|
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. |
|
References |
|
[1] J. Baez and A. Crans, Higher-dimensional algebra VI: Lie 2-algebras, TAC12(2004), |
|
492–528. Also available as arXiv:math/0307263. |
|
[2] J. Baez, A. Hoffnung, and C. Rogers, Categorified symplect ic geometry and |
|
the classical string, Comm. Math. Phys. 293(2010), 701–715. Also available as |
|
arXiv:0808.0246. |
|
[3] J. Baez and C. Rogers, Categorified symplectic geometry a nd the string Lie 2-algebra, |
|
available as arXiv:0901.4721. |
|
[4] P. Bressler and A. Chervov, Courant algebroids, J. Math. Sci. (N.Y.) 128(2005), |
|
3030–3053. Also available as arXiv:hep-th/0212195. |
|
[5] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantiz ation, |
|
Birkhauser, Boston, 1993. |
|
[6] M.J. Bowick and S.G. Rajeev, Closed bosonic string theor y,Nuc. Phys. B 293(1987), |
|
348–384. |
|
[7] H. Bursztyn, G.R. Cavalcanti, and M. Gualtieri, Reducti on of Courant algebroids |
|
and generalized complex structures, Adv. Math. 211(2007), 726–765. Also available |
|
as arXiv:math/0509640. |
|
[8] F. Cantrijn, A.Ibort, andM. DeLeon, Onthegeometryofmu ltisymplectic manifolds, |
|
J. Austral. Math. Soc. (Series A) 66(1999), 303–330. |
|
[9] T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319(1990), 631–661. |
|
[10] T. DeDonder, Theorie Invariantive du Calcul des Variations , Gauthier–Villars, Paris, |
|
1935. |
|
[11] M. Gotay, J. Isenberg, J. Marsden, and R. Montgomery, Mo mentum maps and classi- |
|
cal relativistic fields.Part I:covariantfieldtheory, avai lable as arXiv:physics/9801019. |
|
[12] F. H´ elein, Hamiltonian formalisms for multidimensio nal calculus of variations and |
|
perturbation theory, in Noncompact Problems at the Intersection of Geometry , eds. |
|
A. Bahri et al, AMS, Providence, Rhode Island, 2001, pp. 127–148. Also ava ilable as |
|
arXiv:math-ph/0212036. |
|
[13] J. Kijowski, A finite-dimensional canonical formalism in the classical field theory, |
|
Commun. Math. Phys. 30(1973), 99–128. |
|
[14] T. Lada and J. Stasheff, Introduction to sh Lie algebras f or physicists, Int. Jour. |
|
Theor. Phys. 32(7) (1993), 1087–1103. Also available as hep-th/9209099. |
|
[15] C. Rogers, L∞-algebras from multisymplectic geometry, in preparation. |
|
[16] D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie |
|
algebras, Lett. Math. Phys. 46(1998), 81–93. Also available as arXiv:math/9802118. |
|
[17] D. Roytenberg, Courant Algebroids, Derived Brackets and Even Symplectic S uper- |
|
manifolds , Ph.D. thesis, UC Berkeley, 1999. Also available as arXiv:m ath/9910078. |
|
[18] D. Roytenberg, On the structure of graded symplectic su permanifolds and Courant |
|
algebroids, in Quantization, Poisson Brackets and Beyond , ed. T. Voronov, Con- |
|
temp. Math. ,315, AMS, Providence, RI, 2002, pp. 169–185. Also available as |
|
arXiv:math/0203110.18 CHRISTOPHER L. ROGERS |
|
[19] D. Roytenberg, On weak Lie 2-algebras, in: P. Kielanows kiet al(eds.) XXVI Work- |
|
shop on Geometrical Methods in Physics. AIP Conference Proc eedings956, pp. 180- |
|
198. American InstituteofPhysics, Melville (2007). Alsoa vailable as arXiv:0712.3461. |
|
[20] P. ˇSevera, Letter to Alan Weinstein, available at |
|
http://sophia.dtp.fmph.uniba.sk/ ~severa/letters/ |
|
[21] P.ˇSevera, and A. Weinstein, Poisson geometry with a 3-form bac kground, Prog. |
|
Theor. Phys. Suppl. 144(2001), 145–154. Also available as arXiv:math/0107133. |
|
[22] P.ˇSevera, Some title containing the words ‘homotopy’ and ‘sym plectic’, e.g. this one, |
|
available as arXiv:math/0105080 |
|
[23] Y. Sheng and C. Zhu, Semidirect products of representat ions up to homotopy, avail- |
|
able as arXiv:0910.2147. |
|
[24] H. Weyl, Geodesic fields in the calculus of variation for multiple integrals, Ann. Math. |
|
36(1935), 607–629. |
|
E-mail address :[email protected] |
|
Department of Mathematics, University of California, Rive rside, Califor- |
|
nia 92521, USA |