We prove the first nontrivial reconstruction theorem for modular tensor categories: the category associated to any twisted Drinfeld double of any finite group, can be realised as the representation ...category of a completely rational conformal net. We also show that any twisted double of a solvable group is the category of modules of a completely rational vertex operator algebra. In the process of doing this, we identify the 3-cocycle twist for permutation orbifolds of holomorphic conformal nets: unexpectedly, it can be nontrivial, and depends on the value of the central charge modulo 24. In addition, we determine the branching coefficients of all possible local (conformal) extensions of any finite group orbifold of holomorphic conformal nets, and identify their modular tensor categories.
Much ado about Mathieu Gannon, Terry
Advances in mathematics (New York. 1965),
10/2016, Letnik:
301
Journal Article
Recenzirano
Eguchi, Ooguri and Tachikawa observed that the coefficients of the elliptic genus of type II string theory on K3 surfaces appear to be dimensions of representations of the largest Mathieu group. ...Subsequent work by several people established a candidate for the elliptic genus twisted by each element of M24. In this paper we prove that the resulting sequence of class functions are true characters of M24, proving the Eguchi–Ooguri–Tachikawa ‘Mathieu Moonshine’ conjecture. The integrality of multiplicities is proved using a small generalisation of Sturm's Theorem, while positivity involves a modification of a method of Hooley, for finding an effective bound on a family of Selberg–Kloosterman zeta functions at s=3/4. We also prove the evenness property of the multiplicities, as conjectured by several authors, and use that to investigate the proposal of Gaberdiel–Hohenegger–Volpato that Mathieu Moonshine lifts to the Conway groups Co0 and Co1. We identify the role group cohomology plays in both Mathieu Moonshine and Monstrous Moonshine; in particular this gives a cohomological interpretation for the non-Fricke elements in Norton's Generalised Monstrous Moonshine conjecture, and gives a lower bound for H3(BCo1;C×).
The quantum double of the Haagerup subfactor, the first irreducible finite depth subfactor with index above 4, is the most obvious candidate for exotic modular data. We show that its modular data
...fits into a family
, where
n
≥ 0 and
. We show
is related to the subfactors Izumi hypothetically associates to the cyclic groups
. Their modular data comes equipped with canonical and dual canonical modular invariants; we compute the corresponding alpha-inductions, etc. In addition, we show there are (respectively) 1, 2, 0 subfactors of Izumi type
and
, and find numerical evidence for 2, 1, 1, 1, 2 subfactors of Izumi type
(previously, Izumi had shown uniqueness for
and
), and we identify their modular data. We explain how
(more generally
) is a
graft
of the quantum double
(resp. the twisted double
) by affine so(13) (resp. so
) at level 2. We discuss the vertex operator algebra (or conformal field theory) realisation of the modular data
. For example we show there are exactly 2 possible character vectors (giving graded dimensions of all modules) for the Haagerup VOA at central charge
c
= 8. It seems unlikely that any of this twisted Haagerup-Izumi modular data can be regarded as exotic, in any reasonable sense.
We present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a – not necessarily ...semisimple – modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories.
We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: First, for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data. For rational theories our proposal reproduces the familiar result for the boundary states of the Cardy case. Further, with the help of sewing we compute annulus amplitudes. Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.
The two pillars of rational conformal field theory and rational vertex operator algebras are modularity of characters, and the interpretation of its category of modules as a modular tensor category. ...Overarching these pillars is the Verlinde formula. In this paper we consider the more general class of logarithmic conformal field theories and C2-cofinite vertex operator algebras. We suggest logarithmic variants of those pillars and of Verlinde's formula. We illustrate our ideas with the Wp-triplet algebras and the symplectic fermions.
A pariah finds a home Gannon, Terry
Nature,
10/2017, Letnik:
550, Številka:
7675
Journal Article
Recenzirano
Odprti dostop
Pariahs are fundamental building blocks in a branch of mathematics called group theory, but seem to be unconnected to both physics and other areas of mathematics. Such a connection has now been ...identified.
The theory of subfactors connects diverse topics in mathematics and mathematical physics such as tensor categories, vertex operator algebras, quantum groups, quantum topology, free probability, ...quantum field theory, conformal field theory, statistical mechanics, condensed matter physics and, of course, operator algebras. We invited an international group of researchers from these areas and many fruitful interactions took place during the workshop.
The orbifold construction
A
↦
A
G
for a finite group
G
is fundamental in rational conformal field theory. The construction of
Rep
(
A
G
) from
Rep
(
A
) on the categorical level, often called ...gauging, is also prominent in the study of topological phases of matter. Given a non-degenerate braided fusion category
C
with a
G
-action, the key step in this construction is to find a braided
G
-crossed extension compatible with the action. The extension theory of Etingof–Nikshych–Ostrik gives two obstructions for this problem,
o
3
∈
H
3
(
G
)
and
o
4
∈
H
4
(
G
)
for certain coefficients, the latter of which depends on a categorical lifting of the action and is notoriously difficult to compute. We show that in the case where
G
≤
S
n
acts by permutations on
C
⊠
n
, both of these obstructions vanish. This verifies a conjecture of Müger, and constitutes a nontrivial test of the conjecture that all modular tensor categories come from vertex operator algebras or conformal nets.
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