# Anomaly in QFT physics v.s. determinant line bundle

In a quantum field theory (QFT) lecture, a math-physics professor explains the anomaly in physics, say the non-invariance of the partition function of an anomalous theory under background field transformation, can be regarded as a section of a complex line bundle over the space of background field.

In math terminology, he explained that the so-called anomaly in physics is the determinant line bundle in math.

• How precise is the analogy:

anomaly in QFT physics v.s. determinant line bundle

• can you provide a few examples between the twos?
• The answer by Severin Bunk is pretty good but I think more answers could be useful too, hopefully someone could go further on the parallel between physics and maths terminology or share another perspective or give a illustrative example, etc. Apr 27 '20 at 16:39
• Almost-duplicate question on physics:SE: physics.stackexchange.com/q/254984/50583 Apr 27 '20 at 18:51
• I’m addition to the complete works of Dan Freed mentioned below, you might like Moore and Nelson’s [The Aetiology of Sigma Model Anomalies] (repository.upenn.edu/cgi/…) Apr 28 '20 at 2:00
• To be a little more specific on Dan’s stuff, start with Determinants, Torsion and Strings (projecteuclid.org/download/pdf_1/euclid.cmp/1104116145) and On Determinant Line Bundles (projecteuclid.org/download/pdf_1/euclid.cmp/1104116145) Apr 28 '20 at 2:17
• @AaronBergman: Thanks for pointing out Moore's paper -- I had not seen it before, but it looks great! Apr 28 '20 at 6:21

The partition function should assign to each possible field configuration $$\Phi$$ (or field history) in your quantum field theory a number $$Z(\Phi)$$. That is, it should be a function on the collection of fields configurations, and from that function you can derive lots of quantities in the field theory. It can happen, however, that in order to come up with, or write down, the number $$Z(\Phi)$$, you need to make some auxiliary choices. Often these choices work only for certain fields, rather than for all fields at the same time. For instance, in a gauge theory with fermions you might need to choose a real number $$\lambda$$ which is not in the spectrum $$\sigma(D_A)$$ of the Dirac operator coupled to the gauge field $$A$$. In general, such a choice cannot be made for all gauge potentials $$A$$ simultaneously, and hence exists "only locally" on the collection of fields. Different local choices of auxiliary information will lead to different values of what you compute as $$Z(\Phi)$$, and it usually turns out that the transformation law between the values of $$Z(\Phi)$$ for different auxiliary choices is that of a section of a line bundle on the collection of fields.
What people normally want in gauge theory is for the partition function to be well-defined on the space $$\mathcal{A}/\mathcal{G}$$ of gauge potentials modulo gauge transformations. While any line bundle on $$\mathcal{A}$$ is trivialisable (since $$\mathcal{A}$$ is an affine space), this is not true on the quotient $$\mathcal{A}/\mathcal{G}$$. Let us say that our partition function can be understood as a section $$Z$$ of a line bundle $$L \to \mathcal{A}/\mathcal{G}$$. Then, any trivialisation of $$L$$ allows us to translate $$Z$$ into a function on $$\mathcal{A}/\mathcal{G}$$, and hence into an actual partition function. The QFT anomaly can hence be described as the obstruction to the existence of a trivialisation of $$L$$ -- this is a class in $$H^2(\mathcal{A}/\mathcal{G};\mathbb{Z})$$. Often this class can be computed, like in the case of the Dirac anomaly.
• Thanks voted up, so in one sentence, how is the anomaly in QFT in physics = determinant line bundle? Or is that a class in $H^2(\mathcal{A}/\mathcal{G};\mathbb{Z})$? Apr 27 '20 at 15:11
• Yes, you can interpret the QFT anomaly, or at least the chiral anomaly, as the Chern class in $H^2(\mathcal{A}/\mathcal{G};\mathbb{Z})$. There are more anomalies which have different descriptions, for instance they do not have to live on a space $\mathcal{A}/\mathcal{G}$ of gauge equivalence classes. Apr 28 '20 at 6:20