Wall crossing, string networks and quantum toroidal algebras

TL;DR

The work connects BPS spectra in 4d ${\mathcal N}=4$ SYM to Type IIB string networks, showing that line operators form a representation of the quantum toroidal algebra ${U_{\frak q,\frak t}(\widehat{\widehat{gl}}_1)}$ acting on ${(\,\mathcal V_{\frak q}^{*})^{\otimes N}}$ and that wall-crossing is implemented by Drinfeld twists of the coproducts. The phase ${\zeta}$ selects a specific coproduct (irregular slope), with rational walls ${W_{n,m}}$ producing elementary twists that reproduce the Kontsevich-Soibelman spectrum generator via the Khoroshkin-Tolstoy ${R}$-matrix. Framed BPS states are realized as networks with semi-infinite strings, and their protected spin characters are encoded in UV-IR expansions that match the PBW generators ${P_{(n,m)}}$ in the tensor-product representation; for ${N=2}$ the construction illustrates how line operators decompose into vector-representation components. The results place BPS wall-crossing in a precise representation-theoretic framework, connecting line operators, dualities, and quantum toroidal structures, and suggest avenues for categorification and broader brane configurations.

Abstract

We investigate BPS states in 4d N=4 supersymmetric Yang-Mills theory and the corresponding (p, q) string networks in Type IIB string theory. We propose a new interpretation of the algebra of line operators in this theory as a tensor product of vector representations of a quantum toroidal algebra, which determines protected spin characters of all framed BPS states. We identify the SL(2,Z)-noninvariant choice of the coproduct in the quantum toroidal algebra with the choice of supersymmetry subalgebra preserved by the BPS states and interpret wall crossing operators as Drinfeld twists of the coproduct. Kontsevich-Soibelman spectrum generator is then identified with Khoroshkin-Tolstoy universal R-matrix.

Figures (2)

• Figure 1: An example of a string network in the $\mathbb{R}^2_{xy}$ plane consisting of three $(p,q)$ strings (drawn as wavy lines) stretched between three D3 branes located at points $(x_1,y_1)$, $(x_2,y_2)$ and $x_3, y_3$. The relative angles of the $(p,q)$ string segments are fixed by their charges and the value of $\tau$ (see Eq. \ref{['eq:3']}). We set $\mathop{\mathrm{Re}}\nolimits \tau = 0$ in the figure, so that $(1,0)$ and $(0,1)$ strings are orthogonal. The overall angle of the network $\alpha$ is equal to the phase of the central charge of the corresponding $\frac{1}{4}$-BPS state.
• Figure 2: String network $\mathcal{P}_{\mathrm{SY}}(\vec{r}, \vec{s})$ corresponding to a Stern-Yi bound state of dyons.