coordinate, and note in passing the Legendre transform ψ * (τ ) = τ (log τ − 1) of ψ, in the sense that ε L = exp(dψ * ).
Functions on ε L are identified with C[τ ± ]; this is the U 1 -equivariant symplectic cohomology of L, rather than its quantum cohomology C[τ ]. We can recover the full quantum cohomology by gluing, onto the open set τ = 0 in (2.6), a second copy T ∨ C × , with coordinates τ and z ′ = z/τ . This gluing is compatible with projection to the τ -coordinate and leads to the space C 2 \ {0}, with coordinates (x, y) = (z, τ /z), living over the line τ = xy. The section ε L closes now to the line y = 1, identified by projection with the full τ -axis.
In [BDG, BFN], C 3 (U 1 ; L ⊕ L ∨ ) is taken to be the affine completion C 2 = Spec C[x, y]. The following characterisation is now obvious: 2.7 Proposition. C[x, y] is the subring of regular functions f (τ, z) on T ∨ C × with the property that f (τ, zτ ) is also regular.
Our Lagrangian ε L is related to the Euler class of the index bundle as follows. Denote by Pic(P 1 ) the moduli stack of holomorphic line bundles on P 1 ; its equivariant homotopy type is the stack BU 1 × ΩU 1 implicit in (2.6). Over P 1 × Pic(P 1 ) lives the universal line bundle, with fibre the standard representation L. Its index along P 1 , with a simple vanishing constraint at a single marked point, is a virtual bundle Ind L over Pic(P 1 ), with equivariant Euler class e L ∈ H * Pic(P 1 ) [τ −1 ] in the localised equivariant cohomology ring. Specifically, Ind L = L ⊕n and e L = τ n on the component Pic n , n ∈ Z = π 1 U 1 . The following is clear from these constructions.
These propositions capture the rôle of C (G; L ⊕ L ∨ ) in quantum cohomology: the condition of regularity under capping with the Euler class picks out precisely those equivariant Seidel shift operators which act on QH * U 1 (L). More generally, we have the following 2.9 Proposition. The subring C[x, y] ⊂ C[τ, z ± ] acts on QH * U 1 (M ×L) for any compact U 1 -Hamiltonian symplectic manifold M , and it is the largest subring with that property.
Proof. The subring C[τ ] ∼ = H U 1 * (point) acts in the natural way. Recall now (for instance, [OP, I]) that the Seidel element σ n associated with z n (which is a co-character of the original U 1 ) is the following "twisted 1-point function": namely, the element in QH * U 1 (X) defined by the evaluation ev ∞ at ∞ of stable sections of the X-bundle over P 1 associated to O(−n). All is well when X is compact: σ n is a unit in QH * U 1 (X), with inverse σ −n . (Without equivariance, this goes back to Seidel's original paper [S].) For X = M × L though, we have a problem when n < 0: equivariant integration along the fibres of ev ∞ incorporates integration along Ind L , the kernel of H 0 P 1 ; O(−n) ⊗ C × L → L, with dimension (−n); the operation contributes its Euler class as a denominator, a factor of τ n . The factor τ in y = τ z −1 precisely cancels the denominator.
(2.10) Generalisation. Propositions 2.7-2.9 extend to all G and representations V , as Theorems 1 and 3 in §4 below; Theorem 2 is the K-theory analogue. Non-commutative versions of Coulomb branches are described in §7. One required change throughout is the inclusion in the ground ring of an additional equivariant parameter µ, from the natural C × -scaling of V . The need for this will become evident in the example that follows. One can indeed include the full G-automorphism group of V (the flavor symmetries), but any single scaling symmetry that is compactifying -fully expanding or fully contracting -suffices. I will spell out the case of the overall scaling.
(2.11) Example II: U 1 with a general representation V . For a d-dimensional representation V of U 1 with weights n 1 , . . . , n d ∈ Z, the super-potential ψ V : C × → C for its mirror is computed by the following adaptation of the Givental-Hori-Vafa recipe. 2 The defining homomorphism ρ V : "pushes down" to the multi-valued function ψ V (z) on the target C × whose multi-values are the critical values of Ψ along the fibers of ρ ∨ V . A clean restatement is that the Legendre transform ψ * V (τ ) is the restriction, under the infinitesimal representation dρ V , of the Legendre transform of Ψ: in obvious notation, The reader should meet no difficulty in comparing this ε V with the Euler class e V of the respective index bundle over Pic, as in Proposition 2.8. It should be equally clear how to extend this prescription to the case of a higher-rank torus and a general representation.
However, literal application of the lesson from Example 2.5 runs into trouble, already for U 1 with V = L ⊕ L ∨ . In the GHV construction, the super-potential Ψ = z 1 + z 2 has no critical points along the fibres of ρ V (z 1,2 ) = z 1 /z 2 . We have better luck with the Legendre transform, which identifies ε V with the cotangent fibre over exp(πi) = −1 ∈ C × , and induces the automorphism z ↔ (−z) of T ∨ C × . While this does match Proposition 2.8, thanks to the Euler class cancellation e L⊕L ∨ = e L ∪ e L ∨ = (−1) n on Pic n , raw application of Proposition 2.7 would falsely predict that The remedy incorporates scaling-equivariance into the Euler index class, converting it into the µ-homogenized total Chern class. As a Laurent series in µ −1 , the latter is defined for arbitrary virtual bundles. For the index bundles over G ⋉ ΩG of representations of general compact groups G, we will always find rational functions. With V = L ⊕ L ∨ , we get (µ + τ ) n (µ − τ ) −n on Pic n , and the earlier cancellation in the Euler class is now seen to be 'fake', arising from premature specialisation to µ = 0. The Coulomb branch is spelt out in Example 5.2.
Algebraically, µ is to be treated as an independent parameter. It changes the super-potential Ψ by subtracting µ log z k ; this adds scale-equivariance to the mirror of C d . The Legendre transform Ψ * is modified by the substitution τ k → τ k + µ, and the topological origin as a scale-equivariant promotion of the Chern class is now clearly displayed. For a general V , the remedied Lagrangian is defined by z = k (µ + n k τ ) n k ; in particular, it determines the representation.
Extension to a higher-rank torus, with arbitrary representations, is now a simple matter, and it should also be clear how to incorporate the entire flavor symmetry group (the G-automorphism group of V ), if desired, by equivariant enhancements of the Lie algebra coordinates τ k . There is a characterisation of C 3 analogous to Proposition 2.7, as the subring of regular functions on C 3 (T ; 0) which survive translation by the newly µ-remedied ε V , and it is easy to relate it with the abelian presentations in [N, BDG]. The contribution of this paper is the non-abelian generalisation.
2.12 Remark. The remedy of scale-equivariance should not surprise readers versed in toric mirror symmetry: naïve application of the GHV recipe is problematic for toric actions with non-compact quotients -which is when our fake cancellations can happen -and the recipe can be corrected by including equivariance under the full torus.

Background on Coulomb branches
We recall here the construction and properties of Coulomb branches; this mostly condenses material from [BFM, BF, BFN]. I will write C 3,4 for C 3,4 (G; 0) when no confusion arises. Denote by H ⊂ G a maximal torus and by H ∨ , G ∨ the Langlands dual groups, g, h the Lie algebras, W the Weyl group.
(3.1) The basic Coulomb branches [BFM]. The space C 3 := Spec H G * (ΩG; C) is an affine symplectic resolution of singularities of the Weyl quotient T ∨ H ∨ C /W . It arises by adjoining to T ∨ H ∨ C , prior to Weyl division, the functions (e α ∨ − 1)/α for all root-coroot pairs α, α ∨ of G. The C × -action on the cotangent fibres arises from the homology grading and scales the symplectic form. The underlying Poisson structure is the leading term of a non-commutative deformation over C[h] = H * (BR), obtained by incorporating in to C 3 the equivariance under the loop-rotation circle R. The loop rotation is revealed by writing ΩG ∼ = LG/G.
For simply connected G, the spectrum of K G * (ΩG; C) is also a symplectic manifold giving an affine resolution of (H C × H ∨ C )/W . This is now accomplished by adjoining the functions (e α ∨ − 1)/(e α − 1) before Weyl division. However, the space has singularities when π 1 G has torsion. Write G =G/π for the torsion subgroup π ⊂ π 1 G, H =H/π. As a subgroup of Z(G), π acts by automorphisms of KG(X) ⊗ C for any G-space X: to see this, decompose a class in KG(X) into π-eigen-bundles, and multiply each of them by the corresponding character of π, before re-summing to a complex K-class. We adopt the smooth symplectic orbifold π ⋉ Spec KG * (ΩG; C) as the definition of C 4 .
3.2 Remark (Sphere topology). Some features of C 3,4 are explained by Chas-Sullivan theory in dimension 3, one higher than usual. The underlying topological object is the mapping space from S 2 to the stack BG; it has a natural E 3 structure, which turns out to correspond to the Poisson form on C 3,4 (G; 0). Loop rotation is seen in the presentation as the two-sided groupoid G ⋉ LG ⋊ G, with Hecke-style product (see Remark 3.9 below). Tracking the loop rotation breaks E 3 down to E 1 , because rotating spheres in an ambient R 3 may be strung together linearly as beads on the rotation axis, but can no longer move around each other. This leads to the non-commutative Coulomb branches we shall review in §7.
When π 1 G has torsion, the second base is an affine orbifold whose ring of functions is K G (point). (The abelian property is a piece of characteristic-zero good fortune: the correct commutativity structure is E 3 , as explained in Remark 3.2, but this decouples into a strictly commutative and a graded Poisson structure.) These maps define integrable systems: χ is a partial completion of the classical Toda system 3 [BF], whereas κ is its finite-difference version.
3.5 Remark (Adjoint and Whittaker descriptions). As an algebraic symplectic manifold, C 3 is the algebraic symplectic reduction T ∨ reg G ∨ C //G ∨ C of the fibrewise-regular part of the cotangent bundle under conjugation. There is a simliar description of C 4 using the Langlands dual Kac-Moody group (not the loop group of G ∨ ), capturing the holomorphic (but not algebraic) symplectic structure.
The space C 3 has another description as the two-sided symplectic reduction of T ∨ G ∨ C by N , at the regular nilpotent character. Clearly, this is algebraic symplectic; much less obviously, it is hyper-Kähler, thanks to work of Bielawski on the Nahm equation [B]. The non-commutative deformation has a corresponding description in terms of N × N monodromic differential operators on G ∨ C [BF]. In both descriptions, multiplication along the group G ∨ induces the group scheme structure of §3.3. Commutativity is more evident in the adjoint description, where the Toda fibers are the centralizers of regular co-adjoint orbits in g ∨ C .
(3.6) Coulomb branches for E = V ⊕V ∨ . To build the spaces C 3,4 (G; E), we follow [BFN], to which we refer for full details, and replace ΩG in the original C by a linear space L V → Ω a G, a stratified space whose fibres are vector bundles over the Schubert strata of the algebraic model Ω a G := G C ((z))/G C [[z]] of ΩG. The fibre of L V over a Laurent loop γ ∈ Ω a G is the kernel of the difference ] were zero. The normalised grading is compatible with the multiplication defined by the following correspondence diagram on the fibres of L V , which lives over the multiplication of two loops γ, δ ∈ Ω a G: ( The map g glues the bottom sheet of the first disk to the top sheet of the second, while i hits the outer centres of the triple-centred disk. The E 3 property comes from sliding the multiple centres around, as in Chas-Sullivan sphere topology. With rotation-equivariance, this freedom is lost and we are reduced to an E 1 multiplication. On G-bundles, the Hecke operation is represented by multiplication of transition functions, once we identify, on the left side, the top bundle on its bottom sheet with the bottom bundle on its top sheet. Next, associated to the representation V is a vector bundle over −:−, whose space of global sections is L V . The correspondence (3.8) arises by retaining those pairs of global sections on the left which match on the glued pair of sheets, and then restricting them to the top and bottom sheets of the triple-centred disk.
(3.10) Massive versions. We enhance the Coulomb branches by the addition of a symmetry in which C × ⊃ S 1 scales the fibres of L V : The projections to the massive Toda bases are defined as in (3.4), and denoted by χ(µ), κ(m), with generators µ ∈ H 2 (BS 1 ), m ± ∈ K S 1 (point). The fibres over fixed values of the parameters µ, m are total spaces of (usually singular) integrable systems; this will follow from flatness of the projections ( §5.8). The scaling is trivial when E = 0 and L V = Ω a G, but it will couple to the Euler class of the index bundle over G ⋉ ΩG, promoting it to the total Chern class. The notation is subtly abusive: the C • depend on the polarisation V and not just on E. For instance, switching V ↔ V ∨ leads an isomorphic space only if we also change the orientation of the rotating circle. This V -dependence disappears at µ = 0 or m = 1. We will see in §6 that the C • (G; E) are flat over C[µ], C[m ± ], and that the same spaces C 3,4 (G; E), as defined earlier in this section, appear by specialising to µ = 0 or m = 1, independently of the choice of V .

Main results
We are finally in position to state Theorems 1-3; the non-commutative analogues of Theorems 1 and 2 will wait until §7. First, I describe the Lagrangians generalising the massive ε V of Example 2.5. Their Euler class interpretation, already mentioned following Proposition 2.7, will be spelt out in §6 below.
(4.1) The Euler Lagrangians. For w ∈ C × and ν a weight of H, w ν := exp(ν log w) determines a point in H ∨ C . Consider the following rational maps from h C × C and H C × C × to H ∨ C , defined in terms of the weights ν of V , which are to be included with their multiplicities: (In parsing each formula, note the double use of ν, first as infinitesimal character of H and then as co-character of H ∨ .) The maps are Weyl-equivariant and their graphs are regular, away from a co-dimension 2 locus over their domains (cf. §5.1 below); their closures define Lagrangian sub-varieties

Remark (Broader picture).
For generic µ and m (but most meaningfully, near µ, m = ∞), the maps (4.2) are the exponentiated differentials of the following functions, in which ξ ∈ g C and x ∈ G C are the arguments while µ, m are treated as parameters: The first function appeared as the "Σ log Σ Landau-Ginzburg B-model mirror" of the abelian GLSM on V : [W], and see also Remark 7.7. The Lagrangian λ V and its primitive appeared 4 in the index formula for Kähler differentials over the moduli of G-bundles on curves [TW, Eqn. 6.2 and Thm. 6.4], with the powers of m −1 tracking the degree of the forms. The relation with Coulomb branches was not known at the time. Today, we would express that index formula in terms of Lagrangian calculus in C • 4 (G, 0), namely the intersection of λ V with the graphs of certain isogenies H C → H ∨ C , defined from the levels of central extensions of the loop group LG. Those isogenies correspond to the Theta line bundles on the moduli of G C -bundles on curves; they are semiclassical limits of Theta-functions -in the same sense that the Lagrangians ε V , λ V are semiclassical Γ-functions, see §7 -and are also twists of the unit section by the discrete Toda Hamiltonian of C 4 .
(4.4) Algebraic description of the Coulomb branches. The first two results generalise to non-Abelian G the explicit presentations of Coulomb branches given in [N, BDG] for torus groups. Their proofs, in §6, are straightforward; more intriguing are the the non-commutative generalisations in §7. To state the theorems, note that translation on the group schemes by the section ε V , respectively λ V , gives a rational symplectomorphism of C • 3 , C • 4 , relative to the massive Toda projection of §3.10.
Theorem 1. The space C • 3 (G; E) → h C /W × C is the affinisation of two copies of C • 3 (G; 0) glued together by means of ε V -translation. In other words: regular functions on C • (G; E) are those regular functions on C • (G; 0) which remain regular after translation by ε V . Theorem 2. The orbifold C • 4 (G; E) → π ⋉ (H C /W ) × C × is the relative affinisation of two copies of C • 4 (G; 0) glued together by means of λ V -translation.
Abstractly, the spaces are the quotients, in affine schemes over the massive Toda bases, of an equivalence relation on C • C • defined fromε V ,λ V . The relation is not very healthy, being neither proper nor open. Concretely, note that the surviving condition can equally well be imposed prior to Weyl division, giving the following moderately explicit description: 4.5 Corollary. The regular functions on C • 3,4 (G; E) are those Weyl-invariant elements of (ranging over the roots α) which survive translation by ε V , respectively by λ V .
Survival can be restated in terms of growth constraints along the Toda fibres over the locus of zeroes and poles of ε V , λ V ; we shall do that in the next section, as we review more of the algebraic geometry. Meanwhile, the next theorem, characterising the regular functions C • (G; E) in terms of quantum cohomology, is simple enough to prove here. Proof of Theorem 3. Away from the root hyperplanes on the massive Toda base (or the singular conjugacy locus, respectively), the statement follows by abelianisation from the calculation of Proposition 2.9. On the other hand, away from µ = 0 (or m = 1), the fixed-point theorem allows us to ignore E and V , and we are reduced to the action of H G * (ΩG) on equivariant quantum cohomology (see [T2, T3]). The remaining locus has co-dimension 2 on the base, over which QH * G×S 1 (M ) is finite and free as a module.

Some consequences
We discuss briefly some geometry of the Coulomb branches as it emerges from their description in §4. Flatness and normality were already established in [BFN], but it may be helpful to review them in the new construction.
(5.1) Generic geometry of the Coulomb branches. The divisor S of singularities of the section ε V , resp. λ V is the unions of hyperplanes S ν defined by the monomial factors in (4.2). The pairwise intersections of the S ν contain the indeterminacy locus I. Away from I, each C • (G; E) is the affinisation of a smooth space, obtained by gluing two open charts C • with a vertical relative shift over the Toda base. Away from S, the glued space is of course isomorphic to the original C • ; whereas, near each S ν \ I, the Toda fibres undergo a nodal degeneration along the C × factor C ν , modeled on C × C ⊔ 0 C in the fibres of the A n−1 -singularity (x, y) → t = (xy) 1/n . (The number n is computed from the divisibility and the multiplicities of the weight ν.) The appearance of the nodal locus, along which C • (G; E) is singular when n > 1, is a consequence of affinisation: the smooth charts C • cover the complement, as in Example 2.5. From here, Hartog's theorem determines C • (G; E) completely; but we can be more specific in concrete cases. Thus, some fibres of C • (G; 0) are crushed in co-dimension 2, over I.
(5.3) Example: SU 2 with the standard representation. Consider the Weyl double cover of C • 3 of C • 3 , defined from Corollary 4.5 before Weyl division. In the z, τ -notation already used for the maximal torus of SU 2 , the functions over C • 3 are generated over C[µ, τ ] by u = z−1 τ and v = 1−1/z τ , with the single relation u − v = τ uv. The Weyl action switches u and v and changes the sign of τ . Translation by ε V sends z to µ+τ µ−τ z. Let x := µu − z, y := µv − z −1 , w = (x − y)/τ ; the surviving subring is described by generators and relations over C[µ, τ ] as {x, y, w} , with relations x − y = τ w, xy = 1 + µw.
(5.6) Checking the SU 2 example. Let A be the surviving subring, and A ′ ⊂ A the subring generated by (5.5); let us check that A ′ = A. This is clear with τ inverted, by reduction to the case of U 1 , when zφ − and z −1 φ + generate C • 3 over C[µ, τ ± ]. Upon formal completion near τ = 0, the statement is equally clear with µ inverted, when φ ± become units. This shows that A/A ′ is a quasi-coherent torsion sheaf on the (µ, τ )-plane supported at µ = τ = 0. But such a sheaf would yield a Tor 2 group against the sky-scraper at µ = τ = 0, which is forbidden, because (I claim) both A ′ and A are flat over C [τ, µ]. Flatness A ′ is checked easily from the 3-step resolution built from (5.5); that of A is discussed below.
(5.7) Normality. Our description of C • (G; E) implies its normality: indeed, if a function f is integral over the surviving subring, then f • (ε V ·) is integral over C • , so it is regular, and so f survives. Alternatively, granting flatness of the Toda projections (to be discussed below), one sees the desired regularity in co-dimension 1 from the generic geometric behaviour described in §5.1.
Normality of the massless specialisation can be extracted from the flatness discussion below, where we build the C • (G; E) from C • by blow-ups and contractions along loci transversal to µ = 0. Alternatively, granting flatness, we can again check regularity in co-dimension 1: the generic Abelian description applies away from the root hyperplanes, while on the generic part of a root hyperplane the SU 2 description of Example 5.4 takes its place.
(5.8) More geometry. Flatness of the Coulomb branches over the massive Toda bases (freedom, in fact) is wrapped into the proof of Theorem 1 in the next section. However, we can also extract it from our algebraic description; we outline the argument here, as it points a way to a more geometric description of their Weyl covers. It does suffice to treat the Weyl cover: the Toda base for the Cartan subalgebra h ⊂ g is flat 5 over that of G, and extracting Weyl invariants does not spoil flatness (or normality).
Choose a smooth Weyl-invariant toric compactificationH ∨ C of H ∨ C , requiring that the weights ν appearing in (4.2) should define boundary divisors B ν . 6 The latter assemble to an ample relative boundary divisor B for the compactified projectionχ : We will create a space leading to C • 3 (G; E) by blowing up products of pairs divisors (in the base and fiber), then remove a boundary, and finally perform an affinising contraction over the Toda base.
First, we prepare to create the Weyl cover C • by blowing up the loci (exp α ∨ = 1, α = 0)or rather, their successive proper transforms in a chosen sequence. 7 Then, we prepare the surviving growth constraints by performing further blow-ups along (the successive proper transforms of) products B ν × S ν , with an appropriate multiplicity on the hyperplane S ν . Cyclic singularities appear here. Nonetheless, the co-dimension (1, 1) of the blowing up loci and with respect to the Toda projection, and of their successive proper transforms, ensures that we get a flat modificationχ of our projection χ.
The final step is the removal of the boundary and the collapse of proper components by taking fibre-wise global functions along the projectionχ. The boundary comprises the proper transformsB of B andR of the root hyperplanes (the latter is to produce the original C • ). The resulting ring is the colimit, as N → ∞, of Rχ * O(NB + NR) . The sheaf in the total direct image is a line bundle 8 with no higher direct images, because of its quasi-positivity and the negativity of K. Thus, each term in the limit is free over the base and so the colimit is flat.

Proof of Theorems 1 and 2
We use the Schubert stratification of Ω a G into G[[z]]-orbits. Even-dimensionality collapses the associated spectral sequences and leads to ascending filtrations on the rings C[C • (G; E)]. The associated graded components are easily described ( §6.7 below), and are locally free over the Toda bases. This makes the original rings locally free as well; in particular, they are flat over C[µ], C[m ± ].
I write out the proof for C • 3 ; the K-theory case is entirely parallel. Call A V the ring, implied in Theorem 1, of regular functions on C • 3 which survive ε V -translation. We will see from topology how this last operation is compatible with the Schubert filtration, so that we can also define the subring Σ V ⊂ Gr C[C • 3 ] of symbols which remain regular after ε V -translation. Clearly, GrA V ⊂ Σ V . The theorems will follow from two observations: 1) The index bundle. Over the stack Bun G (P 1 ) of principal G C -bundles over P 1 , there lives the virtual index bundle Ind V , the holomorphic Euler characteristic of the sheaf of sections of V over P 1 with simple vanishing condition at one marked point ∞. It is a class in K 0 G×S 1 (ΩG), after incorporating the mass parameter µ (equivariance under scaling of V ). Call e V its equivariant Euler class, more accurately defined as the µ-homogenised G-equivariant total Chern class of Ind V . The following two propositions are understood after suitable localisation on the massive Toda base h C /W C × C.
6.2 Proposition. Translation by ε V on C 3 corresponds to cap-product with e V on H G×S 1 * (ΩG).
6.3 Remark. Cap-product with e V must a priori correspond to translation by some rational section: the index bundle is additive for the sphere multiplication in G⋉ΩG, so its Euler class is multiplicative. As a group-like element in the dual Hopf algebra, it represents a (rational) section of the group scheme C • over its Toda base. We identify this section by abelianization.
Proof. Localise to the complement of the root hyperplanes on the Toda base to reduce, by the fixedpoint theorem, to the case of a torus, where Proposition 2.8 applies (as enhanced in Example 2.11).
6.4 Corollary. The Schubert filtration is preserved by ε V -translation.
(6.5) Two embeddings of C[C • 3 (G; E)]. Refer to the notation in §3.6 and Remark 3.9. The Hecke construction at 0 ∈ P 1 maps the stack Bun G (− : −) = G [[z]]\Ω a G of G C -bundles over the doublecentered disk to Bun G (P 1 ). This gives an equivariant homotopy equivalence and in particular a (K-)homology equivalence. The key observation is that, More precisely, note the two inclusions ι l,r : ], and recall that over any finite union of strata, L V contains a finite co-dimension sub-bundle. Quotienting it out regularises the difference of V [[z]]-bundles into a class in K G×S 1 (ΩG). A moment's thought identifies this with Ind V , as the index of the Hecke transform of the trivial V -bundle on P 1 , minus that of the trivial V -bundle.
Each inclusion ι l,r defines a graded ring homomorphism ϕ l,r : , by intersecting with the zero-section in the ambient bundle. Per our discussion, ϕ l = e V ∩ ϕ r . Using ϕ r to pin down C • 3 (G; E), Proposition 6.2 now settles Observation (i).
(6.6) Working out Σ V . For a 1-parameter subgroup z η ∈ ΩH, with Schubert stratum C η and Levi centraliser Z(η) ⊂ G, split V = V + ⊕ V 0 ⊕ V − following the sign of the η-eigenvalue. The index bundle then splits as Ind V = I + (η) ⊖ I − (η), with the ν-weight space of V ± appearing ± ν|η times in I ± (η). The Euler class e V factors at z η as There is a (degree-shifting) isomorphism (6.7) Working out Gr C[C • 3 (G; E)]. Collapse of the Schubert spectral sequence implies that Now, the the homology group is generated over H Z(η)×S 1 * (point) by the fundametal class of the total space of L V over C η , whose complement in the right V [[z]] of (3.7) is precisely I − (η); therefore in agreement with the η-component of Σ V above. This settles Observation (ii).

Non-commutative Coulomb branches
Incorporating the loop rotation circle R in the previous constructions leads to non-commutative deformations N C • 3,4 (G; E) of the Coulomb branches over the ground rings C[h] = H * (BR) and C[q ± ] = K R (point), respectively. The geometric objects exist in the formal neighbourhoods of h = 0 and q = 1; away, only their function rings A 3,4 survive. Nonetheless, we sometimes keep the convenient conversational pretence of underlying spaces N C . The calculation in §6 for their description applies with only minor changes: we are only missing the good statements, which we summarise below before spelling out the argument.
This section is rather sketchy; a development spelling out the rôle of our non-commutative solutions, the Γ-functions, in connection with the GLSM, is planned for a follow-up paper.
(7.1) Summary. The integrable abelian group structure of the C • over their Toda bases deforms to a symmetric tensor structure 9 on A -modules, induced from the diagonal inclusion ΩG ΩG × ΩG. Restricting the module structure to the Toda base, this is the ordinary tensor product, with tensor unit the structure sheaf O 1 of the identity section. For C • 3 in the Whittaker presentation (Remark 3.5), the operation comes from convolution of D-modules on the Langlands dual group G ∨ : from this stance, the symmetric monoidal structure is developed in [BZG].
The Lagrangians ε V , λ V deform to modules E V , Λ V over A 3,4 , and the (rational) automorphisms of C defined by ε V , λ V -translation become, on A -modules, the functors of convolution with E V , Λ V . The Hamiltonian nature of the translations renders these functors (generically) trivialisable by (singular) inner automorphisms of A . In Theorem 4, I characterise the Coulomb branches N C • (G; E) as the subrings of elements of A which survive these inner automorphisms.
While this loose description of the N C • (G; E) appears uniform, a distinction arises between formal and genuine deformations. Formally, the modules E V and Λ V are generically invertible, analogous to flat line bundles with singularities, with the latter located on the singular loci of the sections ε V , λ V . If, following the language of D-modules, we call solutions the A -module morphisms to O 1 , then the super-potentials that were introduced in Remark 4.3 are the leading h → 0 asymptotics of the logarithms of the solutions (cf. Remark 7.7).
With the deformation parameters turned on, these asymptotics become meromorphic solutions that are easily found. For E V on C 3 , a solution is the Γ-function of the representation V (recalled in §7.9 below), while a q-analogue solves Λ V on C 4 . Conjugation by these solutions impose the defining regularity constraints for the N C • (G; E). Out of range the formal limit, the modules E V , Λ V are described by these solutions, which become the primary objects. We may prefer, for convenience, the (tensor) inverse modules and their holomorphic solutions; thus, E −1 V is the quotient of A 3 by the annihilator I V of the solution Γ −1 V : If we view the quotient A 3 /I V as an analytic sheaf over the Toda base, the solution map Γ −1 V is an isomorphism. (Otherwise, its infinitely many zeroes prevent it from surjecting onto O 1 .) We can then characterise N C • (G; E) in three equivalent ways, the last two of which are Γ-conjugate: (i) the subring of A 3 which survive conjugation by Γ −1 V , (ii) the subring of A 3 whose action preserves the inclusion O 1 ⊂ E −1 V , (iii) the subring of A 3 whose action preserves the inclusion Γ −1 V O 1 ⊂ O 1 . There is a parallel story for N C 4 . Before spelling out the details, let us revisit the case of U 1 . 9 I thank David Ben-Zvi for pointing out to me the generality of this statement.
7.10 Remark (Massless specialisation.). The correct massless specialisation is at µ = 1 2 h (not µ = 0). In the construction of [BFN], this should be interpreted as inserting a square root of the canonical bundle on the doubled disk −:−. The same insertion within the index bundle does away with the vanishing condition at ∞ in the constructions of §6. The specialisation is illustrated by the identity Γ( h 2 + τ ; h)Γ( h 2 − τ ; h) = π h sec πτ h ; the product is therefore anti-central in N C (U 1 ) -that is, it conjugates z to (−z) -generalizing the identity e L ∪ e L ∨ = (−1) n of Example II in §2.

Remark.
Interpreting h as the equivariant parameter of the loop rotation group R, the Weierstraß product expansion portrays Γ −1 V as a regularised Euler class of the space of Taylor loops V [[z]]. This interpretation also makes sense over certain stacks with a circle action, such as G [[z]]\Ω a G: a reasonable demand is that their R-equivariant homology is free over C[h], so that extension of scalars to functions of h holomorphic off the negative real axis (and allowing poles in µ, ξ as needed) is a faithful operation. For a torus, we can always pretend that h is a numerical parameter, because the R-action on the stack Pic(P 1 ) is trivial (even though it is not canonically so).
Proof. From (7.13) we see that the listed generators survive, and (7.14) shows that their nth power generates the summand of degree ±n over the Toda base. Fix now n > 0 say. The need for the z n e − (n) factor in a surviving element follows from unique factorisation in the ground ring C[µ, τ, h]. Namely, z n f (µ, τ, h) conjugates to e + (n)z n e − (n) −1 f . The linear factors of z −n e + z n have the form (µ + kτ − ph) with p > 0, and are prime to the denominator e − (n), whose factors carry non-negative multiples of h going with µ; so all canceling factors must come from f . Localising on the Toda base, we find from the Abelian calculation, formally close to h = 0: 7.16 Corollary. Γ V conjugates the unit module of N C • 3 into a module E V with supportε V .
Away from formal h = 0, we can define the convolution-inverse module E −1 V as the quotient of A 3 by the annihilator of Γ −1 V . Sending 1 ∈ A 3 to Γ −1 V identifies it with O 1 .
Proof. Working over the common Toda base h/W , H-fixed point localisation shows that the map induced by the named morphisms is an isomorphism away from the root hyperplanes; whereas, generically on the root hyperplanes, the SL 2 calculation of §5.4 confirms isomorphy. This settles the matter, because the algebras are free O-modules over the Toda base and agree in co-dimension 2.
8.1 Remark. The calculation for C 3 of SL 2 was seen to hold more generally, for all but a few choices of E. This generalizes to all groups, by the argument above: however, the formulation of the right-hand side needs more care. Exploiting the local descriptions of the C 4 Toda bases in terms of C 3 , one can then push this to an awkward but effective calculation of most C 4 Coulomb branches. It would be truly useful to find the formulation which dispenses with all constraints on E: this might allow an abelianised calculation of Coulomb branches with non-linear matter.