nLab geometry of physics -- BPS charges

this page is one chapter of geometry of physics

previous chapters: manifolds and orbifolds, WZW terms, supergeometry

---

Motivation and results

Consider $(X,g)$ a super-spacetime and $\omega$ a degree-$(p+2)$ differential form on $X$ which is a WZW curvature form definite on the super Lie algebra cocycle

$$\overline \psi \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_p} \in CE(\mathbb{R}^{d-1,1\vert N})$$

for Green-Schwarz super p-brane sigma model with target space $X$. Then the Polyvector extensions

$$[Q_\alpha, Q_\beta] = (\Gamma^a C)_{\alpha\beta} P_a + (\Gamma^{a_1\cdots a_p}C)_{\alpha \beta} Z_{a_1\cdots a_p}$$

of the super Lie algebra of super-isometries of $(X,g)$ by charges $Z$ of Noether currents of the super p-brane sigma model are known as algebras of BPS charges. (The spacetime $(X,g,\omega)$ is called a supergravity $\frac{1}{k}$-BPS state if the dimension of the space of supercharges $Q$ in the kernel of the above bracket is $\frac{1}{k}$th of that of super Minkowski spacetime).

This is well understood in the literature (Azcárraga-Gauntlett-Izquierdo-Townsend 89) for the case that $X$ is locally modeled on an ordinary super Minkowski spacetime and that the $p$-brane species is in the old brane scan (e.g the type II superstrings, the heterotic superstring and the M2-brane, also e.g. the super 1-brane in 3d and the 3-brane in 6d, but not the D-branes and not the M5-brane).

For the full story of string theory this needs to be refined in three ways (see Fiorenza-Sati-Schreiber 13), and this has been left open in the literature, for previous lack of a higher differential geometry that could handle this:

1. For a genuine global definition of the Green-Schwarz super p-brane sigma model with target $(X,\omega)$, the WZW curvature form $\omega$ needs to be prequantized to a globally well-defined WZW term, a genuine cocycle in Deligne cohomology (a circle (p1+1)-bundle with connection).

   (The need for this has broadly been ignored, one place where it is mentioned is (Witten 86, p. 17).)

2. For the inclusion of charges of $p_2$-branes on which $p_1$-branes may end (for $p_1 = 1$: type II strings ending on D-branes, for $p_1= 2$ and $p_2 = 5$ M2-branes ending on M5-branes) then $X$ is to be locally modeled on an extended super Minkowski spacetime, hence on a super orbispace, hence a curved spacetime now is an object in higher Cartan geometry and one needs to make sense of Noether currents there.

   (Arguments in this direction for the D-branes have been given in (Hammer 97) and for the M5-brane in (Sorokin-Townsend 97).)

3. For inclusion of non-infinitesimal isometries one needs the global structure of the full supergroup of BPS charges, not just its super Lie algebra.

Here we discuss how to solve these problems in full generality (Sati-Schreiber 15). Specified to the situation in 11-dimensional supergravity with M2-branes and M5-branes we find that the BPS charges traditionally seen in the M-theory super Lie algebra as living in ordinary cohomology $H^2(X) \oplus H^5(X)$ of spacetime $X$ receive corrections by $d_4$-differentials of a Serre spectral sequence given by cup product with the class of the supergravity C-field. This is in higher analogy to how D-brane charges are well known (Maldacena-Moore-Seiberg 01) to be in ordinary cohomology only up to corrections of the $d_3$-differential (and higher) in an Atiyah-Hirzebruch spectral sequence for twisted K-theory, given by cup product with the class of the B-field. This supports the conjecture (Sati 10) that M5-brane charge should really be in twisted elliptic cohomology, since this is what is canonically twisted by these degree-4 classes (Ando-Blumberg-Gepner 10). (Realizing this fully amounts to refining the term $\mathbf{L}_{M5}^X$ that we construct in ordinary differential cohomology below to ellitptic differential cohomology. Discussion of that refinement is beyond the scope of this page here.)

We also close a gap in (AGIT89): what is strictly derived there from the Noether theorem is extension of the supersymmetry algebra by differential forms, while the argument that it is only the de Rham cohomology class of these forms that matters relies on physics intuition. We find here that the Lie algebra of conserved currents extending the (super-)isometry algebra is naturally not just a (super-)Lie algebra but a (super-)Lie (p+1)-algebra including higher order symmetries of Noether symmetries. It is by quotienting these out when restricting the current Lie $n$-algebra to its lowest Postnikov stage that current forms pass to their de Rham equivalence classes. Accordingly, the fully globalized current groups that we find are really (super-)smooth n-groups. For instance the M-theory super Lie algebra is refines to a super Lie 6-group, where $6 = 5+1$ is the dimension of the M5-brane worldvolume.

$\,$

We formulate the theory general abstractly in the context of higher differential geometry given by an (∞,1)-topos $\mathbf{H}$ equipped with cohesion and differential cohesion. The application to supergravity takes place in the model $\mathbf{H} =$ SuperFormalSmooth∞Grpd.

Prerequisites

For ease of reference, we recall here some definitions and propositions form previous chapters of geometry of physics which we need for the discussion of BPS charge groups below.

$\infty$-Representations and associated $\infty$-bundles

recalled from geometry of physics – representations and associated bundles

Definition

Given $V,E\in \mathbf{H}$, a $V$-fiber ∞-bundle $E$ over $X$ is a bundle $E \in \mathbf{H}_{/X}$ such that there exists a cover (i.e. a 1-epimorphism) $U \longrightarrow X$ and a homotopy pullback diagram of the form

$$\array{ U \times V &\longrightarrow& E \\ \downarrow && \downarrow \\ U &\longrightarrow& X } \,.$$

Definition

Given an ∞-group $G$ and a $G$-∞-action on $V$, and given an $G$-principal ∞-bundle $P \in \mathbf{H}_{/X}$ modulated by $\mathbf{c} \colon X \longrightarrow \mathbf{B}G$, then the associated ∞-bundle is $V$-fiber ∞-bundle $E = P \times_G V$ which is the homotopy pullback in

$$\array{ P \times_G V &\longrightarrow& V/G \\ \downarrow && \downarrow \\ X &\stackrel{\mathbf{c}}{\longrightarrow}& \mathbf{B}G } \,.$$

A $V$-fiber bundle realized this way is said to have structure group $G$.

Proposition

Every $V$-fiber ∞-bundle is the associated ∞-bundle, def. , of some $\mathbf{Aut}(V)$-principal ∞-bundle.

Definition

Given a $G$-principal ∞-bundle $P$ modulated by some $\mathbf{c}\colon X \longrightarrow \mathbf{B}G$, and given a homomorphism of ∞-groups $H \hookrightarrow$, then a reduction/lift of the structure group is a lift $\hat {\mathbf{c}}$ in

$$\array{ && \mathbf{B}G \\ &{}^{\mathllap{\hat{\mathbf{c}}}}\nearrow& \downarrow \\ X &\stackrel{\simeq}{\longrightarrow}& \mathbf{B}G } \,.$$

Similarly for $V$-fiber ∞-bundles via def. , prop. .

Homotopy stabilizer groups

recalled from geometry of physics – representations and associated bundles

For $\mathbf{H}$ an (∞,1)-topos, $G\in \mathbf{H}$ an object equipped with ∞-group structure, hence with a delooping $\mathbf{B}$G, and for $\rho$ an ∞-action of $G$ on some $V$, exhibited by a homotopy fiber sequence of the form

$$\array{ V &\stackrel{i}{\longrightarrow}& V/G \\ && \downarrow^{\mathrlap{p_\rho}} \\ && \mathbf{B}G } \,.$$

Definition

Given a global element of $V$

$$x \colon \ast \to X$$

then the stabilizer $\infty$-group $Stab_\rho(x)$ of the $G$-action at $x$ is the loop space object

$$Stab_\rho(x) \coloneqq \Omega_{i(x)} (X/G) \,.$$

Remark

Equivalently, def. , gives the loop space object of the 1-image $\mathbf{B}Stab_\rho(x)$ of the morphism

$$\ast \stackrel{x}{\to} X \to X/G \,.$$

As such the delooping of the stabilizer $\infty$-group sits in a 1-epimorphism/1-monomorphism factorization $\ast \to \mathbf{B}Stab_\rho(x) \hookrightarrow X/G$ which combines with the homotopy fiber sequence of prop. to a diagram of the form

$$\array{ \ast &\stackrel{x}{\longrightarrow}& X &\stackrel{}{\longrightarrow}& X/G \\ \downarrow^{\mathrlap{epi}} && & \nearrow_{\mathrlap{mono}} & \downarrow \\ \mathbf{B} Stab_\rho(x) &=& \mathbf{B} Stab_\rho(x) &\longrightarrow& \mathbf{B}G } \,.$$

In particular there is hence a canonical homomorphism of $\infty$-groups

$$Stab_\rho(x) \longrightarrow G \,.$$

However, in contrast to the classical situation, this morphism is not in general a monomorphism anymore, hence the stabilizer $Stab_\rho(x)$ is not a sub-group of $G$ in general.

Higher Kostant-Souriau extensions

recalled from geometry of physics – prequantum geometry, following (FRS 13a)

Throughout, let $\mathbb{G} \in Grp(\mathbf{H})$ be a braided ∞-group equipped with a Hodge filtration. Write $\mathbf{B}\mathbb{G}_{conn}\in$ for the corresponding moduli stack of differential cohomology.

Example

For $\mathbf{H} =$ Smooth∞Grpd we have $\mathbb{G} = \mathbf{B}^p (\mathbb{R}/\Gamma)$ for $\Gamma = \mathbb{Z}$ is the circle (p+1)-group. Equipped with its standard Hodge filtration this gives $\mathbf{B}\mathbb{G}_{conn} = \mathbf{B}^p U(1)_{conn}$ presented via the Dold-Kan correspondence by the Deligne complex in degree $(p+2)$.

Definition

For $X \in \mathbf{H}$, for write

$$conc \colon [X,\mathbf{B}\mathbb{G}_{conn}] \longrightarrow \mathbb{G}\mathbf{Conn}(X)$$

for the differential concretification of the internal hom.

This is the proper moduli stack of $\mathbb{G}$-principal ∞-connections on $X$ in that a family $U \longrightarrow \mathbb{G}\mathbf{Conn}(X)$ is a vertical $\mathbb{G}$-principal $\infty$-connection on $U \times X\to U$.

Example

For $\mathbf{H} =$Smooth∞Grpd or =FormalSmooth∞Grpd, for $\mathbb{G} = \mathbf{B}^p U(1)$ the circle (p+1)-group with its standard Hodge filtration as in example , then for $X$ any smooth manifold or formal smooth manifold, $(\mathbf{B}^p U(1))\mathbf{Conn}(X)$ is presented via the Dold-Kan correspondence by the sheaf $U \mapsto Ch_\bullet$ of vertical Deligne complexes on $U \times X$ over $U$.

Proposition

For $\mathbb{G} \simeq \mathbf{B}\mathbb{G}'$ then the loop space object of the moduli stack of $\mathbb{G}$-principal $\infty$-connections on $X$ is the moduli stack of flat ∞-connections with gauge group $\Omega \mathbb{G}$

$$\Omega_0 (\mathbb{G}\mathbf{Conn}(X)) \simeq (\Omega\mathbb{G})\mathbf{FlatConn}(X) \,.$$

Proposition

The canonical precomposition ∞-action of the automorphism ∞-group $\mathbf{Aut}(X)$ on $[X,\mathbf{B}\mathbb{G}_{conn}]$ passes along $conc$ to an ∞-action on $\mathbb{G}\mathbf{Conn}(X)$.

Definition

Given a $\mathbb{G}$-principal ∞-connection $\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$ there are the following concepts in higher geometric prequantum theory.

1. The quantomorphism ∞-group is the stabilizer ∞-group of $\nabla \in \mathbb{G}\mathbf{Conn}(X)$, def. , under the $\mathbf{Aut}(X)$-action of ;

   $$\mathbf{QuantMorph}(X,\nabla) \coloneqq \mathbf{Stab}_{\mathbf{Aut}(X)}(conc(\nabla)) \,.$$

2. The Hamiltonian symplectomorphism ∞-group

   $$\mathbf{HamSymp}(X,\nabla) \longrightarrow \mathbf{Aut}(X)$$

   is the 1-image of the canonical morphism $\mathbf{QuantMorph}(X,\nabla) \longrightarrow \mathbf{Aut}(X)$ from remark .

3. A Hamiltonian action of an ∞-group $G$ on $(X,\nabla)$ is an ∞-group homomorphism

   $$\rho \colon G \longrightarrow \mathbf{HamSymp}(X,\nabla) \;$$

4. An ∞-moment map is an $\infty$-group homomorphism

   $$G \longrightarrow \mathbf{QuantMorph}(X,\nabla)$$

5. The Heisenberg ∞-group for a given Hamiltonian $G$-action $\rho$ is the homotopy pullback

   $$\mathbf{Heis}_G(X,\nabla) \coloneqq \rho^\ast \mathbf{QuantMorph}(X,\nabla) \,.$$

Example

For $\mathbf{H} =$ Smooth∞Grpd, for $X \in SmoothMfd \hookrightarrow \mathbf{H}$ a smooth manifold and for $\nabla$ a prequantum line bundle on $X$, then $\mathbf{QuantMorph}(X,\nabla)$ is Soriau’s quantomorphism group covering the Hamiltonian diffeomorphism group. In the case that $(X, F_\nabla)$ is a symplectic vector space $X = V$ regarded as a linear symplectic manifold with Hamiltonian action on itself by translation, then $\mathbf{Heis}_{V}(X,\nabla)$ is the traditional Heisenberg group.

Remark

Since $\mathbf{HamSymp}(X,\nabla)\hookrightarrow \mathbf{Aut}(X)$ is by construction a 1-monomorphism, given any $G$-action $\rho \colon G \longrightarrow \mathbf{Aut}(X)$ on $X$, not necessarily Hamiltonian, then the homotopy pullback $\rho^\ast \mathbf{QuantMorph}(X,\nabla)$ is the Heisenberg ∞-group of the maximal sub-$\infty$-group of $G$ which does act via Hamiltonian symplectomorphisms. Therefore we will also write $\mathbf{Heis}_G(X,\nabla)$ in this case.

The following is the refinement of the Kostant-Souriau extension to higher differential geometry

Proposition

Given a $\mathbb{G}$-principal ∞-connection $\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$, there is a homotopy fiber sequence of the form

1. if $\mathbb{G}$ is 0-truncated then

   $$\array{ \mathbb{G}\mathbf{ConstFunct}(X) &\longrightarrow& \mathbf{QuantMorph}(X,\nabla) \\ && \downarrow \\ && \mathbf{HamSymp}(X,\nabla) &\stackrel{\mathbf{KS}}{\longrightarrow}& \mathbf{B} (\mathbb{G}\mathbf{ConstFunct}(X)) }$$

2. if $\mathbb{G} \simeq \mathbf{B}\mathbb{G}'$ then

   $$\array{ (\Omega \mathbb{G})\mathbf{FlatConn}(X) &\longrightarrow& \mathbf{QuantMorph}(X,\nabla) \\ && \downarrow \\ && \mathbf{HamSymp}(X,\nabla) &\stackrel{\mathbf{KS}}{\longrightarrow}& \mathbf{B} ((\Omega \mathbb{G})\mathbf{FlatConn}(X)) }$$

exhibiting the quantomorphism ∞-group as an ∞-group extension of the Hamiltonian symplectomorphism ∞-group by the moduli stack of $\Omega \mathbb{G}-$flat ∞-connections, classified by a cocycle $\mathbf{KS}$.

(FRS13a)

Example

In $\mathbf{H} =$ Smooth∞Grpd, let $\mathbb{G} = \mathbf{B}^p U(1)$ be the circle (p+1)-group and let $X \in SmoothMfd \hookrightarrow Smooth \infty Grpd$ be p-connected, then $(\Omega\mathbf{B}^p U(1))\mathbf{FlatConn}(X)\simeq \mathbf{B}^{p}U(1)$. Hence here prop. gives

$$\array{ \mathbf{B}^{p}U(1) &\longrightarrow& \mathbf{QuantMorph}(X,\nabla) \\ && \downarrow \\ && \mathbf{HamSymp}(X,\nabla) &\stackrel{\mathbf{KS}}{\longrightarrow}& \mathbf{B}^{p+1}U(1) }$$

Corollary

Given a $\mathbb{G}$-principal ∞-connection $\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$, and for $\rho \colon G \longrightarrow \mathbf{HamSymp}(X,\nabla)$ a $G$-Hamiltonian action, then there is a homotopy fiber sequence

1. if $\mathbb{G}$ is 0-truncated then

   $$\array{ \mathbb{G}\mathbf{ConstFunct}(X) &\longrightarrow& \mathbf{Heis}_G(X,\nabla) \\ && \downarrow \\ && G &\stackrel{\mathbf{KS}(\rho)}{\longrightarrow}& \mathbf{B} (\mathbb{G}\mathbf{ConstFunct}(X)) }$$

2. if $\mathbb{G} \simeq \mathbf{B}\mathbb{G}'$ then

   $$\array{ (\Omega \mathbb{G})\mathbf{FlatConn}(X) &\longrightarrow& \mathbf{Heis}_G(X,\nabla) \\ && \downarrow \\ && G &\stackrel{\mathbf{KS}(\rho)}{\longrightarrow}& \mathbf{B} ((\Omega \mathbb{G})\mathbf{FlatConn}(X)) }$$

exhibiting the Heisenberg ∞-group as an ∞-group extension of the $G$ by the moduli stack of $\Omega \mathbb{G}-$flat ∞-connections, classified by a cocycle $\mathbf{KS}(\rho)$.

The class of the cocycle $\mathbf{KS}(\rho)$ is the obstruction to prequantizing $\rho$ to a moment map (the classical anomaly of $\rho$); and the the Heisenberg ∞-group extension of $G$ is the universal cancellation of this anomaly.

Definite forms

The concept of extending a closed differential form defined on a Cartesian space $\mathbb{R}^n$ to a definite form on an $n$-dimensional manifold is familiar from special holonomy manifolds. For instance a definite globalization of the associative 3-form on $\mathbb{R}^7$ to a 7-manifold induces and is induced by a G₂-structure. But by the discussion at geometry of physics – prequantum geometry, whenever we see a closed differential form we have to ask whether it is the curvature of a cocycle in differential cohomology, hence we have to ask for a higher prequantization. Here we consider the concept of definite forms prequantized to such definite globalizations of WZW terms.

Proposition

Given a $V$-fiber ∞-bundle $E$ over $X$, def. , and given any coefficient $A$, there is a natural equivalence beween

• morphisms $E \longrightarrow A$;

• sections of the canonically associated ∞-bundle $P \times_{\mathbf{Aut}(V)} [V,A]$ over $X$.

Definition

Given a $V$-fiber ∞-bundle $E$ over $X$, and a global element $x\colon \ast \to V$ then a section $\sigma$ of $E$ is definite on $x$ if there exists a 1-epimorphism $U \to X$ and a diagram

$$\array{ U &\longrightarrow& \ast \\ \downarrow & \swArrow & \downarrow^{\mathrlap{x}} \\ X & \stackrel{\sigma}{\longrightarrow} & V/\mathbf{Aut}(V) \\ & \searrow & \downarrow \\ && \mathbf{B}\mathbf{Aut}(V) } \,.$$

Proposition

Choices of sections definite on $x$ are equivalent to reductions of the structure group, def. , along the stabilizer group map $Stab_\mathbf{Aut(V)}(x)\longrightarrow \mathbf{Aut}(V)$.

Definition

Given $\mathbf{c} \colon V \longrightarrow A$, and given a $V$-fiber ∞-bundle $E$ over $X$, then a definite parameterization of $\mathbf{c}$ over $E$ is a $\mathbf{c}^E \colon E \longrightarrow A$ such that the section $\sigma_{\mathbf{c}^X}$ coresponding to it via prop. is definite on $\mathbf{c}$ in the sense of def. .

Definition

For $V$ an ∞-group, $\mathbf{L}\colon V \longrightarrow \mathbf{B}\mathbb{G}_{conn}$ a WZW term, and for $X$ a V-manifold, then a definite globalization of $\mathbf{L}$ over $X$ is

1. A reduction of the structure group, def. , of the frame bundle of $X$ along

$$\mathbf{Aut}_{Grp}(\mathbb{D}^V) = \mathbf{Aut}^{\ast/}(\mathbf{B}\mathbb{D}^V) \longrightarrow \mathbf{Aut}(\mathbb{D}^V) = GL(V) \,,$$

for $\mathbb{D}^V$ the infinitesimal disk in $V$;

1. an $\mathbf{L}^X \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$ such that its pullback $T_{inf} X \to X \stackrel{\mathbf{L}^X}{\longrightarrow} \mathbf{B} \mathbb{G}_{conn}$ to the infinitesimal disk bundle of $X$ is definite, def. , on $\mathbf{L}|_{\mathbb{D}^V} \in \mathbb{G}\mathbf{Conn}(\mathbb{D}^V)$, def. .

Since, according to prop. , the second item in def. implies a lift/reduction of the structure group to $\mathbf{QuantMorph}(\mathbf{L}|_{\mathbb{D}^V})$, in total this requires a reduction/lift to the Heisenberg ∞-group

$$G = \mathbf{Heis}_{\mathbf{Aut}_{grp}(\mathbb{D}^V)}(\mathbf{L}|_{\mathbb{D}^V}) \coloneqq \mathbf{QuantMorp}(\mathbf{L}|_{\mathbb{D}^V}) \underset{\mathbf{Aut}(\mathbb{D}^V)}{\times} \mathbf{Aut}_{Grp}(\mathbb{D}^V) \,.$$

This G-structure we require to be first order integrable (with respect to the canonical left-invariant framing of $V$.)

Example

Let $\mathbf{H} =$ Smooth∞Grpd.

1. For $\mathbf{L}\colon T^\ast \mathbb{R}^n \to \mathbf{B}U(1)_{conn}$ the Liouville-Poincaré 1-form $\theta = \sum_{i = 1}^n p_i d q^i$ (regarded as a principal connection on the trivial circle bundle), then a definite globalization of $\mathbf{L}$ is a symplectic manifold equipped with a prequantum line bundle.

2. for $\mathbf{L}\colon \mathbb{R}^7 \longrightarrow \mathbf{B}^3 U(1)_{conn}$ a potential for the associative 3-form, then a definite globalization is a manifold with G₂-structure $(X,\omega_3)$ equipped with a bundle gerbe with connection whose 3-form curvature is $\omega_3$.

Example

In $\mathbf{H} =$ SuperFormalSmooth∞Grpd and for $V$ being super Minkowski spacetime of bosonic dimension $d = 3,4,10,11$ regarded as the supersymmetry super-translation group in that dimension, and for $\mathbf{L} = \mathbf{L}$ the WZW term induced by differential Lie integration (here) from the super Lie algebra cocycles of the brane scan in these dimensions, then the Heisenberg ∞-group in def. is a $\mathbf{B}(\mathbb{R}/\Gamma)$-∞-group extension of the Lorentz group in these dimensions.

This means that a choice of definite globalization of $\mathbf{L}_{string}$ over a supermanifold $X$ is in particular a choice of super-orthogonal structure, hence a choice of graviton and of a gravitino field.

The condition that this G-structure be first-order integrable with respect to the canonical left-invariant framing of super Minkowski spacetime then means that the supertorsion of this orthogonal structure vanishes.

For $d = 1$ this is the torsion constraint of supergravity. By (Candiello-Lechner 93, Howe 97) this implies that the above graviton and gravitino field satisfy the Einstein equations for bosonic backgrounds of 11-dimensional supergravity.

This in turn implies in particular that the curvature of the WZW term $\mathbf{L}$ is the fermionic component of the supergravity C-field field strength. This finally means that $\mathbf{L}$ itself is a consistent choice of prequantization of this hence a genuinely globally defined WZW term for the Green-Schwarz sigma model for the M2-brane with target space $X$.

BPS Charges

Once a $V$-manifold $X$ is equipped with a definite globalization $\mathbf{L}^X$ of a WZW term $\mathbf{L}$, according to , and hence also with a G-structure $\mathbf{g}$ for $G$ the suitable homotopy stabilizer group of $\mathbf{L}$ on infinitesimal disks, then the automorphism ∞-group $\mathbf{Aut}(X)$ is naturally “broken” to the homotopy stabilizer group of this extra data. The stabilizer of the $G$-structure itself yields the isometry group $\mathbf{Iso}(X,\mathbf{g})$, but since the higher WZW term has in general higher gauge symmetries, the total homotopy stabilizer of the triple $(X,\mathbf{g},\mathbf{L}^{X})$ is a Heisenberg ∞-group extension of that. Since for the case of applications to supergravity (examples , below) the 0-truncation of this ∞-group extension turns out to be the extension by BPS charges, we here speak, for lack of any other established term, generally of BPS charge groups for homotopy stabilizers of definitely globalized higher WZW terms.

For a single $p$-brane species

Definition

Given a definite globalization $\mathbf{L}^X \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}$ of a WZW term, def. , hence in particular a G-structure $\mathbf{g}$ on $X$ for $G = \mathbf{Heis}_{\mathbf{Aut}_{grp}(\mathbb{D}^V)}(\mathbf{L}|_{\mathbb{D}^V})$, then the corresponding BPS charge group is the Heisenberg n-group, def. , of $\mathbf{L}^X$ over the isometry group of this $G$-structure:

$$BPS(X,\mathbf{g}, \mathbf{L}) \coloneqq \mathbf{Heis}_{\mathbf{Iso}(X,\mathbf{g})}(X,\mathbf{L}^X) \,.$$

Remark

For $V$ super Minkowski spacetime, then the super L-∞ algebra of $BPS(X,\mathbf{g}, \mathbf{L})$, def. is the Poisson bracket L-∞ algebra $\mathfrak{Pois}(X,\omega)$ of $\omega = F_{\mathbf{L}}$ regarded as a pre-(p+1)-plectic form on $X$. See the discussion in (FRS 13b, section 4).

Accordingly we find $L_\infty$-algebraic versions of the higher Kostant-Heisenberg $\infty$-extensions of prop. :

Proposition

There is a homotopy fiber sequence in the homotopy theory of L-∞ algebras of the form

$$\array{ \mathbf{H}(X,\flat \mathbf{B}^p \mathbb{R}) &\longrightarrow& \mathfrak{Pois}_{Iso(X,g)}(X,\omega) \\ && \downarrow \\ && HamIso(X,g,\omega) &\stackrel{ks}{\longrightarrow}& \mathbf{B}\mathbf{H}(X,\flat \mathbf{B}^p \mathbb{R}) }$$

which exhibits the Poisson bracket L-∞ algebra as an L-∞ algebra extension of the Lie algebra of $\omega$-Hamiltonian Killing vectors and Killing spinors by the truncated de Rham complex

$$\mathbf{H}(X,\flat \mathbf{B}^p \mathbb{R}) \simeq (\Omega^0(X)\stackrel{d}{\to} \Omega^1(X)\stackrel{d}{\to} \cdots \stackrel{d}{\to} \Omega^p_{cl}(X))$$

in degree $p$, regarded as an abelian $L_\infty$-algebra.

(FRS 13b, theorem 3.3.1)

Remark

By (FRS 13b, theorem 4.2.2) $\mathfrak{Pois}(X,\omega)$ has a model by the dg-Lie algebra (FRS 13b, def./prop. 4.2.1). Its bracket in degree-0 (FRS 13b, equation (4.2.1)) is the bracket of Noether currents for $\omega$ regarded as a WZW curvature as considered in (AGIT 89).

But $\mathfrak{Pois}(X,\omega)$ encodes also the higher order currents between these currents, which get quotiented out when passing to its degree-0 Postnikov stage:

Proposition

For connective L-∞ algebras, 0-truncation yields a functor

$$\tau_0 \colon L_\infty Alg_{\geq 0} \longrightarrow LieAlg$$

to Lie algebras. Under this functor this higher Kostant-Soriau extension of prop. becomes a Lie algebra extension

$$0 \to H^p_{dR}(X) \longrightarrow \tau_0 \mathfrak{Pois}(X,\omega) \longrightarrow Ham(X,\omega) \to 0$$

of the Hamiltonian vector fields by the degree-$p$ de Rham cohomology group of $X$, regarded as an abelian Lie algebra.

For $p_1$-branes ending on $p_2$-branes

Consider now two consecutive WZW terms

$$\array{ \widetilde{\hat V} &\stackrel{\mathbf{L}_2}{\longrightarrow}& \mathbf{B}(\mathbb{G}_2)_{conn} \\ \downarrow \\ V &\stackrel{\mathbf{L}_1}{\longrightarrow}& \mathbf{B}(\mathbb{G}_1)_{conn} }$$

with $\mathbf{L}_2$ defined on the differential refinement of the ∞-group extension

$$\array{ \mathbb{G} &\longrightarrow& \hat V \\ && \downarrow \\ && V &\stackrel{}{\longrightarrow}& \mathbf{B}\mathbb{G} }$$

which is the underlying $\mathbb{G}$-principal ∞-bundle underlying $\mathbf{L}_1$.

$$\array{ \widetilde{\hat V} &\stackrel{}{\longrightarrow}& \Omega^1(-,\mathbb{G}) \\ \downarrow &(pb)& \downarrow \\ V &\stackrel{\mathbf{L}_1}{\longrightarrow}& \mathbf{B}(\mathbb{G}_1)_{conn} }$$

Proposition

Given two consecutive WZW terms, $(\mathbf{L}_1,\mathbf{L}_2)$ and given a define globalization, def. , of $\mathbf{L}_1$ over a $V$-manifold $X$ then

1. the isometry action canonically lifts from $X$ to to the extended spacetime $\widetilde{\hat X}$

$$\array{ \widetilde {\hat X} &\longrightarrow& \Omega^1(-\mathbb{G}) \\ \downarrow &(pb)& \downarrow \\ X &\stackrel{\mathbf{L}_1}{\longrightarrow}& \mathbf{B}\mathbb{G}_{conn} } \,;$$

1. the infinitesimal disks in $\widetilde {\hat X}$ are equivalent to those of $\widetilde {\hat V}$.

Definition

By this proposition it is consistent to ask for a consecutive definitite globalization of two consecutive WZW term

$$\array{ \widetilde{\hat X} &\stackrel{\mathbf{L}_2^X}{\longrightarrow}& \mathbf{B}(\mathbb{G}_2)_{conn} \\ \downarrow \\ X &\stackrel{\mathbf{L}_1^X}{\longrightarrow}& \mathbf{B}(\mathbb{G}_1)_{conn} } \,.$$

The BPS charge $\infty$-group of this setup is

$$\mathbf{BPS}(X,\mathbf{g}, \mathbf{L}_1^X, \mathbf{L}_2^X) \coloneqq \mathbf{Heis}_{\mathbf{Iso}(X,\mathbf{g})}(\widetilde {\hat X},\mathbf{L}_2^X) \,.$$

By cor. this is an ∞-group extension of $\mathbf{Iso}(X,\mathbf{g})$ by $\mathbb{G}_2\mathbf{FlatConn}(\widetilde{\hat X})$. Forgetting the differential part of the twist, this extension group receives a map from $\mathbb{G}_2\mathbf{FlatConn}({\hat X})$.

Example: M5-brane charges in an M2-brane condensate

Example

Consider again $\mathbf{H}=$ SuperFormalSmooth∞Grpd as in example .

From the super point $\mathbb{R}^{0|1} \in \mathbf{H}$ there emanates a bouquet of consecutive super L-∞ algebra extensions, part of which looks as follows (FSS 13):

We now concentrate on the branch of this classified by the cocycles $\mu_4$ for the M2-brane, and $\mu_7$ for the M5-brane:

$$\array{ \mathfrak{m}5\mathfrak{brane} \\ \downarrow \\ \mathfrak{m}2\mathfrak{brane} &\stackrel{\mu_7\coloneqq\overline{\psi}\Gamma^{a_1 \cdots a_5}\wedge \psi \wedge e_{a_1}\wedge \cdots \wedge e_{a_5} + c_3\wedge \mu_4}{\longrightarrow}& b^6 \mathbb{R} \\ \downarrow \\ \mathbb{R}^{10,1\vert \mathbf{32}} &\stackrel{\mu_4 \coloneqq \overline{\psi}\Gamma^{a_1 a_2}\wedge \psi \wedge e_{a_1}\wedge e_{a_2}}{\longrightarrow}& b^2 \mathbb{R} }$$

Here $\mathfrak{m}2\mathfrak{brane}$ denotes the “supergravity Lie 3-algebra” regarded as an extended super Minkowski spacetime and $\mathfrak{m}5\mathfrak{brane}$ denotes the “supergravity Lie 6-algebra”. Both hooks $\array{\downarrow \\ & \rightarrow}$ in the diagram are homotopy fiber sequences in the homotopy theory of super L-∞ algebras.

By the discussion at geometry of physics – WZW terms – Consecutive WZW terms, following (FSS 13) applying differentially refined Lie integration to this yields two consecutive higher WZW terms of the form

$$\array{ \mathbf{B}^{2}(\mathbb{R}/\Gamma_1)_{conn} &\longrightarrow& \widetilde {\hat{\mathbb{R}}}^{10,1\vert \mathbf{32}} &\stackrel{\mathbf{L}_{M5}}{\longrightarrow}& \mathbf{B}^{5} (\mathbb{R}/\Gamma_2)_{conn} \\ && \downarrow \\ && \mathbb{R}^{10,1\vert \mathbf{32}} &\stackrel{\mathbf{L}_{M2}}{\longrightarrow}& \mathbf{B}^2 (\mathbb{R}/\Gamma_1)_{conn} } \,.$$

(Here by the van Est isomorphism we do not notationally distinguish super Minkowski spacetime regarded as a super Lie algebra or a super Lie group.)

By example a choice of definite globalization of $\mathbf{L}_{M2}$ is equivalent to bosonic solution $(X,\mathbf{g})$ to the Einstein equations of 11-dimensional supergravity equipped with a compatible globally defined WZW term $\mathbf{L}_{M2}^X$ for the M2-brane Green-Schwarz sigma model with target space $X$.

By prop. this defines an extended superspacetime $\widetilde{\hat X}$ which is a higher Cartan geometry of locally modeled on the supergravity Lie 3-algebra on which the local M5-brane Green-Schwarz sigma model is defined, and hence may ask for a choice of definite globalization of that

$$\array{ \widetilde {\hat{X}} &\stackrel{\mathbf{L}^X_{M5}}{\longrightarrow}& \mathbf{B}^{5} (\mathbb{R}/\Gamma_2)_{conn} \\ \downarrow \\ X &\stackrel{\mathbf{L}^X_{M2}}{\longrightarrow}& \mathbf{B}^2 (\mathbb{R}/\Gamma_1)_{conn} } \,.$$

This is now a globally well defined background for the M5-brane sigma model and def. determines its BPS-char super-6-group. By corollary this is a super 6-group extension of the superisometry group of $(X,\mathbf{g})$ by $(\mathbf{B}^5 (\mathbb{R}/\Gamma_2))\mathbf{FlatConn}(\widetilde{\hat X})$.

By the discussion at geometry of physics – WZW terms – Consecutive WZW terms the extended spacetime $\widetilde {\hat X}$ here is such that smooth maps into it

$$\Sigma \longrightarrow \widetilde{\hat X}$$

(which are the fields of the M5-brane sigma model with WZW term $\mathbf{L}^X_{M5}$) are equivalently pairs, consisting of

1. a smooth function $\phi \colon \Sigma \longrightarrow X$ into the actual spacetime $X$;

2. a cocycle $\nabla$ in $\phi$-twisted degree-3 Deligne cohomology on $\Sigma$, hence a 2-form gauge field on $\Sigma$, subject to certain compatibility conditions with the function $\phi$.

The first item here is the evident sigma model field, the second 2-form field is part of the “tensor multiplet” on the M5-brane, exhibiting the Green-Schwarz sigma-model for the M5-brane as a higher gauged WZW model.

Now to consider the BPS charge group of $\mathbf{L}^X_{M5}$, def. . By corollary this is an ∞-group extension of the super-isometry group of the 11-dimensional super spacetime by the moduli stack $(\mathbf{B}^5 U(1))\mathbf{FlatConn}(\widetilde{\hat X})$ of flat 5-form connection on the extended spacetime.

This receives a map $(\mathbf{B}^5 U(1))\mathbf{FlatConn}(\hat X) \longrightarrow (\mathbf{B}^5 U(1))\mathbf{FlatConn}(\widetilde{\hat X})$ from the moduli of 5-form connections of the extended spacetime $\hat X$ (which is $\widetilde{\hat X}$. This consists of the cohomological data without the differential cohomologica data in $\mathbf{L}_{M2}^X$): it is the $\mathbf{B}^2 (\mathbb{R}/\Gamma_1)$-principal ∞-bundle which sits in the homotopy fiber sequence of the form

$$\array{ \mathbf{B}^2 (\mathbb{R}/\Gamma_1) &\longrightarrow& \hat X \\ && \downarrow \\ && X &\stackrel{\mathbf{DD}(\mathbf{L}_{M2}^X)}{\longrightarrow}& \mathbf{B}^3 (\mathbb{R}/\Gamma_1) } \,.$$

Under Lie differentiation as in prop. $(\mathbf{B}^5 U(1))\mathbf{FlatConn}(\widetilde{\hat X})$ turns into $(\mathbf{B}^5 \mathbb{R})\mathbf{FlatConn}(\widetilde{\hat X})$ hence into $\mathbf{H}(\hat X, \flat \mathbf{B}^5 \mathbb{R})$. Under the adjunction between shape modality $\int$ and flat modality $\flat$, this is the degree-5 real cohomology of the geometric realization of $\hat X$. This in turns is a K(Z,3)-fibration $\int \hat X \to \int X$ over the underlying bare homotopy type of spacetime $X$ which is classified by the integral degree-4 class which is the higher Dixmier-Douady class $DD(\mathbf{L}_{M5}^X)$ of $\mathbf{L}_{M2}^{X}$.

The degree-5 real cohomology of such a fibration is computed by a Serre spectral sequence. By the discussion at Eilenberg-MacLane space – cohomology of EM spaces only very few entries in this spectral sequence contribute, and the result is the middle cohomology of this sequence

$$H^1(X) \stackrel{(0,d_4)}{\longrightarrow} H^2(X) \oplus H^5(X) \stackrel{(d_4,0)}{\longrightarrow} H^6(X)$$

where $d_4 \propto (-)\cup DD(\mathbf{L}_{M2}^X)$ is given by taking the cup product with the class of the M2-WZW term.

This is the group of M2-brane and M5-brane charges with corrections by global effects, in the corrected M-theory super Lie algebra for the superspacetime $(X,g)$.

References

• José de Azcárraga, Jerome Gauntlett, J.M. Izquierdo, Paul Townsend, Topological Extensions of the Supersymmetry Algebra for Extended Objects, Phys. Rev. Lett. 63 (1989) 2443 (spire)

• Edward Witten, Twistor-like transform in ten dimensions, Nuclear Physics B266 (1986) (spire)

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory (arXiv:1304.0236)

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, L-∞ algebras of local observables from higher prequantum bundles, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 (arXiv:1304.6292)

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields, International Journal of Geometric Methods in Modern Physics, Volume 12, Issue 02 (2015) 1550018 (arXiv:1308.5264)

• Hisham Sati, Urs Schreiber, Lie n-algebras of BPS charges (arXiv:1507.08692)

All details and proofs for the above are in

• Urs Schreiber, differential cohomology in a cohesive topos (pdf)