[Paper Review] An exact string representation of 3d SU(2) lattice Yang--Mills theory

Florian Conrady, Igor Khavkine|ArXiv.org|Jun 23, 2007

Black Holes and Theoretical Physics32 references3 citations

TL;DR

This paper establishes an exact string representation of 3D SU(2) lattice Yang-Mills theory by reformulating its spin foam dual as a sum over self-avoiding worldsheets of fundamental strings. By introducing a framed lattice composed of cubes and truncated rhombic dodecahedra, the authors prove a bijection between spin foams and non-branching, unlabelled worldsheets, enabling the partition function and Polyakov loop expectation values to be expressed as explicit sums over these string worldsheets with derived amplitude factors.

ABSTRACT

We show that 3d SU(2) lattice Yang--Mills theory can be cast in the form of an exact string representation. The derivation starts from the exact dual (or spin foam) representation of the lattice gauge theory. We prove that every dual configuration (or spin foam) can be equivalently described as a self--avoiding worldsheet of strings on a framing of the lattice. Using this correspondence, we translate the partition function into a sum over closed worldsheets that are weighted with explicit amplitudes. The expectation value of two Polyakov loops with spin j becomes a sum over worldsheets that are bounded by 2j strings along a framing of the loops.

Motivation & Objective

• To establish a fully explicit and exact string representation of 3D SU(2) lattice Yang-Mills theory.
• To resolve the long-standing conjecture that gauge theories may be dual to string-like degrees of freedom.
• To overcome limitations of previous approaches by constructing a bijection between spin foams and non-branching, self-avoiding worldsheets.
• To provide a new computational framework for the dual representation using geometric string worldsheets instead of spin-foam graphs.
• To clarify the physical interpretation of flux lines in terms of fundamental strings bounded by Wilson loops of spin $j$.

Proposed method

• Replace the standard cubic lattice with a tiling of cubes and truncated rhombic dodecahedra to ensure exactly three faces meet at each edge.
• Framed (thickened) the 2-skeleton of this lattice to allow for the construction of multiple, distinct surfaces per face.
• Assign $2j_f$ sheets to each face labeled with spin $j_f$, enabling the formation of connected, non-branching worldsheets.
• Define worldsheets as compact, self-avoiding surfaces embedded in the framed lattice that satisfy topological and combinatorial constraints.
• Prove a bijective correspondence between admissible worldsheets and spin foam configurations via a mapping that preserves spin coupling conditions.
• Derive explicit amplitude factors for worldsheets, including contributions from $6j$-symbols and face occupation numbers $N_{f'}$, which generalize the Nambu-Goto action.

Experimental results

Research questions

• RQ1 Can 3D SU(2) lattice Yang-Mills theory be exactly reformulated in terms of fundamental string worldsheets?
• RQ2 What topological and geometric modifications to the lattice are required to ensure a one-to-one correspondence between spin foams and non-branching worldsheets?
• RQ3 How do the amplitude factors for worldsheets relate to the original spin foam amplitudes, particularly the $6j$-symbols?
• RQ4 What is the boundary structure of worldsheets that correspond to the expectation value of two Polyakov loops of spin $j$?
• RQ5 To what extent does the resulting string representation generalize or differ from the Nambu-Goto action, especially when multiple intersections occur?

Key findings

• A bijection is established between spin foams in the original lattice and self-avoiding, non-branching worldsheets in a framed, modified lattice composed of cubes and truncated rhombic dodecahedra.
• The partition function of 3D SU(2) lattice Yang-Mills theory is exactly rewritten as a sum over closed worldsheets, each weighted by an explicit amplitude factor involving $6j$-symbols and face occupation numbers.
• The expectation value of two Polyakov loops of spin $j$ is expressed as a sum over worldsheets bounded by $2j$ strings along each loop’s framing, with the string between quarks remaining unbroken for $j=1/2$.
• When a worldsheet intersects a face only once ($N_{f'} o 1$), the amplitude reduces to a form proportional to the area of the worldsheet, resembling the Nambu-Goto action.
• For higher face occupations ($N_{f'} > 1$), nonlinear dependencies on $N_{f'}$ introduce new interactions between neighboring strings, beyond simple merging and splitting.
• The string representation is the first exact and fully explicit realization of gauge-string duality in 3D SU(2) lattice gauge theory, differing from both abelian 2-chain formulations and effective Nambu-Goto models.