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Generalized Charges, Part II: Non-Invertible Symmetries and the Symmetry TFT
by Lakshya Bhardwaj, Sakura Schäfer-Nameki
Submission summary
| Authors (as registered SciPost users): | Lakshya Bhardwaj |
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| Preprint Link: | https://arxiv.org/abs/2305.17159v2 (pdf) |
| Date submitted: | 2023-10-17 11:08 |
| Submitted by: | Bhardwaj, Lakshya |
| Submitted to: | SciPost Physics |
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| Academic field: | Physics |
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Abstract
Consider a d-dimensional quantum field theory (QFT) $\mathfrak{T}$, with a generalized symmetry $\mathcal{S}$, which may or may not be invertible. We study the action of $\mathcal{S}$ on generalized or $q$-charges, i.e. $q$-dimensional operators. The main result of this paper is that $q$-charges are characterized in terms of the topological defects of the Symmetry Topological Field Theory (SymTFT) of $\mathcal{S}$, also known as the ``Sandwich Construction''. The SymTFT is a $(d+1)$-dimensional topological field theory, which encodes the symmetry $\mathcal{S}$ and the physical theory in terms of its boundary conditions. Our proposal applies quite generally to any finite symmetry $\mathcal{S}$, including non-invertible, categorical symmetries. Mathematically, the topological defects of the SymTFT form the Drinfeld Center of the symmetry category $\mathcal{S}$. Applied to invertible symmetries, we recover the result of Part I of this series of papers. After providing general arguments for the identification of $q$-charges with the topological defects of the SymTFT, we develop this program in detail for QFTs in 2d (for general fusion category symmetries) and 3d (for fusion 2-category symmetries).
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The paper discusses finite generalized global symmetries and their representations from the perspective of symmetry topological field theory (SymTFT). The main proposal is that there is a correspondence between the set of representations for $q$-dimensional defects, which are referred to as "$q$-charges" in the manuscript, and the set of $(q+1)$-dimensional topological defects in the corresponding SymTFT.
The authors convincingly argue why such a correspondence is physically natural, especially through the discussions in Section 3.1. The proposal is made for arbitrary spacetime dimensions, and explicit examples are given in 1+1 and 2+1 dimensions.
Perhaps it needs to be emphasized, however, that such an observation and the main physical insight in the context of fusion category symmetries of 1+1d quantum field theories had already appeared in the literature previously, in particular in https://arxiv.org/pdf/2208.05495.pdf (please see more on this below).
The main technical advance made in the current manuscript in that regard is that the similar structure is also demonstrated in the 2+1d examples with fusion 2-category symmetries discussed in Section 5.
The referee believes that the main proposal made by the authors is convincing and expected to be true in general spacetime dimensions (with the caveat that some of the necessary mathematical concepts may need to be defined properly in higher dimensions, please see below). Moreover, the paper sets a natural stage for further developments in the study of infrared phases of condensed matter systems having generalized global symmetries, some of which have already appeard in the literature. The paper therefore deserves to be published in SciPost, once the comments below are addressed.
Requested changes
(1) Throughout the paper, fusion $n$-categories and their Drinfeld centers for arbitrary values of $n$ are mentioned frequently. However, it is not very clear if such mathematical concepts have actually been properly defined for $n>2$ in the mathematics community, although the physics picture provided by the authors is clear.
If they have been defined mathematically even for $n>2$ (relevant for 3+1d and higher), it would be helpful for the interested readers if the corresponding math papers are cited. If they are not yet defined mathematically, then explicitly acknowledging this somewhere in the manuscript would help providing the readers with a better idea on the current state of the arts on the math side.
(2) In page 61, there is a following sentence: "However the perspective of 0-charges of the symmetry S as topological line defects in the SymTFT is new."
However, to the best of referee's knowledge, such an obeservation was first made in the physics literature in an earlier paper https://arxiv.org/pdf/2208.05495.pdf, based on results from the math literature in the 90's and early 2000. Maybe the above sentence needs to be modified accordingly.
In referee's opinion, the above paper by Lin et al deserves to be mentioned even around the Main Statement (i.e. around Eq. (1.1)) of the current manuscript in the Introduction, as the first paper where the same statement has been explained in physics terms for the special case of $d=2$.
(3) The normalization for the condensation defects in (1.17) and (1.18) are incorrect, which is important for the locality of the defect (e.g. to have a well-defined defect Hilbert space). The denominator must be either $|H^0 (M_2,\mathbb{Z}_N)|$ or $\sqrt{ |H_1 (M_2, \mathbb{Z}_N)| }$. The two choices are related by an Euler counterterm on the defect.
Below are minor typos:
- In the second line on page 14, the 2-morphisms must be "Topological defects of codimension-3" rather than codimension-2.
- In Eq. (1.4), maybe an $i$ is missing compared to (1.3)?