In this paper, we construct two sets of vertex operators S + and S ? from a direct sum of two sets of Heisenberg algebras. Then by calculating the vacuum expectation value of some products of vertex operators, we get Macdonald function in special variables x i = t i-1 ( i = 0,1, 2,
). Hence we obtain the operator product formula for a special Macdonald function
P
λ (1,
t,
,
t
n-1;
q,
t ) when
n is finite as well as when n goes to infinity.
The study of topological string on Calabi-Yau manifolds is interested in mathematical physics for many years. It was found that gauge theories with certain gauge groups can be geometrically engineered from some Calabi-Yau threefolds, and the topological string partition functions on such spaces are related to instanton sums in gauge theories [
The topological vertex formalism provides a powerful method to calculate the topological string partition function for non-compact toric Calabi-Yau 3-fold. By transfer matrix approach, A. Okounkov, N. Reshetikhin and C. Vafa proposed the topological vertex C λ μ ν using Schur and skew Schur functions [
C λ μ ν ( q ) = q κ ( μ ) 2 s ν t ( q − ρ ) ∑ η s λ t / η ( q − ν − ρ ) s μ / η ( q − ν t − ρ )
where λ , μ , ν are Young diagrams, λ t denotes the transpose of λ, and ρ = ( − 1 / 2 , − 3 / 2 , − 5 / 2 , ⋯ ) . The topological vertex C λ μ ν has a nice interpretation by statistical mechanics of the melting crystal model [
On the other hand, gauge theory partition function is a function with two equivariant parameters. In 2007, based on the arguments of geometric engineering, concerning the K-theoretic lift of the Nekrasov partition functions, A. Iqbal, C. Kozçaz and C. Vafa introduced a refined version of topological vertex [
C λ μ ν ( t , q ) = ( q t ) ‖ μ ‖ 2 + ‖ ν ‖ 2 2 t κ ( μ ) 2 P ν t ( t − ρ ; q , t ) × ∑ η ( q t ) | η | + | λ | − | μ | 2 s λ t / η ( t − ρ q − ν ) s μ / η ( t − ν t q − ρ )
where ‖ λ ‖ 2 = ∑ i λ i 2 . Moreover H. Awata and H. Kanno proposed another formula [
C μ λ ν ( q , t ) = P λ ( t ρ ; q , t ) f ν ( q , t ) − 1 × ∑ σ ι P μ t / σ t ( − t λ t q ρ ; t , q ) P ν / σ ( q λ t ρ ; q , t ) ( q 1 / 2 / t 1 / 2 ) | σ | − | ν | ,
where f λ ( q , t ) = ( − 1 ) | λ | q n ( λ t ) + | λ | / 2 t − n ( λ ) − | λ | / 2 and ι is the involution on the algebra of symmetric functions defined by ι ( p n ) = − p n , here p n ( x ) = ∑ i = 1 ∞ x i n . Although C λ μ ν ( t , q ) and C μ λ ν ( q , t ) have different expressions, they are supposed to give the same result.
Therefore it seems that the key problem is to change Schur function for the unrefined case to Macdonald function for the refined one. Hence to find a vertex operator formalism for the refined topological vertex will be interesting. The essential step is to realize the special Macdonald function P λ ( t − ρ ; q , t ) . However a vertex operator formalism for P λ ( t − ρ ; q , t ) does not exist so far.
In this paper, we get the operator product formula for the special Macdonald function P λ ( 1, t , ⋯ , t n − 1 ; q , t ) . We also extend this formula to the case when n goes to infinity.
• ℚ : the set of rational numbers;
• ℚ ( q , t ) : the field of rational functions of q, t over ℚ ;
• The q infinite product: ( x ; q ) ∞ : = ∏ n ≥ 0 ( 1 − x q n ) .
A partition is any (finite or infinite) sequence λ = ( λ 1 , λ 2 , ⋯ , λ r , ⋯ ) of non-negative in decreasing order: λ 1 ≥ λ 2 ≥ ⋯ ≥ λ r ≥ ⋯ and containing only finitely many non-zero terms. We denote by | λ | the size of the partition, i.e. | λ | = ∑ i λ i and by l ( λ ) the number of non-zero λ i . The set of all partitions is denoted by P .
A pictorial representation of a partition λ is called 2D Young diagram , it can be obtained by placing λ i boxes at the i-th row. For example,
The transpose of λ is denoted by λ t , λ t = ( λ 1 t , λ 2 t , ⋯ ) , here λ j t = Card { i | λ i ≥ j } . For example, the transpose of λ = ( 5 , 4 , 4 , 1 ) is λ t = ( 4 , 3 , 3 , 3 , 1 ) .
We denote by s = ( i , j ) ∈ ℤ 2 for each square of a partitionλ, here
The numbers
define a scalar product
here
Macdonald function
1)
2)
When
In particular,
here
Let
We introduce an algebra
Let
The bosonic Fock space
The dual vacuum state
There is a paring
To construct the vertex realization for
We define
Since
we can obtain
We define another set of vertex operators
Since
likewise we obtain
With the help of the vertex operator
and
We propose operator product formula
After some careful computation via the commutative relation (6) and (9), the Formula (12) is equal to
Using the identity (we will prove it in the appendix)
we get the vacuum expectation value of this operator product formula
In other words,
Therefore we get the operator product formula for
Similarly, by using the identity
we get
Hence we get the operator product formula for
The operator product formula for a special Macdonald function
Firstly, we will proof the identity (13).
Suppose
For the first part
For the second part
For the third part
Next, we will simplify the left hand side of the (13).
Since
Before combing them all, we can check
In conclusion,
To show the identity (13), we need to use some properties of Young diagram λ, namely we need to interpret those powers of q in terms of arm lengths, leg lengths, arm co-lengths and leg co-lengths of those squares of Young diagram λ.
Now let us take i-th row as an example. We can classify all the arm lengths denoted as
For those squares which have leg length
Similarly for leg co-length
Now from previous computation of
Next we will prove the identity (16).
we notice that if n goes to infinity
So when n goes to infinity,
From previous analysis about the properties of Young diagram, we can deduce the identity (16).
This work is partially supported by National Natural Science Foundation of China (No. 11475116, 11401400, 11626084, 11647123).
Wang, L.F., Wu, K. and Yang, J. (2018) Operator Product Formula for a Special Macdonald Function. Applied Mathematics, 9, 459-471. https://doi.org/10.4236/am.2018.94033