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March 4, 2008

Charges and Twisted n-Bundles, II

Posted by Urs Schreiber

Last time I recalled how the historically big insight

\;\;\bullet an electromagnetic field is a line bundle with connection

has to actually be replaced, more generally, by the statement

\;\;\bullet an electromagnetic field is a twisted line bundle, i.e. a “gerbe module” or “2-section” of the magentic charge line 2-bundle.

This time I recall Freed’s description of the Euclidean action for electromagnetism in the presence of electric currents. Then, again, I rephrase everything in the language of L L_\infty-connections (blog, arXiv) and the arrow-theoretic Σ\Sigma-model (slide 11).

I’ll do so for the very simple case where all nn-bundles appearing are actually trivial, so that only their connection forms matter. This makes most of the differential cohomology/nn-bundle terminology overkill, but allows to nicely see how the action functional on configuration space arises from transgression of a “background field”, following the general tao.

For simplicity and definiteness, I’ll assume throughout that the dimension

d:=dim(Y) d := \mathrm{dim}(Y)

of the manifold YY on which we are studying electromagnetism is even. (Otherwise some trivial signs will change in some formulas.)

Our electric bundle, assumed to be trivial, has a connection 1-form AA, a twisting 2-form BB of its curvature F=dA+BF = d A + B such that

dF=j E d F = j_E

with j 3j_3 the closed electric current 3-form. (It may happen that I say “charge” instead of “current”, being sloppy.)

This we read as saying that FF is the curvature 2-form of a twisted 1-bundle with connection, the twist being the the trivial 2-bundle with connection whose curvature 3-form is j Ej_E.

Similarly, there is a twisted magnetic bundle, whose (d11)(d-1-1)-form curvature

d(F)=j E d (\star F) = j_E

trivializes a 3-form called the electric current. (I write F\star F to indicate that this is supposed to be the Hodge star of the original FF, but for what I will say here F\star F could be a symbol denoting any d2d-2-form trivializing j Ej_E.)

A choice of this data is a field configuration of electromagnetism on YY. In order to phrase this as the field configurations of a generalized Σ\Sigma-model, such that each such configuration is a morphism

Y(someclassifyingspaceforsuchformdata) Y \to (some classifying space for such form data)

we identity the relevant L L_\infty-algebra as

g=(u(1)u(1))(b d3u(1)b d3u(1)). g = (u(1) \to u(1)) \oplus (b^{d-3} u(1) \to b^{d-3} u(1)) \,.

This is such that smooth maps

YX CE(g) Y \to X_{CE(g)}

from YY to the smooth space X CE(g)X_{CE(g)} (as described in Impressions on Lie-infinity theory) are precisely form data as above: since DGCA morphisms

W(g) (A,B,F,j B),(F,j E) Ω (Y) \array{ W(g) \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;\downarrow^{(A,B,F, j_B),(\star F, j_E)} \\ \Omega^\bullet(Y) }

are given by

A Ω 1(Y) F=dA+B Ω 2(Y) dF=:j B Ω closed 3(Y) d(F)=j E Ω closed d1(Y). \array{ A & \in \Omega^1(Y) \\ F = d A + B & \in \Omega^2(Y) & \\ d F =: j_B & \in \Omega^3_{closed}(Y) \\ \\ d(\star F) = j_E & \in \Omega^{d-1}_{closed}(Y) } \,.

The first line isthe connection 1-form.

The second line is the curvature 2-form.

The third line is the electric Bianchi identity twisted by magnetic current 3-form.

The fourth line is the magnetic Bianchi identity twisted by electric current (d-1)-form.

(Compare maybe with the description of L L_\infty-connections on twisted bundles here and with Freed’s discussion p. 33, p. 34.)

For this field content of our theory, we now obtain an action functional by choosing a background field on X CE(g)X_{CE(g)}: a (d+1)(d+1)-bundle with connection and “with section” (i.e. trivialized by a twisted dd-bundle) which “couples” to our parameter space YY “propagating” in X CE(g)X_{CE(g)}.

As for the Chern-Simons bundle on BGB G on p. 90 of L L_\infty-connections, this will be given by a diagram

CE(g) CE(b d1u(1)b d1u(1)) W(g) W(b d1u(1)b d1u(1)) inv(g) inv(b d1u(1)b d1u(1)). \array{ CE(g) &\leftarrow& CE(b^{d-1} u(1)\to b^{d-1} u(1)) \\ \uparrow && \uparrow \\ W(g) &\stackrel{}{\leftarrow}& W(b^{d-1} u(1) \to b^{d-1} u(1)) \\ \uparrow && \uparrow \\ inv(g) &\stackrel{}{\leftarrow}& inv(b^{d-1} u(1) \to b^{d-1} u(1)) } \,.

Since we assume all nn-bundles in question to be trivial, I can get away with just looking at the morphism

W(g)W(b d1b d1). W(g) \stackrel{}{\leftarrow} W( b^{d-1} \to b^{d-1}) \,.

By definition (since, as we shall see, this will reproduce the ordinary familiar action functional for electromagnetism coupled to electric currents) we define this morphism by

W(g) W(b db d) (A,F,j B),(F,j E) (),(A,F A) Ω (Y×U) = Ω (Y×U) \array{ W(g) &\stackrel{}{\leftarrow} & W( b^{d} \to b^{d}) \\ ^{(A,F, j_B),(\star F, j_E)}\downarrow\;\;\;\;\;\;\; && \;\;\;\;\;\;\;\downarrow^{(\cdots),(\mathbf{A},\mathbf{F}_{\mathbf{A}})} \\ \Omega^\bullet(Y \times U) &\stackrel{=}{\rightarrow}& \Omega^\bullet(Y \times U) } with curvature (d+2)(d+2)-form F A=j Ej B \mathbf{F}_{\mathbf{A}} = j_E \wedge j_B and corresponding connection (d+1)(d+1)-form A=12((F)B+j EA), \mathbf{A} = \frac{1}{2}(\;\, (\star F) \wedge B + j_E \wedge A \;\;) \,, which is hence also the curvature (d+1)(d+1)-form of the twisted dd-bundle whose form data I won’t spell out.

Notice that our “dd-particle” YY (meaning: our dd-dimensional parameter space) couples here to a possibly twisted dd-bundle background field. This means that as we transgress that background field to a 0-bundle on configuration space as usual, we might end up with a twisted 0-bundle. A 0-bundle is nothing but a function, so this would be our action functional – but a twisted 0-bundle is nothing but a section of an ordinary bundle.

If that twisting bundle on configuration space is non-trivial, there is no way to think of our section as a function, and hence no way to think of the “action” we get really as a function. This is one type of anomaly: the action function may fail to actually be a function.

To find out if that is the case, we need to compute the transgression.

By the general formalism from section 9.2 of L L_\infty-conections, we do so by essentially forming the inner hom out of parameter space, meaning here that we hit the background field morphism

W(g)((F)B+j EA,j Ej B)W(b d1b d1) W(g) \stackrel{((\star F) \wedge B + j_E \wedge A, j_E \wedge j_B)}{\leftarrow} W( b^{d-1} \to b^{d-1})

with the functor

maps(,Ω (Y)) maps(--, \Omega^\bullet(Y))

to get

maps(W(g),Ω (Y)) Y((F)B+j EA,j Ej B)maps(W(b d1b d1),Ω (Y)), maps(W(g),\Omega^\bullet(Y)) \stackrel{\int_Y((\star F) \wedge B + j_E \wedge A, j_E \wedge j_B)}{\leftarrow} maps(W( b^{d-1} \to b^{d-1}), \Omega^\bullet(Y)) \,,

where I denoted the transgression operation on the morphism somewhat sloppily by its image after doing integration without integration.

The main point is this: the TT-parameterized families that Freed considers on p. 21 arise automatically here from forming the inner hom, since a degree 0 element in

maps(W(g),Ω (Y)):=Ω (UHom DGCA(W(g),Ω (Y)Ω (U))) maps(W(g),\Omega^\bullet(Y)) := \Omega^\bullet( U \mapsto Hom_{DGCA}(W(g), \Omega^\bullet(Y)\otimes \Omega^\bullet(U)) )

comes from a 0-form on each plot UU, natural in UU.

Suffice it to say that as a result we find that the transgressed background field is indeed a 0-form on configuration space, possibly being not a function but a section of a line bundle on config space, which is the transgression of the twisting (d+1)(d+1)-bundle on X CE(g)X_{CE}(g) that we started with.

I realize that this deserves more details spelled out than I do here and more than should be squeezed in here, so I end simply by pointing out how the familiar action functional is recovered:

suppose the twist vanishes (no magnetic charges), then the twisted curvature of our action becomes simply

Yj EF= Yj EdA=d Yj EA. \int_Y j_E \wedge F = \int_Y j_E \wedge d A = d \int_Y j_E \wedge A \,.

So the action itself, if we take the electric current j Ej_E to be Poincaré-dual to the worldline γ:S 1Y \gamma : S^1 \to Y of a charged particle (the prefactor given by the charge, qq), becomes

S 1γ *A, \int_{S^1} \gamma^* A \,,

the usual coupling of the electromagnetic field to electric charges.

(And in case you are wondering: I simply suppressed the remaining Maxwell-term YFF\int_Y F \wedge \star F from the discussion, since it does not participate in the twisting business.)

Posted at March 4, 2008 2:00 PM UTC

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