Physical interpretation of TFTs

1. Defining TFTs

Let $n$ be a positive integer and $mathbb k$ be a field.

In my lecture I was introduced to TFTs using the following definition going back to Atiyah (around 1988):

A $n$-dimensional, oriented TFT is a symmetric monoidal functor $$Z: Cob_n rightarrow vect(mathbb k).$$

The cobordism category $C=Cob_n$ is defined as follows:

• Objects are $(n-1)$-dimensional closed, smooth oriented manifolds.
• A morphism from $M in Obj(C)$ to $Nin Obj(C) $ is a certain equivalence class of $n$-bordisms from $M in Obj(C)$ to $N in Obj(C).$
• A $n$-bordism from $M in Obj(C)$ to $N in Obj(C)$ is a tuple $(B, M, N, phi_B)$ with $B$ a $n$-dimensional smooth oriented manifold with boundary (empty ir non-empty) together with an orientation preserving diffeomorphism $phi_B: overline M coprod N rightarrow partial B$. Here, $overline M$ denotes the manifold $M$ with opposite orientation.
• We call two bordisms $(B, M, N, phi_B)$ and $(B’, M, N, phi_{B’})$ equivalent if there exists an orientation preserving diffeomorphism $phi$ such that $phi circ phi_B= phi_{B’}$. One checks that this defines indeed an equivalence relation on all $n$-bordisms from $M$ to $N$ for given $M,N in Obj(C)$.
• Composition of bordisms is defined by "gluing along the common boundary." (A formal exposition of "gluing along a boundary" can apparently be found in Lee’s Introduction to Topological Manifolds.) For a given $M in Obj(C)$ the identity morphism is given by the cylinder $(M times [0,1], M, M)$.
• One checks that this defines a category. In particular, one shows that the composition is well-defined, i.e. independent of the choice of representative.
• The disjoint union of manifolds endows $C$ with the structure of a monoidal category. (The monoidal unit is given by the empty set considered as a $(n-1)$-dimensional manifold. Associator, as well as left and right unit constraint are defined appropriately.)

2. My background

I am a mathematics undergraduate student with very little background in physics.
Apparently, TFT’s have a strong relation to physics, in particular condensed matter physics and dynamics. Wikipedia even states that "TFT’s were inveted by physicists."

3. Question

• What are interpretations and applications of TFTs in physics?

EDIT: Motivating my question is the following statement from my (maths) lecture notes. After a study of TFTs in dimension 1 the notes propose "In physical language, we have a quantum mechanical system which has only ground states and is thus trivial Hamiltonian. Then the only invariant of the system is the degeneracy $dim(V)$ of the space of ground states."