1. Identify substance A

   The statement says:

   • On heating A at $573\,\text{K}$, we get red phosphorus.

   A well-known allotrope conversion is: $\text{white phosphorus} \xrightarrow[\;573\,K\;]{} \text{red phosphorus}$

   Hence, $A = \text{white phosphorus}$

2. Identify substance B

   The statement further says:

   • Red phosphorus on heating at $803\,\text{K}$ under pressure converts to B.

   Another standard allotrope conversion is: $\text{red phosphorus} \xrightarrow[\text{under pressure}]{803\,K} \text{black phosphorus}$

   The black phosphorus obtained under these conditions is $\alpha$-black phosphorus.

   Hence, $B = \alpha\text{-black phosphorus}$

3. Match with options

   So,

   • $A =$ white phosphorus
   • $B = \alpha$-black phosphorus

   This corresponds to Option D.

4. Check all options briefly

   • A: $\beta$-black phosphorus, white phosphorus $\rightarrow$ incorrect
   • B: white phosphorus, $\beta$-black phosphorus $\rightarrow$ incorrect
   • C: $\alpha$-black phosphorus, white phosphorus $\rightarrow$ incorrect
   • D: white phosphorus, $\alpha$-black phosphorus $\rightarrow$ correct

Therefore, the correct answer is: $\boxed{\text{D}}$