Schubert line defects in 3d GLSMs. Part I. Complete flag manifolds and quantum Grothendieck polynomials

A bstract We construct new half-BPS line defects in 3d $$ \mathcal{N}=2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> supersymmetric quiver gauge theories whose Higgs branches are complete flag manifolds X = Fl( n ). Upon circle compactification, the bulk theory flows to a non-linear sigma model (NLSM) with target space X and the line defects flow to objects supported on Schubert varieties X w ⊆ X . These Schubert line defects form an important basis of the quantum K-theory of X . They are realized as $$ \mathcal{N}=2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> supersymmetric quantum mechanics (SQM) quivers coupled to the 3d gauge theory. We show that the insertion of the Schubert line defect restricts the target space of the 3d gauged linear sigma model (GLSM) to the Schubert variety X w , with the 1d degrees of freedom physically realizing a Bott-Samelson resolution of X w . Moreover, we verify in examples that the 1d flavored Witten index of the quiver SQM reproduces the (equivariant) Chern character of the structure sheaf $$ {\mathcal{O}}_{X_w} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>w</mml:mi> </mml:msub> </mml:msub> </mml:math> as a (double) quantum Grothendieck polynomial, generalizing previous results for X a Grassmannian manifold. Our construction thus provides a more direct realization of the 3d GLSM/quantum K-theory correspondence for complete flag manifolds. Finally, in the small-circle limit, we obtain a 0d-2d coupled system that realizes the Schubert classes [ X w ] in the quantum cohomology ring of X .