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August 22, 2007

Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

Posted by Urs Schreiber

Today at ESI in Vienna, S. Lyakhovich kindly pointed me to lots of his work. While most everybody was gone hiking, I spent the afternoon reading his articles.

These all develop two main threads:

A) A very clear-sighted description of classical and quantum, Lagrangian and Hamiltonian, gauge and constrained mechanics, closely related to, but going beyond, BV-formalism. Lyakhovich gives the nicest description of the BRST operator which I remember having seen.

B) Holography. While Lyakhovich doesn’t mention that word, he does discuss the underlying issues. In particular, this work does provide some nice insights into the relation between Chern-Simons theory in nn-dimensions and the coresponding theories on the (n1)(n-1)-dimensional boundary.

I’d expect that various aspects of this body of work will give the impression of familiarity to various experts. But I am struck by the clarity with wich these concepts are understood and ordered to a coherent whole.

Among the many talks we had related to BV-AKSZ and holographic phenomena, all apparently pointing to a deeper story waiting at our fingertips to be fully unraveled, the one on this work stands out as having the clear intent not only to plough through labyrinths of impressive formulas, but to actually increase the level of conceptual understanding.

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 pp-Form Gauge Theory, I

Freed, Moore, Segal on pp-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 nn-particle (a particle, a string, a membrane, etc.) propagating on some space XX where it is charged under an nn-bundle with connection. With the worldvolume of the nn-particle – an nn-dimensional space – denoted Y Y we get, using the connection on the nn-bundle, a way to assign a number to each map f:YX, 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 nn-bundle on XX along [Y,X]×Y ev X p [Y,X] \array{ [Y,X] \times Y &\stackrel{\mathrm{ev}}{\to} & X \\ \downarrow^p \\ [Y,X] } to the space of histories, M=[Y,X]=X YM = [Y,X] = X^Y, we get a 0-bundle with connection on the space of “field configurations” X YX^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 S : X^Y \to \mathbb{R} - a 0-form.

But now recall all the trouble I was lately going through of understanding nn-curvature. Recall that the important lesson of these struggles is that, fundamentally, we should better think of our nn-bundle with connection in terms of its curvature (n+1)(n+1)-bundle: the nn-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)(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 SS proper, studying the classical mechanics of this system will make us want to look at its differential dS. 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)(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 YY as the nn-dimensional boundary of something (n+1)(n+1)-dimensional, we might get a theory of an (n+1)(n+1)-particle which does come from a Lagrangian – and which essentially knows everything about the nn-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 nn-categorical structure of configuration space which we were chatting about with here:

let M=X Y 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 0E kE 1RTMJEZ 0 \to E_{-k} \to \cdots \to E_{-1} \stackrel{R}{\to} T M \stackrel{J}{\to} E \stackrel{Z}{\to}

over MM 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 QQ, whose nilpotency Q 2=0 Q^2 = 0 encodes entirely what they call a Lagrange structure on MM (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 QQ 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 nn-categorical interpretation which we mentioned before, using the equivalence between differential graded structures and Lie nn-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 kk-morphisms of order kk isomrophisms of physical histories.

The positively graded bundles here – starting with EE 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 MM.

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 [dϕ]exp(iS(ϕ)) \int [d\phi] \exp(\frac{i}{\hbar} S(\phi)) crucially involves the action functional SS, clearly. Notice that the Fourier-transform of the exponentiated action Ψ(ϕ):=exp(iS(hi)) \Psi(\phi) := \exp(\frac{i}{\hbar}S(\hi)) is the generating functional Z(J)Z(J) for the correlators, mentioned above Z(J):=[dϕ]exp(i(S(ϕ)Jϕ)). 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 (Sϕ+iϕ)Ψ[ϕ]=0 \left( \frac{\partial S}{\partial \phi} + i \hbar \frac{\partial }{\partial \phi} \right) \Psi[\phi] = 0 or, equivalently, (Sϕ(iJJ))Z[J]=0 \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 Ψ\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 Ψ\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 iS(ϕ)e^{\frac{i}{\hbar}S(\phi)} in a local theory with action SS) 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=0T_a = 0 and the Lagrangian theory with action S[ϕ,J,λ]= 0 1dt(ϕ˙ iJ iλ aΘ a). 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 ton[0,1]t \on [0,1]. The solution to the SD equation can be formally represented by the path integral Ψ(ϕ)=[dϕ][dJ][dλ]exp(iS[ϕ,J,λ]), \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=4n+2d = 4 n + 2 dimensions in terms of the (4n+3)(4 n + 3)-dimensional Chern-Simons theory

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

Posted at August 22, 2007 8:12 PM UTC

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15 Comments & 7 Trackbacks

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

Do any of these links discuss the homology of framed discs operad, from which BV algebras arise?

Posted by: Kea on August 23, 2007 5:17 AM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

Do any of these links discuss the homology of framed discs operad

No. None.

from which BV algebras arise?

Please use the chance to share with us this piece of knowledge and provide the link which I failed to provide! Thanks!

Posted by: Urs Schreiber on August 23, 2007 12:16 PM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

Try these:

I don’t understand this stuff very well!

More precisely:

If you take the space of multivector fields VV on a manifold MM, and think of VV equipped with its wedge product and Schouten bracket, you get the easiest example of a Gerstenhaber algebra.

A Gerstenhaber algebra is an associative supercommutative graded algebra AA together with a bracket of degree 1-1 which makes AA into a kind of ‘graded Poisson algebra with bracket of degree 1-1’. All the usual Poisson algebra axioms hold, but sprinkled with minus signs according to the usual conventions.

Now if your manifold MM is a Poisson manifold, then the space VV of multivector fields comes equipped with a differential given by taking the Schouten bracket with the Poisson bivector field ΠV\Pi \in V.

Axiomatizing this mess, we get the definition of a Batalin–Vilkovisky algebra: a Gerstenhaber algebra with differential that’s compatible with the other structure in a certain way.

There are also lots of Batalin–Vilkovisky algebras that don’t come from Poisson manifolds. But just like Poisson manifolds, we can still think of these as describing phase spaces in classical mechanics — in a clever algebraic way. And, that’s what BV quantization is all about: figuring out how to treat these Batalin–Vilkovisky algebras as classical phase spaces and quantize them!

All this makes some sense to me. But then it gets weird and mystical…

First, thanks to an old result of Fred Cohen, a Gerstenhaber algebra is the same as an algebra of the operad H *(D)H_*(D) — the homology of the little disks operad!

Did I just hear some of you say “Huh?”

Well, let me sketch what that means. The little disks operad is a gadget with a bunch of nn-ary operations corresponding to ways of sticking nn little disks in a big one. For each nn there’s a topological space of these nn-ary operations. Taking the homology of this topological space, we get a graded vector space. These are the nn-ary operations of the operad I’m calling H *(D)H_*(D). I explained this result by Cohen in week220, based on a nice lecture by Dev Sinha. Sinha eventually wrote a nice expository paper on this subject, which he submitted to the Bulletin of the American Mathematical Society — who said it was too expository.

Anyway, while I roughly follow how this works, I don’t understand the deep inner meaning. It seems amazing: there’s a mystical relation between ways of sticking little 2d disks in bigger ones, and operations you can do on the space of multivector fields on a manifold!

I don’t know if the connections to 2d topological and conformal field theory (described in the articles I cite) actually explain this mystical relation, or merely exploit it.

Now, as I said, a Batalin–Vilovisky algebra is a Gerstenhaber algebra with an extra operation. And, Getzler showed that this extra operation corresponds to our ability to twist a little disk 360°. More precisely, he showed that a Batalin–Vilkovisky algebra is the same as an algebra of the framed little discs operad.

This extra twist of the knife only makes me more curious to know what’s really going on here.

Here’s a clue that could help. As I explained to Urs a couple days ago, this business of ‘taking homology’ is really some sort of procedure for turning weak \infty-groupoids (i.e. spaces) into stable strict \infty-groupoids (i.e. chain complexes) — followed by taking the homology of the chain complex, which in principle loses even more information, but doesn’t in this particular example. That suggests that these Gerstenhaber (and Batalin–Vilkovisky) algebras are really just watered-down chain complex versions of spaces equipped with nn-ary operations corresponding to ways of sticking nn (framed) little disks into a big disk.

But still: what’s really going on? What do classical phase spaces have to do with little 2-dimensional disks???

As far as I’m concerned, the Rosetta Stone on the third page of Getzler’s paper only serves to heighten the mystery further!

Posted by: John Baez on August 23, 2007 3:38 PM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

This might be totally off base, but could this be a different version of the statement that 2d TFTs correspond to Calabi-Yaus? (e.g. in Costello’s version)

As far as I understand a BV algebra is a formalization of a Calabi-Yau. Given a space with a volume form (which I will think of formally as a CY, though of course in the any Riemannian manifold will do) we combine the de Rham differential on forms and the Schouten bracket on the exterior algebra of vector fields to get a BV algebra (i.e. take d and transfer it to polyvector fields using contraction against the volume form). I think all geometric examples come this way, except that we can mildly generalize this by replacing vector fields by any Lie algebroid, equipped again with a volume form.

The simplest example is take any Lie algebra g with a volume form, then the exterior algebra of g (i.e. the Chevalley complex calculating Lie algebra homology) gets a BV structure. (The Chevalley differential for Lie algebra homology is not a derivation of the wedge product, like the cohomological one, but a second order operator, i.e. a BV operator).

This is why BV algebras come up in gauge theory I think — when you’re doing hamiltonian reduction (classical or quantum) you need to take Lie algebra coinvariants (for g with zero bracket or the regular bracket, respectively) to impose the moment map condition.

Anyway, the point is these are all versions of forms or polyvector fields on a CY (which are the same), and we know very generally that forms on a CY should form a homological version of a 2d TFT. The tree level part of this (with marked boundary circle) is precisely an action of (chains on) the framed little discs operad…

Posted by: David Ben-Zvi on August 23, 2007 7:40 PM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

This might be totally off base, but could this be a different version of the statement that 2d TFTs correspond to Calabi-Yaus? (eg in Costello’s version)

I’ve been bemused by Kevin adopting the CY name for his stuff. To me, CY gives rise to a very special kind of BV.

As far as I understand a BV algebra is a formalization of a Calabi-Yau.

That’s certainly not where they came from according to the history as I know it.

Given a space with a volume form (which I will think of formally as a CY, though of course in the any Riemannian manifold will do)

That’s why I think CY is more special.

we combine the de Rham differential on forms and the Schouten bracket on the exterior algebra of vector fields to get a BV algebra (ie take d and transfer it to polyvector fields using contraction against the volume form).


I think all geometric examples come this way, except that we can mildly generalize this by replacing vector fields by any Lie algebroid, equipped again with a volume form.


I don’t think of that as mild. Geometric intuition may help, but it’s really `just’ homological algebra.

Posted by: jim stasheff on August 24, 2007 1:06 AM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

2d TFTs correspond to Calabi-Yaus?

While it might or might not be directly related to your comment here:

it seems to me that the term “Calabi-Yau category” as used by Kontsevich and Costello is not so optimal. These categories, unless I am misremembering, should really be called “Frobenius categories” or the like, since they are the obvious many-object generalization (the oidazation) of a Frobenius algebra.

We expect any 2-d QFT to be related to such a Frobenius category, in one way or another. The objects always characterizing the allowed boundary conditions, and the morphisms characterizing the admissable staes of the QFT stretching between two given boundary conditons.

While it is true that the popular method of twisting a 2d SCFT sigma-model to get a 2d TFT which is then described by such a category happens to work if and only if the target of the sigma model is a CY, this more or less special (though of course important) case of a 2d QFT shouldn’t induce the name for the general structure.

Other people say “cyclic category” for essentially the same concept as a CY-category. Some comments on the relation are in Lazaroiu’s article. (I assume David Ben-Zvi knows this better than I do. I am just mentioning this for the record.)

I’d think that in the end we might even want to do away with the condition that our categories have to be linear. We can express the crucila “CY property”, which is really a Frobenius property, in a much nicer abstract way:

we want to be talk about

- categories

- which are at the same time co-categories - such that composition and co-composition are compatible in the Frobenius sense .

Somebody famous should make that definition and publish it.

Posted by: Urs Schreiber on August 24, 2007 9:09 AM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

John wrote:

It seems amazing: there’s a mystical relation between ways of sticking little 2d disks in bigger ones, and operations you can do on the space of multivector fields on a manifold!

I don’t know if the connections to 2d topological and conformal field theory (described in the articles I cite) actually explain this mystical relation, or merely exploit it.

David wrote:

This might be totally off base, but could this be a different version of the statement that 2d TFTs correspond to Calabi-Yaus? (e.g. in Costello’s version)

You can’t be totally off base here — this stuff must be closely related! But this relation between 2d topological conformal field theories and Batalin–Vilvovisky algebras was known long before Costello’s work, and back then the word ‘Calabi–Yau’ wasn’t part of the game. So, I think something simpler and more general is at work, and the Calabi–Yau business is a slightly red herring. A pink herring, let’s say.

Jim wrote:

I’ve been bemused by Kevin adopting the CY name for his stuff. To me, CY gives rise to a very special kind of BV.

Okay, that’s reassuring. I’m really trying to understand the bare, simple essence of what’s going on here. The fewer bells and whistles our machinery is bedecked with, the happier I’ll be!

Right now I’m hoping that something like this is at work. Since a phase space in classical mechanics is typically a Poisson manifold, and the multivector fields on a Poisson manifold form the simplest example of a BV algebra, I should think of a BV algebra as a sophisticated algebraic version of a classical phase space XX!

And, given any such thing, I can try to cook up a 2d field theory where the fields are maps from Riemann surfaces into XX — a “Poisson sigma model” or some generalization thereof. Technically, this field theory should be a topological conformal field theory, or TCFT. The geometry of our phase space XX get cleverly encoded in this field theory… but we can recover it, using Getzler’s result that any TCFT gives a BV algebra.

If this is right, what I need is some intuition about why the various operations in the BV algebra:

  1. product,
  2. bracket,
  3. differential
correspond to various things we can do in our TCFT, ultimately coming from various operations in the homology of little disks operad:
  1. sticking two little disks in a big one
  2. switching two little disks
  3. twisting a little disk 360°

But, before I can make any progress at understanding this stuff, I have to come out and say: it seems rather weird to probe a classical phase space by mapping Riemann surfaces into it, since it was designed to have 1-manifolds mapped into it! Classical states trace out 1d trajectories as time passes! Paths! How the heck do surfaces get into the game?

However, I’ve recently met precisely this same “dimension shift” when trying to learn about the string topology of symplectic manifolds. And there, the physical meaning became clear to me. From week255:

Cohen’s talk described some cool relations between string topology and symplectic geometry! In physics we use symplectic manifolds to describe the space of states - the so-called “phase space” - of a classical system. So, if you have a loop in a symplectic manifold MM, it can describe a periodic orbit of some classical system. In particular, if we pick a periodic time-dependent Hamiltonian for this system, a loop will be a solution of Hamilton’s equations iff it’s a critical point for the “action”.

But, we can also imagine letting loops move in the direction of decreasing action, following the “gradient flow”. They’ll trace out 2d surfaces which we can think of as string world-sheets! This is just what string topology studies, but now we can get “Morse theory” into the game: this studies a space (here LML M) by looking at critical points of a function on this space, and its gradient flow.

So, we get a nice interaction between periodic orbits in phase space, and the string topology of that space, and Morse theory! For more, try this:

16) Ralph Cohen, The Floer homotopy type of the cotangent bundle, available as arXiv:math/0702852.

So you see, we start by doing something perfectly sensible: studying periodic orbits in our classical phase space XX. But then, we notice that such orbits are physical when they’re critical points of the action. The gradient flow with respect to this action defines a new “meta-dynamics” for loops — that is, strings mapped into XX. So before you know it, we’re studying Riemann surfaces mapped into a classical phase space… hmm, and yes, it looks like this gives some sort of topological conformal field theory.

So, maybe the overall physics picture is starting to makes some sense to me. But, I still don’t understand the correspondence between this trinity:

  1. product,
  2. bracket,
  3. differential
and this one:
  1. sticking two little disks in a big one
  2. switching two little disks
  3. twisting a little disk 360°

To dig further I’ll probably need (among other things) to understand what Kontsevich means when he writes:

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization…. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals.

Urs will probably tell me that this relation between particle mechanics and string theory is a case of holography (see the blog entry that spawned this thread) — states in an (n+1)(n+1)-dimensional theory corresponding to correlators in an nn-dimensional theory. I wish I could really see that at work here.

Posted by: John Baez on August 24, 2007 10:58 AM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

If I literally think of a periodic orbit in some phase space as the boundary of a little disk™, then rotating that little disk would seem to simply correspond to following the time evolution.

This, in turn, would – infinitesimally – just correspond to acting with {H,}\{H,\cdot\}.

Now, the differential we are talking about is supposed to be action with something people write (S,)(S,\cdot).

Looks suspicious to me. But I don’t quite know yet what to make of it.

Posted by: Urs Schreiber on August 24, 2007 2:56 PM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

John writes: I should think of a BV algebra as a sophisticated algebraic version of a classical phase space X!

Jim: almost - that would at least be a dg Gerstenhaber algebra, but BV also wants a second order diff op or you can play the volume form game

On the other hand, the BFV construction is very very much a sophisticated algebraic version of a classical phase space X!

Posted by: jim stasheff on August 24, 2007 3:13 PM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

No doubt these papers of Tamarkin are important too.

Posted by: David Corfield on August 23, 2007 5:20 PM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

and lots and lots of other papers - for BV algebras, for the BV construction, for the BFV construction (the Hamiltonian analog) as well as for little disks operads, little interval operads (cf. open strings),…

It’s best to supply a list of papers ONLY for the particular aspect anyone out there wants to learn more about, unless John wants to assemble a maximal list.

Posted by: jim stasheff on August 24, 2007 1:11 AM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

I certainly don’t want to assemble a maximal list of papers! I want to figure out the secret relation between classical mechanics (and its quantization) and 2d field theory — the secret relation that’s underlying so many of these papers.

It seems there’s something quite simple going on, which nobody has bothered to tell me. It’s possible that reading more papers won’t help at all!

Kontsevich certainly drops some heavy hints:

In this paper it is proven that any finite-dimensional Poisson manifold can be canonically quantized (in the sense of deformation quantization)… The solution presented here uses in an essential way ideas of string theory. Our formulas can be viewed as a perturbation series for a topological two-dimensional quantum field theory coupled with gravity.

Unfortunately he doesn’t seem to expand on that remark.

Hmm, maybe this paper gives it all away:

I like the abstract:

We give a quantum field theory interpretation of Kontsevich’s deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. We give a quantum field theory interpretation of Kontsevich’s deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory.

Hmm, they thank Jim Stasheff… I like this sentence, too:

Although the non-rigorous quantum field theory arguments of this paper are of course no substitute for the proofs in [Kontsevich’s paper], this approach offers an explanation for why Kontsevich’s construction works, and puts it in the context of Feynman’s original picture of quantization.

Yes, I think this may really illuminate things for me…

Posted by: John Baez on August 24, 2007 6:32 PM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

Maxim: The solution presented here uses in an essential way ideas of string theory.

One of the ideas that pops out at me his use of the punctured upper half-plane and it’s compactification - punctures could be at the centers of little disks
or, on the real line, half disks

Posted by: jim stasheff on August 25, 2007 12:38 AM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

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.

Does this set you against the Gelfand Principle

which asserts that whenever you state a new concept, definition, or theorem, (and better still, right before you do) give the SIMPLEST possible non-trivial example?

I suppose it all hinges on the term ‘non-trivial’. Opting for complicated cases increases the chances you won’t meet with triviality, and in some cases you may need to go a long way to avoid it.

Posted by: David Corfield on August 23, 2007 11:33 AM | Permalink | Reply to this

Re: Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III)

Does this set you against the Gelfand Principle

which asserts that whenever you state a new concept, definition, or theorem, (and better still, right before you do) give the SIMPLEST possible non-trivial example?

I perfectly agree with that. When you state a definition, give the simplest possible example.

But when you make up your definition, have the most generic example in mind.

Posted by: Urs Schreiber on August 23, 2007 12:09 PM | Permalink | Reply to this
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