Classification of Metaplectic Modular Categories

0.11 MB·

by  Eddy Ardonne

CLASSIFICATION OF METAPLECTIC MODULAR
CATEGORIES
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EDDY ARDONNE, MENG CHENG, ERIC C. ROWELL, AND ZHENGHAN WANG
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n Abstract. Weobtainaclassificationofmetaplectic modularcategories: every
a metaplectic modular category is a gauging of the particle-hole symmetry of a
J
cyclic modular category. Our classification suggests a conjecture that every

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weakly-integral modular category can be obtained by gauging a symmetry of a
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pointed modular category.

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A
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h. 1. Introduction
t
a Achieving a classification of modular categories analogous to the classification
m
of finite abelian groups is an interesting mathematical problem [3, 4]. In this note,
[
we classify metaplectic modular categories. Our classification suggests a close con-

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nection between finite abelian groups and weakly integral modular categories via
v
0 gauging, thus leads to a potential approach to proving the Property F conjecture
6 for weakly integral modular categories [2, 5, 12].
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A simple object X is weakly-integral if its squared quantum dimension d2 is an
5 X
0 integer. A modular category is weakly-integral if every simple object is weakly-
.
1 integral. Inspired by the applications to physics and topological quantum com-
0 putation, we focus on weakly-integral modular categories [6, 7]. An important
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class of weakly-integral modular categories is the class of metaplectic modular
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: categories—unitary modular categories with the fusion rules of SO(N) for some
v 2
i odd integer N > 1 [10, 11]. The metaplectic modular categories first appeared in
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the study of parafermion zero modes, which generalize the Majorana zero modes.
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a The name metaplectic comes from the fact that the resulting braid group rep-
resentations from the generating simple objects in SO(N) are the metaplectic
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representations, which are the symplectic analogues of the spinor representations.
Our main result is a classification of metaplectic modular categories: every meta-
plectic modular category is a gauging of the particle-hole symmetry of a cyclic
modular category.
The property F conjecture says that all braid group representations afforded
by a weakly-integral simple object have finite images. For SO(N) , the property
2
F conjecture follows from [13]. It is possible that all weakly-integral modular
The first author is supported in part by the Swedish research council. The third and fourth
authors are partially supported by NSF grants DMS-1410144 and DMS-1411212 respectively.
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2 EDDY ARDONNE,MENG CHENG, ERICC. ROWELL,ANDZHENGHANWANG
categories can be obtained from gauging of pointed modular categories—modular
categories with all simple objects having their quantum dimension equal to 1. Our
classification supports this possibility. If this is true, then a potential approach to
the property F conjecture for all weakly-integral modular categories would be to
prove that gauging preserves property F.
2. Cyclic modular categories
Definition 2.1. Let Z be the cyclic group of n elements. A Z -cyclic modular
n n
category is a modular category whose fusion rule is the same as the cyclic group
Z for some integer n.
n
A Z -cyclic modular category is determined by a non-degenerate quadratic form
n
q : Z U(1). We will denote the Z -cyclic modular category determined by such
n n
a quad→ratic form q as C(Z ,k) for q(j) = e2πisj,s = kj2,0 j n 1,(k,n) = 1.
First for M,N relativelny prime, C(Z ,k) isj a dinrect≤prod≤uct−of C(Z ,kN)
MN M
and C(Z ,kM). The simple object types, j, of C(Z ,k) can be labelled by pairs
N MN
(a,b), where j = aM +bN and 0 a N 1,0 b M 1. The fusion rules
≤ ≤ − ≤ ≤ −
are
(2.1) j j = (a ,b ) (a ,b ) = ([a +a ] ,[b +b ] ),
1 2 1 1 2 2 1 2 N 1 2 M
× ×
and the topological twists are θ = e2πisj:
j
kj2 k(aM +bN)2 kMa2 kNb2
(2.2) s = = = + +2abk.
j MN MN N M
Therefore, we have shown that C(Z ,k) = C(Z ,kN)⊠C(Z ,kM).
MN M N
Next we find all distinct Z -cyclic modular categories, where p is an odd prime.
pa
For C(Z ,k), write k = plm, where p ∤ m. Note that if l 1, the resulting
pa
categoryisnot modular (since the formq(x) = e2πikx2/pa isdegen≥erate). Therefore,
we must assume (k,p) = 1. The twist of the j-th simple object is e2pπaikj2. If for n
1
and n , the categories are isomorphic, it means that one can solve the congruent
2
equation
n n j2
(2.3) 1 2 (mod 1),
pa ≡ pa
for some j such that p ∤ j (so that j is a generator of Z ). We need to solve
pa
j2 n−1n (mod pa), which is solvable when n1 = n2 . Therefore, there are
≡ 2 1 (cid:16)pa(cid:17) (cid:16)pa(cid:17)
two distinct theories.
Braided tensor auto-equivalences of the Z -cyclic-modular categories are group
n
isomorphisms of Z which preserve the topological twists. The particle-hole
n
symmetry of a Z -cyclic modular category with n odd is the Z -braided tensor
n 2
auto-equivalence that maps j to n j.
−
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3. Metaplectic modular categories
The unitary modular categories SO(N) for odd N > 1 has 2 simple objects
2
X ,X of dimension √N, two simple objects 1,Z of dimension 1, and N−1 objects
1 2 2
Y , i = 1,..., N−1 of dimension 2. The fusion rules are:
i 2
(1) Z Y = Y , Z X = X (modulo 2), Z⊗2 = 1,
i ∼ i i ∼ i+1 ∼
⊗ ⊗
(2) X⊗2 = 1 Y ,
i ∼ ⊕Li i
(3) X X = Z Y ,
1 ⊗ 2 ∼ ⊕Li i
(4) Y Y = Y Y , for i = j and Y⊗2 = 1 Z Y .
i⊗ j ∼ min{i+j,N−i−j}⊕ |i−j| 6 i ⊕ ⊕ min{2i,N−2i}
The fusion rules for the subcategory generated by Y (with simple objects 1,Z and
1
all Y ) are precisely those of the dihedral group of order 2N.
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Definition 3.1. A metaplectic modular category is a unitary modular category C
with the same fusion rules as SO(N) for some odd N > 1.
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Theorem 3.2. (1) Suppose C is a metaplectic modular category with fusion
rules SO(N) , then C is a gauging of the particle-hole symmetry of a Z -
2 N
cyclic modular category.
(2) For N = pα1 pαs with distinct odd primes p , there are exactly 2s+1 many
1 ··· s i
inequivalent metaplectic modular categories.
To prove the theorem, we start with two lemmas.
Lemma 3.3. The object Z is a boson: θ = 1.
Z
Proof. Let Y be any of the N−1 simple objects of dimension 2. By orthogonality
2
of the rows of the S-matrix, we find that S = 2. Observing that Y Z = Y,
YZ ∼
⊗
we apply the balancing equation (see e.g. [1]):
r−1
S θ θ = Nk d θ
ij i j X i∗j k k
k=0
to obtain: 2θ θ = S θ θ = θ d = 2θ . It follows that θ = 1. (cid:3)
Y Z YZ Y Z Y Y Y Z
Since Z is a boson (i.e. dim(Z) = 1 and θ = 1), we may condense (“de-
Z
equivariantize” in the categorical language) to obtain a Z -graded category [9].
2
Since Z interchanges 1 Z andX X and fixes the Y the resulting condensed
1 2 i
category D := CZ2 has↔N objects of↔quantum dimension 1 in the identity sector
D and one object of dimension √N in the non-trivial sector D (see [2]). Clearly,
0 1
the fusion rules of D must be identical to those of some abelian group A of order
0
N. In the following, we show that A = Z . As an aside, we point out that the
∼ N
category D is a Tambara-Yamagami category [14].
Lemma 3.4. The fusion rules of D are the same as Z .
0 N
4 EDDY ARDONNE,MENG CHENG, ERICC. ROWELL,ANDZHENGHANWANG
Proof. It is enough to find a tensor generator for D , that is, a simple object U
0
so that U⊗i : i 0 contains all simple objects in D . Now under condensation
0
each obj{ect Y be≥com}es the sum of two invertible simple objects in D . The image
i 0
of Y under condensation is Y1 Y2, a sum of invertible simple objects in D . We
i i ⊕ i 0
denote by 1 the image of 1 and Z under condensation (i.e. the unit object in
0
D ). We will proceed to show that Y1 is a tensor generator for D .
0 1 0
From Y⊗2 = 1 Z Y we obtain
1 ∼ ⊕ ⊕ 2
(Y1)⊗2 (Y2)⊗2 2Y1 Y2 = 21 Y1 Y2.
1 ⊕ 1 ⊕ 1 ⊗ 1 0 ⊕ 2 ⊕ 2
This implies Y1∗ = Y2, so that Y2 appears as some tensor power of Y1. Thus Y1 is
1 1 1 1 1
a tensor generator provided each Y(j) appears in some tensor power of (Y1 Y2).
Since every Y appears in some tenisor power of Y the result follows. 1 ⊕ 1(cid:3)
i 1
Proof of Theorem 3.2: (1) By Lemmas 3.3,3.4, each metaplectic modular
categoryisobtainedfromgaugingaZ symmetryofacyclicmodularcategory. But
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the particle-hole symmetry is the only non-trivial Z symmetry of a cyclic modular
2
category. (2): As discussed above, there are exactly two cyclic modular categories
for each prime power factor in N. When gauging the particle-hole symmetry, there
is an additional choice parameterized by H3(Z ;U(1)) = Z [8, 2, 5]. Therefore,
2 ∼ 2
the number of metaplectic modular categories is 2s+1.
4. Witt classes and open problems
Gauging preserves Witt classes [5]. Therefore, the Witt classes of metaplectic
modular categories are the same as those of the corresponding cyclic modular
categories.
Proposition 4.1. The modular category C(Zp2a,q) is a quantum double Z(VecωZpa).
Proof. It is easy to see that regardless of the quadratic form q, the simple objects
[npa] are all bosons, for n = 0,1,...,pa 1. They form a Z fusion category. In
pa
tfahcatt,Con(Zepc2aan,qd)eifisninedaeeLdagaraqnugainatnusmubdaolug−ebbler.aNLownpa=−l0e1t[nupsa]coofndCe(nZspe2at,hqi)s. sTuhbiaslgsehborwas,
which identifies [j] with [j + npa]. Therefore, one can label the distinct simple
objects after condensation by [j],j = 0,1,...,pa 1. Hence C(Zp2a,q) must be a
quantum double of Z , generally twisted by a cla−ss in H3 [9]. (cid:3)
pa
One open question is to prove property F for all metaplectic modular categories.
Another one is to construct universal computing models from metaplectic modular
categories by supplementing braidings with measurements [7].
References
[1] B. Bakalov and A. Kirillov, Jr., Lectures on Tensor Categories and Modular Functors,
University Lecture Series, vol. 21, Amer. Math. Soc., 2001.
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[2] M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, Symmetry, defects, and gauging of
topological phases. arXiv preprint arXiv:1410.4540(2014).
[3] P. Bruillard, S. H. Ng, E. Rowell, and Z. Wang, Rank-finiteness for modular categories,
Journal of the American Mathematical Society (to appear).
[4] P. Bruillard, S. H. Ng, E. Rowell, and Z. Wang, On classification of modular categories by
rank. arXiv preprint arXiv:1507.05139.
[5] S. X. Cui, C. Galindo, J.Y. Plavnik, and Z. Wang, On Gauging Symmetry of Modular
Categories. arXiv preprint arXiv:1510.03475.
[6] S. X. Cui, S.M. Hong, and Z. Wang. Universal quantum computation with weakly integral
anyons. Quantum Information Processing (2014): 1-41.
[7] S.X.Cui,andZ.Wang.Universal quantum computation with metaplectic anyons. Journal
of Mathematical Physics 56.3 (2015): 032202.
[8] P. Etingof, D. Nikshych, V. Ostrik, Fusion categories and homotopy theory. With an ap-
pendix by Ehud Meir. Quantum Topol. 1 (2010), no. 3, 209273.
[9] V. Drinfeld, S. Gelaki,D. Nikshych,and V. Ostrik.On braided fusion categories I. Selecta
Mathematica, 16(1), 1-119.
[10] M. B. Hastings; C. Nayak;Z. Wang, On metaplectic modular categories and their applica-
tions. Comm. Math. Phys. 330 (2014), no. 1, 45–68.
[11] M. B. Hastings; C. Nayak; Z. Wang, Metaplectic anyons, Majorana zero modes, and their
computational power Phys. Rev. B 87, (2013) 165421.
[12] D. Naidu, and E. C. Rowell. A finiteness property for braided fusion categories. Algebras
and representation theory 14.5 (2011): 837-855.
[13] E. C. Rowell, and H. Wenzl. SO(N)2 Braid group representations are Gaussian. arXiv
preprint arXiv:1401.5329(2014).
[14] D. TambaraandS. Yamagami,Tensor categories with fusion rules of self-duality for finite
abelian groups, J. Algebra 209 (1998), no. 2, 692–707.
E-mail address: [email protected]
Department of Physics, Stockholm University, Albanova University Center,
SE-106 91 Stockholm Sweden
E-mail address: [email protected]
Microsoft Station Q, University of California, Santa Barbara, CA 93106-6105,
U.S.A.
E-mail address: [email protected]
DepartmentofMathematics, TexasA&MUniversity,CollegeStation,TX77843,
U.S.A.
E-mail address: [email protected]
MicrosoftStationQandDeptofMathematics, UniversityofCalifornia,Santa
Barbara, CA 93106-6105, U.S.A.