Real representation of smallest dimension of Clifford Algebra with $d$ generators

Question

I'm trying to understand the model described in this paper. I have a question about a claim they make. From page 2:

To describe the fermionic degrees of freedom let, as a preliminary

\begin{align*} \gamma^i &= (\gamma^i_{\alpha \beta})_{\alpha, \beta = 1, ... ,d}, \end{align*}

be the real representation of smallest dimension, called $s_d$, of the Clifford Algebra with $d$ generators: $\{\gamma^s, \gamma^t\} = 2\delta^{st} \mathbb{1}_d$. On the representation space, $Spin(d)$ is realized through the matrices $R \in SO(s_d)$, so that we may view

\begin{align*} Spin(d) \hookrightarrow SO(s_d) \end{align*} as a simply connected subgroup. We recall that

$$ s_d = \left\{ \begin{array}{ll} 2^{\lfloor\frac{d}{2}\rfloor} &, d = 0, 1, 2, mod(8) \\ 2^{\lfloor\frac{d}{2}\rfloor+ 1} & else \\ \end{array} \right. $$

where $ \lfloor\cdot\rfloor $ denotes the integer part.

I have checked the appendix in Polchinski's string theory book, and various more mathematical notes I could find online. I cannot find any resource that will explain this result for $s_d$. I would appreciate if someone can explain it, and hopefully give some reference for this.

EDIT: I don't see why this question was closed. I don't see how the linked post answers the question.

I'm asking about real irreducible representations with positive definite metric. The linked post seems to deal with complex and/or Lorentzian cases. And I am asking about both odd and even dimensional cases.
And I'm asking about the specific formula above.
Linking to that other post, since it is somewhat related seems reasonable, and is appreciated. But please reopen my question.

Answer 1

You need to put two things together here: First, the classification of the complex irreps of the Clifford algebra, and then the question when these have a real structure.

The unique complex irreducible representation of the Clifford algebra in $d$ dimensions has dimension $2^{p}$, where $p = \lfloor \frac{d}{2}\rfloor$, see also this answer of mine. This representation, if we forget the complex structure, is a real representation of dimension $2^{p+1}$.

It reduces to a real representation of dimension $2^p$ when Majorana spinors exist, and the question of when exactly the Clifford algebra of signature $(p,q)$ has Majoranas is a bit of a subtle question and depends on what exactly one means by this, see this question of mine. The claim in the source in the OP that such real representations exist for $d = 0,1,2 \mod 8$ fit with O'Farrill's convention in the linked question that Majoranas exist for $p-q = 0,6,7 \mod 8$, since for $p = 0,q = d$ we have that $-d = 0,6,7\mod 8$ implies $d = 0,1,2 \mod 8$.