3d Partition Function as Overlap of Wavefunctions

TL;DR

This work computes and interprets the S^3 partition functions of 3d N=4 theories arising from 4d N=4 SU(N) SYM on segments and junctions. The authors show that these partition functions arise as overlaps of boundary-condition–determined wavefunctions, with the 3d theories acting as 1d kernels mediated by an S-duality operator $\mathcal{S}$, i.e. $Z[T^{\sigma}_{\rho}[\mathrm{SU}(N)]] = \langle \rho,\zeta|\mathcal{S}|\sigma,m\rangle$, and similarly for the trivalent $T_N^{\rho_1,\rho_2,\rho_3}$ theory as a triple overlap. They connect these 3d results to the zero-radius limit of the 4d superconformal index and to $q$-deformed 2d Yang-Mills, revealing how representation labels and holonomies become continuous Lie-algebra data in 3d and how Weyl-group symmetries become explicit kernel symmetries. The study provides a concrete 1d–3d–4d dictionary for dual boundary conditions and duality kernels, with implications for understanding non-Lagrangian theories (via their 3d mirrors) and for future work on finite-length segments and Hamiltonian dynamics.

Abstract

We compute the partition function on S^3 of 3d N=4 theories which arise as the low-energy limit of 4d N=4 super Yang-Mills theory on a segment or on a junction, and propose its 1d interpretation. We show that the partition function can be written as an overlap of wavefunctions determined by the boundary conditions. We also comment on the connection of our results with the 4d superconformal index and the 2d q-deformed Yang-Mills theory.

Figures (14)

• Figure 1: A graph with two boundary conditions $\rho$ and $\sigma$. The wavy line represents an S-duality wall.
• Figure 2: A trivalent graph with three boundary conditions $\rho_1$, $\rho_2$ and $\rho_3$. $|T_N\rangle\!\rangle\!\rangle$ acts as a boundary condition at the center of the trivalent vertex.
• Figure 3: Mirror symmetry between $T_\rho^\sigma$ and $T_\sigma^\rho$ theories. The low-energy theory is called $T_\rho^\sigma$ when the S-duality wall is close to the right boundary $\rho_X$, while it is called $T_\sigma^\rho$ when the wall is close to the left boundary $\sigma_Y$.
• Figure 4: The quiver for $T^\sigma_\rho(\mathrm{SU}(N))$ theory. A circle represents $\mathrm{U}(v_i)$ gauge groups, a box represents $\mathrm{U}(w_i)$ flavor symmetry, and the line connecting two unitary groups represents hypermultiplets in the bifundamental representation.
• Figure 5: An example of brane construction of $T_\rho^\sigma[\mathrm{SU}(N)]$ theory with $\rho=[1,1,1,1]$ and $\sigma=[2,1,1]$. The circle "$\otimes$" stands for an NS5-brane, and the dotted and solid lines denote D5- and D3-branes, respectively. When 5-branes are rearranged, some of D3-branes are annihilated due to the Hanany-Witten effect Hanany:1996ie.