See [Mor94, sec.2.2].
One may want wonder what differential equations does 1-matrix model
satisfty?
In this case we have enough counterterms so that under analytic change of integration variables preserves its integral form, just with some changed arguments . But of course change of dummy integration variable doesn't change the result of integration, so . Generators of such analytic changes of variables are
where is some infinitesimal matrix. So, for every we can introduce auxiliary argument in the definition of such that and and obtain Ward identity
or
Due to sufficient amount of counterterms we will be able to rewrite the derivative through ones and will get countably many Ward identities
This is the idea. Below follows its realization.
The deformed version of is
Let's first expand the exponent
The measure as always transforms by multiplying on Jacobian
where double index notation was used to express Jacobian matrix indices in this case. By the rules of matrix calculus we have
and we also know that
so
Now let's put everything back to (6)
and according to (4) we have
what can actually be rewritten in terms of derivatives in as it was promised in (5)
with
have turned out to be Virasoro operators.
Power sum variables
If one wish to switch to power sum variables
the simple substitution won't do the trick because of lost . So, we'd better first note that
and replace action of explicitly with this relation in (16)
where it's now assumed that
Everything is now prepared for substitution . In new variables becomes
First sum should be taken from , as it implicitly was done in (19).
We can also write symbolically
Pay attention to the lower sum limit. In that case
and (16) can be symbolically rewritten as
where every combination is treated as multiplication by . Lower bound of the first sum now is raised to 1 because even in (16) summation went from due to factor.