Urs Schreiber (CAS Prague & MPI Bonn)
Higher Structures in Mathematics and Physics
an introductory talk held at:
Meeting of Maths@CAS Brno, 2016 Nov 9-11
Oberwolfach Workshop 1651a, 2016 Dec. 18-23
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The term “higher structures” is short for
mathematical structures on higher homotopy types.
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We first explain what this means.
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Abstract homotopy
Consider a set
Given two elements
there is the proposition
Either $x$ is equal to $y$, 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
for one such way.
Call this a homotopy from $x$ to $y$.
Let’s write
for the set of homotopies from $x$ to $y$ in $X$.
Now the same reasoning applies to homotopies: given
two homotopies, there is the proposition
Either they are equal or not. But again there may be more than one way/reason/proof $\kappa$ that exhibits their equality:
This may be called a higher homotopy.
And it keeps going this way:
And so on.
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Taking into account all these higher homotopies, the original “set” $X$ is in general richer than a set in the sense of ZFC.
To avoid clash of terminology, $X$ is called a type, or homotopy type.
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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”.
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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).
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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 $X$ space to a topological space $Y$
hence given two points in the mapping space
then a homotopy $\eta$ between them
is a continuous family of continuous paths between the points
(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 $n$.
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Hence an ordinary set is equivalently a homotopy 0-type.
The rest is “higher homotopy types”.
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Now a “higher structure” is a structure on higher homotopy types:
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Higher structures
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A mathematical structure, à la Bourbaki, is
a collection of sets
equipped with functions between them,
subject to equations between these functions.
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Example. A group is
a set $G$
equipped with
multiplication: a function $\;\;(-)\cdot(-) \colon G \times G \to G\;\;$
neutral element: a function $\;\;e \colon \ast \to G\;\;$
inverses: a function $\;\;(-)^{-1} \colon G \to G\;\;$
such that this satisfies the equations for
associativity: $(x \cdot y) \cdot z = x \cdot (y \cdot z)$
unitality: $x \cdot e = e \cdot x = x$
invertibility: $x^{-1} \cdot x = e$.
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A higher structure is like a Bourbakian mathematical structure but
replacing sets by higher homotopy types;
replacing equations by homotopies;
adding higher order homotopies to enforce coherence.
Here “coherence” is the condition that “iterated structure homotopies are unique up to homotopy”.
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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
called the associator.
The coherence to be imposed on this is the condition that the composite homotopy
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:
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Coherence theorem for 2-groups (MacLane 63): All homotopies of re-bracketing any expression, using the given associators, coincide.
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Example (delooping 2-group)
For $G$ a discrete group, there is its classifying space denoted
or
(an Eilenberg-Mac Lane space).
For $X$ a (paracompact) topological space, then homotopy classes of continuous functions into $B G$ classify $G$-principal covering spaces of $X$:
If $G$ happens to be an abelian group, then $B G$ is a 2-group. The multiplication
is the map which classifies the tensor product of $G$-coverings regarded as group ring-module bundles.
The existence of inverses reflects the fact that as such each $G$-covering corresponds to a line bundle.
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Higher geometric groups
Just like
a group may be equipped with geometry to make it a
or Lie group
or super group
etc.
so
a 2-group may be equipped with higher geometry. The result is called
etc.
and generally a group stack.
and generally
an infinity-group may be equipped with higher geometry. The result may be called a
(e.g. Schreiber 13)
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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 $G$-principal bundle – is a space
equipped with a $G$-action
over some space $X$
such that over some open cover $\{U_i \to X\}$ then there are $G$-equivariant equivalences
over each chart $U_i$.
For example the frame bundle $Fr(X)$ of a smooth manifold $X$ of dimension $n$ is an general linear group-principal bundle
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 $\mathbf{B}A$ as above
is what is often called a bundle gerbe with band $A$.
We will see examples of higher principal bundles naturally appear in the following.
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Higher categorical structures
All higher homotopies are invertible, up to homotopy.
For example the inverse-up-to-homotopy of a topological homotopy
is the “reversed” map
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But one may generalize further and consider non-invertible homotopies up to some order $\lt n$.
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:
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A higher structure is a mathematical structure internal to an (infinity,n)-category.
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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.
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Here we do not further dwell on such “directed higher structures”.
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Various “shadows” of higher homotopy theory are more widely familiar.
One example is chain complexes.
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Consider a chain complex in non-negative degree.
Let
be two elements in degree 0, whose images in degree-0 chain homology
are equal
This means that there exists an element
in degree 1 such that
(a coboundary).
Hence every such $\gamma$ is a reason for the equality, hence a homotopy
Next, let
be two coboundaries, both between $x$ and $y$. Then an element
with
is a way for them to be equal
Namely this is a way for them to be equal in degree-1 chain homology
in the sense that
witnessed by
Next, an element in $V_3$ defines a third order homotopy
and so on.
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This way each chain complex defines a homotopy type.
This is called the Dold-Kan correspondence (Dold 58, Kan 58).
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Example (Poincaré lemma)
For $X$ a smooth manifold of dimension $n$, its de Rham complex is the chain complex
of differential forms with the de Rham differential between them. If $X$ is an open ball, then there is a homotopy equivalence from the de Rham complex to
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.
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In conclusion, there are higher mathematical structures on chain complexes.
These are often called differential graded structures, or dg-structures for short.
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Here is a key example:
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Higher Lie algebras
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Recall the mathematical structure called a Lie algebra:
Example A Lie algebra is
a vector space $V$
equipped with a function
such that
skew-symmetry: $[x,y] = - [y,x]$
bilinearity: $[k_1 v_1 + k_2 v_2, w] = k_1 [v_1, w] + k_2 [v_2,w]$
Jacobi identity: $[x,[y,z]] = [[x,y], z] + [y,[x,z]]$.
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A higher homotopy Lie algebra structure in the homotopy theory of chain complexes, is called equivalently a
strong homotopy Lie algebra (the “strong” refers to the coherence)
L-infinity algebra (for “L”ie algebra with homotopies up to infinity)
(Lada-Stasheff 92, Lada-Markl 94).
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Such a higher Lie algebra consists, first of all, of a skew-symmetric chain map
hence of a graded-skew symmetric
bilinear map that respects the differential
and which satisfies the Jacobi identity up to a specified homotopy called the Jacobiator
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.
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The following example is trivial but important for the theory.
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Example. For $p \in \mathbb{N}$, there is a unique higher Lie algebra structure on the chain complex
namely the one with vanishing bracket and vanishing higher homotopies. This is the line Lie (p+1)-algebra.
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What are higher Lie algebras good for?
There are several answers:
they are equivalent to “formal moduli problems”
controlling deformation theory (Hinich 97, Pridham 07, Lurie 1-, see Doubek-Markl-Zima 07)
they control closed string field theory (Zwiebach 93)
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Here is yet another answer:
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Higher extensions
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Higher Lie algebras turn out to be the answer to the question:
What do higher Lie algebra cocycles classify?
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A (p+2)-cocycle on a Lie algebra $(V,[-,-])$ is a function
such that
multilinearity holds,
skew-symmetry holds,
cocycle condition: for all $(p+3)$-tuples $(x_0, \cdots, x_{p+2})$
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Classical fact. Given a 2-cocycle $\mu_2$ it corresponds to a central extension Lie algebra extension, namely a Lie algebra structure on
given by
For this bracket the Jacobi identity is equivalently the cocycle condition on $\mu_2$.
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Fact: For $p \in \mathbb{N}$ then a $(p+2)$-cocycle $\mu_{p+2}$ still defines an extension, but this is now a higher Lie algebra structure on the chain complex
whose Jacobiator and all its higher analogs are zero, except for the one of arity $(p+2)$, which is given by $\mu_{p+2}$.
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Example. Every semisimple Lie algebra $\mathfrak{g}$ carries a Killing form pairing
and the resulting trilinear map
is a 3-cocycle.
This classifies a Lie 2-algebra extension of $\mathfrak{g}$,
called the string Lie 2-algebra
(Baez-Crans 03, Baez-Crans-Schreiber-Stevenson 05).
(The reason for this terminology we discuss below.)
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Higher cocycles on higher Lie algebras
There are also $n$-cocycles on higher Lie algebras $\mathfrak{g}$.
In fact a $(p+2)$-cocycle on a higher Lie algebra $\mathfrak{g}$ is equivalently a homomorphism of higher Lie algebras of the form
and the higher Lie algebra extension $\widehat{\mathfrak{g}}$ that this classifies is equivalently the homotopy fiber
(Fiorenza-Sati-Schreiber 13, prop. 3.5)
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This implies that from any higher Lie algebra $\mathfrak{g}$ there emanates a “bouquet” of consecutive higher extensions
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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).
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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.
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Classical physics is governed by the “principle of least action”
(extremal action, really)
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This is formalized in variational calculus as follows.
Let
$\Sigma$ be a smooth manifold of dimension $p+1$,
called the spacetime or worldvolume;
$E \to \Sigma$ be a bundle (smooth map of manifolds),
called the field bundle
so that
a field configuration is a section $\phi \in \Gamma_\Sigma(E)$
Write
This means that if
is a local coordinate chart of $E$, then the corresponding coordinate chart of the jet bundle is
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A local Lagrangian field theory is defined by a local Lagrangian density:
a $(p+1)$-form $L$ on $\Sigma$ which is a function of the fields and some finite number of its partial derivatives
This is a horizontal differential $(p+1)$-form on the jet bundle of $E$.
Here the horizontal differential $d_H$ is the “total derivative” given in the above coordinate chart by
This defines a chain complex called the horizontal de Rham complex
Define the vertical differential on $J^\infty E$ to be
This induces a chain complex of chain complexes, a double complex called the variational bicomplex
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Say that the source forms are the wedge products of horizontal $(p+1)$-forms with vertical derivatives of functions of just the fields
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Euler partial integration:
Source forms are a model for the top horizontal vertical 1-forms modulo horizontal divergences:
Hence the full de Rham differential of any local Lagrangian density $L$ uniquely splits into a source-form and a horizontally exact term
This $\delta_v$ is the variational derivative.
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The Euler-Lagrange equations of motion on field configurations $\phi$ is the partial differential equation
hence
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One checks immediately that
Hence the horizontal de Rham complex continues to a longer chain complex
In fact it continues further:
This is called the Euler-Lagrange complex.
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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).
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This means for instance that
any given equation of motion $EL \in \Omega^{n+1,1}_S(E)$
is locally a “principle of extremal action”
precisely if the Helmholtz operator annihilates it $d_V EL = 0$.
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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!
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This is due to
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For many field theories of interest… | Example $\;\;$ |
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(1) …the Lagrangian density is $\;\phantom{(3)}\;\;\;$ 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 $\;\phantom{(3)}\;\;\;$ a higher connection on a higher bundle. | Wilson loop, Wilson surface, etc., |
(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) |
$\;\phantom{(3)}\;$ …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) |
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We now explain these items in turn.
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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).
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This means that
there exists
on each of the charts,
such that
they have a globally well defined variational derivative
in that
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Example: charged particle
Let
$\Sigma = \mathbb{R}$ (the “worldline”)
$(X,g)$ a 4d spacetime
$E \coloneqq \Sigma \times X$ the field bundle,
so that a field configuration is equivalently a smooth function
i.e. a trajectory of a particle in $X$.
$X$ carrying an electromagnetic field given by a differential 2-form
called the Faraday tensor, which in a local coordinate chart
defines
an electric field $\vec E = \left[ \array{E_1 \\ E_2 \\ E_3} \right]$ and magnetic field $\vec B = \left[ \array{B_1 \\ B_2 \\ B_3} \right]$
via
Then
there exists
an open cover $\{ U_i \to X\}$
a collection of 1-forms $A_i \in \Omega^1(U_i)$
such that
$\omega|_{U_i} = d_{dR} A_i$
and the locally defined Lagrangian density for the charged particle propagating in this background is
The Euler-Lagrange equations for this are the geodesic equations modified by the Lorentz force
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Hence there is an obvious question:
When may we find a globally defined Lagrangian for the charged particle?
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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…
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A collection of chartwise defined Lagrangians is not enough data to make sense of a globally defined exponentiated action functional
Here Planck's constant $\hbar$ coordinatizes the circle group $U(1)$.
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Instead, making sense of this requires to choose
on each $(n+1)$-fold intersection of charts
an order-$n$ higher homotopy
between the local Lagrangian densities:
and so on.
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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:
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Technical aside:
The statement here is that
Consistent assignments of phases in $U(1)$ to field configurations are Cheeger-Simons differential characters
these are classified by ordinary differential cohomology,
which is computed by hyper Cech cohomology with coefficients in the Deligne complex.
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The higher bundle underlying a local Lagrangian
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Underlying any such homotopically enhanced locally variational field theory $\mathbf{L}$ is a Cech cocycle representing an element
in the integral cohomology of the field bundle
This is given by the evident chain homomorphism
Here the chain complex
is a Lie (p+1)-group.
A cocycle as above classifies a higher principal bundle as above with higher structure group $\mathbf{B}^p U(1)$.
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Fact.
The obstruction to there being a globally defined Lagrangian density is the trivialization of this higher principal bundle, hence the vanishing
of its characteristic class.
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Example
In the example of the charged particle above, this is the case precisely if, in physics speak:
there is vanishing magnetic charge enclosed by the spacetime,
hence if the “$U(1)$-instanton number” vanishes
This is the modern incarnation of the old Dirac charge quantization phenomenon (Dirac 31).
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For exposition see: Higher field bundles for gauge fields.
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Given a locally variational Lagrangian gerbe $\mathbf{L}$ as above, then a symmetry is a flow
along some vertical vector field
such that it preserves the Lagrangian density $\mathbf{L}$ in that the transformed Lagrangian density gerbe
coincides with $\mathbf{L}$.
In the spirit of homotopy theory, we need to choose a homotopy for each $t$
An infinitesimal symmetry is obtained by differentiating this with respect to $t$. This yields
where $\mathcal{L}$ denotes the Lie derivative.
By applying
Euler partial integration (above)
this becomes
Here $J$ 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.
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But we could have made a different choice of homotopy
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
(Khavkine-Schreiber, based on Fiorenza-Rogers-Schreiber 14)
Passing to chain homology, this gives ordinary Lie algebra extension.
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).)
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Example
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.
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To understand what is going on here, we now explain how these fundamental $p$-branes work:
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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.
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These have two kinds of applications
they describe the propagation of strings and generally of super p-branes in string theory/M-theory
(Henneaux-Mezincescu 85, Azcarraga-Izquierdo 95, Fiorenza-Sati-Schreiber 14
based on Green-Schwarz 84, Achúcarro-Evans-Townsend-Wiltshire 87).
they are argued to describe symmetry protected topological order in solid state physics
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We close by discussing the archetypical case of a string propagating on a Lie group manifold.
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The fundamental string in a group manifold
Let $G$ be compact simply connected simple Lie group.
Such as
the spin group $Spin(d)$ for $d \geq 3$
the special unitary group $SU(n)$ for $n \geq 2$.
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We consider a “charged string” propagating on $G$, in direct analogy to the charged particle on $X$ from above.
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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
we obtain the closed left invariant differential form.
Pulling this back to the jet bundle along
and projecting to the horizontal component yields the “Euler-Lagrane form for the higher Lorentz force”
By the above obstruction theory we find:
there does not exist a globally defined Lagrangian density $L_{WZW}$ with $\delta_v L_{WZW} = (\mu_3)|_H$.
Instead: there exists a $U(1)$ bundle gerbe
such that
This is the jet bundle version (Khavkine-Schreiber) of a statement that goes back to (Gawedzki 87). For review see (Schweigert-Waldorf 07).
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Theorem The higher Noether current 2-group of this $\mathbf{L}_{WZW}$ sits in a homotopy fiber sequence of smooth infinity-groups
(Fiorenza-Rogers-Schreiber 13)
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This identifies (Schreiber 13)
the higher Noether current group $Cur(\mathbf{L}_{WZW})$ 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
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Green-Schwarz anomaly cancellation
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Now let
be a $G$-principal bundle over some $X$.
We may ask for a Lagrangian that restricts on each fiber to the one above (a “parameterized WZW model”).
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Theorem (Distler-Sharpe 07, Schreiber 13) the obstruction to the parameterized WZW model on $P$ to exist is equivalently
a lift of $P$ to a principal 2-bundle (as above) for the string 2-group (from above)
vanishing of the canonical 4-class of $P$.
If $G = Spin(d) \times SU(n)^{op}$ then this 4-class is
The vanishing of this class is the Green-Schwarz anomaly cancellation in heterotic string theory.
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This is only the first step in a rich story.
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For more on this see the lecture notes
Pointers to original references are given in the above text. Here we just list surveys.
Jim Stasheff, Higher homotopy structures, then and now, talk at Martin Markl et al. (org.) Opening conference of the program on Higher Structures in Geometry and Physics, MPI Bonn, Jan. 2016 (pdf)
Urs Schreiber, Higher Prequantum Geometry, (arXiv:1601.05956, v2, talk recording) chapter in Gabriel Catren, Mathieu Anel (eds.) New Spaces for Mathematics and Physics
See also
Higher Structures in M-Theory 2018, Durham Symposium 13-17 August 2018
Christian Saemann, Lectures on Higher Structures in M-Theory (arXiv:1609.09815)
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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 August 8, 2020 at 17:59:00. See the history of this page for a list of all contributions to it.