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This in reference to this fascinating lecture by Nicolai Reshetikhin-

http://staff.science.uva.nl/~nresheti/Holb-Quant-Gauge.pdf

  • Given what is said on page 13 in section 4.1 its not clear to me why the ``partition function" should be a vector in the same space of states which is being assigned to the boundary of the space-time manifold.

I am confused to see the claim that $Z(M) \in H(\partial M)$.I would have thought that the partition function is a linear function on the space of states which takes in a boundary configuration and gives back a number.

  • I would like to know what is meant by saying (on the same page) that "..(this vector space of states) may depend on the extra structure at the boundary (it can be a vector bundle over the moduli space of such structures).."

  • How does the above relate to the claim on page 47 that for Chern-Simons theory the "..space of states assigned to the boundary is the space of holomorphic sections of the geometric quantization line bundle over the moduli space of flat connections in a trivial principal G-bundle over the boundary (provided we made a choice of complex structure)..."

I would like to know what the above means and I would be happy to get back some further references about this...especially what is a "geometric quantization bundle"?

  • Is the above somehow purely an effect of quantization? Thinking classically intuitively i would have thought that the space of states assigned to the boundary is the space of gauge equivalence classes of flat connections on the manifold which can be extended to a flat connection on the whole space-time. Is the above wrong?

I can't see where the structures of a vector bundle and may be even its sections over the above moduli space seem to be getting involved.

  • Is there a general argument to see that the space of states attached to the boundary is always a symplectic manifold and that the subspace of that will ever be picked up by solving the Euler-lagrangian equations in the interior will be some Lagrangian submanifold of it?

In these lectures some comment seems to be made about how the gauge invariance may complicate the above scenario.I would like to know more about that.

  • At the start of the lecture the author seems to demand that the space-times belong to a category of manifolds where the morphism is that of cobordism. I would like to know what was the intuition behind making this choice. Why not some other simpler morphism?

Does working in this category of cobordisms somehow help in justifying the functoriality demand on the partition function with respect to gluing of manifolds?

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It is perhaps best if you gave the author's surname (Reshetikhin) given that there is a mathematical physicist whose name is (Hermann) Nicolai. Reading this question confused me no end until I downloaded the PDF. –  José Figueroa-O'Farrill May 15 '11 at 19:25
    
@Jose Edited the name. I hope the question framing was not confusing. Apologies for the trouble. –  Anirbit May 15 '11 at 19:38
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It's even more jarring since I've never heard anyone refer to said author as "Nicolai;" everyone who's on a first name basis with him calls him "Kolya." –  Ben Webster May 15 '11 at 20:05
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1 Answer

This question is really many questions, and so it will be hard for anyone to answer all of them succinctly. I will try to answer some of them.

Given what is said on page 13 in section 4.1 its not clear to me why the ``partition function" should be a vector in the same space of states which is being assigned to the boundary of the space-time manifold.

I am confused to see the claim that $Z(M)\in H(\partial M)$. I would have thought that the partition function is a linear function on the space of states which takes in a boundary configuration and gives back a number.

Suppose that I have a functor $(H,Z)$ from the cobordism category to Vect. Then to a manifold $M$ with $\partial M$ decomposed into "incoming boundary" $\partial_{\rm in}M$ and "outgoing boundary" $\partial_{\rm out}M$, I get a morphism of vector spaces $H(\partial_{\rm in}M) \to H(\partial_{\rm out}M)$. In particular, if I take the same manifold but declare the entire boundary to be "incoming", I get what you propose; if I take the entire boundary to be "outgoing", I get what Reshetikhin says. Moreover, consider the two "elbow macaroni" on a fixed codimension-$1$ manifold $Y$ (i.e. the manifolds $M = I \times Y$, where $I$ is the half-circle, topologically $I = [0,1]$, with either both endpoints marked "incoming" or both endpoints marked "outgoing"). The elbows get mapped under $Z$ to maps $H(Y) \otimes H(Y) \to H(\emptyset)$ and $H(\emptyset) \to H(Y) \otimes H(Y)$, and together identify $H(Y)$ with its dual space. Introduce just a little analysis (so that not all spaces must be finite-dimensional), and you decide that $H(Y)$ must be a Hilbert space.

To avoid the setting in the previous paragraph, one method is to have objects and morphisms in your cobordism category be oriented manifolds. Denote orientation-reversed $Y$ by $\overline Y$. Then any component of $\partial M$ has an induced ("outward", say) orientation, and you declare that a manifold with marked incoming-and-outgoing boundary is a morphism $\overline{\partial_{\rm in}M} \to \partial_{\rm out}M$. Then you do not quantize manifolds to Hilbert spaces but just to vector spaces, but you do have (via the elbows) that $H(\overline{Y})$ is the dual vector space to $H(Y)$.

In fact, most quantum field theories in nature are not define on the cobordism category of oriented manifolds, but only on the category of framed manifolds.

I would like to know what is meant by saying (on the same page) that "..(this vector space of states) may depend on the extra structure at the boundary (it can be a vector bundle over the moduli space of such structures).."

Most quantum field theories are not "topological", in the sense that they depend on geometric structure on spacetime. For example, you and I are particles in the Standard Model QFT, but the way that we interact depends sensitively on the distance between us. There are at least two equivalent ways to understand this dependence. One is to use a cobordism category whose objects and morphisms come equipped with extra structure. Another is to use the topological cobordism category, but to take as target not the category of vector spaces, but rather a category of vector bundles. Then the trick is to say: VectorBundles is fibered over Manifolds, and I have a morphism from the cobordism category to Manifolds that sends every object to the moduli space of geometric structures, and I can ask that my QFT intertwine these two maps to Manifolds.

How does the above relate to the claim on page 47 that for Chern-Simons theory the "..space of states assigned to the boundary is the space of holomorphic sections of the geometric quantization line bundle over the moduli space of flat connections in a trivial principal G-bundle over the boundary (provided we made a choice of complex structure)..."

I would like to know what the above means and I would be happy to get back some further references about this...especially what is a "geometric quantization bundle"?

Is the above somehow purely an effect of quantization? Thinking classically intuitively i would have thought that the space of states assigned to the boundary is the space of gauge equivalence classes of flat connections on the manifold which can be extended to a flat connection on the whole space-time. Is the above wrong?

I can't see where the structures of a vector bundle and may be even its sections over the above moduli space seem to be getting involved.

Is there a general argument to see that the space of states attached to the boundary is always a symplectic manifold and that the subspace of that will ever be picked up by solving the Euler-lagrangian equations in the interior will be some Lagrangian submanifold of it?

Yes, you are wrong here. Note that your proposed "assignment" to a boundary manifold depends on what space it is the boundary of.

Glossing some details, classical Chern-Simons theory assigns to a 2-manifold $Y$ the (symplectic --- inherited from CS action) manifold $H(Y)$ of flat $G$-bundles (mod gauge transformations) over $Y$. For a fixed manifold $M$ with $\partial Y = M$, the space of flat bundles that extends over $M$ is a coisotropic submanifold of the space of flat bundles over $Y$. When $M$ is compact, this coisotropic is actually a Lagrangian $Z(M)$.

An analogy may be valuable. Classical Mechanics with configuration space $N$ assigns to a point the symplectic manifold ${\rm T}^* N$, and to an interval a Lagrangian inside $\overline{{\rm T}^* N} \times {{\rm T}^* N}$, where by $\overline \Box$ for $\Box$ a symplectic manifold I mean the same manifold with negative symplectic structure. For well-behaved systems, this Lagrangian correspondence is actually the graph of a symplectomorphism (the "time evolution function").

Upon quantization, the symplectic manifold $H(Y)$ deforms to a vector space ($C^\infty(H(Y))$ deforms to $\operatorname{End}(H(Y))$), and one way to find/define this vector space is via geometric quantization.

At the start of the lecture the author seems to demand that the space-times belong to a category of manifolds where the morphism is that of cobordism. I would like to know what was the intuition behind making this choice. Why not some other simpler morphism?

Does working in this category of cobordisms somehow help in justifying the functoriality demand on the partition function with respect to gluing of manifolds?

This idea goes back at least to Segal and Atiyah. Ultimately, the notion of "cobordism category" is probably wrong, because it doesn't handle well the more complicated style of cutting and gluing that can occur in higher-dimensional manifolds. (So you should read, e.g., Lurie's classification of TQFTs, and also "Blob Complex" by Morrison and Walker.) In any case, the cobordism category is a good approximation.

The point is simply that you do expect that whatever the "partition function" is, it should behave well under cutting and gluing of manifolds. Let's say you and I are physicists in adjoining laboratories. You do some experiments, so that you completely understand the physics inside your lab: this means in particular that you understand the expectation value of any boundary configuration for your laboratory. (Maybe you mark one end of your lab "incoming" and the other end "outgoing", and fully understand the relationship between incoming and outgoing states. Actually, maybe your lab is in a universe with a distinguished "time" axis, and then "incoming" and "outgoing" are pretty well defined.) This data of expectation values is "the partition function".

Now let's say that my laboratory is next to yours, and I have similar knowledge of expectation values for boundary configurations. Now we get a joint grant and want to merge labs: do we already know the physics for the two labs thought of as a single lab? Certainly yes if the labs do not touch: we just take the tensor product of laboratories. But if they have a common boundary, then when we merge the labs, that common boundary is now part of the interior. How should we work out the expectation values for field configurations on the boundary of the merged lab? We should integrate from our expectation values out all the possible fields on the common boundary.

This makes pretty good sense physically. The problem is to make is make sense mathematically. The cobordism category is well-suited to this (especially if instead you use a cobordism $\infty$-category that allows corners).

Two good warm-ups are:

  • quantum mechanics with configuration space $N$ ($H({\rm pt}) = L^2(N)$ and $Z([0,t]) = \exp( i\hat E / \hbar)$, where $\hat E$ is the differential operator acting on $L^2$ normally called the "hamiltonian"; when the action of the system is (momentum)$^2$, $\hat E$ is the Laplacian. Note that in this case, "spacetime" is the interval, and it is a cobordism from a point to a point. Note that $\exp( i\hat E / \hbar)$ "is" the partition function for quantum mechanics.

  • classical string theory with configuration space $N$ (with Riemannian metric). Spacetimes are surfaces with boundary. $H(S^1) = {\rm T}^*(\operatorname{Maps}(S^1 \to N))$. (Classical theory, so valued in symplectic manifolds and Lagrangian correspondences, not Hilbert spaces and operators.) A cobordism $\Sigma$ ($\partial \Sigma = \coprod S^1$) is mapped to the Lagrangian correspondence that finds the locally-energy(=surface tension)-minimizing immersion $\Sigma \to N$ with given boundary $\coprod S^1 \to N$. Note that this Lagrangian is the graph of exp of a Hamilton-Jacobi "function", which assigns to any point in the Lagrangian the total energy of the corresponding surface. Note also that energy of a surface is additive, so exp of energy is multiplicative, along cuttings of a surface.

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@Theo Thanks a lot for our fabulous explanations. I can't say that I have understood everything you said but I guess somethings got clarified. I guess you would want the $Z$ to be maps from $H(\partial_{\rm in}M) \to H(\partial_{\rm out}M)$ .. right? In the Chern-Simon's example, I guess your $Y$ is a 3-manifold..right? –  Anirbit May 17 '11 at 7:04
    
@Theo Can you say something more about the motivation for sometimes shifting the target category from that of vector spaces to that of vector bundles? Like it would be a vector bundle over some "moduli space"..How does this moduli space arise? I guess in the classical CS theory this would be the moduli space of flat connections on the boundary 2-manifold. But why was just assigning to it the space of flat connections on the boundary not enough? –  Anirbit May 17 '11 at 7:08
    
@Theo In the classical CS case you seem to want to assign to your 2-boundary $Y$ the space of ``flat G-bundles" (mod gauge) over it. What do you mean by that? I would have thought one wants to assign flat connections on the fixed G-bundle. The G is fixed to something to start off with and generally trivial bundles are chosen. –  Anirbit May 17 '11 at 7:13
    
In CS theory, the top-dimensional manifolds (the cobordisms) are 3-dimensional, and the boundaries are 2-dimensional. When I say "the space of flat G-bundles on Y" for Y in any dimension, I mean the "space" of pairs (a G-bundle over Y, a flat connection on said G-bundle). Of course, this collection doesn't even form a set. So what I really mean is the groupoid of such things --- this is where you mod-out by gauge action. You topologize this "space" by saying that a "smooth curve" is a G-bundle on $Y\times I$ that has on each slice $Y\times \{t\}$ a flat connection, smoothly varying in $t$. –  Theo Johnson-Freyd May 17 '11 at 16:18
    
There's a much more hands-on way to do it, of course. The space of flat G-bundles mod gauge over Y is equivalent to the space of group homomorphisms $\pi_1(Y) \to G$, modulo the action of conjugation by $G$. (If $g\in G$ and $\phi: H \to G$ is a homomorphism, then $h \mapsto g\phi(h)g^{-1}$ is a homomorphisms.) In particular, automorphisms of a G-bundle are those conjugations that centralize the homomorphism. In most cases of interest (compact manifolds) $\pi_1(Y)$ is finitely presented; then the space of homomorphisms (mod gauge) is (a quotient of) an affine variety. –  Theo Johnson-Freyd May 17 '11 at 16:26
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