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I have expanded Green-Schwarz mechanism a fair bit
added pointer to the original:
added pointer to today’s
I’m sure this was sorted out long time ago but where exactly does the anomaly polynomial $I_{n+2}$ live?
That’s a very good question:
The anomaly polynomial is a generalized element of the sheaf internal hom $[X, \Omega^{n+2}]$, where $X$ is the $n$-dimensional spacetime manifold, regarded as a smooth set, and $\Omega^{n+2}$ is the “smooth set of forms”, hence the sheaf of forms on $CartSp$.
More concretely, this means that for $U$ any Cartesian space and considering fields on $X$ in families parameterized by $U$, then the anomaly polynomial for that family of fields is an ordinary $n+2$-form on $X \times U$.
That’s how the anomaly polynomial can have degree higher than the dimension of spacetime without trivially vanishing.
I thought that I had note on this point somewhere on the nLab, but right now I don’t find it.
right now I don’t find it.
Oh, it’s right here in this entry, under The anomaly line bundle.
(Also, if only most most briefly, footnote 2 in Twisted Cohomotopy implies M5-brane anomaly cancellation.)
By the way, you can see a hint in this direction in Freed 2002 Exp. 2.28 (p. 21), where the manifold indexing a given family of fields is denoted “$T$” instead of “$U$”.
The question left open there is which $T$ to choose, if any, and how to relate these. The answer is to work over the site of all manifolds $T$ (or just the Cartesian spaces among them, as that is sufficient and equivalent) and consider the resulting system of anomaly forms as a single form on a moduli stack.
Wait, so how standard or widespread is this formulation? I’m only finding this in dcct (and whatever paper this was first worked out).
As I said in #7, it’s implicit in Freed 2002. Of course he thinks that “the s-word conjures up demons” (p. 3 here) and hence won’t use it.
Funny, that explains the nature of 2209.07471…
So why do we only use higher U(1) connections to cancel anomalies? Or is it that in principle one could use bundles with connection of other higher groups to cancel anomalies (with some appropriate map to U(1), such as exp(Tr…)) but the U(1) case is easier because its eom’s are simpler? Obviously, in those other cases you would require something different than $I\wedge j$.
Right, analogous constructions would apply more generally.
Of course, from the perspective of anomalies produced by a fermionic path integral, the group $U(1)$ is fundamentally the group of phases in which the exponentiated action functionals take values. This basic fact of quantum physics is ultimately the “reason” why all this discussion is about higher $U(1)$-bundles.
(One could of course consider using other/higher “groups of phases”…)
Also, this is clearly a global perspective for the anomaly. But what (I’ve seen that) people call a global perspective (e.g. the recent 2310.06895), instead of using $U$-parameterised fields so as to allow higher-degree forms, they use manifolds of one or two dimensions more than spacetime so as to support the anomaly polynomial, from where they argue that one only needs to compute some cobordism invariant to check the anomaly is zero. How are these two pov’s compatible?
I think that essentially comes down to using just a fixed $U = \mathbb{R}^2$ and crossing your fingers that no information is lost.
So just to clarify, the anomaly polynomial acts as a magnetic source for the prequantum circle n-bundle, right? Or in other words, it makes the pre-n-plectic form not closed?
That’s not quite how it’s usually thought of, at least:
The complex line bundle on bosonic field space (after having performed the fermionic path integral, which is well-defined) of which the GS-type anomaly polynomial is (supposed to be) the curvature 2-form (after integration over spacetime) is meant to be that object of which the exponentiated action functional $exp(\mathrm{i} S)$ is a section.
The idea is that to even start constructing the phase space (pre-quantumly or even classically) or to even start making sense of the path integral, the expression $exp(\mathrm{i} S)$ must be an actual function instead of a section, hence that line bundle must be trivial (and trivial-ized), hence the class of that bundle (which is the anomaly, by definition) must vanish.
Only with the anomaly vanishing, so that the expression $exp(\mathrm{i} S)$ is turned into an actual function can we ask that it comes from a Lagrangian density, from which we then get the phase space with its (pre-)symplectic form, which then we can pre-quantize to a prequantum line bundle.
But wouldn’t what I mentioned be the detransgressed picture? Of course we wouldn’t call $\omega:Fields\to \Omega^n$ a phase space since $d\omega=I\neq 0$, but in this sense the Lagrangian density would be a twisted circle n-bundle with connection, and perhaps one can still investigate those morphisms $\mathbf{B}^n U(1)_{tw}\to \mathbf{B}GL_1 (E)$ in quantization?
Maybe you are thinking by analogy of the case of a charged particle propagating on a manifold $X$ subject to an electromagnetic line bundle on that manifold, in which case the canonical symplectic form on the particle’s phase space $T^\ast X$ gets shifted by the pullback of the curvature 2-form of that line bundle.
That story will go through in higher generalization along the lines you are indicating: The higher pre-symplectic phase space of the string propagating in a background with a higher GS-gauge field (whose curvature satisfies $\mathrm{d} H_3 \propto \langle R \wedge R \rangle - \langle F \wedge F\rangle$) should carry the canonical 2-symplectic form shifted by a “twisted-closed” 3-form being the pullback of $H_3$ from spacetime.
But here we are talking about the phase space of the single string, not the phase space of its target space field theory. Maybe that’s what you have in mind?
Indeed I’m talking about the target space field theory. Say for an n-dim’l target space theory, we would want to have a circle $n$-bundle with connection $L:Fields\to B^n U(1)_{conn}$ that would be a Lagrangian density. The description in the current nlab entry is phrased in terms of an action $S$ on an $n$-dim’l spacetime $X$, where in the case of no anomalies this is a well-defined function to $U(1)$ obtained as
$\exp(i S(-)) \coloneqq \exp(2 \pi i \int_{X} [X, \mathbf{L}]) \;\; \colon \;\; Conf= [X, Fields] \stackrel{[X, \mathbf{L}]}{\to} [X, \mathbf{B}^n U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{X}(-))}{\to} U(1) \,.$The existence of anomalies means that there is no well-defined action $exp(i S(-)): Conf\to U(1)$. We usually start with a local Lagrangian, say as in d=10 N=1 sugra + SYM. If this Lagrangian was indeed the local form of a circle n-bundle then we would have a well-defined function to $U(1)$ via fiber integration. But since we don’t have this function then it seems to me one can regard $L$ as being the local expression of a circle n-bundle twisted by some form $I$ s.t. $dF_L=I$, where this $I$ is a $n+2$ form, the anomaly polynomial, which is why I said it acts as a magnetic source for a higher prequantum bundle. Or is this not correct?
Aren’t you still conflating the anomaly bundle whose sections are action functionals with the prequantum bundle whose sections are quantum states?
Yes, in a sense I’m asking whether the two come together as a twisted circle n-bundle where the twist is the anomaly. Or here’s a different question, what does it mean to twist a higher prequantum bundle? Is this not introducing an anomaly?
Right, so the term “anomaly” refers to various different phenomena. All it means in full generality is when in the long cooking recipe for quantization of field theories, some conditions necessary for some step fail.
The terminology reflects the physicist’s attitude towards how the world works:
Contrary to mathematicians who first check the assumptions of their theorem and then work out the implications, the physicist storms ahead, fingers crossed.
If things don’t work out as expected because tacit assumptions are not actually satisfied, the physicist grudgingly acknowledges that the situation is “not normal” (anomalous), to his mind, because he feels that: normally this works (last time it worked!, so).
Now he backtracks his steps to see what one could fiddle with to save the day. Once such a hack has been implmented and the machine seems back on track again, he proclaims, with satisfaction, that he has “canceled the anomaly”.
One could call it the drama-approach to mathematical physics.
$\,$
The Green-Schwarz-type anomaly of this thread is one specific such situation, namely the case where one considers the effective bosonic action functional of a field theory with chiral fermions (or other chiral fields) by performing the fermionic path integral, noticing that the result is not quite a function but only locally a section of a Pfaffian line bundle, observing that however the class of that line bundle is the transgression of a factorizable class of the same form as an electro-magnetic anomaly, and then “cancelling” the problem by introducing the required opposite magnetic charge.
But there are other kinds of anomalies than this.
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