The non-abelian self-dual string

Abstract

We argue that the relevant higher gauge group for the non-abelian generalization of the self-dual string equation is the string 2-group. We then derive the corresponding equations of motion and discuss their properties. The underlying geometric picture is a string structure, i.e. a categorified principal bundle with connection whose structure 2-group is the string 2-group. We readily write down the explicit elementary solution to our equations, which is the categorified analogue of the 't Hooft-Polyakov monopole. Our solution passes all the relevant consistency checks; in particular, it is globally defined on R 4 and approaches the abelian self-dual string of charge one at infinity. We note that our equations also arise as the BPS equations in a recently proposed six-dimensional superconformal field theory and we show that with our choice of higher gauge structure, the action of this theory can be naturally reduced to four-dimensional supersymmetric Yang-Mills theory.

Higher Structures, Self‐Dual Strings and 6d Superconformal Field Theories

“…Interestingly, our Lie 4-algebra string ω sk (g) satisfies these axioms. [45,50] This is very encouraging, as it gives us a link between supergravity and our action and a similar link was also present in the case of the Chern-Simons matter actions of M2-brane models.…”

“…Interesting topological structures can indeed be found, and the equations can be compactified on S 4 . [45] From a physics perspective, we note that our equations nicely reduce to the Bogomolny monopole equation on R 3 , [45] as one would expect from the general M-theory picture. We can also use the usual Bogomolny trick to obtain a Bogomolny bound.…”

“…The last point implies that we also need a formulation of string structures for string (g). This was developed in [45] and we have the following data. The gauge potential decomposes into…”

“…As shown in [45], gauge equivalence classes for both models of the string Lie 2-algebra are mapped into each other under the quasi-isomorphism linking both. Note that this form of string structure can certainly be extended to include contributions from the tangent bundle as in (15).…”

Twisted Weil Algebras for the String Lie 2‐Algebra

“…Here, however, we will adopt a third way of describing n-term L ∞ -algebras which arises when dualizing the above concept 1 . The graded symmetric coalgebra ∨ • g[−1] becomes the graded symmetric algebra ∧ • g * [1], where, in the process of dualizing all gradings acquire a minus sign. The coderivation D then induces a degree 1 differential Q : ∧ • g * [1] → ∧ • g * [1], acting on the graded symmetric algebra ∧ • g * [1].…”