The n-Category Café

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

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

of the manifold $Y$ 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 $A$, a twisting 2-form $B$ of its curvature $F = d A + B$ such that

$$d F = j_E$$

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

This we read as saying that $F$ 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 $j_E$.

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

$$d (\star F) = j_E$$

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

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

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

we identity the relevant $L_\infty$-algebra as

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

This is such that smooth maps

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

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

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

are given by

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

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 $L_\infty$-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 $X_{CE(g)}$: a $(d+1)$-bundle with connection and “with section” (i.e. trivialized by a twisted $d$-bundle) which “couples” to our parameter space $Y$ “propagating” in $X_{CE(g)}$.

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

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

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

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

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

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

Notice that our “$d$-particle” $Y$ (meaning: our $d$-dimensional parameter space) couples here to a possibly twisted $d$-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 $L_\infty$-conections, we do so by essentially forming the inner hom out of parameter space, meaning here that we hit the background field morphism

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

with the functor

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

to get

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

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 $T$-parameterized families that Freed considers on p. 21 arise automatically here from forming the inner hom, since a degree 0 element in

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

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

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 $(d+1)$-bundle on $X_{CE}(g)$ 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

$$\int_Y j_E \wedge F = \int_Y j_E \wedge d A = d \int_Y j_E \wedge A \,.$$

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

$$\int_{S^1} \gamma^* A \,,$$

the usual coupling of the electromagnetic field to electric charges.

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