Higher Structures


Urs Schreiber (CAS Prague & MPI Bonn)

Higher Structures in Mathematics and Physics

an introductory talk held at:

  1. Meeting of Maths@CAS Brno, 2016 Nov 9-11

  2. Oberwolfach Workshop 1651a, 2016 Dec. 18-23


Higher structures


The term “higher structures” is short for

mathematical structures on higher homotopy types.


We first explain what this means.


Abstract homotopy

Consider a set


Given two elements

x,yX x,y \in X

there is the proposition

x=y x = y

Either xx is equal to yy, or it is not.

But there may be more than one

  • way how they are equal,

  • i.e. reason that they are equal,

  • i.e. proof that they are equal.

Let’s write

xγy x \stackrel{\gamma}{\longrightarrow} y

for one such way.

Call this a homotopy from xx to yy.

Let’s write

Id X(x,y) Id_X(x,y)

for the set of homotopies from xx to yy in XX.

Now the same reasoning applies to homotopies: given

γ 1,γ 2Id X(x,y) \gamma_1, \gamma_2 \in Id_X(x,y)

two homotopies, there is the proposition

γ 1=γ 2 \gamma_1 = \gamma_2

Either they are equal or not. But again there may be more than one way/reason/proof κ\kappa that exhibits their equality:

γ 1κγ 2. \gamma_1 \stackrel{\kappa}{\longrightarrow} \gamma_2 \,.

This may be called a higher homotopy.

And it keeps going this way:

And so on.


Taking into account all these higher homotopies, the original “set” XX is in general richer than a set in the sense of ZFC.

To avoid clash of terminology, XX is called a type, or homotopy type.


One formalization of this picture is due to

This is mostly known as the theory of “model categories”,

short for “categories of models for homotopy types”.


Another formalization of this picture is Martin-Löf type theory (Martin-Löf 73), a kind of intuitionistic constructive set theory.

That this is secretly a formalization of abstract homotopy theory was only understood more recently (Hofmann-Streicher 98, Awodey-Warren 07, Kapulkin-Lumsdaine 12)

Ever since it is also called homotopy type theory (UFP 13).


Topological homotopy

The classical model that motivated abstract homotopy theory is the homotopy theory of topological spaces.

Given two continuous functions from a topological space XX space to a topological space YY

f,g:XY f,g \;\colon\; X \longrightarrow Y

hence given two points in the mapping space

f,gMaps(X,Y) f,g \in Maps(X,Y)

then a homotopy η\eta between them

fηg f \stackrel{\eta}{\longrightarrow} g

is a continuous family of continuous paths between the points

X×{0} f X×[0,1] η Y g X×{1} \array{ X \times \{0\} \\ \downarrow & \searrow^{\mathrlap{f}} \\ X \times [0,1] &\overset{\eta}{\longrightarrow}& Y \\ \uparrow & \nearrow_{\mathrlap{g}} \\ X \times \{1\} }

(graphics grabbed from J. Tauber here)

In this perspective topological spaces are called homotopy types.

A homotopy n-type is a space inside which there exist higher homotopies that are non-trivial up to higher homotopy only up to order nn.


Hence an ordinary set is equivalently a homotopy 0-type.

The rest is “higher homotopy types”.


Now a “higher structure” is a structure on higher homotopy types:


Higher structures


A mathematical structure, à la Bourbaki, is

  1. a collection of sets

  2. equipped with functions between them,

  3. subject to equations between these functions.


Example. A group is

  1. a set GG

  2. equipped with

    1. multiplication: a function ()():G×GG\;\;(-)\cdot(-) \colon G \times G \to G\;\;

    2. neutral element: a function e:*G\;\;e \colon \ast \to G\;\;

    3. inverses: a function () 1:GG\;\;(-)^{-1} \colon G \to G\;\;

  3. such that this satisfies the equations for

    1. associativity: (xy)z=x(yz)(x \cdot y) \cdot z = x \cdot (y \cdot z)

    2. unitality: xe=ex=xx \cdot e = e \cdot x = x

    3. invertibility: x 1x=ex^{-1} \cdot x = e.


A higher structure is like a Bourbakian mathematical structure but

Here “coherence” is the condition that “iterated structure homotopies are unique up to homotopy”.


Example. A “first step higher group”, called a 2-group (Sinh 73, Baez-Lauda 03), is a group structure on a homotopy 1-type.

So the associativity equality is to be replaced by a choice of homotopy

(xy)za x,y,zx(yz) (x \cdot y) \cdot z \stackrel{a_{x,y,z}}{\longrightarrow} x \cdot (y \cdot z)

called the associator.

The coherence to be imposed on this is the condition that the composite homotopy

((wx)y)zw(x(yz)) ((w \cdot x) \cdot y) \cdot z \longrightarrow w \cdot (x \cdot (y \cdot z))

is unique, in that the following two ways of composing associators to achieves this are related by a homotopy-of-homotopies, hence an equality (by assumption that we have just a 1-type).

This coherence condition is called the pentagon identity.

It implies coherence in the following sense:


Coherence theorem for 2-groups (MacLane 63): All homotopies of re-bracketing any expression, using the given associators, coincide.


Example (delooping 2-group)

For GG a discrete group, there is its classifying space denoted



K(G,1) K(G,1)

(an Eilenberg-Mac Lane space).

For XX a (paracompact) topological space, then homotopy classes of continuous functions into BGB G classify GG-principal covering spaces of XX:

{XBG}{G-principal coverings ofX}. \left\{X \longrightarrow B G \right\} \;\; \simeq \;\; \left\{ \; G\text{-principal coverings of} \; X \; \right\} \,.

If GG happens to be an abelian group, then BGB G is a 2-group. The multiplication

BG×BGBG B G \times B G \longrightarrow B G

is the map which classifies the tensor product of GG-coverings regarded as group ring-module bundles.

The existence of inverses reflects the fact that as such each GG-covering corresponds to a line bundle.


Higher geometric groups

Just like


and generally

(e.g. Schreiber 13)



Higher principal bundles

All the usual constructions done with ordinary groups now have their higher homotopy theoretic analogs.

For instance a “parameterized group” – called a GG-principal bundle – is a space


equipped with a GG-action

G×PP G \times P \longrightarrow P

over some space XX

P X \array{ P \\ \downarrow \\ X }

such that over some open cover {U iX}\{U_i \to X\} then there are GG-equivariant equivalences

P| U iU i×G P|_{U_i} \stackrel{\simeq}{\longrightarrow} U_i \times G

over each chart U iU_i.

For example the frame bundle Fr(X)Fr(X) of a smooth manifold XX of dimension nn is an general linear group-principal bundle

GL(n) Fr(X) X \array{ GL(n) &\hookrightarrow& Fr(X) \\ && \downarrow \\ && X }

The higher coherent homotopy version of this definition yields

the concept of principal infinity-bundles (Nikolaus-Schreiber-Stevenson 12).

For instance a principal 2-bundle over a delooping 2-group of the form BA\mathbf{B}A as above

BA P X \array{ \mathbf{B}A &\hookrightarrow& P \\ && \downarrow \\ && X }

is what is often called a bundle gerbe with band AA.

We will see examples of higher principal bundles naturally appear in the following.


Higher categorical structures

All higher homotopies are invertible, up to homotopy.

For example the inverse-up-to-homotopy of a topological homotopy

X×[0,1]ηY X \times [0,1] \stackrel{\eta}{\longrightarrow} Y

is the “reversed” map

X×[0,1](x,t)(x,1t)X×[0,1]ηY. X \times [0,1] \stackrel{(x,t) \mapsto (x,1-t)}{\longrightarrow} X \times [0,1] \stackrel{\eta}{\longrightarrow} Y \,.


But one may generalize further and consider non-invertible homotopies up to some order <n\lt n.

The result might be called “directed homotopy theory”.

But one speaks of higher categories, specifically (infinity,n)-categories.

This is more complicated than plain higher homotopy theory. The latter is the special case of (infinity,1)-categories.

More generally then:


A higher structure is a mathematical structure internal to an (infinity,n)-category.


This is also called categorification (Crane-Yetter 96, Baez-Dolan 98).

For a long time the relevant theory had been mostly missing, but now it exists (Lurie 1-).

A key application and motivation for this are extended topological quantum field theories.


Here we do not further dwell on such “directed higher structures”.




Various “shadows” of higher homotopy theory are more widely familiar.

One example is chain complexes.


Consider a chain complex in non-negative degree.

[V nV 1V 0],=0. \left[ \cdots \stackrel{\partial}{\to} V_n \stackrel{\partial}{\to} \cdots \stackrel{\partial}{\to} V_1 \stackrel{\partial}{\to} V_0 \right] \;\;\;\;\;\,,\;\;\; \partial \circ \partial = 0 \,.


x,yV 0 x,y \in V_0

be two elements in degree 0, whose images in degree-0 chain homology

[x],[y]H 0(V )V 0/im(| V 1) [x], [y] \;\in H_0(V_\bullet) \coloneqq V_0/im(\partial|_{V_1})

are equal

[x]=[y]. [x] = [y] \,.

This means that there exists an element

γV 1 \gamma \in V_1

in degree 1 such that

γ=yx \partial \gamma = y - x

(a coboundary).

Hence every such γ\gamma is a reason for the equality, hence a homotopy

(γ=yx)(xγy). \left( \; \partial \gamma = y - x \; \right) \;\;\Leftrightarrow\;\; \left( x \stackrel{\gamma}{\longrightarrow} y \right) \,.

Next, let

γ 1,γ 2V 1 \gamma_1, \gamma_2 \in V_1

be two coboundaries, both between xx and yy. Then an element

κV 2 \kappa \in V_2


κ=γ 2γ 1 \partial \kappa = \gamma_2 - \gamma_1

is a way for them to be equal

Namely this is a way for them to be equal in degree-1 chain homology

H 1(V )ker(| V 1)/im(| V 2) H_1(V_\bullet) \coloneqq ker(\partial|_{V_1})/im(\partial|_{V_2})

in the sense that

[0]=[γ 2γ 1] [0] = [\gamma_2 - \gamma_1]

witnessed by

0κ(γ 2γ 1). 0 \stackrel{\kappa}{\longrightarrow} (\gamma_2 -\gamma_1) \,.

Next, an element in V 3V_3 defines a third order homotopy

and so on.


This way each chain complex defines a homotopy type.

This is called the Dold-Kan correspondence (Dold 58, Kan 58).



Example (Poincaré lemma)

For XX a smooth manifold of dimension nn, its de Rham complex is the chain complex

Ω (X)[C (X)=Ω 0(X)d dRΩ 1(X)d dRd dRΩ n(X)] \Omega^\bullet(X) \;\coloneqq\; \left[ C^\infty(X) = \Omega^0(X) \stackrel{d_{dR}}{\to} \Omega^1(X) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n(X) \right]

of differential forms with the de Rham differential between them. If XX is an open ball, then there is a homotopy equivalence from the de Rham complex to

[n][degn00000]. \mathbb{R}[n] \;\coloneqq\; \left[ \underset{deg \, n}{\underbrace{ \mathbb{R} }} \stackrel{0}{\to} 0 \stackrel{0}{\to} \cdots \stackrel{0}{\to} 0 \right] \,.

namely chain homotopies of the form

This is the statement of the Poincaré lemma. The second chain homotopy is traditionally known as the homotopy operator used in the standard proof of the Poincaré lemma.



In conclusion, there are higher mathematical structures on chain complexes.

These are often called differential graded structures, or dg-structures for short.


Here is a key example:


Higher Lie algebras


Recall the mathematical structure called a Lie algebra:

Example A Lie algebra is

  1. a vector space VV

  2. equipped with a function

    • Lie bracket: [,]:V×VV[-,-] \;\colon\; V \times V \longrightarrow V
  3. such that

    1. skew-symmetry: [x,y]=[y,x][x,y] = - [y,x]

    2. bilinearity: [k 1v 1+k 2v 2,w]=k 1[v 1,w]+k 2[v 2,w][k_1 v_1 + k_2 v_2, w] = k_1 [v_1, w] + k_2 [v_2,w]

    3. Jacobi identity: [x,[y,z]]=[[x,y],z]+[y,[x,z]][x,[y,z]] = [[x,y], z] + [y,[x,z]].


A higher homotopy Lie algebra structure in the homotopy theory of chain complexes, is called equivalently a

(Lada-Stasheff 92, Lada-Markl 94).


Such a higher Lie algebra consists, first of all, of a skew-symmetric chain map

[,]:VVV [-,-] \;\colon\; V \otimes V \longrightarrow V

hence of a graded-skew symmetric

[x,y]=(1) deg(x)deg(y)[y,x] [x,y] = -(-1)^{deg(x)deg(y)}\, [y,x]

bilinear map that respects the differential

[x,y]+(1) deg(x)[x,y]=[x,y] [\partial x, y] + (-1)^{deg(x)} [x, \partial y] = \partial[x,y]

and which satisfies the Jacobi identity up to a specified homotopy called the Jacobiator

[x,[y,z]]j x,y,z[[x,y],z]+(1) deg(x)deg(y)[y,[x,z]]. [x,[y,z]] \;\;\overset{j_{x,y,z}}{\longrightarrow}\;\; [[x,y],z] + (-1)^{deg(x)deg(y)} [y,[x,z]] \,.

Then there is a coherence condition which says that the two possible ways of re-bracketing four elements are homotopic

(graphics grabbed from Baez-Crans 04, p. 19)

If the chain complex is concentrated in degrees 0 and 1 then this is all the data there is. This is a Lie 2-algebra (Baez-Crans 04)

In general there is a second-order homotopy filling this diagram, which in turn satisfies its own coherence condition up to a third-order homotopy, and so on.



The following example is trivial but important for the theory.


Example. For pp \in \mathbb{N}, there is a unique higher Lie algebra structure on the chain complex

B p[0in degreep000] B^p \mathbb{R} \;\coloneqq\; \left[ \cdots \stackrel{0}{\to} \underset{\text{in degree} \atop {p}}{\underbrace{\mathbb{R}}} \stackrel{0}{\to} \cdots \stackrel{0}{\to} 0 \right]

namely the one with vanishing bracket and vanishing higher homotopies. This is the line Lie (p+1)-algebra.


What are higher Lie algebras good for?

There are several answers:

  1. they are equivalent to “formal moduli problems

    controlling deformation theory (Hinich 97, Pridham 07, Lurie 1-, see Doubek-Markl-Zima 07)

  2. they control closed string field theory (Zwiebach 93)


Here is yet another answer:


Higher extensions


Higher Lie algebras turn out to be the answer to the question:

What do higher Lie algebra cocycles classify?


A (p+2)-cocycle on a Lie algebra (V,[,])(V,[-,-]) is a function

μ p+2:V××Vp+2copies \mu_{p+2} \;\colon\; \underset{p+2\;\text{copies}}{\underbrace{V \times \cdots \times V}} \longrightarrow \mathbb{R}

such that

  1. multilinearity holds,

  2. skew-symmetry holds,

  3. cocycle condition: for all (p+3)(p+3)-tuples (x 0,,x p+2)(x_0, \cdots, x_{p+2})

    σpermutations(1) sgn(σ)μ p+2([x σ 0,x σ 1],x σ 2,,x σ p+2)=0. \underset{\sigma\in permutations}{\sum} (-1)^{sgn(\sigma)} \, \mu_{p+2}([x_{\sigma_0},x_{\sigma_1}], x_{\sigma_2}, \cdots, x_{\sigma_{p+2}}) = 0 \,.


Classical fact. Given a 2-cocycle μ 2\mu_2 it corresponds to a central extension Lie algebra extension, namely a Lie algebra structure on

V V \oplus \mathbb{R}

given by

[(x 1,c 1),(x 2,c 2)]([x 1,x 2],μ 2(x 1,x 2)). [\, (x_1,c_1),\,(x_2,c_2) \,] \; \coloneqq \; (\, [x_1,x_2], \, \mu_2(x_1,x_2) \,) \,.

For this bracket the Jacobi identity is equivalently the cocycle condition on μ 2\mu_2.


Fact: For pp \in \mathbb{N} then a (p+2)(p+2)-cocycle μ p+2\mu_{p+2} still defines an extension, but this is now a higher Lie algebra structure on the chain complex

[0indegreep00V] \left[ \cdots \overset{0}{\to} \underset{{in\, degree} \atop p}{\underbrace{\mathbb{R}}} \overset{0}{\to} \cdots \overset{0}{\to} V \right]

whose Jacobiator and all its higher analogs are zero, except for the one of arity (p+2)(p+2), which is given by μ p+2\mu_{p+2}.


Example. Every semisimple Lie algebra 𝔤\mathfrak{g} carries a Killing form pairing

,:𝔤×𝔤𝔤 \langle -,- \rangle \;\colon\; \mathfrak{g} \times \mathfrak{g} \longrightarrow \mathfrak{g}

and the resulting trilinear map

μ 3,[,] \mu_3 \coloneqq \langle -, [-,-] \rangle

is a 3-cocycle.

This classifies a Lie 2-algebra extension of 𝔤\mathfrak{g},

called the string Lie 2-algebra

(Baez-Crans 03, Baez-Crans-Schreiber-Stevenson 05).

(The reason for this terminology we discuss below.)

\, \,

Higher cocycles on higher Lie algebras

There are also nn-cocycles on higher Lie algebras 𝔤\mathfrak{g}.

In fact a (p+2)(p+2)-cocycle on a higher Lie algebra 𝔤\mathfrak{g} is equivalently a homomorphism of higher Lie algebras of the form

μ p+2:𝔤B p+1 \mu_{p+2} \;\colon\; \mathfrak{g} \longrightarrow B^{p+1}\mathbb{R}

and the higher Lie algebra extension 𝔤^\widehat{\mathfrak{g}} that this classifies is equivalently the homotopy fiber

𝔤^ hofib(μ p+2) 𝔤 μ p+2 B p+1 \array{ \widehat{\mathfrak{g}} \\ {}^{\mathllap{hofib(\mu_{p+2})}}\downarrow \\ \mathfrak{g} &\stackrel{\mu_{p+2}}{\longrightarrow}& B^{p+1}\mathbb{R} }

(Fiorenza-Sati-Schreiber 13, prop. 3.5)


This implies that from any higher Lie algebra 𝔤\mathfrak{g} there emanates a “bouquet” of consecutive higher extensions


Fact. Such a higher homotopy bouquet controls the classification of super p-branes in string theory/M-theory (Fiorenza-Sati-Schreiber 13, Fiorenza-Sati-Schreiber 16).


This shows that something deep relates higher structures with physics.

We come back to this at the end.

First we now explain how higher structures appear in physics.



A higher structure at the heart of physics


Classical physics is governed by the “principle of least action

(extremal action, really)


This is formalized in variational calculus as follows.


so that

  • a field configuration is a section ϕΓ Σ(E)\phi \in \Gamma_\Sigma(E)

    E ϕ Σ = Σ \array{ && E \\ &{}^{\mathllap{\phi}}\nearrow& \downarrow \\ \Sigma &=& \Sigma }


This means that if

( x i, v a ) spacetimecoordinates fieldcoordinateschart ofE \array{ (& x^i, & v^a & ) \\ & {spacetime \atop coordinates} & {field \atop coordinates} } \;\;\;\;\;\;\;\;\;\;\;\;\; \text{chart of} \; E

is a local coordinate chart of EE, then the corresponding coordinate chart of the jet bundle is

( x i, v a, v i a,v ij a, ) spacetimecoordinates fieldcoordinates partialderivativesoffieldcoordinateschart ofJ E \array{ (& x^i, & v^a, & v^a_i, v^a_{i j}, \cdots & ) \\ & {spacetime \atop coordinates} & {field \atop coordinates} & {{partial\; derivatives}\atop {of\;field\; coordinates}} } \;\;\;\;\;\;\;\;\;\;\;\;\; \text{chart of} \; J^\infty E



A local Lagrangian field theory is defined by a local Lagrangian density:

a (p+1)(p+1)-form LL on Σ\Sigma which is a function of the fields and some finite number of its partial derivatives

L({fields},{partial derivatives of fields},)Ω p+1(Σ). L(\,\{\text{fields}\}, \{\text{partial derivatives of fields}\}, \cdots \,) \in \Omega^{p+1}(\Sigma) \,.

This is a horizontal differential (p+1)(p+1)-form on the jet bundle of EE.

LΩ H p+1(E)Ω p+1(J E). L \in \Omega^{p+1}_H(E) \subset \Omega^{p+1}(J^\infty E) \,.

Here the horizontal differential d Hd_H is the “total derivative” given in the above coordinate chart by

d H=dx i(x i+v i av a+v ij av j a+) d_H = d x^i \left( \frac{\partial}{\partial x^i} + v^a_i \frac{\partial}{\partial v^a} + v^a_{i j} \frac{\partial}{\partial v^a_j} + \cdots \right)

This defines a chain complex called the horizontal de Rham complex

[Ω H 0(E)d HΩ H 1(E)d Hd HΩ H p+1(E)] \left[ \Omega^0_H(E) \overset{d_H}{\longrightarrow} \Omega^1_H(E) \overset{d_H}{\longrightarrow} \cdots \overset{d_H}{\longrightarrow} \Omega^{p+1}_H(E) \right]

Define the vertical differential on J EJ^\infty E to be

d Vd dRd H. d_V \coloneqq d_{dR} - d_H \,.

This induces a chain complex of chain complexes, a double complex called the variational bicomplex


Say that the source forms are the wedge products of horizontal (p+1)(p+1)-forms with vertical derivatives of functions of just the fields

Ω S p+1,1(E)Ω H p+1(E)tophorizontald VΩ 0(E)verticaldiff.offield. \Omega^{p+1,1}_S(E) \;\coloneqq\; \underset{top \atop horizontal}{\underbrace{\Omega^{p+1}_H(E)}} \wedge \underset{{vertical \; diff.} \atop {of \; field}}{\underbrace{d_V \Omega^0(E)}} \,.


Euler partial integration:

Source forms are a model for the top horizontal vertical 1-forms modulo horizontal divergences:

Ω S p+1,1(E)Ω p+1,1(E)/im(d H). \Omega^{p+1,1}_S(E) \simeq \Omega^{p+1,1}(E)/im(d_H) \,.

Hence the full de Rham differential of any local Lagrangian density LL uniquely splits into a source-form and a horizontally exact term

dL=δ vLsourceformd Hθhorizontallyexact d L = \underset{source \atop form}{\underbrace{\delta_v L}} - \underset{horizontally \atop exact}{\underbrace{d_H \theta}}

This δ v\delta_v is the variational derivative.


The Euler-Lagrange equations of motion on field configurations ϕ\phi is the partial differential equation

(δ vL)(ϕ)=0 (\delta_v L)(\phi) = 0


(v ax iv i a+ 2x ix jv ij a)L(ϕ, iϕ, ijϕ,)=0 \left( \frac{\partial}{\partial v^a} - \frac{\partial}{\partial x^i} \frac{\partial}{\partial v^a_i} + \frac{\partial^2}{\partial x^i \partial x^j} \frac{\partial}{\partial v^a_{i j}} - \cdots \right) L\left( \phi, \partial_i \phi, \partial_{i j}\phi, \cdots \right) \;\; = \;\; 0



One checks immediately that

δ vd H=0. \delta_v \circ d_H = 0 \,.

Hence the horizontal de Rham complex continues to a longer chain complex

Ω H 0(E) d H Ω H 1(E) d H d H Ω H p(E) d H Ω H p+1(E) δ v Ω S p+1,1(E) \array{ \Omega^0_H(E) &\overset{d_H}{\longrightarrow}& \Omega^1_H(E) &\overset{d_H}{\longrightarrow}& \cdots &\overset{d_H}{\longrightarrow}& \Omega^p_H(E) &\overset{d_H}{\longrightarrow}& \Omega^{p+1}_H(E) \\ && && && && \downarrow^{\mathrlap{\delta_v}} \\ && && && && \Omega^{p+1,1}_S(E) }

In fact it continues further:

totaldivergences Ω H 0(E) d H Ω H 1(E) d H d H Ω H p(E) d H Ω H p+1(E) d V Euler-Lagrange variational derivative Ω n+1,1(E)/im(d H) d V Helmholtz operator Ω p+1,2(E)/im(d H) d V . \array{ && && && & {\text{total} \atop \text{divergences}} & \\ \Omega^0_H(E) &\overset{d_H}{\longrightarrow}& \Omega^1_H(E) &\overset{d_H}{\longrightarrow}& \cdots &\overset{d_H}{\longrightarrow}& \Omega^p_H(E) &\overset{d_H}{\longrightarrow}& \Omega^{p+1}_H(E) \\ && && && && \downarrow^{\mathrlap{d_V}} & \text{Euler-Lagrange variational derivative} \\ && && && && \Omega^{n+1,1}(E)/im(d_H) \\ && && && && \downarrow^{\mathrlap{d_V}} & \text{Helmholtz operator} \\ && && && && \Omega^{p+1,2}(E)/im(d_H) \\ && && && && \downarrow^{\mathrlap{d_V}} \\ && && && && \vdots } \,.

This is called the Euler-Lagrange complex.


Theorem. (Vinogradov 78, Tulczyjew 80)

The Euler-Lagrange complex is locally exact. In other words, it participates in a variational version of the Poincaré lemma as above.

The Helmholtz operator on linear differential equations was found all the way back in (Helmholtz 1887).



This means for instance that

any given equation of motion ELΩ S n+1,1(E)EL \in \Omega^{n+1,1}_S(E)

is locally a “principle of extremal action

precisely if the Helmholtz operator annihilates it d VEL=0d_V EL = 0.



Hence the Euler-Lagrange complex is a higher structure at the heart of physics.

Question? What more do we gain by making it explicit this higher structure?

Surprising Answer: Many central examples of field theories may only be understood using this!


This is due to


Three well-kept secrets

For many field theories of interest…Example \;\;
(1) …the Lagrangian density is
(3)\;\phantom{(3)}\;\;\; not globally defined.
charged particle in electromagnetic background field,
WZW model,
string in Kalb-Ramond field,
membrane in supergravity C-field,
D-brane in Kalb-Ramond field
(2) …the Lagrangian density is
(3)\;\phantom{(3)}\;\;\; a higher connection on a higher bundle
Wilson loop,
Wilson surface,
(3) …the field bundle is itself a higher bundle.gauge theory such as Yang-Mills theory, Chern-Simons theory,
higher gauge theory such as AKSZ sigma-model
(Fiorenza-Rogers-Schreiber 11),
7d Chern-Simons theory of String 2-connection field
(Fiorenza-Sati-Schreiber 14)
(3)\;\phantom{(3)}\; …all three of these apply.electromagnetism with electron source fields,
RR-fields with D-brane source fields
(amplified in Freed 00)
super p-brane with tensor multiplet
such as
- D-branes with Chan-Paton gauge fields
- M5-brane with worldvolume gerbe
(Fiorenza-Sati-Schreiber 13)



We now explain these items in turn.


First secret: Locally variational theories


Many local Lagrangian densities of interest are not actually globally defined.

These field theories are only locally variational

(in the terminology ofAnderson-Duchamp 80, Ferraris-Palese-Winterroth 11).


This means that

  1. there exists

    1. an open cover of the field bundle by coordinate charts
    {U iE} \{U_i \to E\}
    1. a collection of Lagrangian densities
    L iΩ H p+1(U i) L_i \in \Omega^{p+1}_H(U_i)

    on each of the charts,

  2. such that

    • they have a globally well defined variational derivative

      ELΩ S p+1,1(E) EL \in \Omega^{p+1,1}_S(E)

      in that

      EL| U i=δ vL i. EL|_{U_i} \, = \, \delta_v L_i \,.


Example: charged particle


  • Σ=\Sigma = \mathbb{R} (the “worldline”)

  • (X,g)(X,g) a 4d spacetime

  • EΣ×XE \coloneqq \Sigma \times X the field bundle,

    so that a field configuration is equivalently a smooth function

    ϕ:X\phi \colon \mathbb{R} \longrightarrow X

    i.e. a trajectory of a particle in XX.

  • XX carrying an electromagnetic field given by a differential 2-form

    ωΩ cl 2(X) \omega \in \Omega^2_{cl}(X)

    called the Faraday tensor, which in a local coordinate chart

    3,1X \mathbb{R}^{3,1} \hookrightarrow X


    an electric field E=[E 1 E 2 E 3]\vec E = \left[ \array{E_1 \\ E_2 \\ E_3} \right] and magnetic field B=[B 1 B 2 B 3]\vec B = \left[ \array{B_1 \\ B_2 \\ B_3} \right]


    ω =E 1dx 1dx 0+E 2dx 2dx 0+E 3dx 3dx 0 +B 1dx 2dx 3+B 2dx 3dx 1+B 3dx 1dx 2 \begin{aligned} \omega & = E_1 \; d x^1 \wedge d x^0 + E_2 \; d x^2 \wedge d x^0 + E_3 \; d x^3 \wedge d x^0 \\ & + \;\; B_1 \; d x^2 \wedge d x^3 + B_2 \; d x^3 \wedge d x^1 + B_3 \; d x^1 \wedge d x^2 \end{aligned}


  1. there exists

    1. an open cover {U iX}\{ U_i \to X\}

    2. a collection of 1-forms A iΩ 1(U i)A_i \in \Omega^1(U_i)

  2. such that

    ω| U i=d dRA i\omega|_{U_i} = d_{dR} A_i

and the locally defined Lagrangian density for the charged particle propagating in this background is

L i(g μνϕ t μϕ t ν(L kin) i+(A i) μϕ t μ(L int) i)dt. L_i \coloneqq \left( \underset{(L_{kin})_i}{\underbrace{ g_{\mu \nu} \phi^\mu_t \phi^\nu_t }} + \underset{(L_{int})_i}{\underbrace{(A_i)_\mu \phi^\mu_t}} \right) d t \,.

The Euler-Lagrange equations for this are the geodesic equations modified by the Lorentz force

δ vL i = δ v(L kin) i + δ v(L int) i = ϕ tt μ+Γ μ αβϕ t αϕ t βforce of gravity + F μ νϕ t νLorentz forceof electromagnetic field=0 \array{ \delta_v L_i & = & \delta_v (L_{kin})_i &+& \delta_v (L_{int})_i \\ & = & \underset{\text{force of gravity}}\underbrace{ \phi^\mu_{t t} + \Gamma^\mu{}_{\alpha \beta} \phi^\alpha_t \phi^\beta_t } & +& \underset{\text{Lorentz force} \atop \text{of electromagnetic field}}{\underbrace{ F^\mu{}_\nu \phi^\nu_t }} = 0 }



Hence there is an obvious question:

When may we find a globally defined Lagrangian for the charged particle?


This is the kind of question that tools from homotopy theory naturally apply to.

We obtain the answer below,

after a closer look at the following…



Second secret: Global action functionals

A collection of chartwise defined Lagrangians is not enough data to make sense of a globally defined exponentiated action functional

exp(i ΣL):Γ X(E)/ U(1). \exp\left( \tfrac{i}{\hbar} \int_{\Sigma} L \right) \;\; \colon \Gamma_X(E) \longrightarrow \mathbb{R}/_{\hbar} \mathbb{Z} \simeq U(1) \,.

Here Planck's constant \hbar coordinatizes the circle group U(1)U(1).

0/ U(1)0 0 \to \mathbb{Z} \stackrel{\hbar}{\longrightarrow} \mathbb{R} \stackrel{}{\longrightarrow} \underset{U(1)}{\underbrace{\mathbb{R}/_{\hbar} \mathbb{Z}}} \to 0


Instead, making sense of this requires to choose

  • on each (n+1)(n+1)-fold intersection of charts

  • an order-nn higher homotopy

between the local Lagrangian densities:

and so on.


Such a system of local Lagrangians with higher order homotopies between them is called

a cocycle in hyper Cech cohomology

with coefficients in the horizontal Deligne complex:

[Ω H 0d HΩ H 1d Hd HΩ H p+1]. \left[ \mathbb{Z} \overset{\hbar}{\hookrightarrow} \Omega^0_H \overset{d_H}{\longrightarrow} \Omega^1_H \overset{d_H}{\longrightarrow} \cdots \overset{d_H}{\longrightarrow} \Omega^{p+1}_H \right] \,.



Technical aside:

The statement here is that



The higher bundle underlying a local Lagrangian


Underlying any such homotopically enhanced locally variational field theory L\mathbf{L} is a Cech cocycle representing an element

c(L)H p+2(E,) c(\mathbf{L}) \;\in\; H^{p+2}(E,\mathbb{Z})

in the integral cohomology of the field bundle

This is given by the evident chain homomorphism

[ Ω H 0 d H Ω H 1 d H d H Ω H p+1 ] id [ Ω 0 0 0 ]. \array{ [ & \mathbb{Z} &\hookrightarrow& \Omega^0_H &\overset{d_H}{\longrightarrow}& \Omega^1_H &\overset{d_H}{\longrightarrow}& \cdots &\overset{d_H}{\longrightarrow}& \Omega^{p+1}_H & ] \\ & \downarrow^{\mathrm{id}} && \downarrow && \downarrow && && \downarrow \\ [ & \mathbb{Z} &\hookrightarrow& \Omega^0 &\overset{}{\longrightarrow}& 0 &\overset{}{\longrightarrow}& \cdots &\overset{}{\longrightarrow}& 0 & ] } \,.

Here the chain complex

B p+1U(1)[ Ω 0 0 0 ] \mathbf{B}^{p+1}U(1) \;\coloneqq\; \array{ [ & \mathbb{Z} &\hookrightarrow& \Omega^0 &\overset{}{\longrightarrow}& 0 &\overset{}{\longrightarrow}& \cdots &\overset{}{\longrightarrow}& 0 & ] }

is a Lie (p+1)-group.

A cocycle as above classifies a higher principal bundle as above with higher structure group B pU(1)\mathbf{B}^p U(1).



The obstruction to there being a globally defined Lagrangian density is the trivialization of this higher principal bundle, hence the vanishing

c(L)=0H p+2(E,) c(\mathbf{L}) = 0 \;\;\;\;\in H^{p+2}(E,\mathbb{Z})

of its characteristic class.



In the example of the charged particle above, this is the case precisely if, in physics speak:

This is the modern incarnation of the old Dirac charge quantization phenomenon (Dirac 31).



Third secret: Higher field bundles


For exposition see: Higher field bundles for gauge fields.




Higher Noether theorem


Given a locally variational Lagrangian gerbe L\mathbf{L} as above, then a symmetry is a flow

texp(tv):EE t \mapsto \exp(t v) \;\colon\; E \longrightarrow E

along some vertical vector field

vT vertJ E v \in T_{vert} J^\infty E

such that it preserves the Lagrangian density L\mathbf{L} in that the transformed Lagrangian density gerbe

exp(tv) *L \exp(t v)^\ast \mathbf{L}

coincides with L\mathbf{L}.

In the spirit of homotopy theory, we need to choose a homotopy for each tt

exp(tv) *Lγ(t)L. exp(t v)^\ast \mathbf{L} \underoverset{\simeq}{\;\;\;\gamma(t)\;\;\;}{\longrightarrow} \mathbf{L} \,.

An infinitesimal symmetry is obtained by differentiating this with respect to tt. This yields

vL=d Hγ \mathcal{L}_v L = d_H \gamma'

where \mathcal{L} denotes the Lie derivative.

By applying

  1. Cartan's magic formula

  2. Euler partial integration (above)

this becomes

d H(γ+ι vθJ)=ι vδ vL d_H \,(\, \underset{J}{\underbrace{\gamma' + \iota_v \theta}} \,) \;=\; \iota_v \delta_v \mathbf{L}

Here JJ is a conserved current

in that whenever the equations of motion hold

then its horizontal differential (total divergence) vanishes.

Hence symmetries of field theories induce conserved currents.


But we could have made a different choice of homotopy

(exp(tv)) *Lγ 2(t)L (\exp(t v))^\ast L \underoverset{\simeq}{\;\;\;\gamma_2(t)\;\;\;}{\longrightarrow} L

and we could have a higher order homotopy between them.

Higher Noether theorem There is an extension of the Lie algebra of symmetries by the higher Lie algebra of higher currents

[Ω 0(E) d H d H Ω H p(E) cl]Cur(L)Sym(L) \left[ \array{ \Omega^0(E) \\ \downarrow^{d_H} \\ \cdots \\ \downarrow^{d_H} \\ \Omega^{p}_H(E)_{cl} } \right] \;\longrightarrow\; Cur(\mathbf{L}) \;\longrightarrow\; Sym(\mathbf{L})

(Khavkine-Schreiber, based on Fiorenza-Rogers-Schreiber 14)

Passing to chain homology, this gives ordinary Lie algebra extension.

0H cl n1(E)H 0(Cur(L))Sym(L)0 0 \to H^{n-1}_{cl}(E) \longrightarrow H_0(Cur(\mathbf{L})) \longrightarrow Sym(\mathbf{L}) \to 0

This is the cohomological version of the Noether theorem.

(Technically aside: The conserved currents here are the “conserved currents in characteristic form” in the terminology of Olver 86, after (4.29).)



Applied to Green-Schwarz sigma model for fundamental super p-branes this characterizes BPS states in supergravity. (Azcárraga-Gauntlett-Izquierdo-Townsend 89, Sati-Schreiber 15). This plays a major role in string theory.



To understand what is going on here, we now explain how these fundamental pp-branes work:

Example: Fundamental pp-branes


There are classes of examples of local variational local field theories for whose full analysis tools from higher structures are inevitable.

One such class is the generalization of the charged particle from above to higher dimensions,

called higher dimensional WZW models.


These have two kinds of applications



We close by discussing the archetypical case of a string propagating on a Lie group manifold.


The fundamental string in a group manifold

Let GG be compact simply connected simple Lie group.

Such as


We consider a “charged string” propagating on GG, in direct analogy to the charged particle on XX from above.


The higher analog of the Faraday tensor now is a differential 3-form.

Recall the Lie algebra 3-cocycle ,[,]\langle -,[-,-]\rangle from above above. By evaluating it on the Maurer-Cartan form

θΩ 1(G,𝔤) \theta \in \Omega^1(G,\mathfrak{g})

we obtain the closed left invariant differential form.

ω 3=θ[θ,θ] \omega_3 = \langle \theta \wedge [\theta, \theta]\rangle

Pulling this back to the jet bundle along

J (Σ×G)G J^\infty(\Sigma \times G) \longrightarrow G

and projecting to the horizontal component yields the “Euler-Lagrane form for the higher Lorentz force”

ω 3| H \omega_3|_H

By the above obstruction theory we find:

there does not exist a globally defined Lagrangian density L WZWL_{WZW} with δ vL WZW=(μ 3)| H\delta_v L_{WZW} = (\mu_3)|_H.

Instead: there exists a U(1)U(1) bundle gerbe

L WZWH(G,[Ω H 0Ω H 1Ω H 2]) \mathbf{L}_{WZW} \in H(G, [\mathbb{Z} \to \Omega_H^0 \to \Omega_H^1 \to \Omega_H^2] )

such that

δ vL WZW=(μ 3)| H \delta_v \mathbf{L}_{WZW} = (\mu_3)|_H

This is the jet bundle version (Khavkine-Schreiber) of a statement that goes back to (Gawedzki 87). For review see (Schweigert-Waldorf 07).


Theorem The higher Noether current 2-group of this L WZW\mathbf{L}_{WZW} sits in a homotopy fiber sequence of smooth infinity-groups

BU(1) Cur(L WZW) G Ωc 2 B 2U(1). \array{ \mathbf{B} U(1) &\longrightarrow& \mathbf{Cur}(\mathbf{L}_{WZW}) &\longrightarrow& G \\ && && \downarrow^{\mathrlap{\Omega \mathbf{c}_2}} \\ && && \mathbf{B}^2 U(1) } \,.

(Fiorenza-Rogers-Schreiber 13)


This identifies (Schreiber 13)

the higher Noether current group Cur(L WZW)Cur(\mathbf{L}_{WZW}) of the string

as the “smooth string 2-group” (Baez-Crans-Schreiber-Stevenson 05)

whose Lie 2-algebra is the string Lie 2-algebra from above above




Green-Schwarz anomaly cancellation


Now let

G P X \array{ G &\hookrightarrow& P \\ && \downarrow \\ && X }

be a GG-principal bundle over some XX.

We may ask for a Lagrangian that restricts on each fiber to the one above (a “parameterized WZW model”).


Theorem (Distler-Sharpe 07, Schreiber 13) the obstruction to the parameterized WZW model on PP to exist is equivalently

  1. a lift of PP to a principal 2-bundle (as above) for the string 2-group (from above)

  2. vanishing of the canonical 4-class of PP.

If G=Spin(d)×SU(n) opG = Spin(d) \times SU(n)^{op} then this 4-class is

12p 1c 2 \tfrac{1}{2}p_1 - c_2

The vanishing of this class is the Green-Schwarz anomaly cancellation in heterotic string theory.


This is only the first step in a rich story.


For more on this see the lecture notes


Pointers to original references are given in the above text. Here we just list surveys.

See also


Acknowledgement This note profited from suggestions by Igor Khavkine. Generally, I profited from collaborating with Igor Khavkine on higher structures in physics, please see the references in the main text above.

Last revised on January 22, 2019 at 00:47:28. See the history of this page for a list of all contributions to it.