Wilson Loops, Geometric Transitions and Bubbling Calabi-Yau's

TL;DR

This work constructs multiple bulk descriptions of Wilson loop operators in Chern-Simons theory within the Gopakumar–Vafa/topological string framework. It shows that Wilson loops can be realized by configurations of D-branes or anti-branes in the deformed conifold, with the representation encoded in holonomy data, and that these configurations undergo geometric transitions to bubbling Calabi–Yau geometries whose topology encodes the knot data. The unknot is checked explicitly to all orders in the genus, and the formalism relates open Gopakumar–Vafa invariants to the closed invariants of the bubbling CY, offering a novel, exact representation of knot invariants via Calabi–Yau topology. Collectively, these results illuminate a holography-like structure for topological strings and provide concrete, quantum-corrected bulk descriptions of Wilson loops and knot invariants.

Abstract

Motivated by recent developments in the AdS/CFT correspondence, we provide several alternative bulk descriptions of an arbitrary Wilson loop operator in Chern-Simons theory. Wilson loop operators in Chern-Simons theory can be given a description in terms of a configuration of branes or alternatively anti-branes in the resolved conifold geometry. The representation of the Wilson loop is encoded in the holonomy of the gauge field living on the dual brane configuration. By letting the branes undergo a new type of geometric transition, we argue that each Wilson loop operator can also be described by a bubbling Calabi-Yau geometry, whose topology encodes the representation of the Wilson loop. These Calabi-Yau manifolds provide a novel representation of knot invariants. For the unknot we confirm these identifications to all orders in the genus expansion.

Figures (6)

• Figure 1: A Young tableau with $P\leq N$ rows and $M$ columns labeling a representation of $U(N)$. $R_i$ is the number of boxes in the $i$-th row and satisfies $R_i\geq R_{i+1}$. $R^T$ is the tableau conjugate to $R$, obtained by exchanging rows with columns.
• Figure 2: Brane configuration in resolved conifold describing Wilson loop in a representation given by Figure \ref{['young-param-1']}.
• Figure 3: Brane configuration in resolved conifold describing Wilson loop in a representation given by Figure 1.
• Figure 4: The correspondence of the Wilson loop in representation $R$ and the bubbling geometry. In (a) the Young tableau $R$, shown rotated, is specified by the lengths $l_i$ of all the edges. $l_{2m+1}$ is $N$ minus the number of rows. Equivalently, $l_i$ are the lengths of black and white regions in the Maya diagram. The Wilson loop in representation $R$ (a) is equivalent to the toric Calabi-Yau manifold given by the web diagram (b). There are a total of $2m+1$ bubbles of ${\bf P}^1$ in the geometry. The sizes of the ${\bf P}^1$'s are given by $t_i=g_sl_i,~i=1,...,2m+1$.
• Figure 5: The simplest example of the correspondence between a Wilson loop and a bubbling geometry. The Wilson loop along the unknot in the $U(N)$ representation specified by the Young tableau (a) is equivalent to the toric Calabi-Yau manifold given by the web diagram (b). The Kähler moduli in (b) are given by $t_1=g_sm,~t_2= g_s l,~t_3=g_s (N-m)$. The bubbling CY manifold (b) arises from geometric transition of $l$ branes in (c) as well as $m$ anti-branes in (d).