Heterotic tadpole I have a question related to the heterotic string theory. I have seen many references that are talking about heterotic tadpole but no one explains why it is an important quantity. The questions are that


*

*What is a heterotic tadpole?

*Why is it important?

*Does a non-zero heterotic tadpole mean that we consider the heterotic theory in some background? 
 A: I don't have a good reference to hand, but I can provide some context. Tadpoles in general are Feynman diagrams that look like this:

They are not specific to string theory, let alone heterotic strings and were in fact introduced by Coleman in a QFT context. The string diagram involves surfaces rather than lines but the basic idea remains the same.
They are important for the following reason: If we take the time direction to increase upwards in the diagram, then the diagram says that a pair of fermions is created out of nothing and then decays into a scalar. In other words, a stable  scalar particle was just created out of nothing.
There are strong implications for theories that allow such diagrams. For example, it means that the vacuum is unstable. Particles can be created out of nothing. For that reason, "tadpole diagrams" and "vacuum instability" are often discussed together.
Whether such theories are "good" or "bad" is up to debate. For example, one viewpoint is that this should not be allowed and this can be used as a motivation to introduce supersymmetry. In supersymmetric heterotic theories there is an exact cancellation of such diagrams.
On the other hand, the contribution of these diagrams is also related to the cosmological constant $\Lambda$, so in theories were tadpoles are allowed one can calculate the cosmological constant explicitly. In most non-supersymmetric heterotic models that I am aware of, this calculation gives a cosmological constant that is wrong by many orders of magnitude. Nevertheless, one of the selling points of string theory is that at least we can perform these calculations and provide an explicit number for $\Lambda$, something which is not the case in different frameworks. 
It would of course be very desirable to find ways to improve on these models and find ways to suppress the predicted cosmological constant and this is an area of active research.
