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  How can we define BF theory on a general 4-manifold?

+ 7 like - 0 dislike
1944 views

(I have rewritten the question some, with new understanding)

4d BF theory is classically presented as the TFT arising from the Lagrangian

BF,

where B is an abelian 2-connection (locally a real 2-form) and F is the curvature of a connection A. People often throw about this theory on general 4-manifolds, but it seems the naive definition from the Lagrangian above is anomalous for most situations. If one restricts to the simpler case of A and B being globally defined forms (not considering them as connections) the phase space is H1(Σ4,R)+H2(Σ4,R) with A as the first coordinate and B as the second. A and B are conjugate from the point of view of the above Lagrangian, but this has no chance of being symplectic if the first and second Betti numbers are not equal. If A and B are connections, however, the only solution to the equations of motion (which not only set dA=dB=0 but also their holonomies) is the trivial one, and it is not a problem

However, what I'd really like to consider is nBF. With A and B full-fledged connections. Let us first integrate out B. B is a (2-)connection, so dB can be any integral 3-form. We can thus write it as dβ+kmkλk, where β is an ordinary real 2-form, and λk form a basis of integral harmonic 3-forms. Then the action becomes dAβ+knmkAλk. The first term sets dA=0 after integrating over β and the second term sets the holonomies of A to be nth roots of unity. If our 4-manifold doesn't have torsion, there are precisely nb1 such As.

If we do the same thing, integrating out A instead, we get Dirac deltas setting dB=0 and the holonomies of B to be nth roots of unity. Again, if we have no torsion, there are nb2 such Bs.

It looks like the determinants for these Dirac deltas are all 1, so from the naive calculation above, Zn=nb1 but also Zn=nb2. This is a problem on a general 4-manifold, but like the simpler situation above, there is no issue in quantizing on something of the form Σ3×R. What I think must be happening is that when I am integrating out A or B, I am really only integrating out the non-flat parts. There should still be some more factors that correct the discrepancy above. I think I should be more careful in defining the above integral. Perhaps it is possible to define it sensibly on 4-manifolds which bound a 5-manifold a la Chern-Simons theory.

How does one define this theory on general 4-manifolds?

Thanks.

This post imported from StackExchange Physics at 2014-04-05 03:00 (UCT), posted by SE-user Ryan Thorngren
asked Sep 5, 2012 in Theoretical Physics by Ryan Thorngren (1,925 points) [ no revision ]

2 Answers

+ 6 like - 0 dislike

BF theory secretly has another name - Zn gauge theory in the deconfined limit. The parameter n is what appears in front of the BF action. Zn gauge theory can be defined on any manifold you want - just introduce a lattice approximation of the manifold and compute the lattice gauge theory partition function. By taking the extreme deconfined limit of the lattice gauge theory one can verify that this construction is independent of the way you approximated the manifold.

Basic examples

In the deconfined limit the Zn flux through every plaquette of the lattice gauge theory is zero. (This is like the constraint imposed by integrating out B.) We have a residual freedom to specify the holonomy around all non-contractible loops. Hence Zn(M)=Zn(S4)nb1(M) where Zn(S4) is a normalization constant. Requiring that Zn(S3×S1)=1 gives Zn(S4)=1/n. This condition, Zn(S3×S1)=1, is the statement that the theory has one unique ground state on S3. In general, Zn(Σ3×S1) is tr(eβH), but since H=0 we are simply counting ground states.

As a quick check, Zn(S2×S1×S1)=n, which is the number of ground states on S2×S1, and Zn(T3×S1)=n3, which is the number of ground states on T3=(S1)3.

Further relations

Another check on the value of Zn(S4): this is the renormalized topological entanglement entropy of a ball in the 3+1d topological phase described by deconfined Zn gauge theory. More precisely, the topological entanglement entropy of a ball is lnZn(S4) which gives logn in agreement with explicit wavefunction calculations.

We can also consider defects. The BF action is n2πBdA. Pointlike particles (spacetime worldlines) that minimally couple to A carry Zn charge of 1. Similarly, string excitations (spacetime worldsheets) that minimally couple to B act like flux tubes carry Zn flux of 2π/n. Now when a minimal particle encircles a minimal flux, we get a phase of 2π/n (AB phase), but this also follows from the BF action. Without getting into two many details, the term in the action like B12tA3 fixes the commutator of A3 and B12 to be [A3(x),B12(y)]=2πinδ3(xy) (flat space). The Wilson-line like operators given by WA=eidx3A3 and WB=eidx1dx2B12 thus satisfy WAWB=WBWAe2πi/n which is an expression of the braiding flux above that arises since the particle worldline pierced the flux string worldsheet.


Comments on the comments

Conservative thoughts

If I understand you correctly, what you want to do is sort of argue directly from the continuum path integral and show how the asymmetry you mentioned arises. I haven't done this calculation, so I can't be of direct help on this point right now. I'll ponder it though.

That being said, it's not at all clear to me that treating the action as AdB leads one to just counting 2-cycles. Of course, I agree that 2-cycles are how you get non-trivial configurations of B, but in a conservative spirit, it's not at all clear to me, after gauge fixing and adding ghosts and whatever else you need to do, that the path integral simply counts these.

The way that I know the path integral counts 1-cycles is that I have an alternative formulation, the lattice gauge theory, that is easy to define and unambiguously does the counting. However, I don't know how it works for the other formulation. I suppose a naive guess would be to look at lattice 2-gauge theory for B, but this doesn't seem to work.

Ground states

One thing I can address is the issue of ground states. In fact, I favor this approach because one doesn't have to worry about so many path integral subtleties. All you really needs is the commutator.

Take the example you raise, S2×S1. There are two non-trivial operators in this case, a WB for the S2 and a WA for the S1. Furthermore, these two operators don't commute, which is why there are only n ground states. If we define |0 by WA|0=|0, then the n states {|0,WB|0,...,Wn1B|0} span the ground state space and are distinguished by their WA eigenvalue. Importantly, you can also label states by WB eigenvalue, but you still only get n states. Poincare duality assures you that this counting of states will always work out no matter how you do it.

Furthermore, the operator WB has a beautiful interpretation in terms of tunneling flux into the non-contractible loop measured by WA. It's easier to visualize the analogous process in 3d, but it still works here.

You can also see the difference between the theory in 4d and 4d. Since both B and A are 1-forms you have different possibilities. The analogue of S2×S1 might be S1×S1 and this space does have n2 states. However, that is because both A and B can wrap both cycles.

The remarkable thing is that the Zn gauge theory formulation always gets it right. You simply count the number of one-cycles and that's it.

Getting at other spaces

The state counting approach gets you everything of the form Zn(Σ3×S1), but even spaces like S2×S2 can be accessed. You can either do the direct euclidean gauge theory computation, or you can regard Zn(S2×S2)=|Z(S2×D2)|2 i.e. the inner product of a state on S2×S1=(S2×D2) generated by imaginary time evolution.

This post imported from StackExchange Physics at 2014-04-05 03:00 (UCT), posted by SE-user Physics Monkey
answered Sep 5, 2012 by Physics Monkey (80 points) [ no revision ]
Thanks, Physics Monkey. This is a good answer so far, but it does not address the inconsistency that I'm talking about. If one takes the other polarization (B as coordinates, integrating out A), the theory is [ U(1)-n->U(1) ] 2-gauge theory (roughly speaking, instead of holonomies around 1-cycles, we have holonomies around 2-cycles, also in Zn). This interpretation gives the calculation Zn=nb2.

This post imported from StackExchange Physics at 2014-04-05 03:00 (UCT), posted by SE-user Ryan Thorngren
Let me be more particular with my complaint. The number of ground states on S2×S1 in my mind should be n2. Once you pick two elements of Zn this uniquely defines for you a flat 2-connection B (that is, a global 2-form with dB=0) and a flat 1-connection A each with the prescribed holonomies around the sphere and circle factor, respectively. That is to say, dB=0 does not imply that B has null periods. I guess this really comes down to what we choose our gauge transformations to be.

This post imported from StackExchange Physics at 2014-04-05 03:00 (UCT), posted by SE-user Ryan Thorngren
Here's what I mean by my last comment. The theory is symmetric under BB+β and AA+α for a flat 2-connection β and a flat 1-connection α. There is a unique ground state up to these gauge transformations.

This post imported from StackExchange Physics at 2014-04-05 03:00 (UCT), posted by SE-user Ryan Thorngren
Ok, I think I understand more what you're getting at. I'll edit my answer above.

This post imported from StackExchange Physics at 2014-04-05 03:00 (UCT), posted by SE-user Physics Monkey
Thanks for the updated answer. I certainly see what you mean for S2×S1 and agree with you now (also, my third comment is wrong, those are not actually symmetries). The A and B coordinates are conjugate is another way to look at why WA and WB must not commute (indeed, the linking number prescription is like a topological Heisenberg algebra). I am continuing to think about more general 4-manifolds. I will get back to you hopefully soon with some resolution or maybe more questions.

This post imported from StackExchange Physics at 2014-04-05 03:00 (UCT), posted by SE-user Ryan Thorngren
+ 2 like - 0 dislike

I figured this out a little while ago, so as not to leave any of you hanging, here is a (almost complete) resolution.

First, there is a lattice formulation of the 2-gauge theory that agrees with the 1-gauge theory up to the topological factor nχ(Σ4). One has a Zn variable for each 2-cell, gauge transformations on 1-cells, and importantly, 2-gauge transformations (gauge transformations between gauge transformations) on 0-cells. The number of configurations is nb2. Then we divide by the volume of the gauge group, which is the number of gauge transformations divided by the number of 2-gauge transformations. So the partition function of this lattice theory is nb2b1+b0. This gives the same answer as the A-theory on any 4-manifold fibered over S1, in particular the 4-torus @Physics Monkey and I were discussing.

To get this answer from the path integral, I gauge fix B using the Lorentz gauge dB=0, introducing a Lagrange multiplier term π1,δB as well as some ghost terms not involving B or π1. π1 has gauge symmetries, being a 1-form, which I also fix, introducing a term E0,δπ1 and some more ghost terms which do not involve B, π1, or E0. Now I scale BB/n, π1nπ1, and E0E0/n to reduce the integrand to the case with n=1 (that integral can be shown to be 1).

I just need to find how the path integral measure scales under these transformations. DB will scale by n to the power of the dimension of the Bs being integrated over. It's important to notice that the zero modes of the integrand are precisely the harmonic 2-forms, a space of dimension b2. Then if B2 is the (infinite) dimension of all 2-forms, DB scales by nb2B2. Terms like nBk can all be removed by some suitable regularization a la Witten's paper on abelian S-duality. What we are left with is the answer above, nb2b1+b0.

Maybe there is some way to understand the topological factor that appears?

This post imported from StackExchange Physics at 2014-04-05 03:00 (UCT), posted by SE-user Ryan Thorngren
answered Sep 26, 2012 by Ryan Thorngren (1,925 points) [ no revision ]
Excuse me, is the k-cycle used in your context is the same as this definition? i.e. The length of a cycle, is the number of elements of its orbit of non-fixed elements. A cycle of length k is also called a k-cycle.

This post imported from StackExchange Physics at 2014-04-05 03:00 (UCT), posted by SE-user mysteriousness
I mean cycle in the homology sense. Let me know if you have more questions. I understand this topic quite a bit more now that it's been over a year. :)

This post imported from StackExchange Physics at 2014-04-05 03:00 (UCT), posted by SE-user Ryan Thorngren
How do you understand the B as a n-form field in terms of ZN gauge group? In the sense of group cohomology, a 1-form field A can be realize as a group element aZN. The derivative of 1-form A as dA can be realize as (a+b)[a+b]modZN. Is that B as n-form field still realize as bZN? or something more?

This post imported from StackExchange Physics at 2014-04-05 03:00 (UCT), posted by SE-user mysteriousness

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