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May 17, 2008

Electric-Magnetic-Duality and Hodge Duality Extended to Differental Cocycles

Posted by Urs Schreiber

On

M. Caicedo, I. Martín and A. Restuccia
Gerbes and Duality
(arXiv:hep-th/0205002).

General context.

As recalled here recently in Fivebrane Structures (blog, arXiv, pdf) and a bit in Charges and twisted nn-bundles (blog) an important concept in (higher) gauge theory is that of electric-magnetic duality:

Given a line line nn-bundle (P,)(P,\nabla) (aka bundle (n1n-1)-gerbe, aka Deligne nn-cocycle, aka differential nn-character) on a dd-dimensional Riemannian space (X,g)(X,g), equipped with a connection \nabla with integral curvature (n+1)(n+1)-form F n+1F_{n+1} it may happen that the Hodge dual form gF n+1Ω dn1(X) \star_g F_{n+1} \in \Omega^{d-n-1}(X) is again closed and integral. If so, it means there is a line (dn2)(d-n-2)-bundle with connection which has this curvature. So somehow the Hodge star in this case extends from a duality operation on deRham classes to a duality operation on differential cohomology classes (not, in general, uniquely so, since there may be several inequivalent nn-bundles with connection but with the same curvature, in general).

This is important, because this duality operation describes electric-magnetic duality in physics. It plays a big role in various contexts, not the least since it was discovered by Witten and Kapustin that Geometric Langlands (I, II) is apparently just a tiny aspect (I, II, III) of this general mechanism.

But it is hard to find any literature concretely on the extension of the Hodge duality on curvature forms to a relation between the full differential cocycles. The bulk of the physics literature concentrates on things that can be described by globally defined differential forms.

Now recently, Hisham Sati pointed me to

M. Caicedo, I. Martín and A. Restuccia
Gerbes and Duality
(arXiv:hep-th/0205002).

While not addressing exactly the lift of Hodge duality from closed integral to differential cocycles, they do discuss some aspects of electric magnetic duality paying close attention to the full differential cocycle structure of line nn-bundles with connection.


The duality-symmetric Lagrangians for higher abelian Yang-Mills theory

The bulk of their article is a review of Deligne cohomology, which I’ll take for granted here. The main argument which I want to talk about is that described for n=1n=1 and d=4d=4 on p.13,14 and then extended to the general case on p. 35,36. (Their main application, to the relation between M2-branes and D2-branes in section 6 shall not concern me here.

We’ll be talking about “path integrals” in gauge theory, which means we’ll not be rigorous. I hope you can deal with that.

As recalled in Charges and twisted nn-Bundles (I, II, III, IV) the exponentiated action functional for abelian nn-Yang-Mills theory on the Riemannian space (X,g)(X,g) in the presence of electric flux is the assignment (F^ A,j^ E)exp(12g XF AF A)exp(i XA^j), (\hat F_A, \hat j_E) \mapsto \exp( - \frac{1}{2 g}\int_X F_{A} \wedge \star F_A) \exp( i \int_X \hat A \cdot j) \,, where F^ A\hat F_A denotes a line nn-bundle with connection (locally given by an nn-form AA) whose curvature (n+1)(n+1)-form is F AF_A, and where j^ E\hat j_E, the electric flux, is itself a (dn1)(d-n-1)-line bundle, which however here we take to be trivial, given by a global curvatrure (dn)(d-n)-form j Ej_E. The dot on the right is the product in differential cohomology.


REMARK on notation: This corresponds to equation (5.92) in CMR. Notice that these authors have the habit of writing X \int_X when they really mean: choose a good open cover U iIU iU \sqcup_{i \in I} U_i of XX, and choose a triangulation TT of XX subordinate to that cover, i.e. with a lift s:{F k (d)}Is : \{F^{(d)}_k \} \to I of each dd-dimensional face F k (d)F^{(d)}_k to UU, a lift of each (d1)(d-1)-dimensional face to U× XUU \times_X U and so on. Then read expressions like XA \int_X A as k F k (d)A s(F k (d)). \sum_k \int_{F^{(d)}_k} A_{s(F^{(d)}_k)} \,. That kind of implicit gymnastics is necessary to make sense of equations (2.23) and (2.25) for instance.


We want to assume further here, for simplicity, that the electric current j Ej_E is Poincaré-dual to an nn-boundary Σ\partial \Sigma of an (n+1)(n+1)-dimensional submanifold ΣX\Sigma \subset X. Then the action functional simplifies to (F^ A,j E)exp(12g XF AF A)exp(i ΣF A), (\hat F_A, j_E) \mapsto \exp( - \frac{1}{2 g}\int_X F_{A} \wedge \star F_A) \exp( i \int_\Sigma F_A ) \,,


The main point now is the following:

the path integral for our abelian nn-Yang-Mills theory

(1)Z= linenbundleswithconn.D[F^ A]exp(12g XF AF A)exp(i ΣF A) Z = \int_{line n-bundles with conn.} D[\hat F_A] \; \exp( - \frac{1}{2 g}\int_X F_{A} \wedge \star F_A) \exp( i \int_\Sigma F_A )

is over all line nn-bundles F AF_A. But only the curvature (n+1)(n+1)-form F AF_A of these nn-bundles actually appears in the integrand. Therefore we could imagine replacing this by an intregral over all (n+1)(n+1)-forms FΩ n+1(X)F \in \Omega^{n+1}(X) and including a suitable factor which I’ll denote δ closed,integral(F) \delta_{closed,integral}(F) in the integrand which restricts the integral back to those (n+1)(n+1)-forms which actually appear as curvature (n+1)(n+1)-forms of line nn-bundles, i.e. those which are closed and integral:

(2)Z= Ω (n+1)(X)D[F]exp(12g XFF)exp(i ΣF)δ closed,integral(F). Z = \int_{\Omega^{(n+1)}(X)} D[F] \; \exp( - \frac{1}{2 g}\int_X F \wedge \star F) \exp( i \int_\Sigma F ) \delta_{closed,integral}(F) \,.

Now CMR argue (that’s the argument between (2.23) and (2.28) generalized to higher nn) that this factor is itself the path integral over dual line (dn2)(d-n-2)-bundles F^ A˜\hat F_{\tilde A} over the natural pairing of their curvature forms:

(3)δ closed,integral(F)= line(dn2)bundleswithconn.D[F^ A˜]exp(i1gg˜ XFF A˜). \delta_{closed,integral}(F) = \int_{line (d-n-2)-bundles with conn.} D[\hat F_{\tilde A}]\; \exp(i \frac{1}{g \tilde g} \int_X F \wedge F_{\tilde A}) \,.

where

(4)gg˜=2πn. g \tilde g = 2 \pi n \,.

I think a hidden assumption for their argument to make sense is that H (dn1)(X,)=, H^{(d-n-1)}(X,\mathbb{Z}) = \mathbb{Z} \,, i.e. that there are no line (dn2)(d-n-2)-line bundles on XX whose class is a torsion element. Because they decompose the “path integral” over line (dn2)(d-n-2)-bundles with connection into a sum over the corresponding bundles, which then reduces to a sum over integers, and a sum over the possible connections one can put on these bundles. I suppose with this non-torsion assumption (2.25) surely implies (2.28) if one gets the path integral measure under control. That the remaining integral over line (dn2)(d-n-2)-bundles with connection implies (2.24) appears plausible, but much more subtle. (But maybe I am missing something).

In any case, let’s assume that (3) is right.

Then (2) becomes

(5)Z= Ω (n+1)(X)D[F] line(dn2)bundleswithconn.D[F A˜]exp(12g XFF)exp(i ΣF)exp(i XFF A˜). Z = \int_{\Omega^{(n+1)}(X)} D[F] \int_{line (d-n-2)-bundles with conn.} D[F_{\tilde A}] \; \exp( - \frac{1}{2g}\int_X F \wedge \star F) \exp( i \int_\Sigma F ) \exp(i \int_X F \wedge F_{\tilde A}) \,.

We now imagine that we can switch the integration order

(6)Z= line(dn2)bundleswithconn.D[F A˜] Ω (n+1)(X)D[F]exp(12g XFF)exp(i ΣF)exp(i XFF A˜), Z = \int_{line (d-n-2)-bundles with conn.} D[F_{\tilde A}] \int_{\Omega^{(n+1)(X)}} D[F] \; \exp( - \frac{1}{2g}\int_X F \wedge \star F) \exp( i \int_\Sigma F ) \exp(i \int_X F \wedge F_{\tilde A}) \,,

and perform the inner integral. The result is supposedly (5.101)

(7)Z= line(dn2)bundleswithconn.D[F A˜]exp(12g˜ XF A˜F A˜)exp(i *ΣF A˜). Z = \int_{line (d-n-2)-bundles with conn.} D[F_{\tilde A}] \; \exp( - \frac{1}{2 \tilde g}\int_X F_{\tilde A} \wedge \star F_{\tilde A}) \exp( i \int_{*\Sigma} F_{\tilde A} ) \,.

This is now manifestly the kind of path integral that we started with, but now for the dual entities: not for line nn-bundles with connection, but for line (dn2)(d-n-2)-bundles with connection.

The path integrals being equal means: the physical theories, the original nn-gauge theory and the dual (dn2)(d-n-2)-gauge theory are equivalent.

Posted at May 17, 2008 1:27 PM UTC

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Re: Electric-Magnetic-Duality and Hodge Duality Extended to Differental Cocycles

Urs wrote:

We’ll be talking about “path integrals” in gauge theory, which means we’ll not be rigorous. I hope you can deal with that.

Since we’re talking about free field theories here — theories where the action is quadratic — there should be no fundamental obstacle to doing everything rigorously, if one is good at analysis. And this would make a very nice project for the right sort of person, especially since you’ve pointed out some interesting subtleties.

Once I wanted my student Miguel Carrión-Álvarez to tackle this sort of problem. He did a very nice thesis on mathematically rigorous pp-form quantum electromagnetism, and turned up all sorts of subtleties that nobody seems to have noticed, even in ordinary electromagnetism. For example, when space is a cylinder ×S 1\mathbb{R} \times S^1, the electromagnetic field has no Bohm–Aharonov modes even though the first deRham cohomology is nontrivial! Why? What matters is not the deRham cohomology, it’s the L 2L^2 cohomology.

Miguel was an expert at analysis. But, he didn’t really enjoy gerbes or differential cocycles, so we didn’t go in that direction.

By the way: he took a Hamiltonian approach, but it’s also possible to make the Lagrangian approach to quantized free electromagnetic fields rigorous.

Posted by: John Baez on May 17, 2008 10:31 PM | Permalink | Reply to this

Re: Electric-Magnetic-Duality and Hodge Duality Extended to Differental Cocycles

Thanks for your reply!

Since we’re talking about free field theories here — theories where the action is quadratic — there should be no fundamental obstacle to doing everything rigorously,

True. Possibly this is even essentially done in Freed-Moore-Segal (I , II ) or some such place. I’d need to remind myself.

Why? What matters is not the deRham cohomology, it’s the L 2L^2 cohomology.

Hm, interesting. Apparently I am not aware of this. Will have to look at Miguel’s thesis again. I did once, but long ago. Did you ever compare this result with Freed-Moore-Segal’s detailed analysis?

he didn’t really enjoy gerbes or differential cocycles

…and line nn-bundles with connection weren’t around yet :-)

I think for the kind of phenomenon discussed above it is crucial to access also the globally nontrivial case. Only by including the sum over these “soliton” configurations does the above duality argument work, for instance.

By the way: he took a Hamiltonian approach, but it’s also possible to make the Lagrangian approach to quantized free electromagnetic fields rigorous.

Maybe the Hamiltonian approach is actually nicer here. Though it seems the nn-holonomy picture is more immediate in the Lagrangian picture.

Posted by: Urs Schreiber on May 18, 2008 12:48 AM | Permalink | Reply to this

Re: Electric-Magnetic-Duality and Hodge Duality Extended to Differental Cocycles

Once I wanted my student Miguel Carrión-Álvarez to tackle this sort of problem.

I am looking at Miguel Álvarez’s thesis again now.

What matters is not the deRham cohomology, it’s the L 2L^2 cohomology.

Ah, now I see what you mean by L 2L^2-cohomology: the differential cohomology under the constraint that everything appearing satisfies an integrabiliy condition. As in (2.3) of Dirac charge quantization and differential cohomology.

In the footnote on p. 2 FMS promise to look at the ca of noncompact space (which would then demand a similar integrability condition) “in the future”.

Posted by: Urs Schreiber on May 18, 2008 11:13 AM | Permalink | Reply to this

Re: Electric-Magnetic-Duality and Hodge Duality Extended to Differental Cocycles

Hi
First - I consider myself to be an applied physicist-engineer. For 45 years, van Dantzig’s work (from before WWII) convinced me that EM was a topological (not geometrical) theory, and stimulated my interest in applications of topology (based on Cartan’s concepts of exterior differential forms) to deduce a better understanding of non-equilibrium thermodynamic systems.
With regard to Hodge duality, be aware that, experimentally, Hodge duality does not explain Optical Activity or Faraday rotation. To a physicist like me, this produces a cloud on the generality and importance of Hodge duality. These optical properties are easily observable experimental facts. The analysis of Singular Solutions to EM equations (that are not constrained by Hodge duality) led to my invention of dual-polarized ring lasers. Phys Rev A 43 5165 (1991).
Although my work is more mundane, you might find interest in my 4 monographs, that I have constructed in my retirement years summarizing 45 years of applied research.
http://www.lulu.com/kiehn
Often, pratical experiments can lead to topological advances. For example, most authors do not seem to be aware of the fact that the exterior differential is a limit point generator. Hence the evolution of closure properties is best expressed not in terms of homology, but in terms of cohomology. Note that the union of a 1-form A and its limit sets F=dA, has limit points, an interior, has no boundary in the Cartan topology.
Moreover, it is rarely appreciated by mathematicians that Cartan’s Magic Formula is in effect a dynamical expression of the First Law of Thermodynamics.
IN addition string theorists fail to recognize the simple experiments that produce “branes” connected by a string, in a swimming pool. Just Google “Falaco Solitons”.
I will believe string theorists when they can explain “Falaco Solitons”.

Regards RMK

Posted by: R. M. Kiehn on August 18, 2008 2:46 PM | Permalink | Reply to this

Re: Electric-Magnetic-Duality and Hodge Duality Extended to Differental Cocycles

Dear R. M. Kiehn,

thanks for your message and for the interesting links!

Hodge duality does not explain Optical Activity or Faraday rotation.

Maybe an issue is here the difference between “fundamental” physics, such as electromagnetism in vacuum, and “complex” physics such as electromagnetism in nontrivial media. The kind of consideration as in the above entry are meant to apply to fundamental physics, where everything is much cleaner and more pure than in the messy and dirty real word which contains such comparatively hugely complex backgrounds as optically active media.

[…] limit point generator […]

I may be one of those who don’t know what what this “limit point” terminology here means. Could you explain?

Posted by: Urs Schreiber on August 19, 2008 10:26 AM | Permalink | Reply to this

Re: Electric-Magnetic-Duality and Hodge Duality Extended to Differental Cocycles

The idea of limit points is a topological concept related to Cohomology where the Closure of a subset is the union of the set and its limit points.
The topological concept of closure can also be used in Homology theory, where the Closure of a subset is union of the interior of the set and its boundaries.
When using differential p-forms on a Kolmogorov topology, the exterior differential of a p-form, say A, generates the “limit” points of the p-form as a p=1 form, dA. The limit points are the domain of support for
dA. (where dA is not zero.)

Get the Schaums Outline on Topology, by Lipshitz.

Posted by: Kiehn on May 8, 2009 9:24 AM | Permalink | Reply to this

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