Prerequisite material for thinking about fermionic topological phases

Transcription

1 Prerequisite material or thinkin about ermionic topoloical phases Ethan Lake October 30, 2016 These notes are a collection o remarks on various thins needed to start doin real work on ermionic topoloical phases. The ull story will appear elsewhere. 1 Backround math 1.1 Supervector spaces and Cliord alebras In order to think about ermionic topoloical phases, it s helpul to think about the objects in tensor cateories in terms o vector spaces. Since we re workin with unitary TQFTs we will be workin over C, and so the simple objects in our cateories can be rearded as simple C-alebras. When our theories arise rom ermionic derees o reedom, the main thin we need to do is to keep track o ermion parity. We can do this by splittin all vector spaces in the theory up into a direct sum o their ermion-parity even and ermion-parity odd sectors, writin V = V 0 V 1. This decomposition turns vector spaces into supervector spaces, which are just Z 2 -raded versions o reular vector spaces. Throuhout we will use the operator P as the ermion parity operator, deined by P v = v i v V 0 (even ermion parity) and P v = v i v V 1 (odd ermion parity). When we take the tensor product o two supervector spaces, we must use a super version o the tensor product which respects the Z 2 radin. Speciically, the super version o the tensor product o two supervector spaces (which we will reer to as the tensor product, and will just denote by ) works as ollows: (V W ) 0 = V 0 W 0 V 1 W 1, (V W ) 1 = V 0 W 1 V 1 W 0. (1) We will use absolute value bars to denote the ermion parity o vectors in supervector spaces. That is, or v V, we write v = 0 i P v = v (i.e. i v V 0 ) and v = 1 i P v = v (i.e. i v V 1 ). Note that since we want the Z 2 radin to be well-deined on supervector spaces, we are orbidden rom addin even ermion parity vectors with odd ermion parity vectors. Since we re workin with a tensor product that respects the ermion parity structure o supervector spaces, it must account or the actors o ( 1) that appear when the relative order o two ermionic vectors in a tensor product is switched. This means that the tensor product o two vectors is supercommutative, in the sense that the wede product is supercommutative: v w = ( 1) v w w v. (2) 1

2 This means that svec (the cateory o supervector spaces, containin just the vacuum and a ermion) naturally comes equipped with the structure o a braided cateory, with the supercommutative nature o the tensor product controllin the 1 braidin o the ermion with itsel. I a supervector space V has dim V 0 = a and dim V 1 = b, we say that V has superdimension a b. We will use the notation C a b to denote the supervector space o superdimension a b, so that C a b is an (a + b)-dimensional vector space with a ermion-parity even enerators and b ermion-parity odd enerators. When we tensor C a b with C c d, the raded dimensions behave in the same way that supervector spaces behave when you tensor them toether. That is, they behave just like you would expect: C a b C c d = C ac+bd ad+bc. (3) Notice in particular that C a b C 1 0 = C a b and C a b C 0 1 = C b a. That is, tensorin with C 1 0 = C does nothin (as it should), and tensorin with C 0 1 is equivalent to lippin ermion parity. The supervector space C 1 1 will turn out to play an important role in what ollows. This is because it is invariant under ermion-parity lips: C 1 1 C 0 1 = C 1 1. It is also an example o a Cliord alebra, namely the Cliord alebra Cl 1. Cliord alebras will be important or us when we consider theories with Majoranas, and so we will quickly review their deinition and basic properties. The complex Cliord alebras Cl n are the supervector spaces enerated by the number 1 and n parity-odd enerators γ 1,..., γ n which satisy the relations {γ i, γ j } = 2Q ij, (4) where {γ i, γ j } = γ i γ j +γ j γ i and Q ij is some quadratic orm. For real Cliord alebras Q ij can enerically be written as a diaonal matrix with ±1s on the diaonal, althouh our workin over C allows us to set Q ij = δ ij without loss o enerality. The simplest (other than Cl 0 = C) Cliord alebra is Cl 1 = 1, γ, with γ 2 = 1 and P γ = γ. Since by deinition Cl 1 has one even enerator (1) and one odd enerator (γ), we have Cl 1 = C 1 1. In terms o representations, we can choose a representation ρ such that ρ(1) = and ρ(γ) = σ x, which is consistent since {σ x, σ x } = and σ x : C 1 0 C 0 1, C 0 1 C 1 0, i.e. σ x is odd. We will see that Cl 1 is the prototypical Majorana supervector space that will appear later on. We will need a ew miscellaeous acts about Cliord alebras, which we will list o here. First, tensorin two Cliord alebras ives a larer Cliord alebra whose associated quadratic orm is the direct sum o the quadractic orms o the smaller alebras, meanin that the larer Cliord alebras can be built rom Cl 1 as Cl n = Cl n 1. Secondly, Cl 2 = End(C 1 1 ), and since the endomorphism rins o matrix alebras always have trivial modules, Cl 2 and C are Morita equivalent (written Cl 2 =M C), meanin that their simple modules are the same. This will be o use later when we look at constructin quasiparticles and their usion rules. Since Cl n = Cl n 1, this implies the Cl n =M Cl n+1 o Bott periodicity ame. 1.2 Modules and Morita equivalence A module over an alebra is a vector space whose scalars are drawn rom the alebra. That is, it s a way o ivin an alebra an action on a vector space. So modules are the 2

3 alebra analoue o representations: representations construct a way or roups to act on vector spaces, and modules do the same thin or alebras. A mathematical concept that will be relevant is the notion o Morita equivalence. Rouhly, Morita equivalence is a way o establishin when two alebras have the same modules, or when their representation theory is the same. The technical deinition is that two alebras A and B are Morita equivalent (written A = M B) when the cateories o their let modules Mod L (A) and Mod L (B) are equivalent, althouh we won t use this deinition much. The reason why the notion o Morita equivalence is useul or us is because quasiparticles in topoloical phases are identiied with simple modules o an alebra tube called the tube alebra, which we ll talk about in more detail later. Usually iurin out the alebra structure o tube is straihtorward, althouh computin its simple modules (aka indin an idempotent decomposition o tube) can be tedious. Since Morita equivalent alebras have the same modules, we can oten replace a complicated tube alebra or a sub-alebra o a tube alebra with a much simpler but Morita equivalent alebra, which can reatly acilitate the determination o its simple modules. The canonical example o Morita equivalent alebras are the matrix alebras. It turns out that we actually have C(n) = M C or all n (where C(n) are complex n n matrices). This can be seen by usin the ollowin proposition: Let A, B be two alebras, and E be an A B bimodule, that is, let E be a let A-module and a riht B-module. Suppose E is such that Then A and B are Morita equivalent. E A E = B, E B E = A. (5) We will do the proo by settin up the equivalence between the modules o A and B explicitly. Suppose M is a module over A, and N a module over B. Construct the unctors F : Mod(A) Mod(B), G : Mod(B) Mod(A) as F : M Hom A (E, M), G : N E B N. (6) This associates M with a B-module and N with an A-module. We claim that this is an isomorphism, namely that F and G are inverses o one another. I this is true, we must have M = E B Hom A (E, M) = (E B E ) A M, N = Hom A (E, E B N) = (E A E) B N, (7) but this holds precisely due to our assumptions on E. The Morita equivalence between C(n) and C then ollows rom settin the bimodule E to be E = C n in the above construction. As a useul nonexample, C n =M C m or any n m. This can be seen by realizin that the centers o two Morita equivalent alebras must always be the same. This is true or the C(n) = M C example considered above, but o course the centers o C n and C m are dierent i n m, and so indeed C n =M C m. One useul act that we will exploit in calculations is that two alebras A and B are always Morita equivalent i there exists a supervector space V such that A = B C End(V ). (8) 3

4 For a useul example o this, we turn to the Cliord alebras. We will use the act that Cl 2 = End(Cl1 ) = End(C 1 1 ), (9) which can be seen just by realizin that End(Cl 1 ) = Cl 1 Cl 1 = Cl2. This means that Cl n+2 =M Cl n (10) or all n, and so Cl n+2 and Cl n always have the same modules (this is Bott periodicity!). In particular, we see that Cl 2 has the same modules as C! This will be very helpul later on when we look at condensin ermions in the Isin theory. There, we will run into subalebras o tube that are isomorphic to End(C 1 1 ). By what we ve just seen these subalebras must have the same modules as C, which is much easier to work with than Cl 2. Since the modules o the tube alebra (and its subalebras) determine the quasiparticles in the theory, which see in particular that subalebras o tube iven by End(C 1 1 ) must ive rise to only a sinle quasiparticle, since we are ree to replace them by the trivial alebra C. 2 Superusion cateories When we o rom usion cateories to superusion cateories, the simple objects become associated with supervector spaces, and understandin what happens to them is airly straihtorward. However, the usion spaces also become superspaces, which or us is very important. In particular, this means that all the Hom spaces (a.k.a usion spaces) carry a Z 2 -radin that keeps track o their ermion parity. In particular, Hom C (, Y ) 0 contains all the morphisms between and Y that are ermion-parity even, and Hom C (, Y ) 1 contains all the morphisms that are ermion-parity odd. So, we can think o Hom(, Y ) 0 as bein a reular bosonic usion space, while Hom(, Y ) 1 is an odd usion space in which the ermion parity o the usion products chanes sin. Because o this, we can think o usion spaces with odd ermion parity as localizin a ermion that lives on the usion vertex. Especially important or us will be the case when our theories come with objects such that Hom(, ) 1 is nontrivial. These objects will be Majorana in some sense, and we will discuss them in more detail later. For any morphism : Y, we will write the parity o as = 0 i is even (preserves ermion parity) and = 1 i is odd (reverses ermion parity). Without loss o enerality, we can consider usion diarams built rom a tensor product o usion spaces o the orm Hom( Y, Z). Formin a usion raph requires choosin basis vectors or all these usion spaces. Followin convention, we will write s Y Z (α) to denote the parity o the basis vector or the usion space Hom( Y, Z), where 1 α NZ Y Y. For notational simplicity we will initally assume NZ 1, so that all the usion spaces are one-dimensional and have a unique ermion parity (althouh some o the Fibonacci-like theories we will want to consider later won t satisy this restriction). The radin o morphisms is important to keep track o, since morphisms satisy their own supercommutativity law, called the superexchane law, which states that or any our morphisms,, h, k C, we have ( ) (h k) = ( 1) h ( ) ( k). (11) 4

5 Let s run throuh how to see this. First o all, recall that in normal usion cateories, the tensor product o two morphisms is the same as horizontal superposition: = In superusion cateories, we need to be more careul, since morphisms (like the uys in the Hom( Y, Z) usion spaces) can carry nontrivial ermion parity. This means that i we place two odd morphisms side-by-side, the relative positions o the ermions they harbor is ambiuous. This ambiuity is actually really easy to ix diarammatically: we just write the tensor product o two morphisms in a displaced way, where the irst morphism in the tensor product is displaced above the second: (12) = Movin two morphisms past each other vertically may then result in a minus sin, since i both morphisms have odd ermion parity, movin them past each other is like exchanin two ermions. That is, (13) = ( 1) With this property, it becomes easy to veriy the superexchane law. Finally, let s quickly mention the F -moves (we ll come back to them in more detail later). Recall that we can write them as the basis chanes [F ijk l ] : Hom(i j, m) Hom(m k, l) = Hom(i n, l) Hom(j k, n). (15) m n The F -moves shouldn t chane the ermion parity o the usion raph, and so must be even morphisms. This means that the ermion parity on both sides o the above isomorphism must be the same. To quantiy this condition, the even-ness o F means that s ij m + s mk l = s jk n (14) + s in l, (16) where s ij m is the parity o the usion space Hom(i j, m) as beore. In particular, i C = Vec G or some inite Abelian G, this is equivalent to the 2-cocycle condition, and so we see that such theories are (partially) classiied by a choice o cohomoloy class s H 2 (BG, Z 2 ). 5

6 2.1 Cateorical caveats A number o ormulae that we re used to usin when workin with usion cateories ail to hold upon passin to superusion cateories, while others o throuh unchaned. The most important thin that needs to be modiied is the way we write resolutions o the identity. Consider the usual coproduct isomorphism Hom(, Y ) = Z C Hom(, Z) Hom(Z, Y ). (17) This amounts to stitchin and Y toether by summin over all the simple objects that can connect to Y. In strin-net diarams, we usually see (??) used to write the resolution o the identity Hom( Y, Y ) = Z C Hom( Y, Z) Hom(Z, Y ), (18) where, Y, and Z are all simple objects. Diarammatically, this looks like Y = Z C Z Y Y (19) However, these isomorphisms do not hold as written in more eneral settins, namely when the simple objects ail to all possess trivial endomorphism alebras. In this scenario, the RHS o (??) will be much bier than the let, since the internal derees o reedom Z may contribute nontrivially to the size o the RHS. I not all the objects have trivial endomorphism alebras, the bi direct sum on the RHS o (??) needs to be replaced with a colimit, althouh we won t need to o into any detail about what this means mathematically. The simple way to ix (??) involves chanin the type o tensor product we use. Until now, we ve tacitly been assumin that all o our tensor products are secretly C, tensor products over C. This works only when we re workin in reular cateories, where the Hilbert spaces associated with worldlines are always C. However, we run into problems i we have objects with End(Z) = C and continue to use C. I End(Z) is bier than C, then the RHS o (??) is bier than the LHS, since the internal Z le o the usion diaram contributes a larer space to the direct sum. This is unphysical thouh, since the internal worldlines o reedom Z shouldn t carry any more inormation than is carried by the incomin and outoin worldlines (since i this were the case, the sizes o the Hilbert spaces o usion diarams would blow up). We can ix this issue by moddin out by the Hilbert spaces o the internal worldlines. This can be done by treatin End(Z) as the tensor unit when we tensor the two usion spaces in (??) toether. The correct ormula is then Hom(, Y ) = Z C Hom(, Z) End(Z) Hom(Z, Y ). (20) 6

7 As a corollary, this means that the superusion F -symbols should be written as [F ijk l ] : m Hom(i j, m) End(m) Hom(m k, l) = n Hom(i n, l) End(n) Hom(j k, n). (21) Finally, on another cautionary note, we should really be writin Hom C (), just to distinuish it rom Hom C (), and likewise or End C. Let s ocus on End C (). End C () is the space o all endomorphisms o that respect the structure o the cateory C. For example, i C = Rep(G), End C () is the space o morphisms that commute with the G-action. This is very dierent rom the more amiliar (and much larer!) space o all C-linear maps rom to itsel, which is End C (). In particular, we have the amiliar End C () =, but this deinitely doesn t hold or End C ()! Likewise, is certainly not deined by Hom C (, C). Instead, we can deine throuh Hom C (Y, Z) = Hom C ( Y, Z). (22) O course, all Hom spaces should be understood as Hom C spaces unless stated otherwise. 2.2 Majorana objects We miht naively think that usin a physical ermion with a worldline, which corresponds to doin C 0 1, would always be an odd morphism on. That is, we miht uess that doin C 0 1 would always chane the radin (ermion parity) o. However, this may not always be true! Addin a ermion can actually be an even operation, provided that worldlines are let invariant ater tensorin with C 0 1. This happens precisely when End() = Cl 1. (23) Indeed, since Cl 1 (whose simple module is C 1 1 ) has one even enerator and one odd enerator, tensorin with C 0 1 merely interchanes these two enerators, and the result is (oddly) isomorphic to what we started with: Cl 1 C 0 1 = Cl1. In terms o representations, tensorin with C 0 1 is like multiplyin the enerators o Cl 1 by σ x. Since we can take the enerators o Cl 1 are and σ x, multiplyin by σ x leaves the set o enerators unchaned, and so tensorin with C 0 1 doesn t do anythin. Pictorially, (lettin ψ denote a physical ermion) End() = i End() = C ψ End() = i End() = Cl 1 = C 1 1 (24) 7

8 Since objects with End() = Cl 1 can absorb ermions without chanin their radin by way o Cl 1 C 0 1 = Cl1 (and as such don t really have a well-deined ermion parity at all), they behave like Majoranas. We thus deine an object to be Majorana i End() = Cl 1. Reular objects with End() = C are simply reerred to as Bosonic. This distinction is especially useul because all objects must be either bosonic or Majorana there are no other possibilities. The proo o this is straihtorward: i is a simple object, it must be simple when rearded as an object in the cateory (i.e. must not admit a direct sum decomposition with more than one nontrivial summand). Schur s lemma then tells us that any morphism between and itsel must be an isomorphism, and so all the elements o End() must be isomorphisms, and hence every element in End() must be invertible. Thus, End() must be a Z 2 -raded division alebra i.e., a Z 2 -raded alebra in which every element is invertible. Then we can realize that C and Cl 1 = C 1 1 are the only Z 2 -raded division alebras. We won t prove this riorously, and will just ive a plausibility arument. Since we need our division alebra needs to be an alebra over C and needs to be Z 2 -raded, the complex Cliord alebras Cl n are the only obvious possibilities. Let s look at Cl 1 irst. Cl 1 is not a division alebra in the unraded sense, because i we inore the radin we can write (1 γ)(1 + γ) = 0, even thouh neither 1 + γ nor 1 γ is zero. This isn t a problem in the Z 2 -raded case, since γ has odd deree while 1 has even deree, and in a Z 2 -raded alebra we are orbidden rom addin two elements o dierent derees. Playin around with the dierent enerators or a while shows that all elements in Cl 1 are invertible, as lon as we take into account the constraints rom the radin. However, none o the other Cliord alebras Cl n, n > 1 are division alebras: or example, or n = 2 we can take the odd enerators γ i in Cl 2 to obey the anticommutation rule {γ i, γ j } = 2σ z, which means that (1 + γ 1 γ 2 )(1 γ 1 γ 2 ) = 0, and so Cl 2 is not a division alebra (we can add 1 and γ 1 γ 2 since γ 1 γ 2 has even parity). Similar aruments rule out the hiher Cl n s, and so Cl 0, Cl 1 are the only obvious possibilities. Thus, our only choices or End() with simple are End() = C or End() = C 1 1 = Cl1. 3 The underlyin usion cateory o a superusion cateory The oal in this section is to switch rom a radin the usion spaces picture to a radin the worldlines picture. Doin this results in a cateory with a larer collection o objects which is bosonic in some sense. We will call the cateory obtained rom this bosonization procedure the underlyin usion cateory o C (since radin the worldlines is a procedure that associates every superusion cateory with a unique underlyin usion cateory). We will write the underlyin usion cateory obtained rom the superusion cateory C as C. The bosonic-ness o this new cateory is helpul since indin the quasiparticle spectrum can likely be done throuh the usual tube alebra methods we will have more to say about the precise relation between the excitations o superusion cateories and their associated underlyin usion cateories later. Objects in C are written as a, where a Z 2 determines their radin. The naieve rule or usin objects in C is a b = ( Y ) a+b, (25) 8

9 where on the let hand side takes place between objects in C and on the riht hand side takes place between objects in C. Instead, we will see that the a + b on the RHS will actually enerically be twisted by a 2-cocycle (or eneralization thereo or non-abelian theories). A very important issue or us is the rule or determinin how to relate morphisms in C to morphisms in C, and vice versa. Suppose we have a morphism : Y, C with parity. When we make the switch to C, we et a morphism a b : a Y b, where = a + b. We would like to use morphisms simply as a b c d = ( ) b+d a+c, but this leads to inconsistencies, and it turns out that this simple usion law or morphisms must be replaced with somethin more twisted. The correct relation is that tensor products o morphisms in C are deined throuh tensor products o morphisms in C by the relation b a d c = ( 1) (c+d+ )a+d ( ) b+d a+c. (26) Note that since = c + d, we can actually just write b a d c = ( 1) d ( ) b+d a+c. (27) This rule will be super important in what ollows, and is what allows us to determine what the F -symbols in C are. However, it looks rather bizarre at irst, so let s work out how to see it. First, we start by updatin our diarammatic notation or morphisms. In particular, worldlines now carry a ermion parity radin, which we keep track o by writin a morphism b a : a Y b in C as b a = b where the black squares which keep track o the worldline ermion parities. I the black squares are labelled by the radin 1, they are ermions, and braid with each other when we slide them around on diarams, while i they are labelled by the radin 0 they are bosons, and we are ree to slide them around at will. This notation us allows to derive a (28) 9

10 our above rule or tensorin morphisms in C by the ollowin sequence o diarams: b a d c = b a d c = ( 1) ad b a d c = ( 1) ad+d b a d c = ( 1) a(c+d+ )+d b a d c = ( 1) (c+d+ )a+d ( ) b+d a+c 3.1 Derivin the F -symbols in C We are now equipped to derive the F -symbols in C, which we will denote as F (maybe it d be better to use F, but there are two other papers that use F, so we ll stick with F or now). Since C is bosonic in the sense that it has no ermions at the usion spaces, the pentaon identity in C holds exactly, and the F symbols satisy the reular pentaon equations (in contrast to the F symbols, which we ll see do not satisy the pentaon identity in C). We ll inore Majorana objects or the moment, and miht come back to them later (to incorporate them we need to think careully about Cl 1 -valued F -symbols) With Majorana objects aside, the F symbols in C are deined by (29) i a j b k c i a j b k c m e = [F ia j b k c l d n e C ] m e n n l d Note that only a, b, c are independent. Explicitly, we have e = a + b + s ij m, d = e + c + s mk l, and = b + c + s jk n, which are consistent constraints since the even-ness o the F -moves orces s ij m + s mk l = s jk n + s in l. Usher (ariv: ) has derived how the F symbols in C are related to the F symbols in C: [F ia j b k c ] l d m e n = m ( 1)csij [F ijk l ] mn. (31) To prove his result, it helps to write down the alebraic content o the usion diarams involved in the previous diarammatic relation or the F move. The LHS o (??) is the map (i a j b ) k c m e k c l d. (32) l d (30) 10

11 Let H ia j b m denote a basis vector in the usion space e Hom(ia j b, m e ) (i we were not assumin multiplicity-ree usion spaces, we would have to choose several dierent basis vectors). With this notation, the mappins on the LHS o (??) are (in the time lows downwards picture) H ia j b m e id k c : (ia j b ) k c m e k c, H me k c l d : m e k c l d. (33) We can now use our rule o tensorin morphisms in C (equation??) to write H ia j b m e id k c = m ( 1)csij (Hm ij id k c) a+b+c, (34) a+b+c+s ij m since the parity o the usion space Hm ij is Hm ij = s ij m, by deinition. We also trivially have H me k c = (H mk l d l ) e+c. e+c+s mk l Now let s look at the RHS o (??). The usion diaram on the RHS is written as the map i a (j b k c ) i a n l e, (35) where the mappins are accomplished by id i a H jb k c n : i a (j b k c ) i a n, H ia n l d : i a n l e. (36) Turnin these into morphisms in C, we see that we actually have t C [F ijk n id i a H jb k c n = (id i H jk n ) a+b+c a+b+c+s jk n, (37) since id i a = 0. Summarizin, we see that translatin the LHS o (??) into C morphisms ives us a actor o ( 1) csij m, and translatin the usion diaram on the RHS can be done or ree, as lon as we replace F with F. This means that the only dierence between F and F is the ( 1) csij m actor, which is exactly Usher s result. We also note that Bhardwaj, Gaiotto, and Kapustin have also derived this result or the case when the superusion cateory C is drawn rom Vec G. They have a nice picture o this whole derivation in terms o pullin out the ermions rom the usion diarams: check out iures 19 and 20 o their hue paper on spin-tqfts (ariv: ). Since the F symbols satisy the pentaon relation, we can express them in terms o F to determine that the pentaon relation holds only projectively (i.e. up to a sin) in C i the usion space radins s ij m are nontrivial. Explicitly, we have ] mt [Fp itk ] ns [Fs jkl ] tq = ( 1) sij ms kl q [F mkl p ] nq [F ijq p ] ms. (38) The proo is straihtorward, i a bit tedious: just take the pentaon relation in C and use [F ia j b k c ] l d m e n = m ( 1)csij [F ijk l ] mn and s ij m + s mk l = s jk n + s in l. 4 svec G theories or inite Abelian G Despite the rather restrictive set o examples studied in this section, they are a bit nontrivial and still contain a lot o physics. They have also already been sorta understood recently by Kapustin and Gaiotto rom a ield theory point o view, but it still miht be nice to come up with our own perspective on these theories. 11

12 First, we will ocus on the case where none o the objects in the theory are Majorana. In the superusion cateory C = svec G, the pentaon equation δf = 0 will not hold i the radins o the usion spaces are nontrivial. Because the objects in C are invertible, we will write s(, h) instead o s,h h. The pentaon equation is then or equivalently, (δf )(, h, k, l) = ( 1) s(,h)s(k,l), (39) δf = ( 1) s s. (40) The condition (??) on s translates into the 2-cocycle relation, so s H 2 (BG, Z 2 ). The obstruction to satisyin the reular bosonic pentaon equations is overned by the 4- cocycle s s, representin the ermion parity o the 4-simplex labelled by (, h, k, l) which encloses the tetrahedra involved in the deinition o the F move. Because o this, we require that s s be trivial when rearded as a our-cochain in H 4 (G, U(1)), so that our theory is well-deinied in strictly (2+1)D. This is a vacuous constraint or the simple examples o G = Z n, as H 4 (Z n, U(1)) = 0 or all n > 0. Followin what Bhardwaj, Gaiotto, and Kapustin have done, let s look at how we o to the radin the objects picture by passin rom the superusion cateory C = svec G to the bosonized underlyin usion cateory C. The underlyin usion cateory is bosonic, in the sense that C = Vec G, (41) where G is a inite roup determined by the 2-cochain s and the associated short exact sequence 1 Z 2 G G 1. (42) More explicitly, G is the roup consistin o the objects a where G and a Z 2, with the usion o two objects determined by the roup multiplication law in G, with the Z 2 part twisted by s: a h b = (h) a+b+s(,h). (43) Usin our earlier relation between the F -symbols in C and C, we see that or, more concisely F( a, h b, k c ) = ( 1) s(,h)c F (, h, k), (44) F = ( 1) s P F, (45) where P H 1 (BG, Z 2 ), P : a a projects onto the ermion parity o its arument. P is related to S throuh δp = s, (46) which can be understood by recallin that s(, h) measures the dierence in ermion parity between the objects h and h, which in G is precisely measured by the cochain δp. Note that this relation holds in G (not in G!), where at a more precise level, we are interpretin s in the above ormula as the imae o s H 2 (BG, Z 2 ) under the inclusion G G induced by the map 0. Explicitly, this just means that (δp )( a, h b ) = P ( a ) + P (h b ) P ((h) a+b+s(,h) ) = a + b a b s(, h) = s(, h). This means that the actions o the superusion and underlyin usion cateory theories are related by a Z 2 Chern-Simons term: F = ( 1) P δp.f (47) 12

13 Likewise, we can write the pentaon identity in C as holdin only up to a Θ-term (think F F ) δf = ( 1) δp δp. (48) We note that the Θ-term has to be exact in H 4 (BG, R/Z) i our theory is to be consistent in strictly (2+1)D, since a nontrivial ourth cohomoloy class would require the presence o nontrivial (3+1)D physics. 5 Spin structures Fermionic topoloical phases need to come with spin structures in order or the ermions that live on them to be well-deined. In this subsection, we ll briely review what spin structures are, and how we can classiy them or the maniolds we ll be workin over. Since ermions carry spin and braidin statistics and as such are not transparent, they cannot be condensed without some additional help. This is provided by the existence o codimension-1 membranes equipped with spin structures, which are devices used to transparentize the ermions, allowin them to be condensed. Very loosely, a spin structure is a bundle over a maniold M that associates the location o each 2π-twisted ermion worldine with a actor o 1, which cancel out the actors o 1 picked up upon untwistin 2π-twisted worldlines and allows or the transparentization process to o throuh. To make this more precise, we rame ermions by thinkin o them rom the ribbon point o view, and look at the vector bundle o ermion ramins over M. I the ermions were bosonic, the structure roup o this vector bundle would just be SO(n), since rotatin the ramin by 2π at a iven point would be equivalent to doin nothin. However, since ermions are (o course) ermionic, a 2π twist in their ramin at a iven point is not equivalent to doin nothin, and instead ives us a 1 sin. This means that relations between SO(n) representations (like Rπ 2 = 1) only hold projectively when applied to ermions. Thereore, we deine the roup Spin(n) thouh the short exact sequence 1 Z 2 Spin(n) SO(n) 1, (49) so that representations o Spin(n) are Z 2 -projective representations o SO(n) representations. Givin a maniold M a spin structure is then equivalent to equippin its vector bundle o ermion ramins a Spin(n) action. This can be encapsulated by the short exact sequence 0 Z 2 F Sp(M) F SO(M) 0, (50) where F SO(M) is the oriented rame bundle on M (with SO(n) as the structure roup) and F Sp(M) is the oriented spin-rame bundle on M, which is like F SO(M) but with structure roup Spin(n). However, not all exact sequences above are physically allowable. We must place a urther constraint on the spin bundle, namely that it must be untwisted. I the bundle were twisted, we could o around contractible loops and et a nontrivial holonomies, which would prevent us rom constructin a well-deined transparentization procedure. The untwistedness o the bundle corresponds precisely to the condition that the second Stieel-Whitney class ω 2 H 2 (M, Z 2 ) used to deine the twistin o the bundle be trivial (ω 2 is the 2-cocycle associated with M s tanent bundle). For the examples we re interested this can shown to always be the case, since we will always have H 2 (M, Z 2 ) = 0. This 13

14 then implies that any 2-cochain ω 2 on M can be written as ω 2 = δη or some 1-cochain η H 1 (M, Z 2 ), meanin that dierent spin structures are parametrized by the roup H 1 (M, Z 2 ), which is usually easy to calculate, and physically corresponds to a choice o ermionic boundary conditions around the noncontrabile loops in M. This arees with our interpretation o a spin structure in terms o the codimension-1 back wall picture, since Poincare duality allows us to associate any element in H 1 (M, Z 2 ) with a codimension-1 submaniold o M. Finally, we can to calculate the number o dierent spin structures that eneric maniolds possess. As mentioned in the last pararaph, H 1 (M, Z 2 ) classiies the dierent types o spin structures. We will be primarily interested in the cases where M is an n-punctured sphere. Let Sn 2 denote the n-punctured sphere. By drawin a cell complex, it s easy to see that H 0 (S 2 n, Z) = Z, H 1 (S 2 n, Z) = Z n 1, H 2 (S 2 n, Z) = 0. (51) Then we can use the universal coeicient theorem, which ives us an exact sequence 0 Ext(H i 1 (S 2 n, Z), Z 2 ) H i (S 2 n, Z 2 ) Hom(H i (S 2 n, Z), Z 2 ) 0. (52) Since Ext(G, H) = 0 i G is ree and Hom(G, H) = H i G is ree, we et H 1 (S 2 n, Z 2 ) = Z n 1 2 H 2 (S 2 n, Z 2 ) = 0. (53) The act that H 2 (Sn, 2 Z 2 ) = 0 tells us that these maniolds are always spin. In particular, there are two spin structures on a cylinder (vortex or no vortex), and our on a pair o pants. From the structure o H 1 (Sn, 2 Z 2 ) and the duality between H 1 (S2, 2 Z 2 ) and H 1 (Sn, 2 Z 2 ), we see that spin structures are determined by pickin a choice o boundary conditions or ermions alon (n 1) o the punctures, with the boundary condition around the nth puncture then bein uniquely determined. Finally, we mention somethin rather counter-intuitive about spin structures: H 1 (M, Z 2 ) can be identiied with the spin structures over M, but not in a canonical way. This is because the identity spin structure may not correspond to the identity element in H 1 (M, Z 2 ): spin structures are an H 1 (M, Z 2 ) torsor. To illustrate this, suppose we re on a cylinder. Because the spin structure orces contractible ermionic loops to have anti-periodic boundary conditions, the anti-periodic spin structure corresponds to the identity in H 1 (M, Z 2 ) = Z 2, and the periodic spin structure corresponds to the nontrivial element in Z 2. This means that we have the somewhat counter-intuitive spin-structure multiplication law o periodic periodic = anti-periodic. 14