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 $\mathbb {R}^4$ 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 reduced to four-dimensional supersymmetric Yang–Mills theory.

Appendix

Below, we collect some definitions, results and hopefully helpful references. While “Appendix A” is a rough overview over the categorification of the mathematical notions involved in the definition of principal bundles, “Appendices B, C and D” explain in detail various aspects of categorified gauge algebras. Potentially useful and more detailed introductions into some of the following material are found in [5, 6].

1.1 Categorified Lie groups, Lie algebras and principal bundles

Let ${\textsf {Mfd}}^\infty$ be the category of smooth spaces with smooth maps between them. A 2-space (e.g., [2]) is formally a category internal to¹⁶${\textsf {Mfd}}^\infty$. That is, a 2-space consists of two manifolds $M_1, M_0$ with smooth source and target maps $\textsf {s},\textsf {t}{:}\,M_1\rightrightarrows M_0$, a smooth embedding $\mathrm {id}{:}\,M_0{\hookrightarrow }M_1$ and a smooth composition of morphisms $\circ {:}\, M_1\times _{M_0} M_1\rightarrow M_1$ satisfying the usual axioms of the structure maps of a category. Morphisms between 2-spaces can be defined as functors internal to ${\textsf {Mfd}}^\infty$, which are smooth maps between the morphism- and object-manifolds of the 2-spaces.¹⁷ Note that any manifold M gives trivially rise to a 2-space $M\rightrightarrows M$ in which the only morphisms are the identity morphisms. Also, Lie groupoids are 2-spaces in which each morphism is invertible.

In its simplest, strict form, a Lie 2-group is a category internal to ${\textsf {Grp}}$, the category of groups. Such strict Lie 2-groups are most conveniently described as crossed modules of Lie groups [65]. More generally, a Lie 2-group is a Lie groupoid which is simultaneously a monoidal category and in which morphisms are strictly invertible and objects are weakly invertible with respect to the monoidal product [65].

There is a higher analogue of Lie differentiation, which takes a Lie 2-group to a Lie 2-algebra [39]. Strict Lie 2-groups differentiate to strict Lie 2-algebras, which are conveniently described by crossed modules of Lie algebras. More generally, a Lie 2-group differentiates to a 2-term $L_\infty$-algebra [38], see also [11] for a detailed discussion of an extremely general case. Since we use these extensively, we present some more details in “Appendices B, C and D”.

Let us now come to the categorification of principal bundles. As explained in Sect. 2.1, one can use any of the equivalent definitions of principal bundles and replace all notions by categorified ones. To describe transition functions of principal $\textsf {G}$-bundles, one can use the approach via functors from the Čech groupoid$\check{\mathscr {C}}(Y)$ of a surjective submersion $\sigma {:}\,Y\twoheadrightarrow M$ over some manifold M to the category $\textsf {B}\textsf {G}=\textsf {G}\rightrightarrows *$. One may imagine Y to be an open cover $Y=\sqcup U_i$, for concreteness sake. There are now two obvious projections from the fiber product $Y^{[2]}=Y\times _M Y$ (which is the space of double overlaps in the case $Y=\sqcup U_i$) to Y as well as a diagonal embedding $Y{\hookrightarrow }Y^{[2]}$, which, together with the composition $(y_1,y_2)\circ (y_2,y_3)=(y_1,y_3)$ we can use to form the 2-space

$$\begin{aligned} \check{\mathscr {C}}(Y)=(Y^{[2]}\rightrightarrows Y). \end{aligned}$$

(A.1)

Since morphisms are invertible with $(y_1,y_2)^{-1}=(y_2,y_1)$, this is in fact a Lie groupoid. The transition functions of a principal $\textsf {G}$-bundle for some Lie group $\textsf {G}$ are given by functors $g{:}\,\check{\mathscr {C}}(Y)\rightarrow \textsf {B}\textsf {G}$ and isomorphism or gauge transformations correspond to natural transformations between functors. This picture readily extends to categorified groups, where we consider higher functors between the Čech groupoid, trivially regarded as a higher Lie groupoid, and the delooping $\textsf {B}\mathscr {G}$ of a higher Lie group $\mathscr {G}$. For $\mathscr {G}=\textsf {B}\textsf {U}(1)$, we recover abelian gerbes, or principal $\textsf {B}\textsf {U}(1)$-bundles in the form of Hitchin–Chatterjee gerbes [66]. These are stably isomorphic to Murray’s more general and more useful bundle gerbes [67, 68]. Abelian gerbes, or principal $\textsf {B}\textsf {U}(1)$ bundles are described by a characteristic class, called the Dixmier–Douady class in $H^3(M,\mathbb {Z})$, which is the analogue of the first Chern class of line bundles in $H^2(M,\mathbb {Z})$.

To add connections, one can either glue together the local description as derived in Sect. 3.2 or use insights from Lie differentiation, as done in [10, 11].

1.2 $L_\infty$-algebras and Lie n-algebras

Strong homotopy Lie algebras or $L_\infty$-algebras comprise Lie algebras and are useful descriptions of all their categorifications. They play important roles in BV quantization, string field theory and higher geometry in general; the original references are [69,70,71].

An $L_\infty$-algebra is an $\mathbb {Z}$-graded vector space $\textsf {L}=\oplus _{k\in \mathbb {Z}}\textsf {L}_k$ which is endowed with a set of totally antisymmetric, multilinear maps $\mu _i{:}\,\wedge ^i\textsf {L}\rightarrow \textsf {L}$, $i\in \mathbb {N}$, of degree $i-2$, which satisfy the higher or homotopy Jacobi relations

$$\begin{aligned} \sum _{r+s=i}\sum _{\sigma }(-1)^{rs}\chi (\sigma ;\ell _1,\dots ,\ell _{r+s})\mu _{s+1}(\mu _r(\ell _{\sigma (1)},\dots ,\ell _{\sigma (r)}),\ell _{\sigma (r+1)},\dots ,\ell _{\sigma (r+s)})\ =\ 0 \end{aligned}$$

(B.1)

for all $\ell _1,\dots ,\ell _{r+s}\in \textsf {L}$, where the second sum runs over all (r, s) unshuffles, i.e., permutations $\sigma$ of $\{1,\dots ,r+s\}$ with the first r and the last s images of $\sigma$ ordered: $\sigma (1)<\dots <\sigma (r)$ and $\sigma (r+1)<\cdot <\sigma (r+s)$. Also, $\chi (\sigma ;\ell _1,\dots ,\ell _i)$ denotes the graded Koszul sign defined through the graded antisymmetrized products

$$\begin{aligned} \ell _1\wedge \dots \wedge \ell _i\ =\ \chi (\sigma ;\ell _1,\dots ,\ell _i)\,\ell _{\sigma (1)}\wedge \dots \wedge \ell _{\sigma (i)}. \end{aligned}$$

(B.2)

If an $L_\infty$-algebra is non-trivial in degrees $0,\dots ,n-1$, we call it an n-term $L_\infty$-algebra. This is a useful and very general notion of a Lie n-algebra. We therefore often use the term Lie n-algebra, when we actually mean an n-term $L_\infty$-algebra.

A cyclic structure on an $L_\infty$-algebra over $\mathbb {R}$ is a graded symmetric, non-degenerate bilinear form

$$\begin{aligned} \langle -,-\rangle {:}\,\textsf {L}\odot \textsf {L}\ \rightarrow \ \mathbb {R}, \end{aligned}$$

(B.3)

such that

$$\begin{aligned} \langle \ell _1,\mu _i(\ell _2,\dots ,\ell _{i+1})\rangle \ =\ (-1)^{i+|\ell _{i+1}|(|\ell _1|+\dots +|\ell _i|)}\langle \ell _{i+1},\mu _i(\ell _1,\dots ,\ell _{i})\rangle \end{aligned}$$

(B.4)

for $\ell _1,\dots ,\ell _{i+1}\in \textsf {L}$.

We are particularly interested in Lie 2- and 3-algebras. The lowest homotopy Jacobi relations are equivalent to the following ones:

$$\begin{aligned} \mu _1(\mu _1(\ell _1))= & {} 0,\nonumber \\ \mu _1(\mu _2(\ell _1,\ell _2))= & {} \mu _2(\mu _1(\ell _1),\ell _2)+(-1)^{|\ell _1|}\mu _2(\ell _1,\mu _1(\ell _2)),\nonumber \\ \mu _1(\mu _3(\ell _1,\ell _2,\ell _3))= & {} -\mu _3(\mu _1(\ell _1),\ell _2,\ell _3)-(-1)^{|\ell _1|}\mu _3(\ell _1,\mu _1(\ell _2),\ell _3)\nonumber \\&-\,(-1)^{(|\ell _1|+|\ell _2|)}\mu _3(\ell _1,\ell _2,\mu _1(\ell _3))+\mu _2(\ell _1,\mu _2(\ell _2,\ell _3))\nonumber \\&-\,\mu _2(\mu _2(\ell _1,\ell _2),\ell _3)-(-1)^{|\ell _1|\,|\ell _2|}\mu _2(\ell _2,\mu _2(\ell _1,\ell _3)).\nonumber \\ \end{aligned}$$

(B.5)

The first two identities are the same as for a differential graded Lie algebra, but the third identity is a controlled lifting of the Jacobi identity.

A cyclic structure on a Lie 3-algebra satisfies

$$\begin{aligned} \begin{aligned} \langle \ell _1,\mu _1(\ell _2)\rangle&= (-1)^{1+|\ell _2|\,|\ell _1|}\langle \ell _2,\mu _1(\ell _1)\rangle ,\\ \langle \ell _1,\mu _2(\ell _2,\ell _3)\rangle&= (-1)^{1+|\ell _3|(|\ell _1|+|\ell _2|)+|\ell _1|\,|\ell _2|}\langle \ell _3,\mu _2(\ell _2,\ell _1)\rangle ,\\ \langle \ell _1,\mu _3(\ell _2,\ell _3,\ell _4)\rangle&= (-1)^{1+|\ell _4|(|\ell _1|+|\ell _2|+\ell _3)+|\ell _3|(|\ell _1|+|\ell _2|)}\langle \ell _4,\mu _3(\ell _2,\ell _3,\ell _1)\rangle ,\\ \langle \ell _1,\mu _4(\ell _2,\ell _3,\ell _4,\ell _5)\rangle&= (-1)^{1+|\ell _5|(|\ell _1|+|\ell _2|+\ell _3+\ell _4)+|\ell _4|(|\ell _1|+|\ell _2|+|\ell _3|)}\langle \ell _5,\mu _4(\ell _2,\ell _3,\ell _4,\ell _1)\rangle \end{aligned} \end{aligned}$$

(B.6)

for all $\ell _1,\dots ,\ell _5\in \textsf {L}$.

1.3 Lie n-algebras as NQ-manifolds

A very useful and elegant definition of $L_\infty$-algebras can be given in terms of NQ-manifolds, which are known to physicists from BRST quantization and string field theory. In this picture, the cyclic structure arises from a symplectic form. Below, we briefly explain this point of view.

An NQ-manifold$(\mathcal {M},Q)$ is an $\mathbb {N}_0$-graded manifold $\mathcal {M}$ endowed with a vector field Q which is of degree 1 and nilquadratic: $Q^2=0$. Due to a similar argument as that for smooth (real) supermanifolds, NQ-manifolds can be regarded as $\mathbb {N}$-graded vector bundles over the body $\mathcal {M}_0$ of the manifold $\mathcal {M}$.

Archetypical examples of NQ-manifolds are $(T[1]M,\mathrm {d})$, the grade-shifted tangent bundle together with the de Rham differential as well as the grade-shifted Lie algebra $(\mathfrak {g}[1],Q)$ with $Q=-\tfrac{1}{2} f_{\alpha \beta }^\gamma \xi ^\alpha \xi ^\beta \frac{\partial }{\partial \xi ^\gamma }$ a vector field of degree 1 in some coordinates $\xi ^\alpha$ on $\mathfrak {g}[1]$, which are necessarily of degree 1. Note that $Q^2=0$ is equivalent to the Jacobi identity in the latter case.

The latter example indicates the relation of NQ-manifolds to Lie n-algebras: Given a Lie n-algebra $\textsf {L}$, we should first grade-shift the underlying graded vector space:

$$\begin{aligned} \textsf {L}=(\textsf {L}_0\leftarrow \dots \leftarrow \textsf {L}_{n-1})\quad \rightarrow \quad \textsf {L}[1]=(*\leftarrow \textsf {L}_0[1]\leftarrow \dots \leftarrow \textsf {L}_{n-1}[1]). \end{aligned}$$

(C.1)

Correspondingly, the degree of the maps $\mu _i$ changes from $i-2$ to $-1$. The sum of these maps forms a codifferential

$$\begin{aligned} D=\mu _1+\mu _2+\mu _3+\cdots , \end{aligned}$$

(C.2)

which acts on the coalgebra $\wedge ^\bullet \textsf {L}[1]$. If we now dualize to the algebra of functions on $\textsf {L}[1]$, the corresponding differential is a vector field of degree 1 on the NQ-manifold

$$\begin{aligned} *\leftarrow \textsf {L}_0[1]\leftarrow \cdots \leftarrow \textsf {L}_{n-1}[1]. \end{aligned}$$

(C.3)

The condition $Q^2=0$ translates to (B.1).

Altogether, we arrived at a description of a Lie n-algebra $\textsf {L}$ as a differential graded algebra$\big (\mathcal {C}^\infty (\textsf {L}[1]), Q\big )$. This differential graded algebra is also called the Chevalley–Eilenberg algebra$\mathrm{CE}(\textsf {L})$ of $\textsf {L}$.

It is not hard to see that a cyclic structure on a Lie n-algebra $\textsf {L}$ corresponds to a symplectic form on $\textsf {L}[1]$ of degree $n+1$, and we use this in Sect. 6.1. For example, the inner product $(x_1,x_2)=g_{\alpha \beta }\xi _1^\alpha \xi _2^\beta$ on a Lie algebra $\mathfrak {g}$ is described by the symplectic form $\omega =\tfrac{1}{2} g_{\alpha \beta }\mathrm {d}\xi ^\alpha \wedge \mathrm {d}\xi ^\beta$ of degree 2 on $\mathfrak {g}[1]$.

1.4 Morphisms and equivalences of Lie n-algebras

Let us now come to categorical equivalence between Lie n-algebras. For simplicity, we first restrict ourselves to Lie 2-algebras $\textsf {L}=W\leftarrow V$. A morphism of Lie n-algebras, being a morphism of graded spaces, is most readily derived in the Chevalley–Eilenberg picture. There, a morphism $\Phi {:}\,\textsf {L}\rightarrow \tilde{\textsf {L}}$ between Lie 2-algebras $\textsf {L}=(W\leftarrow V)$ and $\tilde{\textsf {L}}=\tilde{W}\leftarrow \tilde{V})$ is given by a morphism of differential graded algebras $\Phi ^*$. That is,

$$\begin{aligned} \Phi ^*{:}\,\mathcal {C}^\infty (\tilde{\textsf {L}}[1])\rightarrow \mathcal {C}^\infty (\textsf {L}[1]),\quad \tilde{Q}\circ \Phi ^*= \Phi ^*\circ Q. \end{aligned}$$

(D.1)

Since $\textsf {L}[1]$ is a graded vector space, this morphism is determined by its image on coordinate functions. Because degrees have to be preserved, $\Phi ^*$ (and thus $\Phi$) is encoded in maps

$$\begin{aligned} \phi _0{:}\,W\rightarrow \tilde{W},\quad \phi _0{:}\,V\rightarrow \tilde{V},\quad \phi _1{:}\,W\wedge W\rightarrow \tilde{V}. \end{aligned}$$

(D.2)

The fact that $\Phi ^*$ is a morphism of differential graded algebras implies that

$$\begin{aligned} \begin{aligned} \phi _0\left( \mu _2(w_1,w_2)\right)&=\tilde{\mu }_2\left( \phi _0(w_1),\phi _0(w_2)\right) +\tilde{\mu }_{1}(\phi _1(w_1,w_2)),\\ \phi _0\left( \mu _2(w,v)\right)&=\tilde{\mu }_2(\phi _0(w),\phi _0(v))+\phi _1 (w,\mu _{1}(v)),\\ \phi _0\left( \mu _3(w_1,w_2,w_3)\right)&=\tilde{\mu }_3(\phi _0(w_1),\phi _0(w_2), \phi _0(w_3))+\left[ \phi _1(w_1,\mu _2(w_2,w_3))\right. \\&\quad \left. +\,\tilde{\mu }_2\left( \phi _0(w_1),\phi _1(w_2,w_3)\right) +\text {cyclic}\,(w_1,w_2,w_3)\right] \end{aligned} \end{aligned}$$

(D.3)

for all $w,w_{1,2}\in W$ and $v\in V$. This reproduces the definition of a morphism of Lie 2-algebra from [39].

Two morphisms $\Phi =(\phi _0,\phi _1)$ and $\Psi =(\psi _0,\psi _1)$ compose as follows:

$$\begin{aligned} (\Psi \circ \Phi )_0(\ell )=\psi _0(\phi _0(\ell )),\quad (\Psi \circ \Phi )_{1}(w_1,w_2)=\psi _0(\phi _1(w_1,w_2))+\psi _1(\phi _0(w_1),\phi _0(w_2)) \end{aligned}$$

(D.4)

for all $\ell \in \textsf {L}$ and $w_{1,2}\in W$. The identity morphism reads as $\mathrm {id}_\textsf {L}=(\mathrm {id}_\textsf {L},0)$. The above data of 2-term $L_\infty$-algebras and their morphisms, together with the identity morphism and composition combines to a category of Lie 2-algebras, ${\textsf {Lie2alg}}$.

Note that the inverse of a morphism $\Phi =(\phi _0,\phi _1)$ is defined if and only if $\phi _0$ is invertible:

$$\begin{aligned} (\Phi ^{-1})_0(\ell )=\phi _0^{-1}(\ell ),\quad (\Phi ^{-1})_{-1}(w_1,w_2) =-\phi _0^{-1}(\phi _1(\phi _0^{-1}(w_1),\phi _0^{-1}(w_2))), \end{aligned}$$

(D.5)

again for $\ell \in \textsf {L}$ and $w_{1,2}\in W$.

A 2-morphism between two morphisms $\Phi ,\Psi {:}\,\textsf {L}\rightarrow \tilde{\textsf {L}}$ is a chain homotopy $\chi {:}\,W\rightarrow \tilde{V}$ such that

$$\begin{aligned} \phi _1(w_1,w_2)-\psi _1(w_1,w_2)=\mu _2(w_1,\chi (w_2))+\mu _2(\chi (w_1), \psi _0(w_2))-\chi (\mu _2(w_1,w_2)) \end{aligned}$$

(D.6)

for all $w_1,w_2$.

Finally, an equivalence or a quasi-isomorphism of Lie 2-algebras between Lie 2-algebras $\textsf {L}=W\xleftarrow {~\mu _1~}V$ and $\tilde{\textsf {L}}=\tilde{W}\xleftarrow {~\tilde{\mu }_1~}\tilde{V}$ is a morphism $\Phi _1{:}\,\textsf {L}\rightarrow \tilde{\textsf {L}}$ and a morphism $\Phi _2{:}\,\tilde{\textsf {L}}\rightarrow \textsf {L}$ such that $\Phi _1\circ \Phi _2\cong \mathbb {1}_{\tilde{frg}}$ and $\Phi _2\circ \Phi _1\cong \mathbb {1}_{\textsf {L}}$. An explicit example of an equivalence between Lie 2-algebras is found in Sect. 2.3.

The above discussion readily generalizes to morphisms $\Phi$ between Lie n-algebras $\textsf {L}$ and $\tilde{\textsf {L}}$. These are given by totally antisymmetric maps $\phi _i{:}\,\textsf {L}^{\wedge i+1}\rightarrow \textsf {L}$, $i=0,\dots ,n-1$, of degree $-i$ such that appropriate extensions of (D.3) hold, see [72] for details. An isomorphism of Lie n-algebras is a morphism of Lie n-algebras with $\phi _0$ an isomorphisms. Equivalences between $L_\infty$-algebras are then captured by Lie n-algebra quasi-isomorphisms which are morphisms of Lie n-algebras which induce an isomorphism on the cohomology of the complex underlying the Lie n-algebras.