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I am working through the derivation for a method published in this paper. On page 586 (equation A1) the authors use the notation:

$$ N(\bullet,\bullet) $$

What do the bullets represent? Do they imply any mean and variance will satisfy the following steps, a specific mean and variance previously written should be assumed, or something else entirely?

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The notation serves to shorten the notation for the distribution of $\beta_i$ as the equation is complicated enough. The distribution of $\beta_i$ is given previously on top of page 585. You can verify this by plugging in an Expectation of $0$ and a standard deviation of $\frac{h^2}{Mp}$.

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