<p>For a 3D <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">$ \mathcal{N} $</EquationSource> </InlineEquation> = 4 gauge theory, turning on the Ω-background in <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math display="inline"> <mi>ℝ</mi> <mo>×</mo> <msubsup> <mi>ℝ</mi> <mi>ϵ</mi> <mn>2</mn> </msubsup> </math></EquationSource> <EquationSource Format="TEX">$ \mathbb{R}\times {\mathbb{R}}_{\epsilon}^2 $</EquationSource> </InlineEquation> deforms the Coulomb branch chiral ring into the quantum Coulomb branch algebra, generated by the <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$ \frac{1}{2} $</EquationSource> </InlineEquation>-BPS monopoles together with the complex scalar in the vector-multiplet. We conjecture that for a 3D <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math display="inline"> <mi mathvariant="script">N</mi> </math></EquationSource> <EquationSource Format="TEX">$ \mathcal{N} $</EquationSource> </InlineEquation> = 4 quiver gauge theory with unitary gauge group, the quantum Coulomb branch algebra can be formulated as the truncated shifted quiver Yangian Y(<InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math display="inline"> <mover accent="true"> <mi>Q</mi> <mo stretchy="true">̂</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">$ \hat{Q} $</EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math display="inline"> <mover accent="true"> <mi>W</mi> <mo stretchy="true">̂</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">$ \hat{W} $</EquationSource> </InlineEquation>) based on the triple quiver <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math display="inline"> <mover accent="true"> <mi>Q</mi> <mo stretchy="true">̂</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">$ \hat{Q} $</EquationSource> </InlineEquation> of the original quiver Q with canonical potential <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math display="inline"> <mover accent="true"> <mi>W</mi> <mo stretchy="true">̂</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">$ \hat{W} $</EquationSource> </InlineEquation>. We check this conjecture explicitly for general tree-type quivers Q by considering the action of monopoles on the <InlineEquation ID="IEq9"> <EquationSource Format="MATHML"><math display="inline"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></EquationSource> <EquationSource Format="TEX">$ \frac{1}{2} $</EquationSource> </InlineEquation>-BPS vortex configurations. The Hilbert spaces of vortices approaching different vacua at spatial infinity furnish different representations of the shifted quiver Yangian, and all the charge functions have only simple poles. For quivers beyond tree-type, our conjecture is consistent with known results on special examples.</p>

Quiver Yangians as Coulomb branch algebras

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2026年5月8日

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Tiantai Chen,
Wei Li

摘要

For a 3D $N$ $ \mathcal{N} $ = 4 gauge theory, turning on the Ω-background in $ℝ \times ℝ_{ϵ}^{2}$ $ \mathbb{R}\times {\mathbb{R}}_{\epsilon}^2 $ deforms the Coulomb branch chiral ring into the quantum Coulomb branch algebra, generated by the $\frac{1}{2}$ $ \frac{1}{2} $ -BPS monopoles together with the complex scalar in the vector-multiplet. We conjecture that for a 3D $N$ $ \mathcal{N} $ = 4 quiver gauge theory with unitary gauge group, the quantum Coulomb branch algebra can be formulated as the truncated shifted quiver Yangian Y( $\hat{Q}$ $ \hat{Q} $ , $\hat{W}$ $ \hat{W} $ ) based on the triple quiver $\hat{Q}$ $ \hat{Q} $ of the original quiver Q with canonical potential $\hat{W}$ $ \hat{W} $ . We check this conjecture explicitly for general tree-type quivers Q by considering the action of monopoles on the $\frac{1}{2}$ $ \frac{1}{2} $ -BPS vortex configurations. The Hilbert spaces of vortices approaching different vacua at spatial infinity furnish different representations of the shifted quiver Yangian, and all the charge functions have only simple poles. For quivers beyond tree-type, our conjecture is consistent with known results on special examples.