WIP summary
Two-matrix model generating function with logarithmic sources
for every finite integer satisfy Ward identities
where for are defined via recursion relation
From Ward identities formulation (in case there is enough of them) we see that this matrix model enjoys the following symmetry
for any non-zero and .
Special cases
- allows to be infinite and the number of Ward identities increases drastically as restriction on becomes .
- introduces operators and makes condition on to be relaxed to .
- additionally allows identities.
One-matrix model [Mor94, sec.2.2]
One may wonder what differential equations does 1-matrix model
satisfty?
Derivation via shift of integration variables
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 (13)
and according to (11) we have
what can actually be rewritten in terms of derivatives in as it was promised in (12)
with
have turned out to be Virasoro operators.
Derivation via insertion of full derivative
There is also easier, though less conceptual, method to derive the same result. It relies on the following observation
This integral was secretly understood as "contour" one even before. The contour is chosen such that integral
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 (23)
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 (28).
We can also write symbolically
Pay attention to the lower sum limit. In that case
and (23) 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 (23) summation went from due to factor.
Fayet-Iliopoulos
We can also add logarithmic source to our 1-matrix model and see how it will change our Ward identities
In comparison to , derivative by produces
Why one would need something like that in the context of Ward identities? Let's proceed anyway. The only new thing in comparison to what have been discussed before is the expansion of this exponent
Taking it into account the analogue of (21) in this situation is
in obvious notation
For we have with
where
For
and term cannot be obtained by differentiantion of the exponent. But no one said we need this particular Ward identity to formulate the equivalent of (34) definition on the language of differential equations. case is somewhat nicer
but there are convincing arguments [NNZ16] that it isn't needed for the discussed formulation, too. If we all agree with this fact, then we can say that enters on par with in the sense of (40). If we needed to include case in the consideration, such a simple relation between and wouldn't exist.
Two-matrix models
Formally, the most general two-matrix model is
It's a bit degenerate in variables , but this is not a big deal. What is a big deal is that this is a very complicated construction by itself. So we'll restrict ourself with the simplest scenario
of two one-matrix models with some non-trivial coupling. Two models are of particular importance:
- — the simplest one, minus sign is conventional and will simplify things for us in the future (equation (6), to be precise),
- — model subjected to Zamolodchikov' -constraints.
The algebra of the former model constraints got the name -algebra due to its striking similarity to one. We'll restrict ourselves further to the simplest case
The lore is as follows. First, we turn our attention to the integral over exclusively
Ward identities are, as always, obtained via insertion of the full derivative into the integral
Derivative is easy to take
and to reformulate as derivatives exclusively
It's a matrix equation for now, but invariance of under unitary conjugations allows one to show that only depend on eigenvalues of and, moreover, it's a symmetric function of eigenvalues , . To capture this dependence we only need all the equations fixing it, namely
Or, in expanded form,
These identities are enough to fix dependence of . -dependence still needs special consideration. For this purpose one needs to insert power of in the raw Ward identities form like that
Using elementary matrix calculus rules one obtains
where
It rewrites as
Again, by multiplying obtained result by and taking trace one gets
Now, we are prepared to the discussion of Ward identities. Ward identities for ensure that
The second term is to be integrated by parts producing
where -s acts on everything to the right of them, including the last exponent. The expression
at the end of the day cannot be something other than polynomial in , and , multilplied by the exponent of our source. As while acting on exponent, the whole matrix operator can be rewritten as some operator in only
The choice of operator's indices is made for future convenience. Now, (58) can be rewritten as
For (56) it means
Recursive definition of operators
We have obtained rather implicit definition of operators yet. The main instrument for calculating them is an explicit recursive definition, discussed in this section. Let us try to express operators through . First of all, it is time to introduce convenient notation
For case we have
So, for
Next, one derivative in definition
is explicitly taken
leading to
Now, can be substituted by and both sums can be taken out of the traces
operators on the right-hand side are easily recognizable now
Last step is a shifting of the lower index
This means that
Fayet-Iliopoulos
source
Let us add logarithmic source to our theory
In this case Ward identities for -dependence become
To get rid of the inverse of derivative operator we just take an extra derivative of this expression
Now, we multiply it by and take trace
Thus, Ward identities on are
where definition of -operators is as above. In the same fashion Ward identities for in this case are
On the language of -operators
Rewriting explicitly one obtains
If we abandon identity (as it was prescribed for one-matrix model with logarithmic term) then will enter Ward identities only as a sum .
and sources
Ward identities are just (77), but with -operators now defined for
in a standard way
where restriction ensures that we have polynomial in , in front of the exponent on the RHS. That is all the negative powers of obtained by differentianting logarithm in the exponent, or by differentiating these negative powers of over and over again are compensated by large enough positive power of .
In this setting we have the base of recursion given as
So, for
Next, we need to describe a step of the recursion. Here it is
So we have