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  What is the global symmetry group associated to the C-field?

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The C-field in 11-dimensional supergravity is an elusive object that is not the simple higher U(1)-gauge field one would naively make this out to be. For an overview of possible models for this object, see, for instance, section 3 of "The M-theory 3-form and E8 gauge theory" by Diaconescu, Freed and Moore (henceforth DFM).

However, it is always an object that carries with it a notion of "gauge transformation", and for naive higher U(1)-gauge fields with a transformation law CC+dΛ

for C a p-form and Λ a (p1)-form, one can easily see that there are "gauge transformations" that actually do not change the gauge field at all - those with dΛ=0. However, objects charged under this U(1) would transform by eiΛ, meaning this transformation is non-trivial on the other fields. This means that there is a global symmetry group associated to this gauge transformation law, namely given by all closed (p1)-forms Cp1dR(M) on our spacetime manifold M that does not change the gauge field (hence does not need to be quotiented out in order to "fix the gauge") and therefore remains even after quantization. If we consider that the objects charged under such a symmetry are (p1)-dimensional objects, it is clear that the proper symmetry operator is eiΣΛ where Σ is the charge object, and so the final global symmetry group is in fact Hp1(C,U(1)) since exact forms just act as the identity.1

A similar reasoning seems to be carried out in "M-Theory Dynamics On A Manifold Of G2 Holonomy" by Atiyah and Witten to obtain the total and unbroken global symmetry groups associated to the C-field. However, as I mentioned in the first paragraph of this question, the C-field is not a simple higher gauge field, and its exact "gauge group" is a subtle question.

For instance, in one of DFM's models, the proper notion of a gauge transformation is that "the C-field" is a pair of objects (A,c) where A is an ordinary E8-gauge field and c a 3-form with integral periods, and the gauge transformations are given by AA+αccCS3(A,A+α)+ω,

where CS3(A,A+α) is the relative Chern-Simons invariant 3-form between two connections given by integrating tr(F2) along the straight line between A and A+α in connection space (which is affine, so this is possible). The α is simply a 1-form on ad(P) and ω is a 3-form with integral periods.

There seems to be no evident notion of how such a transformation acts on objects charged under the C-field, nor do there seem to be global transformations in this case or indeed a straightforward relation of this transformation to the Λ considered earlier. Section 7 of DFM defines the proper notion of charge for the E8-model of the C-field, but does not consider how objects charged thusly transform as far as I can see.

What is the proper action of such a gauge transformation in the sense of DFM on charged objects? How can this be reconciled with the analysis of global symmetries of the C-field as done by Atiyah and Witten? Is there a notion of the global symmetries associated to the gauge symmetry of the C-field in the stricter formulation where it is no longer a naive higher U(1)-gauge field?


1The clear analogy in electromagnetism is the global U(1) symmetry that persists even after quantization, corresponding to H0(R4,U(1))=U(1).


This post imported from StackExchange Physics at 2017-11-18 23:21 (UTC), posted by SE-user ACuriousMind

asked Mar 16, 2017 in Theoretical Physics by ACuriousMind (910 points) [ revision history ]
edited Nov 18, 2017 by Dilaton
Is it obvious that there is matter "charged" under the C field? ψexp(iΛ)ψ does not seem to be a very reasonable transformation if Λ is a p-form. So I'm not sure I'd interpret the dΛ as some g1dg with g an element of some group.

This post imported from StackExchange Physics at 2017-11-18 23:21 (UTC), posted by SE-user Toffomat
@Toffomat I mention in the second paragraph that the proper operator is not exp(iΛ) but exp(iΛ), where the integral is over the extent of the charged object. People certainly say things like "the 2-brane is charged under the C-field", so some notion of being charged must exist, and Atiyah and Witten do claim that the global symmetry group is given by H2, which I can only explain by observing that exp(iΛ) is trivial for exact forms, since without that we would just have the group of all closed forms as symmetry.

This post imported from StackExchange Physics at 2017-11-18 23:21 (UTC), posted by SE-user ACuriousMind
Usually, form field couple to branes via some WC or more complicated terms (eg. the Chern-Simons coupling, see Eq. (3) in arxiv.org/abs/hep-th/9910053). The gauge transformations CC+dΛ is still not derived from some group element, and the brane does not transform the way a field does in QFT. In that sense, I wouldn't think there is a gauge roupassociated with C in the usual sense.

This post imported from StackExchange Physics at 2017-11-18 23:21 (UTC), posted by SE-user Toffomat
@Toffomat I'm asking for a global symmetry group, not a gauge group. (For example, in the case of an ordinary SU(N) gauge theory, the answer to "what is the global symmetry group?" is the center Zn, not the gauge group SU(N)). I'm willing to entertain the thought that the exp(C) operator is not the correct way to think about this, but any other answer to the global symmetry group needs to explain why Atiyah/Witten claim the global symmetry group is H2. I also think that the exp(C) is compatible with arxiv.org/abs/1412.5148v2, but I'm not fully certain.

This post imported from StackExchange Physics at 2017-11-18 23:21 (UTC), posted by SE-user ACuriousMind
I guess Atiyah/Wittens Eqns 2.10 - 2-12 seem to indicate that eΛ or somthing like it could be a reasonable guess for a symmetry of the low-energy theory, i.e. after integrating out (part of) the extra dimensions. (But I don't see how such an operator could be a symmetry of the full theory, since its not even local.) Is that what you have in mind?

This post imported from StackExchange Physics at 2017-11-18 23:21 (UTC), posted by SE-user Toffomat
@Toffomat I'm not sure - I think my question is pretty much about what exactly Atiyah and Witten have in mind with that claim that H2 is the global symmetry group coming from the C-field. I wrote down what I think it might be, after reading arxiv.org/abs/1412.5148, but I don't really know.

This post imported from StackExchange Physics at 2017-11-18 23:21 (UTC), posted by SE-user ACuriousMind

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