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In this post, we like to relate the following 3 bosonic TQFTs that can be defined on generic non-spin manifold $M^3$.

Given a non-abelian Chern-Simons (CS) TQFT of a gauge group $G$ and the $k$ named as the level, we name it as 3d $G_k$ CS TQFT.

We write its path integral formally as $$ Z=\int [DA] \exp(i S)= \int [DA] \exp(i \frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A)). $$ with an action: $$S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A).$$

The trace tr is taken as the trace of the matrix representation of the Lie algebra valued gauge field $A$. $A$ is a locally 1-form gauge field or 1-connection of the principle $G$ bundle.

  1. 3d (E$_8$)$_1$ CS TQFT

The first is a 3d Chern-Simons TQFT of for a gauge group $G$=(E$_8$) with a level $k=1$, written as (E$_8$)$_1$ CS TQFT.

  1. 3d SO(16)$_1$ CS $\otimes$ a trivial invertible spin TQFT.

This second one we start now with a 3d Chern-Simons TQFT of a gauge group $G$=SO(16) with a level $k=1$, written as SO(16)$_1$ CS TQFT.

Note that:

  • 3d SO(16)$_1$ CS TQFT is a fermionic TQFT defined on a spin manifold.

  • 3d Spin(16)$_1$ CS TQFT is a bosonic TQFT defined also on a non-spin manifold.

Since 3d SO(16)$_1$ CS TQFT is a fermionic TQFT that can be defined only on a spin-manifold,

I propose a tensor product it with a trivial spin TQFT.

i.e. "3d SO(16)$_1$ CS $\otimes$ a trivial invertible spin TQFT. "

  1. 3d Cartan E$_8$ matrix abelian U(1)$_8$ Chern-Simons theory

The $E_8$ Cartan matrix is given by, $$ K_{E_8}=\begin{pmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & -1 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2 \end{pmatrix}. $$ There is one famous application in physics. Which is that a symmetric bilinear $K_{E_8}$-Chern-Simons theory describes the low energy physics of a so-called $E_8$-quantum Hall state (with a $U(1)^8$ gauge group). The field theory partition function $Z$ is given by $$ Z= \int[DA]\exp(\frac{(K_{E_8})_{IJ}}{2 \pi}\int A_I dA_J). $$ This $E_8$-quantum Hall state occurs in a 2-dimensional spatial condensed matter system. This describes an invertible TQFT. Since the $|Z|$ on any closed 3-manifold of $M^2 \times S^1$ has $|Z|=|\det K_{E_8}|=1$. Which means the Hilbert space on any spatial closed $M^2$ is always 1-dimensional.

Proposal:

3d Chern-Simons TQFT of gauge group (E8)$_1$ = SO(16)$_1 \otimes$ a trivial spin TQFT = Cartan E$_8$ matrix

My questions: Physically, the above three theories should be the same. This means that the low energy TQFT of the above three describe the exact same physical phenomena and same physical observables.

How do we see they are the same (or at least related) from a mathematical rigorous viewpoint? Any References?

If you find what I propose is not obvious or rigorous to mathematicians, please feel free to improve my statements in your answer/comment -- thanks!

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