Different definitions of "charged spinors": "bundle splicing" vs. "twisted spinor bundles"

Question

Currently I study the mathematical formulation of the (classical) standard model of particle physics using the language of gauge theory and spin geometry. One of the central objects in the standard model are "charged spinors", which are fermionic particles, which transform under a non-trivial representation. Now, the point is that I stumbled over two appearently different definitions and I would like to know if they are somehow related.

To fix notation, let us take a (suffiently nice) pseudo-Riemannian manifold $(\mathcal{M},g)$ with signature $(s,t)$ and a principal $G$-bundle $P\stackrel{\pi}{\to}\mathcal{M}$. Furthermore, let $\mathrm{Spin}(\mathcal{M})$ be a spin-structure on $\mathcal{M}$ and $(\rho,V)$ be a finite-dimensional representation of $G$.

1. The first definition seems to be a little bit more standard in the mathematical literature. It is for example used in the textbook "Mathematical Gauge Theory" by M. J. D. Hamilton from Springer. First of all, consider the spinor bundle $$S:=\mathrm{Spin}(\mathcal{M})\times_{\kappa}\Delta_{n},$$ where $(\kappa,\Delta_{n})$ denotes the spinor representation of $\mathrm{Spin}(s,t)$ (the vector space $\Delta_{n}$ is $\Delta_{n}\cong\mathbb{C}^{N}$ where $N=2^{n/2}$ if $n$ is even and $N=2^{(n-1)/2}$ otherwise). Then, a "charged spinor" is defined to be a section of the twisted bundle $$E\otimes S,$$ where $E$ denotes the associated vector bundle $E:=P\times_{\rho}V$.
2. The second definition is used in nlab and also in the book "Gauge Theory and Variational Principles" from D. Bleecker. Here, one defines first of all the concept of "bundle splicing": Take two principal bundles $P_{i}\stackrel{\pi_{i}}{\to}\mathcal{M}$ with structure group $G_{i}$ for $i\in\{1,2\}$. Then, the fibre product $P_{1}\times_{\mathcal{M}}P_{2}$ is a principal $(G_{1}\times G_{2})$-bundle denoted by $P_{1}\circ P_{2}$. Now, take a representation $R:\mathrm{Spin}(s,t)\to\mathrm{Aut}(V)$, which commutes with $\rho$. Then, one defines "charged spinors" to be sections of $$(\mathrm{Spin}(\mathcal{M})\circ P)\times_{(R\times\rho)}V$$.

So, is there any relations between the two? I expect that there should be one, since in the end both of them are used to model the same physical object. However, I cannot see how they are related. If they are different, is one of them more general in some sense?

Answer 1

I'll assume that the vector space "$V$" occuring in constructions (1) and (2) doesn't have to be the same. In that case I'll rename vector space in construction (2) to "$W$."

Then I claim that construction (1) is a special case of construction (2) with $W=V\otimes\Delta_n$. That follows from the general fact that if $V_1$ and $V_2$ are representations of $G_1$ and $G_2$, and $P_1$ and $P_2$ are $G_1$- and $G_2$-principal bundles over $M$, then the tensor product of the vector bundles $P_1\times_{G_1}V_1$ and $P_2\times_{G_2}V_2$ is isomorphic to $(P_1\circ P_2)\times_{(G_1\times G_2)}(V_1\otimes V_2)$.

I think the easiest way to see that is to use classifying spaces. Then the bundle $P_i$ are maps $f_i:M\rightarrow BG_i$, the representations determine maps $\rho_i:BG_i\rightarrow B\text{Aut}(V_i)$, the bundle $P_1\circ P_2$ is the map $M\xrightarrow{\text{diag}} M\times M\xrightarrow{f_1\times f_2}BG_1\times BG_2$, the vector bundle $P_i\times_{G_i}V_i$ is the map $M\xrightarrow{f_i} BG_i\xrightarrow{\rho_i}B\text{Aut}(V_i)$, and we find that the vector bundles we're trying to compare are both given by the map $$M\xrightarrow{\text{diag}} M\times M\xrightarrow{f_1\times f_2}BG_1\times BG_2\xrightarrow{\rho_1\times\rho_2}B\text{Aut}(V_1)\times B\text{Aut}(V_2)\rightarrow B\text{Aut}(V_1\otimes V_2)$$