On the Cobordism Classification of Symmetry Protected Topological Phases

Abstract

In the framework of Atiyah’s axioms of topological quantum field theory with unitarity, we give a direct proof of the fact that symmetry protected topological phases without Hall effects are classified by cobordism invariants. We first show that the partition functions of those theories are cobordism invariants after a tuning of the Euler term. Conversely, for a given cobordism invariant, we construct a unitary topological field theory whose partition function is given by the cobordism invariant, assuming that a certain bordism group is finitely generated. Two theories having the same cobordism invariant partition functions are isomorphic.

A Some Categorical Notions

For completeness, here we reproduce the definitions of symmetric monoidal categories, functors and natural transformations summarized in [46]. We denote categories by ${\mathscr {C}},{\mathscr {D}},\ldots$, functories by $F,G,\ldots$, and natural transformations by $\eta ,\ldots$. The definitions of ordinary categories, functors and natural transformations are explained very briefly in Sect. 2.4.

Symmetric monoidal category. First we define symmetric monoidal category.

Definition A.1

A monoidal category is a category equipped with

• a functor $\otimes : {\mathscr {C}}\times {\mathscr {C}}\rightarrow {\mathscr {C}}$ called the tensor product,

• an object $1 \in \mathrm{obj}({\mathscr {C}})$ called the unit object,

• a natural isomorphism $a_{x,y,z}$ ($x,y,z \in \mathrm{obj}({\mathscr {C}})$) called the associator

  $$\begin{aligned} a_{x,y,z}: (x \otimes y) \otimes z \rightarrow x \otimes (y \otimes z) \end{aligned}$$

  (A.1)

  satisfying the pentagon equation

  

  (A.2)

• natural isomorphisms $\ell _x$ and $r_x$ called the left and right unit laws,

  $$\begin{aligned} \ell _x : 1 \otimes x \rightarrow x, \qquad r_x: x \otimes 1 \rightarrow x \end{aligned}$$

  (A.3)

  satisfying the triangle equations

  

  (A.4)

Roughly speaking, the pentagon equation means that “multiplications can be done in any order”, or “any ways to go from $(( w \otimes x) \otimes y) \otimes z$ to $w \otimes ( x \otimes ( y \otimes z ))$ are the same”. The triangle equation means that “any ways to eliminate the unit 1 are the same”.

Definition A.2

A braided monoidal category is a monoidal category with a natural isomorphism $b_{x,y}$ called the braiding,

$$\begin{aligned} b_{x,y}: x \otimes y \rightarrow y \otimes x \end{aligned}$$

(A.5)

satisfying the hexagon equations

(A.6)

(A.7)

Roughly speaking, the first hexagon equation above means that “moving x all at once from the left to the right of $y \otimes z$ is the same as moving x step by step by first going through y and then z.” The second hexagon equation means a similar thing for z.

Definition A.3

A symmetric monoidal category is a braided monoidal category such that the braiding satisfies $b_{y,x} b_{x,y} = 1_{x \otimes y}$.

Symmetric monoidal functor. Let us next consider functors between monoidal categories. In the following, if an expression like e.g. $a^{\mathscr {D}}_{x,y,z}$ appears with a superscript or subscript ${\mathscr {D}}$, that means (in this particular case) “the associator in the category ${\mathscr {D}}$”. The same remark applies to subscripts/superscripts of other quantities.

Definition A.4

A monoidal functor F between monoidal categories ${\mathscr {C}}$ and ${\mathscr {D}}$ is a functor with

• a natural transformation

  $$\begin{aligned} \mu _{x,y}: F(x) \otimes F(y) \rightarrow F(x \otimes y) \end{aligned}$$

  (A.8)

  satisfying the associativity

  

  (A.9)

• an isomorphism

  $$\begin{aligned} \epsilon : 1_{{\mathscr {D}}} \rightarrow F(1_{\mathscr {C}}) \end{aligned}$$

  (A.10)

  satisfying

  

  (A.11)

  and

  

  (A.12)

Roughly speaking, these equations mean that “the associator $a_{x,y,z}$ and the left, right unit laws $\ell _x$, $r_x$ can be used before or after the application of the functor, giving the same result”.

Definition A.5

A braided monoidal functor between braided monoidal categories is a monoidal functor with the additional condition that

(A.13)

Again, this roughly means that “the braiding can be used before or after the functor”.

Definition A.6

A symmetric monoidal functor between symmetric monoidal categories is a braided monoidal functor without extra conditions.

Symmetric monoidal natural transformation.Finally, we describe natural transformations.

Definition A.7

A monoidal natural transformation $\eta$ between monoidal functors F and G is a natural transformation such that the following diagram commutes:

(A.14)

(A.15)

Definition A.8

A braided (reps. symmetric) monoidal natural transformation between braided (reps. symmetric) monoidal functors is a monoidal natural transformation, without extra conditions.

Involution. For completeness, we also describe the notion of involution following [3].

Definition A.9

An involution on a category ${\mathscr {C}}$ is a pair $(\beta , \xi )$ of a functor $\beta : {\mathscr {C}}\rightarrow {\mathscr {C}}$ and a natural isomorphism $\xi : \mathrm{id}_{\mathscr {C}}\rightarrow \beta ^2$ such that $\beta ( \xi _x) = \xi _{\beta (x)}$ as morphisms $\beta (x) \rightarrow \beta ^3(x)$.

Roughly speaking, the involution functor $\beta$ “squares to the identity functor”.

Definition A.10

Let $\beta _{\mathscr {C}}$ and $\beta _{\mathscr {D}}$ be involutions of categories ${\mathscr {C}}$ and ${\mathscr {D}}$. A functor $F : {\mathscr {C}}\rightarrow {\mathscr {D}}$ is equivariant under the involution pair $(\beta _{\mathscr {C}}, \beta _{\mathscr {D}})$ if there is a natural isomorphism $\phi : F \beta _{\mathscr {C}}\Rightarrow \beta _{\mathscr {D}}F$ such that the following diagram commutes:

(A.16)

Roughly speaking, this means that “the involution $\beta$ and the functor F commutes with each other in the way consistent with the fact that the involution squires to the identity”.

The corresponding notions in symmetric monoidal categories can also be defined by using symmetric monoidal functors and symmetric monoidal natural isomorphisms.