Week 1 (Oct. 3) - How the dynamics of p-branes resembles the statics of (p+1)-branes.

Cool. That touches upon many things that I have been thinking about lately.

Let me propose this general way of looking at the situation:

Let $P_n(X)$ be our geometric $n$-category of $n$-paths in **target space** $X$.

Let $d^{p-1}$ be a **$p$-particle**. This means: let $d^{p-1}$ be the $(p-1)$-category which encodes the internal structure of a $p$-particle.

For $p=1$ we usually choose

(1)$d^0 = \{\bullet\}$

to be the one-object 0-category.

For $p=2$ we usually choose

(2)$d^1 = \{0 \to \pi\}$

to be the poset of the oriented interval $[0,\pi]$ for the “open string”, or, similarly for the “closed string”.

And so on.

The **configuration space** of the $p$-particle with target space $P_n(X)$ is the functor $n$-category

(3)$[d^{p-1},P_n(X)]
\,.$

Objects in here are images of the $p$-particle in target space.

Morphisms in here are worldvolumes traced out by the $p$-particle.

By the
nature of pseudonatural transformations, this automatically ensures that the endpoints of the $p$-particle need not be nailed down. In fact, these worldvolumes are automatically (by the logic of $n$-functor categories) **cobordisms** cobounding the source and target $p$-particle image.

There are also higher morphisms in $[d^{p-1},P_n(X)]$, which encode various gauge invariances (reparameterizations and generalizations thereof) of the worldvolume of a $p$-particle.

Next, we want to associate **phases** to morphisms in $[d^{p-1},P_n(X)]$.

To get that, let

(4)$\mathrm{tra} : P_n(X) \to T$

be an $n$-bundle with connection on $X$, under which our $p$-particle shall be charged.

(Take the trivial bundle with trivial connection if in your application the $p$-particle is not charged.)

Composing with $\mathrm{tra}$ provides us with an $n$-functor

(5)$\mathrm{tra}_* :
[d^{p-1},P_n(X)]
\to
[d^{p-1},T]
\,.$

This functor reads in a $p$-particle in target space and spits out a “fiber” over it.

It reads in a worldvolume of a $p$-particle and spits out the “phase” associated to this due to the charge of the $p$-particle.

Now let $\mathrm{triv}$ be a transport functor which factors through the category $\{\bullet\}$ with a single object and no nontrivial morphisms.

(6)$\mathrm{triv} : P_n(X) \to \{\bullet\} \to T
\,.$

The **space of states** of our $p$-particle is the $n$-category

(7)$[\mathrm{triv}_*, \mathrm{tra}_*]
\,.$

(Recall that $\mathrm{tra}_*$ was the functor from configuration space $[d^{p-1},P_n(X)]$ to phases $[d^{p-1},T]$.)

An object in

(8)$[\mathrm{triv}_*, \mathrm{tra}_*]$

is a **section** of the bundle of “fibers” over the configuration space of the $p$-particle.

In order to be able to study the quantum mechanics of our $p$-particle charged under $\mathrm{tra}$, we need to assume that the category of generalized phases

(9)$[d^{p-1},T]$

is a category *with duals*.

If that is the case, we can form

(10)$(\mathrm{tra},\mathrm{tra})
:
[d^{p-1},P_n(X)]
\stackrel{
\mathrm{tra}_*^\dagger \times
\mathrm{tra}_*
}{\to}
[d^{p-1},T]^\mathrm{op} \times
[d^{p-1},T]
\stackrel{\mathrm{Hom}}{\to}
\tilde T
\,.$

(Here $\tilde T$ is whatever the $\mathrm{Hom}$ takes values in, depending on what $[d^{n-1},T]$ is enriched over.)

Similarly for $\mathrm{triv}$.

The point of this is that given any two sections

(11)$\begin{aligned}
e_1 & : \mathrm{triv}_* \to \mathrm{tra}_*
\\
e_2 & : \mathrm{triv}_* \to \mathrm{tra}_*
\end{aligned}$

we get a morphism

(12)$(\mathrm{triv}_*,\mathrm{triv}_*)
\stackrel{(e_1,e_2)}{\to}
(\mathrm{tra}_*,\mathrm{tra}_*)
\,.$

On objects, this encodes the **scalar product** on the space of sections.

On morphisms, this encodes a scalar product on covariant derivatives

(13)$(d_\mathrm{tra} e_1, d_\mathrm{tra} e_2)
\,.$

Notice that this is the $e_1$, $e_2$ matrix element of the **Hamiltonian**

(14)$\Delta_\mathrm{tra}
=
d_{\mathrm{tra}_*}^\dagger
d_{\mathrm{tra}_*}$

of the charged $p$-particle.

I have more details of this discussion scattered on notes flying around on a couple of tables of the $n$-Café, for instance here and here.

I think I have checked that for ordinary charged 1-particles, the above prescription indeed reproduces the ordinary quantization of the particle.

I am in the process of working out what the above says for strings charged under an abelian gerbe. I think everything works as expected, but this is work in progress.

One further aspect one should be able to discuss along these lines is interaction of $p$-particles for $p \gt 1$, by passing from $d^{p-1}$ to suitable interaction diagrams. For instance for the triangle one gets the multiplicative structure on the space of sections over the configuration space of the open string, as described above.

## Re: Quantization and Cohomology (Week 1)

Cool. That touches upon many things that I have been thinking about lately.

Let me propose this general way of looking at the situation:

Let $P_n(X)$ be our geometric $n$-category of $n$-paths in

target space$X$.Let $d^{p-1}$ be a

$p$-particle. This means: let $d^{p-1}$ be the $(p-1)$-category which encodes the internal structure of a $p$-particle.For $p=1$ we usually choose

to be the one-object 0-category.

For $p=2$ we usually choose

to be the poset of the oriented interval $[0,\pi]$ for the “open string”, or, similarly for the “closed string”.

And so on.

The

configuration spaceof the $p$-particle with target space $P_n(X)$ is the functor $n$-categoryObjects in here are images of the $p$-particle in target space.

Morphisms in here are worldvolumes traced out by the $p$-particle.

By the nature of pseudonatural transformations, this automatically ensures that the endpoints of the $p$-particle need not be nailed down. In fact, these worldvolumes are automatically (by the logic of $n$-functor categories)

cobordismscobounding the source and target $p$-particle image.There are also higher morphisms in $[d^{p-1},P_n(X)]$, which encode various gauge invariances (reparameterizations and generalizations thereof) of the worldvolume of a $p$-particle.

Next, we want to associate

phasesto morphisms in $[d^{p-1},P_n(X)]$.To get that, let

be an $n$-bundle with connection on $X$, under which our $p$-particle shall be charged.

(Take the trivial bundle with trivial connection if in your application the $p$-particle is not charged.)

Composing with $\mathrm{tra}$ provides us with an $n$-functor

This functor reads in a $p$-particle in target space and spits out a “fiber” over it.

It reads in a worldvolume of a $p$-particle and spits out the “phase” associated to this due to the charge of the $p$-particle.

Now let $\mathrm{triv}$ be a transport functor which factors through the category $\{\bullet\}$ with a single object and no nontrivial morphisms.

The

space of statesof our $p$-particle is the $n$-category(Recall that $\mathrm{tra}_*$ was the functor from configuration space $[d^{p-1},P_n(X)]$ to phases $[d^{p-1},T]$.)

An object in

is a

sectionof the bundle of “fibers” over the configuration space of the $p$-particle.In order to be able to study the quantum mechanics of our $p$-particle charged under $\mathrm{tra}$, we need to assume that the category of generalized phases

is a category

with duals.If that is the case, we can form

(Here $\tilde T$ is whatever the $\mathrm{Hom}$ takes values in, depending on what $[d^{n-1},T]$ is enriched over.)

Similarly for $\mathrm{triv}$.

The point of this is that given any two sections

we get a morphism

On objects, this encodes the

scalar producton the space of sections.On morphisms, this encodes a scalar product on covariant derivatives

Notice that this is the $e_1$, $e_2$ matrix element of the

Hamiltonianof the charged $p$-particle.

I have more details of this discussion scattered on notes flying around on a couple of tables of the $n$-Café, for instance here and here.

I think I have checked that for ordinary charged 1-particles, the above prescription indeed reproduces the ordinary quantization of the particle.

I am in the process of working out what the above says for strings charged under an abelian gerbe. I think everything works as expected, but this is work in progress.

One further aspect one should be able to discuss along these lines is interaction of $p$-particles for $p \gt 1$, by passing from $d^{p-1}$ to suitable interaction diagrams. For instance for the triangle one gets the multiplicative structure on the space of sections over the configuration space of the open string, as described above.