usually the term implied volatility refers to Black-Scholes implied volatility (also Log-Normal volatility): it is defined as a quantity which when plugged in the Black-Scholes formula returns the right price. In an article by Roper (2010) certain conditions are given which guarantee that the Black-Scholes implied Volatility Surface is consistent with No Arbitrage.
When working with interest rates it has become convenient to use Bachelier implied volatility (or normal implied volatility) instead of Black-Scholes implied volatility. The Bachelier implied volatility is the parameter which when plugged into to the Bachelier option pricing formula returns the correct price. I was wondering whether there is an article which characterizes No Arbitrage conditions for Normal Implied Volatility. I think that such conditions would differ from the conditions given in Roper, since he uses the Black-Scholes formula to transform the Black-Scholes implied volatility surface to the option price surface and checks for No-Arbitrage there.
If not, I think a simple approach to obtain such conditions would be to transform the No-Arbitrage conditions for (Call)-option prices into conditions for implied volatility.
Roper, M. (2010). Arbitrage free implied volatility surfaces. preprint.