3d $$ \mathcal{N} $$ = 2 brane webs and quiver matrices

Abstract

We discuss 3d brane webs and effective Chern-Simons levels for 3d $$ \mathcal{N} $$ N = 2 gauge theories. We find that turning on real masses for chiral multiplets leads to various equivalent brane webs that are related by flipping positions of D5-branes. We interpret flips as ST-transformations for chiral multiplets. ST-transformations could turn abelian theories into dual theories with mixed Chern-Simons levels that are interpreted as quiver matrices Cij encoding DT-invariants. We notice that each brane web corresponds to a quiver matrix. ST-transformations of holomorphic blocks are discussed to verify results. We also discuss the movement of flavor D5-branes, which leads to double-layer brane webs and manifests fiber-base duality. In the second part, we compute refined vortex partition functions of nonabelian theories with the gauge group U(N) and find corresponding quiver matrices. The computation shows that on Higgs branch nonabelian groups are broken to abelian groups.

3d 𝒩 = 2 theories and plumbing graphs: adding matter, gauging, and new dualities

“…Here we note that this exchange symmetry suggests a duality for abelian 3d theories, which amounts to flipping the signs of bare Chern-Simons levels. We confirm this by considering vortex partition functions, or more completely holomorphic blocks of 3d theories [26,31,38,92].…”

“…The term η contains theta functions, which arise from exchanging boundary conditions for all chiral multiplets [31,38,92,93]. Note that 3d theories should couple to 2d (0, 2) theories on their boundaries to cancel anomalies.…”

“…In [38], the ST -moves (transformations) of holomorphic blocks have been discussed. STmoves could be represented as some operations on effective superpotentials.…”

“…The realm of 3d N = 2 gauge theories is an interesting theoretical laboratory. Such theories enjoy various dualities [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], which can be tested by analysis of their partition functions [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39], and which can also be understood from perspective of string theory and Mtheory. In recent years, various constructions of 3d N = 2 theories have been proposed, for example using M5-branes and compactification [40][41][42][43][44][45][46][47][48], as well as the as...…”

“…The decoupling in (2.7) and (2.8) can also be easily illustrated using brane webs, in which the Chern-Simons level is associated to a relative angle k eff = tan θ between branes. Decoupling matter is equivalent to sending the corresponding D5-brane to either positive infinity or negative infinity, which shifts the effective Chern-Simons levels differently [38].…”