Excitations in 3+1D

A bosonic topological order can only have bosonic and (emergent) fermionic particle ex- citations in 3 + 1D as seen from (1.11). However, now we can also have string-like exci- tations23 _{which opens up new and interesting possibilities. As mentioned previously, we}

already know an example of such a topological order, namely the S-wave superconductor. Here the string-like excitations are nothing but Abrikosov vortex strings [72] (known as Nielsen-Olesen strings in high-energy physics [73]).

In [74] it was shown that the statistics of N unlinked string-like excitations are governed by the Loop Braid Group LBN. This group has two classes of generators, the first corresponds to braiding the ith and i + 1th string worldsheets around each other24

si =

i i+1

. (1.24)

These generators satisfy the the following relations

sisj = sjsi, for |i − j| > 1, sisi+1si = si+1sisi+1, for 1 ≤ i ≤ N − 2,

s2_i = 1, for 1 ≤ i ≤ N − 1,

(1.25)

which forms to a permutation subgroup SN ⊂ LBN. This is intuitive to understand as if we shrink the strings to points, the worldsheets become the worldlines of particles and 23_{Actually we must have string-like excitations in order to have a non-anomalous topological order. In} 2+1D we can have string-like excitations in anomalous topological systems, which are unstable unless on the boundary of a 3 + 1D topological system.

from (1.11) we know that the statistics are given by SN. The other class of generators correspond to string excitations trading places by letting one of them go through the other

σi =

i i+1

. (1.26)

These generators satisfy the relations

σiσj = σjσi, for |i − j| > 1, σiσi+1σi = σi+1σiσi+1, for 1 ≤ i ≤ N − 2,

(1.27) which interestingly form a braiding subgroup of the loop braiding group BN ⊂ LBN. These two subgroups are intertwined inside LBN through the mixed relations

siσj = σjsi, for |i − j| > 1, sisi+1σi = σi+1sisi+1, for 1 ≤ i ≤ N − 2, σiσi+1si = si+1σiσi+1, for 1 ≤ i ≤ N − 2.

(1.28)

The loop braid group LBN only describes braiding processes with unlinked string exci- tations, however in 3 + 1D we can have many other interesting braiding processes. For example we can have braiding processes of strings that are linked with each other in com- plicated ways. Or particle-string braiding, for example of the Aharonov-Bohm type of more nontrivial type where particle worldlines and string worldsheets form complicated links.25

Let us consider one particularly simple, but very important, type of linked string exci- tation. Imagine we have N strings of types a, b, . . . with a base string of type X threading them all, see figure 1.5. Due to the base loop X, it is clear that the other strings cannot shrink into a point and cannot perform braids of the type described by the generators si (1.24). Braid processes of the σi type (1.26) is still possible and this implies that the statistics of these loops are governed by the usual braiding group BN. Besides braiding, 25_{It is very straightforward to write down 3 + 1D topological quantum field theories describing these} linking processes. Any BF-type theory has Aharonov-Bohm type braiding, abelian three-loop braiding exist in A ∧ A ∧ dA twisted theories, non-abelian three-string braiding and abelian four-string braiding in A4_{twisted theories, more complicated processes with one particle and two strings are given by A ∧ A ∧ B} twists. The discussing of these theories has not been included in this thesis as many related results were published by other authors before completion.

a

b

c

X

Figure 1.5: Linked string configuration with strings of type a, b, c, . . . threaded by a string of type X.

we can also fuse the strings a ⊗X b, with the base loop X present, similar to 2 + 1D. This implies that for this type of linked excitations we can associate fusion spaces VX(a, b, . . . ) and inherit the algebraic structure from 2 + 1D such as FX and RX-symbols and topolog- ical spins θa;X [75]. Note that the reference to the base loop X is crucial as which MTC describing the braiding and fusion of the strings a, b, c, . . . can depend on X. This all says that the category theory describing 3 + 1D topological order C3D_{, must have many} non-trivial MTCs C2D _{”embedded” within it.}

In chapter 4, using the universal wavefunction overlaps proposed in chapter 2, we extract representations of the mapping class group of the three-torus, MCG(T3_{) = SL(3, Z),} from ground state wavefunctions of gapped quantum liquids. We show that these universal quantities can be used to extract the following dimensional reduction of topological orders

C3D=M X

C_X2D, (1.29)

where the sum is over string types. Here C2D

X is the MTC that describes the fusion and braiding of strings a, b, c, . . . while penetrated by a base loop X. Such a linked-braiding process is called three-string braiding.

This type of linked-braiding process is one of many possible in 3 + 1D, where many cannot be described using dimensional reduction techniques. The construction of the proper (higher) category C3D _{which provides a complete description of 3 + 1D topological} orders is an active research direction.