Looking for Integrability on the Worldsheet of Confining Strings

TL;DR

This work investigates whether the worldsheet theory of confining strings can be integrable under non-linear target-space Poincaré symmetry. It develops a current-algebra framework to establish double-softness for branons and analyzes both bosonic and supersymmetric string worldsheets, showing that one-loop collinear singularities generally spoil integrability unless the bulk dimension is critical ($D=26$) or the string lives in $D=3$. The authors extend the analysis to include massless worldsheet fermions from broken supersymmetry, showing tree-level integrability aligns with a kappa-symmetric Green–Schwarz action in special dimensions, and revealing a hidden linear supersymmetry that makes certain $N=1$ theories equivalent to $N=2$ GS models for all values of the Wess–Zumino coefficient $c$. Overall, the results imply that truly integrable confining strings in four dimensions require additional massless degrees of freedom, with $D=3$ remaining a unique exceptional case, and provide a framework connecting target-space symmetry, worldsheet dynamics, and integrability.

Abstract

We study restrictions on scattering amplitudes on the worldvolume of branes and strings (such as confining flux tubes in QCD) implied by the target space Poincare symmetry. We focus on exploring the conditions for the string worldsheet theory to be integrable. We prove that for a higher dimensional membrane the scattering amplitudes for the translational Goldstone modes ("branons") are double soft. At one-loop double softness is generically violated for the string worldsheet scattering as a consequence of collinear singularities. Violation of double softness implies in turn the breakdown of integrability. We prove that if branons are the only gapless degrees of freedom then the worldsheet integrability is compatible with target space Poincare symmetry only if the number of space-time dimensions is equal to D = 26 (a critical bosonic string), and for D = 3. We extend the analysis to include massless worldsheet fermions, resulting from spontaneous breakdown of the target space supersymmetry. We check that the tree-level integrability in this case is in one-to-one correspondence with the existence of a kappa-symmetric Green-Schwarz (GS) action. As a byproduct we show that at the leading order in the derivative expansion an N = 1 superstring without kappa-symmetry in D = 3,4,6,10 dimensions exhibits an accidental enhanced supersymmetry and is equivalent to a kappa-symmetric N = 2 GS superstring.

Figures (9)

• Figure 1: Yang-Baxter equation cannot be satisfied in the $SO(N)$-symmetric massless theory in the presence of reflections for $N>2$
• Figure 2: Diagrams contributing to the one-loop branon 6-point function
• Figure 3: A collinear singularity present in the shift current Ward identity
• Figure 4: Coleman-Thun type singularity in the shift current Ward identity
• Figure 5: 4-point interactions between branons and goldstini. The solid line represents a goldstino corresponding to one broken supercharge. Time is flowing upwards (also in all later Figures). The amplitudes containing bosons need to be divided by the norm of the bosonic states to obtain the corresponding $S$-matrix element.