1001.0040.txt

arXiv:1001.0040v2  [math-ph]  16 Sep 2010COURANT ALGEBROIDS FROM CATEGORIFIED
SYMPLECTIC GEOMETRY
CHRISTOPHER L. ROGERS
Abstract. In categorified symplectic geometry, one studies the cate-
gorified algebraic and geometric structures that naturally arise on man-
ifolds equipped with a closed nondegenerate ( n+ 1)-form. The case
relevant to classical string theory is when n= 2 and is called ‘2-plectic
geometry’. Just as the Poisson bracket makes the smooth func tions on
a symplectic manifold into a Lie algebra, there is a Lie 2-alg ebra of
observables associated to any 2-plectic manifold. String t heory, closed
3-forms and Lie 2-algebras also play important roles in the t heory of
Courant algebroids. Courant algebroids are vector bundles which gen-
eralize the structures found in tangent bundles and quadrat ic Lie alge-
bras. It is known that a particular kind of Courant algebroid (called an
exact Courant algebroid) naturally arises in string theory , and that such
an algebroid is classified up to isomorphism by a closed 3-for m on the
base space, which then induces a Lie 2-algebra structure on t he space of
global sections. In this paper we begin to establish precise connections
between 2-plectic manifolds and Courant algebroids. We pro ve that any
manifold Mequipped with a 2-plectic form ωgives an exact Courant
algebroid EωoverMwithˇSevera class [ ω], and we construct an embed-
ding of the Lie 2-algebra of observables into the Lie 2-algeb ra of sections
ofEω. We then show that this embedding identifies the observables as
particular infinitesimal symmetries of Eωwhich preserve the 2-plectic
structure on M.
1.Introduction
The underlying geometric structures of interest in categor ified symplectic
geometry are multisymplectic manifolds: manifolds equipp ed with a closed,
nondegenerate form of degree ≥2 [8]. This kind of geometry originated in
the work of DeDonder [10] and Weyl [24] on the calculus of vari ations, and
more recently has been used as a formalism to investigate cla ssical field the-
ories [11, 12, 13]. In this paper, we call a manifold ‘ n-plectic’ if it is equipped
with a closed nondegenerate ( n+ 1)-form. Hence ordinary symplectic ge-
ometry corresponds to the n= 1 case, and the corresponding 1-dimensional
field theory is just the classical mechanics of point particl es. In general,
examples of n-plectic manifolds include phase spaces suitable for descr ibing
n-dimensional classical field theories. We will be primarily concerned with
Date: October 29, 2018.
This work was partially supported by a grant from The Foundat ional Questions
Institute.
12 CHRISTOPHER L. ROGERS
then= 2case. Thisis the firstreally new case of n-plectic geometry andthe
corresponding 2-dimensional field theories of interest inc lude bosonic string
theory. Indeed, just as the phase space of the classical part icle is a mani-
fold equipped with a closed, nondegenerate 2-form, the phas e space of the
classical string is a finite-dimensional manifold equipped with a closed non-
degenerate 3-form. This phase space is often called the ‘mul tiphase space’
of the string [11] in order to distinguish it from the infinite -dimensional
symplectic manifolds that are used as phase spaces in string field theory [6].
In classical mechanics, the relevant mathematical structu res are not just
geometric, butalsoalgebraic. Thesymplecticformgives th espaceofsmooth
functions the structure of Poisson algebra. Analogously, i n classical string
theory, the 2-plectic form induces a bilinear skew-symmetr ic bracket on
a particular subspace of differential 1-forms, which we call H amiltonian.
The Hamiltonian 1-forms and smooth functions form the under lying chain
complex of an algebraic structure known as a semistrict Lie 2 -algebra. A
semistrict Lie 2-algebra can be viewed as a categorified Lie a lgebra in which
the Jacobi identity is weakened and is required to hold only u p to isomor-
phism. Equivalently, it can be described as a 2-term L∞-algebra, i.e. a
generalization of a 2-term differential graded Lie algebra in which the Ja-
cobi identity is only satisfied up to chain homotopy [1, 14]. J ust as the
Poisson algebra of smooth functions represents the observa bles of a system
of particles, it has been shown that the Lie 2-algebra of Hami ltonian 1-forms
contains the observables of the classical string [2]. In gen eral, ann-plectic
structure will give rise to a L∞-algebra on an n-term chain complex of dif-
ferential forms in which the ( n−1)-forms correspond to the observables of
ann-dimensional classical field theory [15].
Many of the ingredients found in 2-plectic geometry are also found in
the theory of Courant algebroids, which was also developed b y generalizing
structures found in symplectic geometry. Courant algebroi ds were first used
by Courant [9] to study generalizations of pre-symplectic a nd Poisson struc-
tures in the theory of constrained mechanical systems. Roug hly, a Courant
algebroid is a vector bundle that generalizes the structure of a tangent bun-
dle equipped with a symmetric nondegenerate bilinear form o n the fibers.
In particular, the underlying vector bundle of a Courant alg ebroid comes
equipped with a skew-symmetric bracket on the space of globa l sections.
However, unlike the Lie bracket of vector fields, the bracket need not satisfy
the Jacobi identity.
In a letter to Weinstein, ˇSevera [20] described how a certain type of
Courant algebroid, known as an exact Courant algebroid, app ears naturally
when studying 2-dimensional variational problems. In clas sical string the-
ory, the string can be represented as a map φ: Σ→Mfrom a 2-dimensional
parameter space Σ into a manifold Mcorresponding to space-time. The
imageφ(Σ) is called the string world-sheet. The map φextremizes the inte-
gral of a 2-form θ∈Ω2(M) over its world-sheet. Hence the classical string
is a solution to a 2-dimensional variational problem. The 2- formθis calledCOURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 3
the Lagrangian and depends on elements of the first jet bundle of the trivial
bundleΣ ×M. TheLagrangian isnotunique. A solution φremainsinvariant
if an exact 1-form or ‘divergence’ is added to θ. It is, in fact, the 3-form dθ
that is relevant. Inthiscontext, ˇSeveraobserved that the3-form dθuniquely
specifies (up to isomorphism) the structure of an exact Coura nt algebroid
overM. The general correspondence between exact Courant algebro ids and
closed 3-forms on the base space was further developed by ˇSevera, and also
by Bressler and Chervov [4], to give a complete classificatio n. An exact
Courant algebroid over Mis determined up to isomorphism by its ˇSevera
class: an element [ ω] in the third de Rham cohomology of M.
Just as in 2-plectic geometry, the underlying geometric str ucture of a
Courant algebroid has an algebraic manifestation. Roytenb erg and Wein-
stein [16] showed that the bracket on the space of global sect ions induces
anL∞structure. If we are considering an exact Courant algebroid , then
the global sections can be identified with ordered pairs of ve ctor fields and
1-forms on the base space. Roytenberg and Weinstein’s resul ts imply that
these sections, when combined with the smooth functions on t he base space,
form a semistrict Lie 2-algebra [23]. Moreover, the bracket of the Lie 2-
algebra is determined by a closed 3-form corresponding to a r epresentative
of theˇSevera class [21].
Thus there are striking similarities between 2-plectic man ifolds and ex-
act Courant algebroids. Both originate from attempts to gen eralize certain
aspects of symplectic geometry. Both come equipped with a cl osed 3-form
that gives rise to a Lie 2-algebra structure on a chain comple x consisting
of smooth functions and differential 1-forms. In this paper, w e prove that
there is indeed a connection between the two. We show that any manifold
Mequipped with a 2-plectic form ωgives an exact Courant algebroid Eω
withˇSevera class [ ω], and that there is an embedding of the Lie 2-algebra
of observables into the Lie 2-algebra corresponding to Eω. Moreover, this
embedding allows us to characterize the Hamiltonian 1-form s as particular
infinitesimal symmetries of Eωwhich preserve the 2-plectic structure on M.
2.Courant algebroids
Here we recall some basic facts and examples of Courant algeb roids and
then we proceed to describe ˇSevera’s classification of exact Courant alge-
broids. There are several equivalent definitions of a Couran t algebroid found
in the literature. In this paper we use the definition given by Roytenberg
[17].
Definition 2.1. ACourant algebroid is a vector bundle E→Mequipped
with a nondegenerate symmetric bilinear form /an}bracketle{t·,·/an}bracketri}hton the bundle, a skew-
symmetric bracket /llbracket·,·/rrbracketonΓ(E), and a bundle map (called the anchor)
ρ:E→TMsuch that for all e1,e2,e3∈Γ(E)and for all f,g∈C∞(M)the
following properties hold:
(1)/llbrackete1,/llbrackete2,e3/rrbracket/rrbracket−/llbracket/llbrackete1,e2/rrbracket,e3/rrbracket−/llbrackete2,/llbrackete1,e3/rrbracket/rrbracket=−DT(e1,e2,e3),4 CHRISTOPHER L. ROGERS
(2)ρ([e1,e2]) = [ρ(e1),ρ(e2)],
(3) [e1,fe2] =f[e1,e2]+ρ(e1)(f)e2−1
2/an}bracketle{te1,e2/an}bracketri}htDf,
(4)/an}bracketle{tDf,Dg/an}bracketri}ht= 0,
(5)ρ(e1)(/an}bracketle{te2,e3/an}bracketri}ht) =/an}bracketle{t[e1,e2]+1
2D/an}bracketle{te1,e2/an}bracketri}ht,e3/an}bracketri}ht+/an}bracketle{te2,[e1,e3]+1
2D/an}bracketle{te1,e3/an}bracketri}ht/an}bracketri}ht,
where[·,·]is the Lie bracket of vector fields, D:C∞(M)→Γ(E)is the map
defined by /an}bracketle{tDf,e/an}bracketri}ht=ρ(e)f, and
T(e1,e2,e3) =1
6(/an}bracketle{t/llbrackete1,e2/rrbracket,e3/an}bracketri}ht+/an}bracketle{t/llbrackete3,e1/rrbracket,e2/an}bracketri}ht+/an}bracketle{t/llbrackete2,e3/rrbracket,e1/an}bracketri}ht).
The bracket in Definition 2.1 is skew-symmetric, but the first property
implies that it needs only to satisfy the Jacobi identity “up toDT”. (The
notation suggests we think of this as a boundary.) The functi onTis often
referred to as the Jacobiator . (When there is no risk of confusion, we shall
refer to the Courant algebroid with underlying vector bundl eE→MasE.)
Note that the vector bundle Emay be identified with E∗via the bilinear
form/an}bracketle{t·,·/an}bracketri}htand therefore we have the dual map
ρ∗:T∗M→E.
Hence the map Dis simply the pullback of the de Rham differential by ρ∗.
Thereisanalternatedefinitiongiven by ˇSevera[20]forCourantalgebroids
which uses a bilinear operation on sections that satisfies a J acobi identity
but is not skew-symmetric.
Definition 2.2. ACourant algebroid is a vector bundle E→Mtogether
with a nondegenerate symmetric bilinear form /an}bracketle{t·,·/an}bracketri}hton the bundle, a bilinear
operation ◦onΓ(E), and a bundle map ρ:E→TMsuch that for all
e1,e2,e3∈Γ(E)and for all f∈C∞(M)the following properties hold:
(1)e1◦(e2◦e3) = (e1◦e2)◦e3+e2◦(e1◦e3),
(2)ρ(e1◦e2) = [ρ(e1),ρ(e2)],
(3)e1◦fe2=f(e1◦e2)+ρ(e1)(f)e2,
(4)e1◦e1=1
2D/an}bracketle{te1,e1/an}bracketri}ht,
(5)ρ(e1)(/an}bracketle{te2,e3/an}bracketri}ht) =/an}bracketle{te1◦e2,e3/an}bracketri}ht+/an}bracketle{te2,e1◦e3/an}bracketri}ht,
where[·,·]is the Lie bracket of vector fields, and D:C∞(M)→Γ(E)is the
map defined by /an}bracketle{tDf,e/an}bracketri}ht=ρ(e)f.
The “bracket” ◦is related to the bracket given in Definition 2.1 by:
x◦y=/llbracketx,y/rrbracket+1
2D/an}bracketle{tx,y/an}bracketri}ht. (1)
Roytenberg [17] showed that if Eis a Courant algebroid in the sense of
Definition 2.1 with bracket /llbracket·,·/rrbracket, bilinear form /an}bracketle{t·,·/an}bracketri}htand anchor ρ, thenEis
a Courant algebroid in the sense of Definition 2.2 with the sam e anchor and
bilinear form but with bracket ◦given by Eq. 1. Unless otherwise stated, all
Courant algebroids mentioned in this paper are Courant alge broids in the
sense of Definition 2.1. We introduced Definition 2.2 mainly t o connect our
results here with previous results in the literature.COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 5
Example 1.An important example of a Courant algebroid is the standard
Courant algebroid E0=TM⊕T∗Mover any manifold Mwith bracket
/llbracket(v1,α1),(v2,α2)/rrbracket0=/parenleftbigg
[v1,v2],Lv1α2−Lv2α1−1
2d(ιv1α2−ιv2α1)/parenrightbigg
,(2)
and bilinear form
/an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht=ιv1α2+ιv2α1. (3)
In this case the anchor ρ:E0→TMis the projection map, and for a
function f∈C∞(M),Df= (0,df).
The standard Courant algebroid is the prototypical example of anexact
Courant algebroid [4].
Definition 2.3. A Courant algebroid E→Mwith anchor map ρ:E→TM
isexactiff
0→T∗Mρ∗
→Eρ→TM→0
is an exact sequence of vector bundles.
2.1.TheˇSevera class of an exact Courant algebroid. ˇSevera’s clas-
sification originates in the idea that choosing a splitting o f the above short
exact sequence corresponds to defining a kind of connection.
Definition 2.4. Aconnection on an exact Courant algebroid Eover a
manifold Mis a map of vector bundles A:TM→Esuch that
(1)ρ◦A= idTM,
(2)/an}bracketle{tA(v1),A(v2)/an}bracketri}ht= 0for allv1,v2∈TM,
whereρ:E→TMand/an}bracketle{t·,·/an}bracketri}htare the anchor and bilinear form, respectively.
IfAis a connection and θ∈Ω2(M) is a 2-form then one can construct a
new connection:
(A+θ)(v) =A(v)+ρ∗θ(v,·). (4)
(A+θ) satisfies the first condition of Definition 2.4 since ker ρ= imρ∗. The
second condition follows from the fact that we have by definit ion ofρ∗:
/an}bracketle{tρ∗(α),e/an}bracketri}ht=α(ρ(e)) (5)
for alle∈Γ(E) andα∈Ω1(M). Furthermore, one can show that any two
connections on an exact Courant algebroid must differ (as in Eq . 4) by a
2-form on M. Hence the space of connections on an exact Courant algebroi d
is an affine space modeled on the vector space of 2-forms Ω2(M) [4].
The failure of a connection to preserve the bracket gives a su itable notion
of curvature:
Definition 2.5. IfEis an exact Courant algebroid over Mwith bracket /llbracket·,·/rrbracket
andA:TM→Eis a connection then the curvature is a map F:TM×
TM→Edefined by
F(v1,v2) =/llbracketA(v1),A(v2)/rrbracket−A([v1,v2]).6 CHRISTOPHER L. ROGERS
IfFis the curvature of a connection Athen given v1,v2∈TM, it follows
from exactness and axiom 2 in Definition 2.1 that there exists a 1-form
αv1,v2∈Ω1(M) such that F(v1,v2) =ρ∗(αv1,v2). Since Ais a connection,
its image is isotropic in E. Therefore for any v3∈TMwe have:
/an}bracketle{tF(v1,v2),A(v3)/an}bracketri}ht=/an}bracketle{t/llbracketA(v1),A(v2)/rrbracket,A(v3)/an}bracketri}ht.
The above formula allows one to associate the curvature Fto a 3-form on
M:
Proposition 2.6. LetEbe an exact Courant algebroid over a manifold M
with bracket /llbracket·,·/rrbracketand bilinear form /an}bracketle{t·,·/an}bracketri}ht. LetA:TM→Ebe a connection
onE. Then given vector fields v1,v2,v3onM:
(1)The function
ω(v1,v2,v3) =/an}bracketle{t/llbracketA(v1),A(v2)/rrbracket,A(v3)/an}bracketri}ht
defines a closed 3-form on M.
(2)Ifθ∈Ω2(M)is a 2-form and ˜A=A+θthen
˜ω(v1,v2,v3) =/an}bracketle{t/llbracket˜A(v1),˜A(v2)/rrbracket,˜A(v3)/an}bracketri}ht
=ω(v1,v2,v3)+dθ(v1,v2,v3).
Proof.The statements in the proposition are proven in Lemmas 4.2.6 , 4.2.7,
and 4.3.4 in the paper by Bressler and Chervov [4]. In their wo rk they
define a Courant algebroid using Definition 2.2, and therefor e their bracket
satisfies the Jacobi identity, but is not skew-symmetric. In our notation,
their definition of the curvature 3-form is:
ω′(v1,v2,v3) =/an}bracketle{tA(v1)◦A(v2),A(v3)/an}bracketri}ht.
In particular they show that ◦satisfying the Jacobi identity implies ω′is
closed. The Jacobiator corresponding to the Courant bracke t is non-trivial
in general. However the isotropicity of the connection and E q. 1 imply
A(v1)◦A(v2) =/llbracketA(v1),A(v2)/rrbracket∀v1,v2∈TM.
Henceω′=ω, so all the needed results in [4] apply here. /square
Thus the above proposition implies that the curvature 3-for m of an exact
Courantalgebroid over Mgives awell-defined cohomology class in H3
DR(M),
independent of the choice of connection.
2.2.Twisting the Courant bracket. The previous section describes how
to go from exact Courant algebroids to closed 3-forms. Now we describe the
reverse process. In Example 1 we showed that one can define the standard
Courant algebroid E0over any manifold M. The total space is the direct
sumTM⊕T∗M, the bracket and bilinear form are given in Eqs. 2 and 3,
and the anchor is simply the projection. The inclusion A(v) = (v,0) of the
tangent bundle into the direct sum is obviously a connection onE0and it
is easy to see that the standard Courant algebroid has zero cu rvature.COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 7
ˇSevera and Weinstein [20, 21] observed that the bracket on E0could be
twisted by a closed 3-form ω∈Ω3(M) on the base:
/llbracket(v1,α1),(v2,α2)/rrbracketω=/llbracket(v1,α1),(v2,α2)/rrbracket0+ω(v1,v2,·).
This gives a new Courant algebroid Eωwith the same anchor and bilinear
form. Using Eqs. 2 and 3 we can compute the curvature 3-form of this new
Courant algebroid:
/an}bracketle{t/llbracketA(v1),A(v2)/rrbracket,A(v3)/an}bracketri}ht=/an}bracketle{t/llbracket(v1,0),(v2,0)/rrbracket,(v3,0)/an}bracketri}ht
=/an}bracketle{t([v1,v2],ω(v1,v2,·)),(v3,0)/an}bracketri}ht
=ω(v1,v2,v3),
and we see that Eωis an exact Courant algebroid over MwithˇSevera class
[ω].
3. 2-plectic geometry
We nowgive abriefoverview of 2-plectic geometry. Moredeta ils including
motivation for several of the definitions presented here can be found in our
previous work with Baez and Hoffnung [2, 3].
Definition 3.1. A3-formωon aC∞manifold Mis2-plectic , or more
specifically a 2-plectic structure , if it is both closed:
dω= 0,
and nondegenerate:
∀v∈TxM, ιvω= 0⇒v= 0
Ifωis a2-plectic form on Mwe call the pair (M,ω)a2-plectic manifold .
The 2-plectic structure induces an injective map from the sp ace of vector
fields on Mto the space of 2-forms on M. This leads us to the following
definition:
Definition 3.2. Let(M,ω)be a2-plectic manifold. A 1-form αonMis
Hamiltonian if there exists a vector field vαonMsuch that
dα=−ιvαω.
We sayvαis theHamiltonian vector field corresponding to α. The set
of Hamiltonian 1-forms and the set of Hamiltonian vector fiel ds on a 2-
plectic manifold are both vector spaces and are denoted as Ham(M)and
VectH(M), respectively.
The Hamiltonian vector field vαis unique if it exists, but note there may
be 1-forms αhaving no Hamiltonian vector field. Furthermore, two distin ct
Hamiltonian 1-forms may differ by a closed 1-form and therefor e share the
same Hamiltonian vector field.
We can generalize thePoisson bracket of functionsin symple ctic geometry
by defining a bracket of Hamiltonian 1-forms.8 CHRISTOPHER L. ROGERS
Definition 3.3. Givenα,β∈Ham(M), thebracket {α,β}is the
1-form given by
{α,β}=ιvβιvαω.
Proposition 3.4. Letα,β,γ∈Ham(M)and letvα,vβ,vγbe the respective
Hamiltonian vector fields. The bracket {·,·}has the following properties:
(1)The bracket of Hamiltonian forms is Hamiltonian:
d{α,β}=−ι[vα,vβ]ω, (6)
so in particular we have
v{α,β}= [vα,vβ].
(2)The bracket is skew-symmetric:
{α,β}=−{β,α} (7)
(3)The bracket satisfies the Jacobi identity up to an exact 1-form :
{α,{β,γ}}−{{α,β},γ}−{β,{α,γ}}=dJα,β,γ (8)
withJα,β,γ=ιvαιvβιvγω.
Proof.See Proposition 3.7 in [2]. /square
4.Lie2-algebras
Both the Courant bracket and the bracket on Hamiltonian 1-fo rms are,
roughly, Lie brackets which satisfy the Jacobi identity up t o an exact 1-
form. This leads us to the notion of a Lie 2-algebra: a categor y equipped
with structures analogous to those of a Lie algebra, for whic h the usual laws
involving skew-symmetry and the Jacobi identity hold up to i somorphism
[1, 19]. A Lie 2-algebra in which the isomorphisms are actual equalities
is called a strict Lie 2-algebra. A Lie 2-algebra in which the laws govern-
ing skew-symmetry are equalities but the Jacobi identity ho lds only up to
isomorphism is called a semistrict Lie 2-algebra.
Here we define a semistrict Lie 2-algebra to be a 2-term chain c omplex
of vector spaces equipped with structures analogous to thos e of a Lie al-
gebra, for which the usual laws hold up to chain homotopy. In t his guise,
a semistrict Lie 2-algebra is nothing more than a 2-term L∞-algebra. For
more details, we refer the reader to the work of Lada and Stash eff [14], and
the work of Baez and Crans [1].
Definition 4.1. Asemistrict Lie 2-algebra is a2-term chain complex
of vector spaces L= (L1d→L0)equipped with:
•a chain map [·,·]:L⊗L→Lcalled the bracket;
•an antisymmetric chain homotopy J:L⊗L⊗L→Lfrom the chain
map
L⊗L⊗L→L
x⊗y⊗z/mapsto−→[x,[y,z]],COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 9
to the chain map
L⊗L⊗L→L
x⊗y⊗z/mapsto−→[[x,y],z]+[y,[x,z]]
called the Jacobiator ,
such that the following equation holds:
[x,J(y,z,w)] +J(x,[y,z],w) +J(x,z,[y,w]) +[J(x,y,z),w]
+[z,J(x,y,w)] =J(x,y,[z,w]) +J([x,y],z,w)
+[y,J(x,z,w)] +J(y,[x,z],w) +J(y,z,[x,w]).(9)
We will also need a suitable notion of morphism:
Definition 4.2. Given semistrict Lie 2-algebras LandL′with bracket and
Jacobiator [·,·],Jand[·,·]′,J′respectively, a homomorphism fromLto
L′consists of:
•a chain map φ= (φ0,φ1) :L→L′, and
•a chain homotopy φ2:L⊗L→Lfrom the chain map
L⊗L→L
x⊗y/mapsto−→[φ(x),φ(y)]′,
to the chain map
L⊗L→L
x⊗y/mapsto−→φ([x,y])
such that the following equation holds:
J′(φ0(x),φ0(y),φ0(z))−φ1(J(x,y,z)) =
φ2(x,[y,z])−φ2([x,y],z)−φ2(y,[x,z])−[φ2(x,y),φ0(z)]′
+[φ0(x),φ2(y,z)]′−[φ0(y),φ2(x,z)]′.(10)
This definition is equivalent to the definition of a morphism b etween 2-
termL∞-algebras. (The same definition is given in [1], but it contai ns a
typographical error.)
4.1.The Lie 2-algebra from a 2-plectic manifold. Given a 2-plectic
manifold( M,ω), wecanconstructasemistrictLie2-algebra. Theunderlyi ng
2-term chain complex is namely:
L=C∞(M)d→Ham(M)
wheredis the usual exterior derivative of functions. This chain co mplex is
well-defined, since any exact form is Hamiltonian, with 0 as i ts Hamiltonian
vector field. We can construct a chain map
{·,·}:L⊗L→L,
by extending the bracket on Ham( M) trivially to L. In other words, in
degree 0, the chain map is given as in Definition 3.3:
{α,β}=ιvβιvαω,10 CHRISTOPHER L. ROGERS
and in degrees 1 and 2, we set it equal to zero:
{α,f}= 0,{f,α}= 0,{f,g}= 0.
The precise construction of this Lie 2-algebra is given in th e following the-
orem:
Theorem 4.3. If(M,ω)is a2-plectic manifold, there is a semistrict Lie
2-algebraL(M,ω)where:
•the space of 0-chains is Ham(M),
•the space of 1-chains is C∞,
•the differential is the exterior derivative d:C∞→Ham(M),
•the bracket is {·,·},
•the Jacobiator is the linear map JL: Ham(M)⊗Ham(M)⊗Ham(M)→
C∞defined by JL(α,β,γ) =ιvαιvβιvγω.
Proof.See Theorem 4.4 in [2]. /square
4.2.The Lie 2-algebra from a Courant algebroid. Given any Courant
algebroid E→Mwith bilinear form /an}bracketle{t·,·/an}bracketri}ht, bracket /llbracket·,·/rrbracket, and anchor ρ:E→
TM, we can construct a 2-term chain complex
C=C∞(M)D→Γ(E),
with differential D=ρ∗d. The bracket /llbracket·,·/rrbracketon global sections can be ex-
tended to a chain map /llbracket·,·/rrbracket:C⊗C→C. Ife1,e2are degree 0 chains then
/llbrackete1,e2/rrbracketis the original bracket. If eis a degree 0 chain and f,gare degree 1
chains, then we define:
/llbrackete,f/rrbracket=−/llbracketf,e/rrbracket=1
2/an}bracketle{te,Df/an}bracketri}ht
/llbracketf,g/rrbracket= 0.
This extended bracket gives a semistrict Lie 2-algebra on th e complex C:
Theorem 4.4. IfEis a Courant algebroid, there is a semistrict Lie 2-
algebraC(E)where:
•the space of 0-chains is Γ(E),
•the space of 1-chains is C∞(M),
•the differential the map D:C∞(M)→Γ(M),
•the bracket is /llbracket·,·/rrbracket,
•the Jacobiator is the linear map JC: Γ(M)⊗Γ(M)⊗Γ(M)→C∞(M)
defined by
JC(e1,e2,e3) =−T(e1,e2,e3)
=−1
6(/an}bracketle{t/llbrackete1,e2/rrbracket,e3/an}bracketri}ht+/an}bracketle{t/llbrackete3,e1/rrbracket,e2/an}bracketri}ht+/an}bracketle{t/llbrackete2,e3/rrbracket,e1/an}bracketri}ht).
Proof.Theproofthat aCourantalgebroid inthesenseofDefinition 2 .1gives
rise to a semistrict Lie 2-algebra follows from the work done by Roytenberg
on graded symplectic supermanifolds [18] and Lie 2-algebra s [19]. In partic-
ular we refer the reader to Example 5.4 of [19] and Section 4 of [18].COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 11
Onthe other hand, theoriginal construction of Roytenberg a nd Weinstein
[16] gives a L∞-algebra on the complex:
0→kerDι→C∞(M)D→Γ(E),
with trivial structure maps lnforn≥3. Moreover, the map l2(correspond-
ing to the bracket /llbracket·,·/rrbracketgiven above) is trivial in degree >1 and the map
l3(corresponding to the Jacobiator JC) is trivial in degree >0. Hence we
can restrict this L∞-algebra to our complex Cand use the results in [1] that
relateL∞-algebras with semistrict Lie 2-algebras. /square
5.The Courant algebroid associated to a 2-plectic manifold
Now we have the necessary machinery in place to describe how C ourant
algebroids connect with 2-plectic geometry. First, recall the discussion in
Section 2.2 on twisting the bracket of the standard Courant a lgebroid E0by
a closed 3-form. From Definition 3.1, we immediately have the following:
Proposition 5.1. Let(M,ω)be a2-plectic manifold. There exists an exact
Courant algebroid EωwithˇSevera class [ω]overMwith underlying vector
bundleTM⊕T∗M→M, anchor ρ(v,α) =v, and bracket and bilinear form
given by:
/llbracket(v1,α1),(v2,α2)/rrbracketω=/parenleftbigg
[v1,v2],Lv1α2−Lv2α1−1
2d(ιv1α2−ιv2α1)+ιv2ιv1ω/parenrightbigg
,
/an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht=ιv1α2+ιv2α1.
More importantly, the Courant algebroid constructed in Pro position 5.1
not only encodes the 2-plectic structure ω, but also the corresponding Lie
2-algebra constructed in Theorem 4.3:
Theorem 5.2. Let(M,ω)be a2-plectic manifold and let Eωbe its corre-
sponding Courant algebroid. Let L(M,ω)andC(Eω)be the semistrict Lie
2-algebras corresponding to (M,ω)andEω, respectively. Then there exists
a homomorphism embedding L(M,ω)intoC(Eω).
Before we prove the theorem, we introduce some lemmas to ease the
calculations. In the notation that follows, if α,βare Hamiltonian 1-forms
with corresponding vector fields vα,vβ, then
B(α,β) =1
2(ιvαβ−ιvβα). (11)
Also by the symbol /anticlockwisewe mean cyclic permutations of the symbols α,β,γ.
Lemma 5.3. Ifα,β∈Ham(M)with corresponding Hamiltonian vector
fieldsvα,vβ, thenLvαβ={α,β}+dιvαβ.
Proof.SinceLv=ιvd+dιv,
Lvαβ=ιvαdβ+dιvαβ=−ιvαιvβω+dιvαβ={α,β}+dιvαβ.
/square12 CHRISTOPHER L. ROGERS
Lemma 5.4. Ifα,β,γ∈Ham(M)with corresponding Hamiltonian vector
fieldsvα,vβ,vγ, then
ι[vα,vβ]γ+/anticlockwise=−3ιvαιvβιvγω+2/parenleftbig
ιvαdB(β,γ)+ιvγdB(α,β)+ιvβdB(γ,α)/parenrightbig
.
Proof.The identity ι[v1,v2]=Lv1ιv2−ιv2Lv1and Lemma 5.3 imply:
ι[vα,vβ]γ=Lvαιvβγ−ιvβLvαγ
=Lvαιvβγ−ιvβ({α,γ}+dιvαγ)
=ιvαdιvβγ−ιvβιvγιvαω−ιvβdιvαγ,
where the last equality follows from the definition of the bra cket.
Hence it follows that:
ι[vγ,vα]β=ιvγdιvαβ−ιvαιvβιvγω−ιvαdιvγβ,
ι[vβ,vγ]α=ιvβdιvγα−ιvγιvαιvβω−ιvγdιvβα.
Finally, note 2 ιvαdB(β,γ) =ιvαdιvβγ−ιvαdιvγβ. /square
Lemma 5.5. Ifα,β∈Ham(M)with corresponding Hamiltonian vector
fieldsvα,vβ, then
Lvβα−Lvαβ=−2({α,β}+dB(α,β)).
Proof.Follows immediately from Lemma 5.3 and the definition of B(α,β).
/square
Proof of Theorem 5.2. We will construct a homomorphism from L(M,ω) to
C(Eω). LetLbe the underlying chain complex of L(M,ω) consisting of
Hamiltonian 1-forms in degree 0 and smooth functions in degr ee 1. Let
Cbe the underlying chain of C(Eω) consisting of global sections of Eωin
degree 0 and smooth functions in degree 1. The bracket /llbracket·,·/rrbracketωdenotes the
extension of the bracket on Γ( Eω) to the complex Cin the sense of Theorem
4.4. Let φ0:L0→C0be given by
φ0(α) = (vα,−α),
wherevαis the Hamiltonian vector field corresponding to α. Letφ1:L1→
C1be given by
φ1(f) =−f.
Finally let φ2:L0⊗L0→C1be given by
φ2(α,β) =−B(α,β) =−1
2(ιvαβ−ιvβα).
Nowweshow φ2isawell-definedchainhomotopyinthesenseofDefinition
4.2. For degree 0 we have:
/llbracketφ0(α),φ0(β)/rrbracketω=/parenleftbigg
[vα,vβ],Lvα(−β)−Lvβ(−α)+1
2d/parenleftbig
ιvαβ−ιvβα/parenrightbig
+ιvβιvαω/parenrightbigg
= ([vα,vβ],−{α,β}+dφ2(α,β)),COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 13
where the last equality above follows from Lemma 5.5. By Prop osition 3.4,
the Hamiltonian vector field of {α,β}is [vα,vβ]. Hence we have:
/llbracketφ0(α),φ0(β)/rrbracketω−φ0({α,β}) =dφ2(α,β).
Indegree1, thebracket {·,·}istrivial. Henceitfollows fromthedefinition
of/llbracket·,·/rrbracketωand the bilinear form on Eω(given in Proposition 5.1 ) that
/llbracketφ0(α),φ1(f)/rrbracketω=−/llbracketφ1(f),φ0(α)/rrbracketω=1
2/an}bracketle{t(vα,−α),(0,−df)/an}bracketri}ht=φ2(α,df).
Therefore φ2is a chain homotopy.
It remains to show the coherence condition (Eq. 10 in Definiti on 4.2) is
satisfied. We rewrite the Jacobiator JCas:
JC(φ0(α),φ0(β),φ0(γ)) =−1
6/an}bracketle{t/llbracketφ0(α),φ0(β)/rrbracket,φ0(γ)/an}bracketri}ht+/anticlockwise
=−1
6/an}bracketle{t([vα,vβ],−{α,β}−dB(α,β)),(vγ,−γ)/an}bracketri}ht+/anticlockwise
=1
6/parenleftBig
ι[vα,vβ]γ+ιvγιvβιvαω+ιvγdB(α,β)/parenrightBig
+/anticlockwise
=−JL(α,β,γ)+1
2/parenleftbig
ιvγdB(α,β)+/anticlockwise/parenrightbig
.
The last equality above follows from Lemma 5.4 and the definit ion of the
Jacobiator JL. Therefore the left-hand side of Eq. 10 is
JC(φ0(α),φ0(β),φ0(γ))−φ1(JL(α,β,γ)) =1
2/parenleftbig
ιvγdB(α,β)+/anticlockwise/parenrightbig
.
By the skew-symmetry of the brackets, the right-hand side of Eq. 10 can
be rewritten as:
(/llbracketφ0(α),φ2(β,γ)/rrbracketω+/anticlockwise)−(φ2({α,β},γ)+/anticlockwise).
From the definitions of the bracket, bilinear form and φ2we have:
/llbracketφ0(α),φ2(β,γ)/rrbracketω+/anticlockwise=1
2/an}bracketle{t(vα,−α),(0,dφ2(β,γ)/an}bracketri}ht+/anticlockwise
=−1
2ιvαdB(β,γ)+/anticlockwise,
and:
φ2({α,β},γ)+/anticlockwise=−1
2/parenleftBig
ι[vα,vβ]γ−ιvγιvβιvαω/parenrightBig
=−(ιvαdB(β,γ)+/anticlockwise).
The last equality above follows again from Lemma 5.4. Theref ore the right-
hand side of Eq. 10 is
1
2/parenleftbig
ιvγdB(α,β)+/anticlockwise/parenrightbig
.
Hencethemaps φ0,φ1,φ2give ahomomorphismofsemistrictLie2-algebras.
/square14 CHRISTOPHER L. ROGERS
We note that Roytenberg [19] has shown that a Courant algebro id defined
using Definition 2.2 with the bilinear operation ◦induces a hemistrict Lie
2-algebra on the complex Cdescribed in Theorem 4.4 above. A hemistrict
Lie 2-algebra is a Lie 2-algebra in which the skew-symmetry h olds up to
isomorphism, while the Jacobi identity holds as an equality . We have proven
in previous work [2] that a 2-plectic structure also gives ri se to a hemistrict
Lie 2-algebra on the complex described in Theorem 4.3. One ca n show that
all results presented above, in particular Theorem 5.2, car ry over to the
hemistrict case.
6.Hamiltonian 1-forms as infinitesimal symmetries of the
Courant algebroid
Givena2-plecticmanifold( M,ω), theLie2-algebraofobservables L(M,ω)
identifies particular infinitesimal symmetries of the corre sponding Courant
algebroid Eωvia the embedding described in the proof of Theorem 5.2. To
see this, we first recall some basic facts concerning automor phisms of exact
Courant algebroids. The presentation here follows the work of Bursztyn,
Cavalcanti, and Gualtieri [7].
Definition 6.1. LetE→Mbe a Courant algebroid with bilinear form /an}bracketle{t·,·/an}bracketri}ht,
bracket /llbracket·,·/rrbracket, and anchor ρ:E→TM. Anautomorphism is a bundle
isomorphism F:E→Ecovering a diffeomorphism ϕ:M→Msuch that
(1)ϕ∗/an}bracketle{tF(e1),F(e2)/an}bracketri}ht=/an}bracketle{te1,e2/an}bracketri}ht,
(2)F(/llbrackete1,e2/rrbracket) =/llbracketF(e1),F(e2)/rrbracket,
(3)ρ(F(e1)) =ϕ∗(ρ(e1)).
Consider the exact Courant algebroid Eωdescribed in Section 2.2 with
underlyingvector bundle TM⊕T∗M→MandˇSevera class [ ω]∈H3
DR(M).
Given a 2-form B∈Ω2(M), one can define a bundle isomorphism
expB:TM⊕T∗M→TM⊕T∗M
by
expB(v,α) = (v,α+ιvB).
The map exp Bis known as a ‘gauge transformation’. It covers the identity
id:M→Mand therefore is compatible (in the sense of Definition 6.1) w ith
the anchor ρ(v,α) =v. Since Bis skew-symmetric, exp Bpreserves the
bilinear form /an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht=ιv1α2+ιv2α1. However a simple compu-
tation shows that exp Bpreserves the bracket /llbracket·,·/rrbracketω(defined in Eq. 2.2) if
and only if Bis a closed 2-form:
/llbracketexpB(v1,α1),expB(v2,α2)/rrbracketω= expB/parenleftbig
/llbracket(v1,α1),(v2,α2)/rrbracketω+dB/parenrightbig
.
Given a diffeomorphism ϕ:M→Mof the base space, one can define a
bundle isomorphism Φ: TM⊕T∗M→TM⊕T∗Mby
Φ(v,α) =/parenleftBig
ϕ∗v,(ϕ∗)−1α/parenrightBig
.COURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 15
ThemapΦsatisfiesconditions1and3ofDefinition6.1butdoes notpreserve
the bracket in general:
/llbracketΦ(v1,α1),Φ(v2,α2)/rrbracketω= Φ/parenleftBig
/llbracket(v1,α1),(v2,α2)/rrbracketϕ∗ω/parenrightBig
.
Bursztyn, Cavalcanti, and Gualtieri [7] showed that any aut omorphism F
of the exact Courant algebroid Eωmust be of the form
F= ΦexpB, (12)
where Φ is constructed from a diffeomorphism ϕ:M→Msuch that
ω−ϕ∗ω=dB. (13)
This classification of automorphisms allows one to classify the infinitesimal
symmetries as well. Let
Ft= ΦtexptB=/parenleftBig
ϕt∗exptB,(ϕ∗
t)−1exptB/parenrightBig
bea 1-parameter family of automorphismsof the Courant alge broidEωwith
F0= idEω. Letu∈Vect(M) be the vector field that generates the flow ϕ−t.
Then differentiation gives:
dFt
dt(v,α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle
t=0= ([u,v],Luα+ιvB).
Sinceω−ϕ∗
tω=tdB, it follows that uandBmust satisfy the equality:
Luω=dB. (14)
Theseinfinitesimaltransformationsarecalled derivations [7]oftheCourant
algebroid Eω, sincetheycorrespondtolinearfirstorderdifferential oper ators
which act as derivations of the non-skew-symmetric bracket :
(v1,α1)◦ω(v2,α2) =/llbracket(v1,α1),(v2,α2)/rrbracketω+1
2d/an}bracketle{t(v1,α1),(v2,α2)/an}bracketri}ht.(15)
= ([v1,v2],Lv1α2−ιv2dα1+ιv2ιv1ω). (16)
In general, derivations are pairs ( u,B)∈Vect(M)⊕Ω2(M) satisfying Eq.
14. They act on global sections ( v,α)∈Γ(Eω) by:
(u,B)·(v,α) = ([u,v],Luα+ιvB).
Global sections themselves naturally act as derivations vi a anadjoint
action[7]. Given ( u,β)∈Γ(Eω) letBbe the 2-form
B=−dβ+ιuω. (17)
Define ad (u,β): Γ(Eω)→Γ(Eω) by
ad(u,β)(v,α) = (u,B)·(v,α) = ([u,v],Luα+ιv(−dβ+ιuω)).(18)
One can see this is indeed the adjoint action in the usual sens e if one con-
siders the non-skew-symmetric bracket given in Eq. 15:
ad(u,β)(v,α) = (u,β)◦ω(v,α).16 CHRISTOPHER L. ROGERS
Recall that in the proof of Theorem 5.2 we constructed a homom orphism
of Lie 2-algebras using the map φ0: Ham(M)→Γ(Eω) defined by
φ0(α) = (vα,−α),
wherevαis the Hamiltonian vector field correspondingto α. Comparing Eq.
17 to Definition 3.2 of a Hamiltonian 1-form, we see that a sect ion (u,β)∈
Γ(Eω) is in the image of the map φ0if and only if its adjoint action ad (u,β)
corresponds to the pair ( u,0)∈Vect(M)⊕Ω2(M). This implies that ad (u,β)
preserves the 2-plectic structure on Mand that −βis a Hamiltonian 1-form
with Hamiltonian vector field u. Also if uis complete, then Eqs. 12 and 13
imply that the 1-parameter family Ftof Courant algebroid automorphisms
generated by ad (u,β)correspondsto a1-parameter family of diffeomorphisms
ϕt:M→Mwhich preserve the 2-plectic structure:
ϕ∗
tω=ω.
In analogy with symplectic geometry, we call such automorph ismsHamil-
tonian 2-plectomorphisms .
We provide the following proposition as a summary of the disc ussion
presented in this section:
Proposition 6.2. Let(M,ω)be a 2-plectic manifold and let Eωbe its cor-
responding Courant algebroid. There is a one-to-one correspo ndence be-
tween the Hamiltonian 1-forms Ham(M)on(M,ω)and sections (u,β)of
Eωwhose adjoint action satisfies the equality
ad(u,β)(v,α) = (Luv,Luα).
7.Conclusions
We suspect that the results presented here are preliminary a nd indicate
a deeper relationship between 2-plectic geometry and the th eory of Courant
algebroids. For example, the discussion of connections and curvature in
Section 2.1 is reminiscent of the theory of gerbes with conne ction [5], whose
relationship with Courant algebroids has been already stud ied [4, 20]. In 2-
plecticgeometry, gerbeshavebeenconjecturedtoplayarol einthegeometric
quantization ofa2-plecticmanifold[2]. Itwillbeinteres tingtoseehowthese
different points of view complement each other.
In general, much work has been done on studying the geometric struc-
tures induced by Courant algebroids (e.g. Dirac structures , twisted Dirac
structures). Perhaps this work can aid 2-plectic geometry s ince many geo-
metric structures in this context are somewhat less underst ood or remain
ill-defined (e.g. the notion of a 2-Lagrangian submanifold o r 2-polarization).
On the other hand, n-plectic manifolds are well understood in the role
they play in classical field theory [11], and are also underst ood algebraically
in the sense that an n-plectic structure gives an n-termL∞-algebra on a
chain complex of differential forms [15]. Perhaps these insig hts can aid inCOURANT ALGEBROIDS FROM CATEGORIFIED SYMPLECTIC GEOMETRY 17
understanding ‘higher’ analogs of Courant algebroids (e.g . Lien-algebroids)
and complement similar ideas discussed by ˇSevera in [22].
8.Acknowledgments
We thank John Baez, Yael Fregier, Dmitry Roytenberg, Urs Sch rieber,
James Stasheff and Marco Zambon for helpful comments, questi ons, and
conversations.
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E-mail address :chris@math.ucr.edu
Department of Mathematics, University of California, Rive rside, Califor-
nia 92521, USA