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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 5th 2010
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexI was having a look at Witten's recent [A New Look At The Path Integral Of Quantum Mechanics](http://uk.arxiv.org/abs/1009.6032), and was wondering if there's anything in the nPOV which would have suggested that analytic continuation of an integrand to a holomorphic function and integration over a different integration cycle would be a good idea. Or is it that, if this is a good move, this is an indication that there's plenty one can do to reformulate basics in quite a low-tech way?

   I was having a look at Witten’s recent A New Look At The Path Integral Of Quantum Mechanics, and was wondering if there’s anything in the nPOV which would have suggested that analytic continuation of an integrand to a holomorphic function and integration over a different integration cycle would be a good idea. Or is it that, if this is a good move, this is an indication that there’s plenty one can do to reformulate basics in quite a low-tech way?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 5th 2010
   • (edited Oct 5th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarted an entry * [[quantization via the A-model]] Maybe somebody finds the time to collect the remaining references. > and was wondering if there's anything in the nPOV which would have suggested that analytic continuation of an integrand to a holomorphic function and integration over a different integration cycle would be a good idea. Well, there has certainly been some indication that _something_ along such lines should be true, but the exciting thing is how Witten ploughs through it and shows how it is true. This is a special case of the general mechanism we once used to chat about on the nCafe from time to time: a kind of general "holography" in quantum field theory: an $n$-dimensional QFT gives rise on its boundary to an $(n-1)$-dimensional QFT. For Quantum Mechanics = 1-dimensional QFT, a major insight in this direction had been the Kontsevich-Cattaneo-Felder proof of the deformation quantization problem: this showed that the 1-dimensional QFT quantum algebra of any phase space was computed on the boundary of the 2-dimensional [[Poisson sigma-model]] QFT with that phase space as its target. I am not sure if a precise relation has been made manifest, but clearly the story that Witten is and has been telling for a little while now is analogous: he says that the _geometric_ quantization of that phase space is encoded on the boundary of another 2d TQFT with target space the classical phase space: the [[A-model]]. Or rather -- and that's the extra aspects he is amplifying now -- this target is a _complexification_ of the original phase space. As he motivates very nicely in the beginning, once you are in this complexification, it is very natural to look for alternative "integration contours" for the path integral -- where we play the standard trick and behave as if the path integral could be manipulated formally as an ordinary integral. So he shows that the different choices correspond to different choices of boundary conditions of the A-model (A-branes). He indicates that this story repeats at least in dimension 3 and 4. So this is a major further puzzle piece in a general story that has been unforling for a while: the holographic relation of quantum field theories in various dimensions. As for the nPOV: we had discussed elsewhere that there are good indications that with nd QFTs understood as n-functors on $(\infty,n)$-categories of cobordisms, this holograohic principle of QFT is the **holographics principle of higher category theory** which says that a transformation of $n$-functors is itself in components an $(n-1)$-functor depending on the boundary data (on codimension 1 cells) of the original functors. Hence that a transformation between n-dimensional QFTs has a chance of being itself an $(n-1)$-dimensional QFT. On the nCafe there are various traces of me describing this kind of nPOV perspective to some extent. But a major insight making this precise in a special case for the relation between 2d QFT and 3d TQFT with defects was then provided by Chris Schommer-Pries, as a spin--off of his explicit proof of the cobordism hypothesis in d=2. See the last slides <a href="http://sites.google.com/site/chrisschommerpriesmath/Home/Slides-MFO-6-11-09.pdf?attredirects=0">here</a>.

   started an entry

   • quantization via the A-model

   Maybe somebody finds the time to collect the remaining references.

   and was wondering if there’s anything in the nPOV which would have suggested that analytic continuation of an integrand to a holomorphic function and integration over a different integration cycle would be a good idea.

   Well, there has certainly been some indication that something along such lines should be true, but the exciting thing is how Witten ploughs through it and shows how it is true.

   This is a special case of the general mechanism we once used to chat about on the nCafe from time to time: a kind of general “holography” in quantum field theory:

   an $n$-dimensional QFT gives rise on its boundary to an $(n-1)$-dimensional QFT.

   For Quantum Mechanics = 1-dimensional QFT, a major insight in this direction had been the Kontsevich-Cattaneo-Felder proof of the deformation quantization problem: this showed that the 1-dimensional QFT quantum algebra of any phase space was computed on the boundary of the 2-dimensional Poisson sigma-model QFT with that phase space as its target.

   I am not sure if a precise relation has been made manifest, but clearly the story that Witten is and has been telling for a little while now is analogous: he says that the geometric quantization of that phase space is encoded on the boundary of another 2d TQFT with target space the classical phase space: the A-model.

   Or rather – and that’s the extra aspects he is amplifying now – this target is a complexification of the original phase space.

   As he motivates very nicely in the beginning, once you are in this complexification, it is very natural to look for alternative “integration contours” for the path integral – where we play the standard trick and behave as if the path integral could be manipulated formally as an ordinary integral.

   So he shows that the different choices correspond to different choices of boundary conditions of the A-model (A-branes).

   He indicates that this story repeats at least in dimension 3 and 4.

   So this is a major further puzzle piece in a general story that has been unforling for a while: the holographic relation of quantum field theories in various dimensions.

   As for the nPOV: we had discussed elsewhere that there are good indications that with nd QFTs understood as n-functors on $(\infty,n)$-categories of cobordisms, this holograohic principle of QFT is the holographics principle of higher category theory which says that a transformation of $n$-functors is itself in components an $(n-1)$-functor depending on the boundary data (on codimension 1 cells) of the original functors. Hence that a transformation between n-dimensional QFTs has a chance of being itself an $(n-1)$-dimensional QFT.

   On the nCafe there are various traces of me describing this kind of nPOV perspective to some extent. But a major insight making this precise in a special case for the relation between 2d QFT and 3d TQFT with defects was then provided by Chris Schommer-Pries, as a spin–off of his explicit proof of the cobordism hypothesis in d=2.

   See the last slides here.

3. • CommentRowNumber3.
   • CommentAuthorEric
   • CommentTimeOct 6th 2010
   • PermaLink
   Author: Eric
   Format: MarkdownItexNeat stuff. And this seems to resonate with what you told me recently about obtaining YM theory from CS theory. Note: Found the comment (took longer than I'd like to admit)! :) [oo-Chern-Simons theory](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=1874&Focus=16417#Comment_16417) >> By the way, which target space would reproduce "physics", i.e. the kind that can be observed directly by current experimental methods? >That's a good questions. All these $\infty$-Chern-Simons theories have the flavor of topological fielld theories. But >1. we sort of know that various "physical" theories are the boundary theories of these topological CS-theories. Notably the boundary theory of ordinary CS theory is the physical WZW model describing the string on a group manifold. And maybe more importantly: Kontsevich/Cattaneo-Felder in effect showed that the boundary theory of the 2d Poisson $\sigma$-model encodes on its boundary the quantum theory of the ordinary particle whose phase space gives the given Poisson Lie algebroid. Witten has recently published more articles along these lines, getting physical 1d QM from topological strings. There should be a general such "holography" mechanism by which the physical theories sit on the boundary of topological ones. But I don't understand this well enough yet. >1. Another phenomenon is that physical theories arise as In&ouml;n&uuml;-Wigner contractions of Chern-Simons theories. This is the point that Zanelli explores <a href="http://ncatlab.org/nlab/show/Chern-Simons+element#Zanelli">here</a>: how theories of gravity arise as limits of higher Chern-Simons theories.

   Neat stuff. And this seems to resonate with what you told me recently about obtaining YM theory from CS theory.

   Note: Found the comment (took longer than I’d like to admit)! :)

   oo-Chern-Simons theory

   By the way, which target space would reproduce “physics”, i.e. the kind that can be observed directly by current experimental methods?

   That’s a good questions. All these $\infty$-Chern-Simons theories have the flavor of topological fielld theories. But

   1. we sort of know that various “physical” theories are the boundary theories of these topological CS-theories. Notably the boundary theory of ordinary CS theory is the physical WZW model describing the string on a group manifold. And maybe more importantly: Kontsevich/Cattaneo-Felder in effect showed that the boundary theory of the 2d Poisson $\sigma$-model encodes on its boundary the quantum theory of the ordinary particle whose phase space gives the given Poisson Lie algebroid. Witten has recently published more articles along these lines, getting physical 1d QM from topological strings. There should be a general such “holography” mechanism by which the physical theories sit on the boundary of topological ones. But I don’t understand this well enough yet.

   2. Another phenomenon is that physical theories arise as Inönü-Wigner contractions of Chern-Simons theories. This is the point that Zanelli explores here: how theories of gravity arise as limits of higher Chern-Simons theories.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 6th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes. Up to some details. for instance notice that what Witten discusses in the latest article is a holographic pair of the opposite flavor as I was alluding to: he finds CS-theory on the boundary of Yang-Mills theory. What I had said was that possibly there is, in turn, a higher CS-theory, such that Yang-Mills theory is _its_ boundary theory. This is not out of the question that we have such towers of holographic relations. We already know at least one now: 1. The [[WZW-model]] is the boundary theory of 3d [[Chern-Simons theory]]; 1. 3d [[Chern-Simons theory]] may be realized as a boundary theory for 4d [[Yang-Mills theory]]. But I need to understand this better. I don't quite know if these two relations are on par or are different flavors of a general mechanism.

   Yes. Up to some details.

   for instance notice that what Witten discusses in the latest article is a holographic pair of the opposite flavor as I was alluding to: he finds CS-theory on the boundary of Yang-Mills theory.

   What I had said was that possibly there is, in turn, a higher CS-theory, such that Yang-Mills theory is its boundary theory.

   This is not out of the question that we have such towers of holographic relations. We already know at least one now:

   1. The WZW-model is the boundary theory of 3d Chern-Simons theory;

   2. 3d Chern-Simons theory may be realized as a boundary theory for 4d Yang-Mills theory.

   But I need to understand this better. I don’t quite know if these two relations are on par or are different flavors of a general mechanism.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 6th 2010
   • (edited Oct 6th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWell, I should say, clearly there is an evident way in which every [[schreiber:infinity-Chern-Simons theory]] governed by an invariant polynomial $\langle -\rangle$ on an $\infty$-Lie algebroid is a boundary theory for TQFT in one dimension higher whose Lagrangian is the corresponding curvature characteristic form $\langle F_{\nabla} \wedge \cdots \wedge F_\nabla\rangle$. For ordinary Cher-Simons theory this is the ordinary Killing form polynomial leading to the action functional $\langle F_\nabla \wedge F_\nabla \rangle$ of [[topological Yang-Mills theory]]. But in what Witten writes there is also a way that the geometric Yang-Mills term $\langle F_\nabla \wedge \star F_\nabla \rangle$ enters.

   Well, I should say, clearly there is an evident way in which every infinity-Chern-Simons theory (schreiber) governed by an invariant polynomial $\langle -\rangle$ on an $\infty$-Lie algebroid is a boundary theory for TQFT in one dimension higher whose Lagrangian is the corresponding curvature characteristic form $\langle F_{\nabla} \wedge \cdots \wedge F_\nabla\rangle$.

   For ordinary Cher-Simons theory this is the ordinary Killing form polynomial leading to the action functional $\langle F_\nabla \wedge F_\nabla \rangle$ of topological Yang-Mills theory. But in what Witten writes there is also a way that the geometric Yang-Mills term $\langle F_\nabla \wedge \star F_\nabla \rangle$ enters.

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeOct 6th 2010
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexUrs, #2 Thanks for that. Sounds like 'holographics principle of higher category theory ' deserves a page. I see you mentioned it [here](http://ncatlab.org/nlab/show/transfor#discussion_6). From #5, is it that while Witten's work can be seen to be placed in a holographic pattern, it relies on specific details of the particular theories he treats.

   Urs, #2 Thanks for that. Sounds like ’holographics principle of higher category theory ’ deserves a page. I see you mentioned it here. From #5, is it that while Witten’s work can be seen to be placed in a holographic pattern, it relies on specific details of the particular theories he treats.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeOct 6th 2010
   • (edited Oct 6th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> From #5, is it that while Witten's work can be seen to be placed in a holographic pattern, it relies on specific details of the particular theories he treats. Yes. There is clearly a general pattern of "holography" at work here. What Witten does in the article in question is study the specific concrete example where the higher dimensional theory is the A-model. This is one case of what should be a plethora of cases. Well, the best understood other case is also due to Witten, that's what he did at the beginning of his hep-th-career: the observation that the WZW-model is the boundary theory of Chern-Simons theory. (Which is why there is the second "W" there ;-).

   From #5, is it that while Witten’s work can be seen to be placed in a holographic pattern, it relies on specific details of the particular theories he treats.

   Yes. There is clearly a general pattern of “holography” at work here. What Witten does in the article in question is study the specific concrete example where the higher dimensional theory is the A-model. This is one case of what should be a plethora of cases.

   Well, the best understood other case is also due to Witten, that’s what he did at the beginning of his hep-th-career: the observation that the WZW-model is the boundary theory of Chern-Simons theory. (Which is why there is the second “W” there ;-).

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeOct 6th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> Sounds like 'holographics principle of higher category theory ' deserves a page. here is a start * [[holographic principle of higher category theory]]. But I have to interrupt now and prepare for our TMF seminar...

   Sounds like ’holographics principle of higher category theory ’ deserves a page.

   here is a start

   • holographic principle of higher category theory.

   But I have to interrupt now and prepare for our TMF seminar…