The n-Category Café

All the main ideas and structures of relevance here are nicely explained in

P.O. Kazinski, S.L. Lyakhovich, A.A. Sharapov
Lagrange structure and quantization
hep-th/0506093

Remarkably, all this work is mainly motivated by the desire to understand non-Lagrangian physical systems, those for which no action functional exists.

Famous examples of such theories which keep vexing physicists are theories of self-dual differential forms. These theories sit at very interesting points in the space of all field theories, for various reasons.

I once had reported on other very interesting work on these theories in

Freed, Moore, Segal on $p$-Form Gauge Theory, I

Freed, Moore, Segal on $p$-Form Gauge Theory, II

Gomi on Chern-Simons terms and central extensions of gerbe gauge groups.

For one, while these theories themselves are lacking a direct Langrangian description, they may be holographically related to theories in one dimension higher which do have a Lagrangian description. This issue is treated in

S.L. Lyakhovich, A.A. Sharapov
Quantizing non-Lagrangian gauge theories: an augmentation method
hep-th/0612086.

Before I go on, here is
A very simple observation which I consider potentially helpful for understanding all three of i) non-Lagrangian mechanics, ii) BV-quantization and iii) holography:

It seems to me that the followng very simple observation is a good way to get an idea of how non-Lagrangian theories and holography fit into the big frame of things – and in fact also to motivate the description of mechanics by Lyakhovich and collaborators:

Suppose we have an $n$-particle (a particle, a string, a membrane, etc.) propagating on some space $X$ where it is charged under an $n$-bundle with connection. With the worldvolume of the $n$-particle – an $n$-dimensional space – denoted $$Y$$ we get, using the connection on the $n$-bundle, a way to assign a number to each map $$f : Y \to X \,,$$ the corresponding holonomy. Notice that the action functional of the theory consists just of this assignment, together with one extra term – the kinetic contribution, which I will gracefully ignore here.

In other words, then: by transgressing an $n$-bundle on $X$ along $$\array{ [Y,X] \times Y &\stackrel{\mathrm{ev}}{\to} & X \\ \downarrow^p \\ [Y,X] }$$ to the space of histories, $M = [Y,X] = X^Y$, we get a 0-bundle with connection on the space of “field configurations” $X^Y$ (think of painting doorknobs, if that helps).

This 0-bundle with connection, otherwise known as a function, is (up to the kinetic term which, recall, I am ignoring here – officially for pedagogical and inofficially for deeper but secret reasons) is our action functional $$S : X^Y \to \mathbb{R}$$ - a 0-form.

But now recall all the trouble I was lately going through of understanding $n$-curvature. Recall that the important lesson of these struggles is that, fundamentally, we should better think of our $n$-bundle with connection in terms of its curvature $(n+1)$-bundle: the $n$-bundle itself may have problems with existing the way we need it.

Assuming you believe me in that, we find a slight modification to the above story of the action functional: by transgressing the background gauge field now in its incarnation as a curvature $(n+1)$-bundle, we find not an action 0-form, but instead a 1-form, locally, on the space of fields.

This is not entirely unfamiliar: already in the more familiar case where we do have an action function $S$ proper, studying the classical mechanics of this system will make us want to look at its differential $$d S \,.$$ (John Baez talked about this in his lecture Quantization and Cohomology (Week 19)).

The difference – now that we have transgressed not the true holonomy action functional itself, but rather its $(n+1)$-curvature – is that in general this 1-form, while still closed might not be exact.

If it is not, we have a non-Lagrangian system.

But at the same time, this might make us want to suspect that if we realized $Y$ as the $n$-dimensional boundary of something $(n+1)$-dimensional, we might get a theory of an $(n+1)$-particle which does come from a Lagrangian – and which essentially knows everything about the $n$-particle we started with. That’s holography in QFT.

Lyakhovich et al.’s description of mechanics

Often, for understanding something deeply, it turns out to be helpful to consider the most complicated and involved case in which this something may occur. The reason is that, while the simpler cases may be easier to handle, the simplicity is likely to lead to structural degeneracies. And these may hide the full underlying picture.

Precisely this is happening here. By insisting on being able to handle the often disregarded special case of non-Lagrangian mechanics, Lyakhovich and collabroators find a rather deep formulation of mechanics in general.

Since it is very late already, I won’t go through the details, not right now at least, but instead jump immediately to the structure they come up with after a careful analysis of the situation. But the inclined reader may profit from the nice read offered by pages 4 - 13 of Lagrange structure and quantization.

What is so very nice about their description is that it so neatly highlights the $n$-categorical structure of configuration space which we were chatting about with here:

let $$M = X^Y$$ be the space of physical field configurations “over time” (also known as the space of histories). Then Lyakhovich et al. show that, in general, there is a complex of vector bundles $$0 \to E_{-k} \to \cdots \to E_{-1} \stackrel{R}{\to} T M \stackrel{J}{\to} E \stackrel{Z}{\to}$$

over $M$ associated with this space of fields, such that on the exterior algebra of sections of the graded direct sum of these spaces, we have an odd vector field $Q$, whose nilpotency $$Q^2 = 0$$ encodes entirely what they call a Lagrange structure on $M$ (definition 3.1 in their article).

If you like classical mechanics, you might enjoy going through their section 2 to see a nice description to how all the different components of $Q$ correspond to the familiar structures:

- the equations of motion

- the Noether identities

- the gauge transformation .

For the moment, being tired, I will just point out the nice $n$-categorical interpretation which we mentioned before, using the equivalence between differential graded structures and Lie $n$-algebroids that I talked about in On BV Quantization, Part II:

The negatively graded bundles here contain the information about gauge transformations: these are to be thought of as encoding tangent spaces to spaces of $k$-morphisms of order $k$ isomrophisms of physical histories.

The positively graded bundles here – starting with $E$ itself which encodes the constraint surface also known as the physical shell – encode the information we want to divide out in the first place, such as to restrict to only the physically sensible part of $M$.

I should be less vague here, eventually. It’s a beautiful story. But for the time being, you might just notice that this nicely harmonizes with what David Ben-Zvi said here about “stacky and dg-directions” – also here.

Holography

In the thread Making AdS/CFT Precise we recently talked about the true content of the holographic principle in quantum field theory.

The main point is that states of a higher dimensional theory may correspond to correlators in a lower dimensional theory. Analogously, higher dimensional configurations correspond to lower dimensional sources.

Again, Lyakhovich and collaborators find that the right waty to think about this situation is suggested by the problems one runs into when trying to understand non-Lagrangian theories.

Here is this very simple – but in fact quite profound – motivating observation, taken from section 2 of

S.L. Lyakhovich, A.A. Sharapov
Quantization of Donaldson-Uhlenbeck-Yau theory
0705.1871 .

The usual path integral as one imagines it $$\int [d\phi] \exp(\frac{i}{\hbar} S(\phi))$$ crucially involves the action functional $S$, clearly. Notice that the Fourier-transform of the exponentiated action $$\Psi(\phi) := \exp(\frac{i}{\hbar}S(\hi))$$ is the generating functional $Z(J)$ for the correlators, mentioned above $$Z(J) := \int [d\phi] \exp(\frac{i}{\hbar}(S(\phi) - J \phi)) \,.$$

This is precisely what is missing in a non-Lagrangian theory. But there are objects of interest whose definition does make sense even without the existence of the action functional.

One may observe that the Schwinger-Dyson equation which is (obviously) satisfied by this (generating functional for the) correlator $$\left( \frac{\partial S}{\partial \phi} + i \hbar \frac{\partial }{\partial \phi} \right) \Psi[\phi] = 0$$ or, equivalently, $$\left( \frac{\partial S}{\partial \phi}( i \hbar \frac{\partial}{\partial J} - J) \right) Z[J] = 0$$ has quite a resemblance with the quantum equation of motion for a theory for which $\Psi$ is not a (generating functional for the) correlator – but a state.

While that is indeed a hint towards holography, let that be as it may for the moment. Quite independently, one may ask under which conditions we may solve the Schwinger-Dyson equation in a non-Lagrangian theory. As Lyakhovich and Sharapov write on p. 5

it can be a problem to explicitly derive the probability amplitude from the Schwinger-Dyson equation, especially in nonlinear field theories. In many interesting cases the amplitude $\Psi$ is given by an essentially nonlocal functional. More precisely, it can be impossible to represent as a (smooth) function of any local functional of fields (by analogy with the Feynman probability amplitude $e^{\frac{i}{\hbar}S(\phi)}$ in a local theory with action $S$) even though the Schwinger-Dyson equations are local. Fortunately, whatever the field equations and Lagrange anchor may be, it is always possible to write down a path-integral representation for in terms of some enveloping Lagrangian theory.

By now, two such representations are known. The first one, proposed in [1], exploits the equivalence between the original dynamical system described by the classical equations of motion $T_a = 0$ and the Lagrangian theory with action $$S[\phi,J,\lambda] = \int_0^1 d t (\dot \phi^i J_i - \lambda^a \Theta_a) \,.$$ The latter can be regarded as a Hamiltonian action of topological field theory on the space-time with one more (compact) dimension $t \on [0,1]$. The solution to the SD equation can be formally represented by the path integral $$\Psi(\phi) = \int [d\phi] [d J] [d\lambda] \exp(\frac{i}{\hbar} S[\phi,J,\lambda]) \,,$$ […]. In [3], we used such a representation to perform a covariant quantization of the chiral bosons in $d = 4 n + 2$ dimensions in terms of the $(4 n + 3)$-dimensional Chern-Simons theory

This is the beginning of a rather interesting story. But i’ll stop here for tonight.