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Coulomb Branch

A mathematical Coulomb branch for a complex reductive group G and a representation N is the spectrum of the commutative Braverman-Finkelberg-Nakajima convolution algebra. Let O=C[[t]], K=C((t)), and

R={[g,s] in G(K)×_(G(O))N(O):gs in N(O)}.

The equivalence is (g,s)∼(gh^(-1),hs) for h in G(O). For a commuting torus F, its parameter-deformed version is Y_f=SpecA_F, with A_F=H_*^(G(O)×F)(R;C) and convolution multiplication. This is the commutative algebra, without the loop-rotation parameter of the quantized version.

For quiver data G=product_(i)GL(V_i), the topological grading is by pi_1(G)=Z^({i:dimV_i>0}). It defines an algebraic torus with character lattice pi_1(G). A stability character specifies a cocharacter and its scheme-theoretic fixed locus for the Hikita comparison.

See also

Affine Scheme, Algebraic Torus, Equivariant Cohomology, Hikita Conjecture, Nakajima Quiver Variety, Reductive Group

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References

Braverman, A.; Finkelberg, M.; and Nakajima, H. "Towards a Mathematical Definition of Coulomb Branches of 3-Dimensional N=4 Gauge Theories, II." Adv. Theor. Math. Phys. 22, 1071-1147, 2018. https://doi.org/10.4310/ATMP.2018.v22.n5.a1.Dumanski, I. and Krylov, V. "K-Theoretic Hikita Conjecture for Quiver Gauge Theories." 7 Sep 2025. https://arxiv.org/abs/2509.06226.

Cite this as:

Weisstein, Eric W. "Coulomb Branch." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CoulombBranch.html

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