One noteworthy aspect of the “higher” quantization in terms not of symplectic 2-forms but of higher forms (as in Rovelli’s work) is - that it reduces to ordinary quantization after suitable “transgression”.

For instance, quantizing the 2-particle (the string) can lead one to something described by a 3-form. But as we pass from target space to its loop space (for closed strings) the 3-form “transgresses” to a 2-form, and we are left with ordinary (albeit infinite dimensional) mechanics on loop space.

For closed strings this is all very easy to see. For open strings it becomes more subtle. For open/closed membranes it becomes even harder.

People thinking about what is called “extended” QFT are trying to figure out what a $d$-dimensional QFT would assign to a $(n \lt d)$-dimensional piece of parameter space, depending on whether that has a boundary or not. The result are tables of the sort # that Hopkins, Freed et al. write down for Chern-Simons theory, or Simon Willerton for Dijkgraaf-Witten theory #.

The situation might be described as follows:

Let $d^n$ be a category addressed as *parameter space*

(1)$\mathrm{par} = d^n
\,.$

For the point particle this would be just the discrete category on a single element. For the closed string it might be $\Sigma(\mathbb{Z})$. Or $\{a \to b\}$ for the open string. For the membranbe it might be some 2-category.

Let $P$ be a ($n$-)category addressed as *target space*. For the point particle this might be some category of paths in a space $X$. For $G$-Dijkgraaf-Witten theory this would be $\Sigma(G)$. For Chern-Simons theory it might be something like $\mathrm{INN}(\mathrm{String}_G)$.

Now the functor category

(2)$[d^n,P]$

is our *configuration space*

(3)$\mathrm{conf} := [d^n,P]
\,.$

On target space there is usually some $n$-bundle with connection under which our $d^n$ is “charged”. Denote the corresponding parallel transport by

(4)$\mathrm{tra} : P \to n\mathrm{Vect}
\,.$

For the point particle this might be an ordinary bundle with connection. For the string it might be a gerbe with connection. For Dijkgraaf-Witten it would be a pseudo-rep of $G$, for Chern-Simons a rep of $\mathrm{INN}(\mathrm{String}_G)$.

We should call $n\mathrm{Vect}$ the *space of phases* (or the space of amplitudes)

(5)$\mathrm{phas} := n\mathrm{Vect}
\,.$

By what might be called “general transgression”, the parallel transport on target space gives rise to one on configuration space, simply by sending any configuration

(6)$d^n \to P$

to

(7)$d^n \to P \stackrel{\mathrm{tra}}{\to} n\mathrm{Vect}
\,,$

so that we get an assignment

(8)$[d^n,P] \stackrel{\mathrm{tra}_*}{\to}
[d^n,n\mathrm{Vect}]
\,.$

A *state* of the system is a section of the $n$-bundle on configuration space, which again is # a morphism

(9)$\array{
& \nearrow \searrow^{1}
\\
[d^n,P]
&e \Downarrow &
[d^n,n\mathrm{Vect}]
\\
&
\searrow \nearrow_{\mathrm{tra}_*}
}
\,.$

The *space of sections* is hence the category

(10)$\mathrm{sect}
=
1 \stackrel{\mathrm{Hom}(1,\mathrm{tra}_*)}{\to} \mathrm{tra}_*$

Fine. Now the question is: how to we get an extended QFT from this data? In other words, how do we assign (transgressed) spaces of sections to parts of parameter space

(11)$QFT : d^n \to (?)$

in a natural way?

There is one *canonical* way to do so, which uses just the given data and some abstract nonsense.

Notice that our space of sections $\mathrm{sect}$ is a subcategory of the space of all morphism from configuration space to $[d^n,\mathrm{phas}]$:

(12)$\mathrm{sect}
\stackrel{\subset}{\to}
\mathrm{Hom}(conf,[d^n,\mathrm{phas}])
\,,$

hence it is an element of

(13)$\mathrm{Hom}
(\mathrm{sect},\mathrm{Hom}(
\mathrm{conf},
\mathrm{Hom}(\mathrm{par},
\mathrm{phas})
))
\,.$

This is the abstract nonsense for a very familiar statement:

A state is something

that assigns to a configuration something

that assigns to a parameter

an amplitude.

You might want to think of the point particle, where there is no parameter. Then compare this to the string, for instance, where a state is something that assigns an amplitude to every mode of a given configuration of the string. The mode number here would be the parameter.

Technically slightly trickier but conceptually much more transparent is the case where we take the parameter on the string to be literally the parameter $\sigma$ which parameterizes the string as a map

(14)$\sigma \mapsto \gamma(\sigma)
\,.$

In this example the element in the Hom-space

(15)$\mathrm{Hom}
(\mathrm{sect},\mathrm{Hom}(
\mathrm{conf},
\mathrm{Hom}(\mathrm{par},
\mathrm{phas})
))$

that we are talking about would schematically read

(16)$|\psi\rangle
\mapsto
(
\langle \gamma|
\mapsto
(
\sigma
\mapsto
\langle \gamma(\sigma)| \psi\rangle
)
)
\,.$

A state is something that sends a configuration to something that sends a parameter to an amplitude.

The nice thing about the above setup is that it interprets this concept $n$-functorially, with all data assigned to the parameter “space” category taken care of.

In particular, realizing our space of states as an element in

(17)$\mathrm{Hom}
(\mathrm{sect},\mathrm{Hom}(
\mathrm{conf},
\mathrm{Hom}(\mathrm{par},
\mathrm{phas})
))$

allows us to canonically extract a ($n$-)functor which should represent the associated extended QFT.

(18)$QFT : \mathrm{par} \to (?)
\,.$

We simply regard the image of our space of states under the canonical equivalence

(19)$\begin{aligned}
&
\mathrm{Hom}
(\mathrm{sect},\mathrm{Hom}(
\mathrm{conf},
\mathrm{Hom}(\mathrm{par},
\mathrm{phas})
))
\\
\simeq&
\mathrm{Hom}
(\mathrm{par},\mathrm{Hom}(
\mathrm{sect},
\mathrm{Hom}(
\mathrm{conf},
\mathrm{phas})
))
\end{aligned}$

and thus obtain an ($n$-)functor

(20)$\mathrm{par}
\to
[\mathrm{sect},
[\mathrm{conf},\mathrm{phas}]]
\,.$

It assigns to objects of parameter space an image of the space of states in $[\mathrm{conf},\mathrm{phas}]$. It assigns to edges in parameter space natural transformations between these images, and so on.

To convince oneself that this functor is really what we are looking for, one should look at a couple of examples.

In particular, applying this to Dijkgraaf-Witten and Chern-Simons theory, we could check if that functor on the parameter space of the 3-particle (membrane) yields state spaces of an extended 3-d QFT which coincide with correlators of a 2-d QFT.

I have one consistency check for Dijkgraaf-Witten, which however might be too top secret to talk about here.

I have also some ideas about the CS-case. But I need to work them out.

## Re: Quantization and Cohomology (Week 6)

One noteworthy aspect of the “higher” quantization in terms not of symplectic 2-forms but of higher forms (as in Rovelli’s work) is - that it reduces to ordinary quantization after suitable “transgression”.

For instance, quantizing the 2-particle (the string) can lead one to something described by a 3-form. But as we pass from target space to its loop space (for closed strings) the 3-form “transgresses” to a 2-form, and we are left with ordinary (albeit infinite dimensional) mechanics on loop space.

For closed strings this is all very easy to see. For open strings it becomes more subtle. For open/closed membranes it becomes even harder.

People thinking about what is called “extended” QFT are trying to figure out what a $d$-dimensional QFT would assign to a $(n \lt d)$-dimensional piece of parameter space, depending on whether that has a boundary or not. The result are tables of the sort # that Hopkins, Freed et al. write down for Chern-Simons theory, or Simon Willerton for Dijkgraaf-Witten theory #.

The situation might be described as follows:

Let $d^n$ be a category addressed as

parameter spaceFor the point particle this would be just the discrete category on a single element. For the closed string it might be $\Sigma(\mathbb{Z})$. Or $\{a \to b\}$ for the open string. For the membranbe it might be some 2-category.

Let $P$ be a ($n$-)category addressed as

target space. For the point particle this might be some category of paths in a space $X$. For $G$-Dijkgraaf-Witten theory this would be $\Sigma(G)$. For Chern-Simons theory it might be something like $\mathrm{INN}(\mathrm{String}_G)$.Now the functor category

is our

configuration spaceOn target space there is usually some $n$-bundle with connection under which our $d^n$ is “charged”. Denote the corresponding parallel transport by

For the point particle this might be an ordinary bundle with connection. For the string it might be a gerbe with connection. For Dijkgraaf-Witten it would be a pseudo-rep of $G$, for Chern-Simons a rep of $\mathrm{INN}(\mathrm{String}_G)$.

We should call $n\mathrm{Vect}$ the

space of phases(or the space of amplitudes)By what might be called “general transgression”, the parallel transport on target space gives rise to one on configuration space, simply by sending any configuration

to

so that we get an assignment

A

stateof the system is a section of the $n$-bundle on configuration space, which again is # a morphismThe

space of sectionsis hence the categoryFine. Now the question is: how to we get an extended QFT from this data? In other words, how do we assign (transgressed) spaces of sections to parts of parameter space

in a natural way?

There is one

canonicalway to do so, which uses just the given data and some abstract nonsense.Notice that our space of sections $\mathrm{sect}$ is a subcategory of the space of all morphism from configuration space to $[d^n,\mathrm{phas}]$:

hence it is an element of

This is the abstract nonsense for a very familiar statement:

A state is something

that assigns to a configuration something

that assigns to a parameter

an amplitude.

You might want to think of the point particle, where there is no parameter. Then compare this to the string, for instance, where a state is something that assigns an amplitude to every mode of a given configuration of the string. The mode number here would be the parameter.

Technically slightly trickier but conceptually much more transparent is the case where we take the parameter on the string to be literally the parameter $\sigma$ which parameterizes the string as a map

In this example the element in the Hom-space

that we are talking about would schematically read

A state is something that sends a configuration to something that sends a parameter to an amplitude.

The nice thing about the above setup is that it interprets this concept $n$-functorially, with all data assigned to the parameter “space” category taken care of.

In particular, realizing our space of states as an element in

allows us to canonically extract a ($n$-)functor which should represent the associated extended QFT.

We simply regard the image of our space of states under the canonical equivalence

and thus obtain an ($n$-)functor

It assigns to objects of parameter space an image of the space of states in $[\mathrm{conf},\mathrm{phas}]$. It assigns to edges in parameter space natural transformations between these images, and so on.

To convince oneself that this functor is really what we are looking for, one should look at a couple of examples.

In particular, applying this to Dijkgraaf-Witten and Chern-Simons theory, we could check if that functor on the parameter space of the 3-particle (membrane) yields state spaces of an extended 3-d QFT which coincide with correlators of a 2-d QFT.

I have one consistency check for Dijkgraaf-Witten, which however might be too top secret to talk about here.

I have also some ideas about the CS-case. But I need to work them out.