nLab
geometry of physics -- BPS charges

this page is one chapter of geometry of physics

previous chapters: manifolds and orbifolds, WZW terms, supergeometry


Contents

Motivation and results

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

ψ¯Γ a 1a pψe a 1e a pCE( d1,1|N) \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 XX. Then the Polyvector extensions

[Q α,Q β]=(Γ aC) αβP a+(Γ a 1a pC) αβZ a 1a p [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)(X,g) by charges ZZ of Noether currents of the super p-brane sigma model are known as algebras of BPS charges. (The spacetime (X,g,ω)(X,g,\omega) is called a supergravity 1k\frac{1}{k}-BPS state if the dimension of the space of supercharges QQ in the kernel of the above bracket is 1k\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 XX is locally modeled on an ordinary super Minkowski spacetime and that the pp-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,ω)(X,\omega), the WZW curvature form ω\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 2p_2-branes on which p 1p_1-branes may end (for p 1=1p_1 = 1: type II strings ending on D-branes, for p 1=2p_1= 2 and p 2=5p_2 = 5 M2-branes ending on M5-branes) then XX 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)H 5(X)H^2(X) \oplus H^5(X) of spacetime XX receive corrections by d 4d_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 3d_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 L M5 X\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 nn-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+16 = 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 H\mathbf{H} equipped with cohesion and differential cohesion. The application to supergravity takes place in the model H=\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,EHV,E\in \mathbf{H}, a VV-fiber ∞-bundle EE over XX is a bundle EH /XE \in \mathbf{H}_{/X} such that there exists a cover (i.e. a 1-epimorphism) UXU \longrightarrow X and a homotopy pullback diagram of the form

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

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

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

A VV-fiber bundle realized this way is said to have structure group GG.

Proposition

Every VV-fiber ∞-bundle is the associated ∞-bundle, def. 2, of some Aut(V)\mathbf{Aut}(V)-principal ∞-bundle.

Definition

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

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

Similarly for VV-fiber ∞-bundles via def. 2, prop. 2.

Homotopy stabilizer groups

recalled from geometry of physics -- representations and associated bundles

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

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

Given a global element of VV

x:*X x \colon \ast \to X

then the stabilizer \infty-group Stab ρ(x)Stab_\rho(x) of the GG-action at xx is the loop space object

Stab ρ(x)Ω i(x)(X/G). Stab_\rho(x) \coloneqq \Omega_{i(x)} (X/G) \,.
Remark

Equivalently, def. 4, gives the loop space object of the 1-image BStab ρ(x)\mathbf{B}Stab_\rho(x) of the morphism

*xXX/G. \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 *BStab ρ(x)X/G\ast \to \mathbf{B}Stab_\rho(x) \hookrightarrow X/G which combines with the homotopy fiber sequence of prop. \ref{InfinityAction} to a diagram of the form

* x X X/G epi mono BStab ρ(x) = BStab ρ(x) BG. \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 ρ(x)G. 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 ρ(x)Stab_\rho(x) is not a sub-group of GG in general.

Higher Kostant-Souriau extensions

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

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

Example

For H=\mathbf{H} = Smooth∞Grpd we have 𝔾=B p(/Γ)\mathbb{G} = \mathbf{B}^p (\mathbb{R}/\Gamma) for Γ=\Gamma = \mathbb{Z} is the circle (p+1)-group. Equipped with its standard Hodge filtration this gives B𝔾 conn=B pU(1) conn\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)(p+2).

Definition

For XHX \in \mathbf{H}, for write

conc:[X,B𝔾 conn]𝔾Conn(X) 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 𝔾\mathbb{G}-principal ∞-connections on XX in that a family U𝔾Conn(X)U \longrightarrow \mathbb{G}\mathbf{Conn}(X) is a vertical 𝔾\mathbb{G}-principal \infty-connection on U×XUU \times X\to U.

Example

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

Proposition

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

Ω 0(𝔾Conn(X))(Ω𝔾)FlatConn(X). \Omega_0 (\mathbb{G}\mathbf{Conn}(X)) \simeq (\Omega\mathbb{G})\mathbf{FlatConn}(X) \,.
Proposition

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

Definition

Given a 𝔾\mathbb{G}-principal ∞-connection :XB𝔾 conn\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 𝔾Conn(X)\nabla \in \mathbb{G}\mathbf{Conn}(X), def. 6, under the Aut(X)\mathbf{Aut}(X)-action of 2;

    QuantMorph(X,)Stab Aut(X)(conc()). \mathbf{QuantMorph}(X,\nabla) \coloneqq \mathbf{Stab}_{\mathbf{Aut}(X)}(conc(\nabla)) \,.
  2. The Hamiltonian symplectomorphism ∞-group

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

    is the 1-image of the canonical morphism QuantMorph(X,)Aut(X)\mathbf{QuantMorph}(X,\nabla) \longrightarrow \mathbf{Aut}(X) from remark 5.

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

    ρ:GHamSymp(X,) \rho \colon G \longrightarrow \mathbf{HamSymp}(X,\nabla) \;
  4. An ∞-moment map is an \infty-group homomorphism

    GQuantMorph(X,) G \longrightarrow \mathbf{QuantMorph}(X,\nabla)
  5. The Heisenberg ∞-group for a given Hamiltonian GG-action ρ\rho is the homotopy pullback

    Heis G(X,)ρ *QuantMorph(X,). \mathbf{Heis}_G(X,\nabla) \coloneqq \rho^\ast \mathbf{QuantMorph}(X,\nabla) \,.
Example

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

Remark

Since HamSymp(X,)Aut(X)\mathbf{HamSymp}(X,\nabla)\hookrightarrow \mathbf{Aut}(X) is by construction a 1-monomorphism, given any GG-action ρ:GAut(X)\rho \colon G \longrightarrow \mathbf{Aut}(X) on XX, not necessarily Hamiltonian, then the homotopy pullback ρ *QuantMorph(X,)\rho^\ast \mathbf{QuantMorph}(X,\nabla) is the Heisenberg ∞-group of the maximal sub-\infty-group of GG which does act via Hamiltonian symplectomorphisms. Therefore we will also write Heis G(X,)\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 𝔾\mathbb{G}-principal ∞-connection :XB𝔾 conn\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}, there is a homotopy fiber sequence of the form

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

    𝔾ConstFunct(X) QuantMorph(X,) HamSymp(X,) KS B(𝔾ConstFunct(X)) \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 𝔾B𝔾\mathbb{G} \simeq \mathbf{B}\mathbb{G}' then

    (Ω𝔾)FlatConn(X) QuantMorph(X,) HamSymp(X,) KS B((Ω𝔾)FlatConn(X)) \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 Ω𝔾\Omega \mathbb{G}-flat ∞-connections, classified by a cocycle KS\mathbf{KS}.

(FRS13a)

Example

In H=\mathbf{H} = Smooth∞Grpd, let 𝔾=B pU(1)\mathbb{G} = \mathbf{B}^p U(1) be the circle (p+1)-group and let XSmoothMfdSmoothGrpdX \in SmoothMfd \hookrightarrow Smooth \infty Grpd be p-connected, then (ΩB pU(1))FlatConn(X)B pU(1)(\Omega\mathbf{B}^p U(1))\mathbf{FlatConn}(X)\simeq \mathbf{B}^{p}U(1). Hence here prop. 3 gives

B pU(1) QuantMorph(X,) HamSymp(X,) KS B p+1U(1) \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 𝔾\mathbb{G}-principal ∞-connection :XB𝔾 conn\nabla \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn}, and for ρ:GHamSymp(X,)\rho \colon G \longrightarrow \mathbf{HamSymp}(X,\nabla) a GG-Hamiltonian action, then there is a homotopy fiber sequence

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

    𝔾ConstFunct(X) Heis G(X,) G KS(ρ) B(𝔾ConstFunct(X)) \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 𝔾B𝔾\mathbb{G} \simeq \mathbf{B}\mathbb{G}' then

    (Ω𝔾)FlatConn(X) Heis G(X,) G KS(ρ) B((Ω𝔾)FlatConn(X)) \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 GG by the moduli stack of Ω𝔾\Omega \mathbb{G}-flat ∞-connections, classified by a cocycle KS(ρ)\mathbf{KS}(\rho).

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

Definite forms

The concept of extending a closed differential form defined on a Cartesian space n\mathbb{R}^n to a definite form on an nn-dimensional manifold is familiar from special holonomy manifolds. For instance a definite globalization of the associative 3-form on 7\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 VV-fiber ∞-bundle EE over XX, def. 1, and given any coefficient AA, there is a natural equivalence beween

Definition

Given a VV-fiber ∞-bundle EE over XX, and a global element x:*Vx\colon \ast \to V then a section σ\sigma of EE is definite on xx if there exists a 1-epimorphism UXU \to X and a diagram

U * x X σ V/Aut(V) BAut(V). \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 xx are equivalent to reductions of the structure group, def. 3, along the stabilizer group map Stab Aut(V)(x)Aut(V)Stab_\mathbf{Aut(V)}(x)\longrightarrow \mathbf{Aut}(V).

Definition

Given c:VA\mathbf{c} \colon V \longrightarrow A, and given a VV-fiber ∞-bundle EE over XX, then a definite parameterization of c\mathbf{c} over EE is a c E:EA\mathbf{c}^E \colon E \longrightarrow A such that the section σ c X\sigma_{\mathbf{c}^X} coresponding to it via prop. 4 is definite on c\mathbf{c} in the sense of def. 9.

Definition

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

  1. A reduction of the structure group, def. 3, of the frame bundle of XX along
Aut Grp(𝔻 V)=Aut */(B𝔻 V)Aut(𝔻 V)=GL(V), \mathbf{Aut}_{Grp}(\mathbb{D}^V) = \mathbf{Aut}^{\ast/}(\mathbf{B}\mathbb{D}^V) \longrightarrow \mathbf{Aut}(\mathbb{D}^V) = GL(V) \,,

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

  1. an L X:XB𝔾 conn\mathbf{L}^X \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn} such that its pullback T infXXL XB𝔾 connT_{inf} X \to X \stackrel{\mathbf{L}^X}{\longrightarrow} \mathbf{B} \mathbb{G}_{conn} to the infinitesimal disk bundle of XX is definite, def. 10, on L| 𝔻 V𝔾Conn(𝔻 V)\mathbf{L}|_{\mathbb{D}^V} \in \mathbb{G}\mathbf{Conn}(\mathbb{D}^V), def. 6.

Since, according to prop. 5, the second item in def. 11 implies a lift/reduction of the structure group to QuantMorph(L| 𝔻 V)\mathbf{QuantMorph}(\mathbf{L}|_{\mathbb{D}^V}), in total this requires a reduction/lift to the Heisenberg ∞-group

G=Heis Aut grp(𝔻 V)(L| 𝔻 V)QuantMorp(L| 𝔻 V)×Aut(𝔻 V)Aut Grp(𝔻 V). 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 VV.)

Example

Let H=\mathbf{H} = Smooth∞Grpd.

  1. For L:T * nBU(1) conn\mathbf{L}\colon T^\ast \mathbb{R}^n \to \mathbf{B}U(1)_{conn} the Liouville-Poincaré 1-form θ= i=1 np idq i\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 L\mathbf{L} is a symplectic manifold equipped with a prequantum line bundle.

  2. for L: 7B 3U(1) conn\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,ω 3)(X,\omega_3) equipped with a bundle gerbe with connection whose 3-form curvature is ω 3\omega_3.

Example

In H=\mathbf{H} = SuperFormalSmooth∞Grpd and for VV being super Minkowski spacetime of bosonic dimension d=3,4,10,11d = 3,4,10,11 regarded as the supersymmetry super-translation group in that dimension, and for L=L\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. 11 is a B(/Γ)\mathbf{B}(\mathbb{R}/\Gamma)-∞-group extension of the Lorentz group in these dimensions.

This means that a choice of definite globalization of L string\mathbf{L}_{string} over a supermanifold XX 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=1d = 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 L\mathbf{L} is the fermionic component of the supergravity C-field field strength. This finally means that L\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 XX.

BPS Charges

Once a VV-manifold XX is equipped with a definite globalization L X\mathbf{L}^X of a WZW term L\mathbf{L}, according to 11, and hence also with a G-structure g\mathbf{g} for GG the suitable homotopy stabilizer group of L\mathbf{L} on infinitesimal disks, then the automorphism ∞-group Aut(X)\mathbf{Aut}(X) is naturally “broken” to the homotopy stabilizer group of this extra data. The stabilizer of the GG-structure itself yields the isometry group Iso(X,g)\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,g,L X)(X,\mathbf{g},\mathbf{L}^{X}) is a Heisenberg ∞-group extension of that. Since for the case of applications to supergravity (examples 6, 7 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 pp-brane species

Definition

Given a definite globalization L X:XB𝔾 conn\mathbf{L}^X \colon X \longrightarrow \mathbf{B}\mathbb{G}_{conn} of a WZW term, def. 11, hence in particular a G-structure g\mathbf{g} on XX for G=Heis Aut grp(𝔻 V)(L| 𝔻 V)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. 8, of L X\mathbf{L}^X over the isometry group of this GG-structure:

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

For VV super Minkowski spacetime, then the super L-∞ algebra of BPS(X,g,L)BPS(X,\mathbf{g}, \mathbf{L}), def. 12 is the Poisson bracket L-∞ algebra 𝔓𝔬𝔦𝔰(X,ω)\mathfrak{Pois}(X,\omega) of ω=F L\omega = F_{\mathbf{L}} regarded as a pre-(p+1)-plectic form on XX. See the discussion in (FRS 13b, section 4).

Accordingly we find L L_\infty-algebraic versions of the higher Kostant-Heisenberg \infty-extensions of prop. 6:

Proposition

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

H(X,B p) 𝔓𝔬𝔦𝔰 Iso(X,g)(X,ω) HamIso(X,g,ω) ks BH(X,B p) \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 ω\omega-Hamiltonian Killing vectors and Killing spinors by the truncated de Rham complex

H(X,B p)(Ω 0(X)dΩ 1(X)ddΩ cl p(X)) \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 pp, regarded as an abelian L L_\infty-algebra.

(FRS 13b, theorem 3.3.1)

Remark

By (FRS 13b, theorem 4.2.2) 𝔓𝔬𝔦𝔰(X,ω)\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 ω\omega regarded as a WZW curvature as considered in (AGIT 89).

But 𝔓𝔬𝔦𝔰(X,ω)\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

τ 0:L Alg 0LieAlg \tau_0 \colon L_\infty Alg_{\geq 0} \longrightarrow LieAlg

to Lie algebras. Under this functor this higher Kostant-Soriau extension of prop. 6 becomes a Lie algebra extension

0H dR p(X)τ 0𝔓𝔬𝔦𝔰(X,ω)Ham(X,ω)0 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-pp de Rham cohomology group of XX, regarded as an abelian Lie algebra.

For p 1p_1-branes ending on p 2p_2-branes

Consider now two consecutive WZW terms

V^˜ L 2 B(𝔾 2) conn V L 1 B(𝔾 1) conn \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 L 2\mathbf{L}_2 defined on the differential refinement of the ∞-group extension

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

which is the underlying 𝔾\mathbb{G}-principal ∞-bundle underlying L 1\mathbf{L}_1.

V^˜ Ω 1(,𝔾) (pb) V L 1 B(𝔾 1) conn \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, (L 1,L 2)(\mathbf{L}_1,\mathbf{L}_2) and given a define globalization, def. 11, of L 1\mathbf{L}_1 over a VV-manifold XX then

  1. the isometry action canonically lifts from XX to to the extended spacetime X^˜\widetilde{\hat X}
X^˜ Ω 1(𝔾) (pb) X L 1 B𝔾 conn; \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 X^˜\widetilde {\hat X} are equivalent to those of V^˜\widetilde {\hat V}.
Definition

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

X^˜ L 2 X B(𝔾 2) conn X L 1 X B(𝔾 1) conn. \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

BPS(X,g,L 1 X,L 2 X)Heis Iso(X,g)(X^˜,L 2 X). \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. 1 this is an ∞-group extension of Iso(X,g)\mathbf{Iso}(X,\mathbf{g}) by 𝔾 2FlatConn(X^˜)\mathbb{G}_2\mathbf{FlatConn}(\widetilde{\hat X}). Forgetting the differential part of the twist, this extension group receives a map from 𝔾 2FlatConn(X^)\mathbb{G}_2\mathbf{FlatConn}({\hat X}).

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

Example

Consider again H=\mathbf{H}= SuperFormalSmooth∞Grpd as in example 6.

From the super point 0|1H\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):

the brane bouquet

We now concentrate on the branch of this classified by the cocycles μ 4\mu_4 for the M2-brane, and μ 7\mu_7 for the M5-brane:

𝔪5𝔟𝔯𝔞𝔫𝔢 𝔪2𝔟𝔯𝔞𝔫𝔢 μ 7ψ¯Γ a 1a 5ψe a 1e a 5+c 3μ 4 b 6 10,1|32 μ 4ψ¯Γ a 1a 2ψe a 1e a 2 b 2 \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 𝔪2𝔟𝔯𝔞𝔫𝔢\mathfrak{m}2\mathfrak{brane} denotes the “supergravity Lie 3-algebra” regarded as an extended super Minkowski spacetime and 𝔪5𝔟𝔯𝔞𝔫𝔢\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

B 2(/Γ 1) conn ^˜ 10,1|32 L M5 B 5(/Γ 2) conn 10,1|32 L M2 B 2(/Γ 1) conn. \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 6 a choice of definite globalization of L M2\mathbf{L}_{M2} is equivalent to bosonic solution (X,g)(X,\mathbf{g}) to the Einstein equations of 11-dimensional supergravity equipped with a compatible globally defined WZW term L M2 X\mathbf{L}_{M2}^X for the M2-brane Green-Schwarz sigma model with target space XX.

By prop. 8 this defines an extended superspacetime X^˜\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

X^˜ L M5 X B 5(/Γ 2) conn X L M2 X B 2(/Γ 1) conn. \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. 13 determines its BPS-char super-6-group. By corollary 1 this is a super 6-group extension of the superisometry group of (X,g)(X,\mathbf{g}) by (B 5(/Γ 2))FlatConn(X^˜)(\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 X^˜\widetilde {\hat X} here is such that smooth maps into it

ΣX^˜ \Sigma \longrightarrow \widetilde{\hat X}

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

  1. a smooth function ϕ:ΣX\phi \colon \Sigma \longrightarrow X into the actual spacetime XX;

  2. a cocycle \nabla in ϕ\phi-twisted degree-3 Deligne cohomology on Σ\Sigma, hence a 2-form gauge field on Σ\Sigma, subject to certain compatibility conditions with the function ϕ\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 L M5 X\mathbf{L}^X_{M5}, def. 13. By corollary 1 this is an ∞-group extension of the super-isometry group of the 11-dimensional super spacetime by the moduli stack (B 5U(1))FlatConn(X^˜)(\mathbf{B}^5 U(1))\mathbf{FlatConn}(\widetilde{\hat X}) of flat 5-form connection on the extended spacetime.

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

B 2(/Γ 1) X^ X DD(L M2 X) B 3(/Γ 1). \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. 7 (B 5U(1))FlatConn(X^˜)(\mathbf{B}^5 U(1))\mathbf{FlatConn}(\widetilde{\hat X}) turns into (B 5)FlatConn(X^˜)(\mathbf{B}^5 \mathbb{R})\mathbf{FlatConn}(\widetilde{\hat X}) hence into H(X^,B 5)\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 X^\hat X. This in turns is a K(Z,3)-fibration X^X\int \hat X \to \int X over the underlying bare homotopy type of spacetime XX which is classified by the integral degree-4 class which is the higher Dixmier-Douady class DD(L M5 X)DD(\mathbf{L}_{M5}^X) of L M2 X\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)(0,d 4)H 2(X)H 5(X)(d 4,0)H 6(X) 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()DD(L M2 X)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)(X,g).

References

All details and proofs for the above are in

Revised on January 14, 2016 19:30:00 by Urs Schreiber (89.0.80.253)