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Lorentz Symmetry Fractionalization and Dualities in (2+1)d
by PoShen Hsin and ShuHeng Shao
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Authors (as registered SciPost users):  PoShen Hsin · ShuHeng Shao 
Submission information  

Preprint Link:  scipost_201910_00031v1 (pdf) 
Date accepted:  20200129 
Date submitted:  20191018 02:00 
Submitted by:  Shao, ShuHeng 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
We discuss symmetry fractionalization of the Lorentz group in (2+1)d nonspin quantum field theory (QFT), and its implications for dualities. We prove that two inequivalent nonspin QFTs are dual as spin QFTs if and only if they are related by a Lorentz symmetry fractionalization with respect to an anomalous $\mathbb{Z}_2$ oneform symmetry. Moreover, if the framing anomalies of two nonspin QFTs differ by a multiple of 8, then they are dual as spin QFTs if and only if they are also dual as nonspin QFTs. Applications to summing over the spin structures, timereversal symmetry, and level/rank dualities are explored. The Lorentz symmetry fractionalization naturally arises in ChernSimons matter dualities that obey certain spin/charge relations, and is instrumental for the dualities to hold when viewed as nonspin theories.
Published as SciPost Phys. 8, 018 (2020)
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 202016 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_201910_00031v1, delivered 20200106, doi: 10.21468/SciPost.Report.1428
Report
This paper introduces a fractionalization map, and relates this map to
the symmetry fractionalization of the Lorentz group, the 1form Z_2
symmetry anomaly. The paper also proves Lemma 1 and Theorem 1, and
briefly comment on the implications for timereversal symmetry.
Referee will be happy to recommend the paper for publication as long
as the authors can answer/resolve Referee's questions/comments:
1) The authors focus on only the 1form Z_2 symmetry. Referee wonders
whether there can be any generalization of 1form Z_n or 1form U(1)
electric or 0form magnetic symmetry story for the fractionalization map? If
so, can the authors make some comments on this generalization?
2) Also, what is the validity of the main Lemma 1, Theorem 1 and Collorary 1?
It looks that in Abstract it says a stronger claim for nonspin QFT
and spin QFT; while in the main text, it says the more restricted nonspin TQFT and spin TQFT. Some clarifications are required.
3) The authors define a fractionalization map, which can map a QFT to
another QFT. However, the original QFT is known to be welldefined (for example, the bosonic TQFT needs to be given by data of modular tensor category). But how do we know whether the output data is also a welldefined QFT?
4) Is the essence of framing anomaly, simply equal to the chiral
central charge c mod 8?
The entries given in eq.(2.6) and eq.(2.7) on the q charges are
related to the modular S matrix of SL(2,Z). Maybe the authors can add
a comment on this.
5) Suppose the timereversal symmetries are involved, such that we
require the timereversal quantum number for eq.(2.6) and eq.(2.7),
etc. Do we have a generalized fractionalization map? Would this be a
case for lifting Spin to Pin^+ and Pin^, or are there something more
than that for a generalized fractionalization map? (p.s. there are
some generalization of Lemma 1 and Theorem 1 in Sec 4.3 Implications
for timereversal symmetry, but it may be useful to know easier
examples.)
6) In p.8, "In the case of the Lorentz symmetry fractionalization
(2.1), the oneform symmetry anomaly gives rise to the framing
anomaly. Here we discuss the change of the framing anomaly under the
fractionalization map."
1form symmetry can induce framing anomaly, by requiring, for example,
B= \pi w_2. and then P(B) \sim P(w_2) \sim RR which produces the
framing anomaly.
But does a nonzero 1form symmetry anomaly imply a framing anomaly
(chiral central charge c mod 8), in 3d? Is this always true?
7) In eq (2.21), the authors claimed that when p=0 mod 4, F[T]<>T. As
said in the introduction (on the top of page 3), the author claimed
that F, in this case, is a 0form symmetry of the theory. However, this
is clearly not an automorphism of the lines (which is usually
discussed in the anyon theory, e.g. https://arxiv.org/abs/1410.4540)
because the topological spin of the lines are not preserved by F. The
authors may need to clarify more about the differences between the
usual 0form symmetry in the anyon theory (realized as an automorphism of
the lines) and the F map.
8) In p.13, "In the generalization to bosonic quantum field theory,
the framing anomaly in the following discussions can be defined by the
coefficient of the parityodd contact term in the stress tensor
twopoint function."
In this definition, can the framing anomaly still equal to the chiral
central charge c mod 8? Is the parityodd contact term only a mod 2
class or mod N for what N?
9) In eq.(4.4), there is a list of 1form symmetry of Spin(M)_1. Does
this 1form G_{[1]} symmetry also depend on the levelk of Spin(M)_k? What is
the dependence of k and the G_{[1]}?
10) In p.17, "The difference between gauging (−1)^F versus summing over
the spin structures is that the former does not project out the
transparent fermion line, while the latter does. Consequently, gauging
(−1)^F of a spin TQFT gives 8 distinct spin TQFTs, while summing over
the spin structures of a spin TQFT gives 16 distinct nonspin TQFTs. "
Referee does not fully grasp this meaning and gets confused. Shouldn't
the gauging (−1)^F be equivalent to a bosonization procedure?
Shouldn't sum over the spin structures also a bosonization procedure?
Shouldn't both procedures term a spin QFT to a nonspin QFT?
If (−1)^F is gauged, then there is no fermion parity symmetry and thus
no (−1)^F symmetry, shouldn't the theory be bosonic and nonspin? How
is this procedure related to "fermionic SPT phases in higher
dimensions and bosonization in https://arxiv.org/abs/1701.08264"?
11) Referee finds some recent or previous works relevant to this
Lorentz symmetry fractionalization (with or without timereversal on
nonspin manifold) can be useful for readers:
https://arxiv.org/abs/1911.00589
https://arxiv.org/abs/1712.08639
https://arxiv.org/abs/1711.11587
12) In p.16 the the Z_2 lines > typo.
13) In p.13, in Lemma 1, to identify the 1form Z_2 symmetry, do the
1form symmetry matched for both sides of eq (4.1) for all L of
SO(L)_1? Does \pi_1(SO(L)) = Z_2 for L >2 correspond to any
symmetry? Say, 1form symmetry or 0form magnetic symmetry in 3d?
14) Lastly and importantly, given a gauge theory of a gauge group G_{gauge},
is this correct that we can determine that 1form electric symmetry by
looking at the two data: center G_{gauge} and the matter field representation? Are there other hidden 1form symmetry not given by the data, but depending on the ChernSimons level? How about the 0form magnetic symmetry?
Author: PoShen Hsin on 20200108 [id 700]
(in reply to Report 1 on 20200106)1) For 1form symmetry A in d spacetime dimension, the fractionalization maps correspond to the elements in H^2(SO(d),A). For A=Z_n there is nontrivial map for even n and only trivial map for odd n, and similarly A=U(1) has a nontrivial map.
For 0form symmetry A^{(0)} a generalized fractionalization map can be defined for timereversal invariant theories, given by H^1(O(d),A^{(0)}) i.e. the map changes how timereversal symmetry acts on the operators odd under Z_2 subgroups of the 0form symmetry A^{(0)}.
For qform symmetry A^{(q)} the generalized fractionalization map activates the background for the qform symmetry given by the pullback to the spacetime of the elements in H^{q+1}(SO(d), A^{(q)}), or more generally H^{q+1}(O(d), A^{(q)}) when there is timereversal symmetry.
2) They are valid for general nonspin QFTs (explained in the second paragraph of section 4). We will rephrase the word TQFT to be QFT in the theorems.
3) The fractionalization map activates a background for the global symmetry in a welldefined QFT, thus it gives a welldefined QFT (the theory can have an 't Hooft anomaly as in general QFTs).
4) The framing anomaly c can be understood as the chiral central charge c mod 8. The braiding of symmetry line a and general line b can be expressed as S_{a,b}/S_{0,b} where 0 denotes the trivial line and S is the modular S matrix.
5) Yes. For instance, consider the timereversal symmetry that does not permute the lines in the untwisted Z_2 gauge theory. There is a generalized fractionalization map given by activating the following background for the Z_2 1form symmetry generated by the f line B = w1^2. This map changes the e and m particles (that are charged under the 1form symmetry) to be both Kramers doublet. One can also consider Pin^\pm(d) bundles (that are special cases of O(d) bundles), where for Pin^+ (Pin^) one sets w2=0 (w2+w1^2=0) so there are fewer distinct generalized fractionalization maps.
6) A theory with an anomalous 1form symmetry does not necessarily have a framing anomaly. For instance, in the twisted Z_2 gauge theory the Z_2 1form symmetry generated by the semion is anomalous (since it is not a boson), but the Z_2 gauge theory itself has trivial framing anomaly.
7) Unlike unitary symmetry, an antiunitary symmetry satisfies (72) of https://arxiv.org/pdf/1410.4540.pdf with complex conjugation. The introduction refers to the example in section 2.5 of twisted Z_2 gauge theory where the map F with respect to the boson is a timereversal symmetry.
8) The contact term odd under parity transformation also corresponds to the gravitational ChernSimons term (or tr RR term in 4d bulk) and it is related to the chiral central charge. In bosonic theory this contact term has ambiguity of mod 16 in the normalization of https://arxiv.org/pdf/1206.5218.pdf coming from adding properlyquantized bosonic gravitational ChernSimons term. In the discussion of section 4 this coefficient is identified with 2c.
9) The 1form symmetry group of Spin(M)_k is given by the center of Spin(M) and it does not depend on the level k.
The level k only affects the 't Hooft anomaly of the 1form symmetry i.e. the spin of the symmetry lines. For odd M the generator for the Z_2 1form symmetry has spin k/2, for M=0 mod 4 the extra Z_2 generator has spin MK/16, and for M=2 mod 4 the generator for Z_4 1form symmetry has spin MK/16.
10) The difference between gauging (1)^F 0form symmetry and summing over spin structure is that the former retains the transparent fermion line that is present in any spin QFT, while the later does not. Thus the former gives a spin QFT, while the later gives a nonspin QFT. This difference also explains the following properties: As discussed in https://arxiv.org/pdf/1605.01640.pdf and https://arxiv.org/abs/1701.08264, summing over the spin structures in (2+1)d gives a nonspin theory with an anomalous Z2 1form symmetry generated by a fermion line. On the other hand, gauging Z_2 (1)^F 0form symmetry in (2+1)d gives a dual Z_2 1form symmetry that is nonanomalous i.e. the symmetry line is a boson. And one can gauge this 1form symmetry to recover the original theory. This boson is the composite of the transparent fermion line present in spin TQFTs and the fermion line arises from summing over the spin structures.
11) Thanks.
12) Thanks.
13) SO(L)_1 for every L only has the identity line and the transparent fermion line (the Wilson line in the vector representation). Thus the theories for all L have the same 1form symmetry. pi_1(SO(L))=Z_2 for L>2 is the Z_2 magnetic 0form symmetry in (2+1)d and it can be identified with (1)F symmetry in SO(L)1 theory.
14) When a continuous gauge group has nontrivial pi_1, the electric 1form symmetry also depends on the ChernSimons level. In such case there are monopole operators that end on suitable Wilson lines depending on the ChernSimons level and reduces the electric 1form symmetry to be a subgroup. An example of this is U(1)_k with Z_k 1form symmetry instead of U(1). There can also be hidden 1form symmetry not from the center of the gauge group and depends on the ChernSimons level. An example is O(n) ChernSimons theory discussed in https://scipost.org/10.21468/SciPostPhys.4.4.021. The 0form magnetic symmetry is given by pi_1 of the gauge group if there is no magnetic matter.