Honestly...I think this is about how you feel about it, and why - not what anyone else thinks. When I started going out with my OH, he was still in touch with his ex - who I jokingly call 'the one who got away'. They'd split up about three years before, and she lived in another country (another continent, actually), and the chances of them ever seeing each other again were pretty remote. But he wrote her these long emails, that were not sexual, but were playful and teasing, and affectionate. I found one of them, because he didn't have a computer and had drafted it on word and saved it, but left it open. So he wasn't 'hiding' it, but he wasn't being super open about it.
Now, I trust him. I still trust him, and I know that they continued to correspond occasionally - though I don't think that recently now. What I was hurt about at the time, was that he wrote her these affectionate emails, but seemed to struggle to express himself to me in the same way. I didn't think he'd do anything, but it was the absence of that with me. So that's what we talked about when I brought it up - not the fact that I felt he'd been unfaithful, or that I was upset he was still in touch with her.
That said, I think this is about boundaries in your relationship, and not everyone has the same ones. But i think the important thing, when one discovers something like this, is to really probe why you feel the way you feel about it. I think this is the thing about 'emotional affairs' - often what really hurts is that your partner has been able to have a connection with someone else at the expense of the one with you. If your relationship is strong, then often a close relationship with someone else isn't a problem. But it's important to unpick all of this, and work out why you feel the way you feel. Is it the secrecy? Is the close connection? Is it something else? And why are you responding the way you are?