Apr 30, 2018 - By transfer matrix approach, A. Okounkov, N. Reshetikhin and C. Vafa proposed the topological vertex Cλµν using Schur and skew Schur ...
Applied Mathematics, 2018, 9, 459-471 http://www.scirp.org/journal/am ISSN Online: 2152-7393 ISSN Print: 2152-7385
Operator Product Formula for a Special Macdonald Function Lifang Wang2, Ke Wu1, Jie Yang1 School of Mathematical Sciences, Capital Normal University, Beijing, China School of Mathematics and Statistics, Henan University, Kaifeng, China
1 2
How to cite this paper: 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
Abstract
Received: April 4, 2018 Accepted: April 27, 2018 Published: April 30, 2018
function in special variables xi = t i−1 ( i = 0,1, 2, ). Hence we obtain the op-
(
erator product formula for a special Macdonald function Pλ 1, t , , t n−1 ; q, t
Copyright © 2018 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access
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
)
when n is finite as well as when n goes to infinity.
Keywords Macdonald Function, Vertex Operator, Heisenberg Algebra
1. Introduction 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 [1]. 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 Cλµν using Schur and skew Schur functions [2]: Cλµν ( q ) = q
κ(µ ) 2
( ) ∑ s (q
sν t q − ρ
η
λ t /η
−ν − ρ
)s
µ /η
(q ) −ν t − ρ
where λ , µ ,ν are Young diagrams, λ t denotes the transpose of λ, and DOI: 10.4236/am.2018.94033
Apr. 30, 2018
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ρ= ( −1 2, − 3 2, − 5 2,) . The topological vertex Cλµν has a nice interpretation by statistical mechanics of the melting crystal model [2] [3]. In this paper two sets of vertex operators constructed specifically by the annihilation and creation generators of Heisenberg algebra play important roles in realizing Schur and skew Schur functions. 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 [4]. In this refinement, one more parameter t comes in and the theory seems to be deeply related to a Macdonald function with special variables, or what we call a special Macdonald function, Pλ ( t − ρ ; q, t ) : 2
Cλµν
q ( t , q ) = t
2
κ(µ ) 2
t
q × ∑ η t where
2
µ +ν
(
Pν t t − ρ ; q, t
)
η +λ −µ
(
2
)
(
sλ t /η t − ρ q −ν sµ /η t −ν q − ρ t
)
λ = ∑ i λi2 . Moreover H. Awata and H. Kanno proposed another 2
formula [5] which is expressed entirely in terms of the special (skew) Macdonald functions:
(
)
Cνµλ ( q, t ) = Pλ t ρ ; q, t fν ( q, t ) × ∑ι P t σ
µ /σ t
( −t
λt
−1
)
(
)(
q ρ ; t , q Pν /σ q λ t ρ ; q, t q1 2 t1 2
)
σ −ν
,
( )
n λt + λ 2 −n λ − λ 2
and ι is the involution on the f λ ( q, t ) = ( −1) q t ( ) ∞ algebra of symmetric functions defined by ι ( pn ) = − pn , here pn ( x ) = ∑ i=1 xin . Although Cλµν ( t , q ) and Cνµλ ( q, t ) have different expressions, they are λ
where
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 Pλ t − ρ ; q, t . However
(
−ρ
a vertex operator formalism for Pλ t ; q, t
)
does not exist so far.
In this paper, we get the operator product formula for the special Macdonald
(
)
function Pλ 1, t , , t n−1 ; q, t . We also extend this formula to the case when n goes to infinity.
2. Preliminaries 2.1. Notations • : the set of rational numbers;
• ( q, t ) : the field of rational functions of q, t over ; • The q infinite product:
DOI: 10.4236/am.2018.94033
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( x; q= )∞ : ∏ (1 − xq n ) . n ≥0
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2.2. Partitions A partition is any (finite or infinite) sequence λ = ( λ1 , λ2 , , λr ,) of non-negative in decreasing order: λ1 ≥ λ2 ≥ ≥ λr ≥ and containing only finitely many non-zero terms. We denote by λ the size of the partition, i.e. λ = ∑ i λi and by l ( λ ) the number of non-zero λi . The set of all partitions
is denoted by . A pictorial representation of a partition λ is called 2D Young diagram , it can be obtained by placing λi boxes at the i-th row. For example, Figure 1 represents a partition λ = ( 5, 4, 4,1) . The transpose of λ is denoted by λ t , λ t = λ1t , λ2t , , here
(
)
= λ Card {i | λi ≥ j} . For example, the transpose of λ = ( 5, 4, 4,1) is λ = ( 4,3,3,3,1) . s ( i, j ) ∈ 2 for each square of a partitionλ, here 1 ≤ j ≤ λi . We denote by= For each square= s ( i, j ) ∈ λ , let t j t
a (s) = λi − j , l ( s ) = λ tj − i, h ( s ) = a ( s ) + l ( s ) + 1, a′ ( s ) = j − 1, l ′ ( s ) = i − 1.
The numbers a ( s ) and a′ ( s ) may be called respectively the arm-length and the arm-colength of s, and l ( s ) , l ′ ( s ) the leg-length and the leg-colength.
2.3. Macdonald Function We define a scalar product l( λ )
pλ , pµ here
q ,t
1 − q λi , λi i =1 1 − t
= δ λµ zλ ∏
(1)
zλ = ∏i i mi ! , where mi = mi ( λ ) is the number of parts of λ equal to i; m
i ≥1
pλ = pλ1 pλ2 for each partition λ = ( λ1 , λ2 ,) and pr = ∑ i xir .
Macdonald function Pλ ( x; q, t ) depends rationally on two parameters q, t , S i.e. Pλ ( x; q, t ) ∈ Λ ⊗ ( q, t ) , here Λ = [ x1 , , xn ,] n and ⊗ means
tensor product over . They are characterized by the following two properties[6]: mλ + ∑ µ 0) and bn = 0 0 ( n > 0 ) . For a partition λ = ( λ1 , λ2 ,) , we use a short notation aλ 0 = aλ1 aλ2 0 . The bosonic Fock space is generated from the vacuum state:
= : span {a− λ b− µ 0 : λ , µ ∈ } . The dual vacuum state and
0 is defined by the conditions
0= bn 0 ( n < 0 ) . The dual boson Fock space
*
0= an 0 ( n < 0 )
is generated by the dual
vacuum state:
= * : span { 0 aλ bµ : λ , µ ∈ } . There is a paring * × → denoted by spaces, defined by the following properties:
= 0 0 1,= ( u a) v
u (a v
)
( u , v )
u v
between two
for all a ∈Ba ,b .
3.2. The Vertex Operators
(
)
To construct the vertex realization for Pλ 1, t , , t n−1 ; q, t , we propose two sets of vertex operators depending on q and t. We define
∞ 1 A+ ( λi ) exp ∑ 1 − t n = n=1 n
(
)
n 12 −λi i q t an ,
n 1 ∞ 1 1 λi − 2 −i . A− (= q t a λi ) exp −∑ n − n 1 − q n n=1
(4)
(5)
Since n ∞ 1 ∞ 1 − λi 1 1 ∑ 1 − t n q 2 t i an , − ∑ n m 1 − qm n 1= 1 = m
(
DOI: 10.4236/am.2018.94033
)
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m λ j − 12 − j q t a− m
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12 − λi i q t
1 1 1− tn = −∑∑ m n 1m = = 1 n m 1− q ∞
∞
n
λ j − 12 − j t q
m
[ an , a− m ]
λ −λ + m i − j ∞ 1 − q j i t , = ln ∏ λ j − λi + m i − j +1 t m = 0 1 − q
we can obtain
1− q
∞
A+ ( λi ) A− ( λ j ) = ∏
m=0
(q = (q
1− q
λ j − λi + m i − j
t
λ j − λi + m i − j +1
λ j − λi i − j
t
t
λ j − λi i − j +1
t
) ; q)
;q
∞
A− ( λ j ) A+ ( λi ) (6)
A− ( λ j ) A+ ( λi ) .
∞
We define another set of vertex operators ∞ 1 = B+ ( x ) exp ∑ 1 − t n x nbn , n=1 n
(7)
∞ 1 1 B− ( x ) = exp ∑ x nb− n . n n − q 1 n=1
(8)
(
)
Since ∞ ∞ 1 1 1 n n y m b− m ∑ 1 − t x bn , ∑ m n = m 1 m 1− q n 1= n ∞ ∞ 1 1 1− t n m = ∑∑ x y [bn , b− m ] m = = 1 n m 1− q n 1m ∞ 1 − q m txy = ln ∏ , m m = 0 1 − q xy
(
)
likewise we obtain
1 − q m txy B− ( y ) B+ ( x ) m m = 0 1 − q xy ∞
B+ ( x ) B− ( y ) = ∏
(9)
( txy; q )∞ B ( y ) B+ ( x ) . = ( xy; q )∞ −
(
3.3. Operator Product Formula for Pλ 1, t , , t n −1 ; q , t
)
With the help of the vertex operator A+ ( λi ) , A− ( λi ) , B+ ( x ) , B− ( x ) , we define vertex operators S+ ( i ) and S− ( i ) as follows:
∞ 1 S+ ( i ) = exp ∑ 1 − t n n=1 n
(
)
n 1 −λi n 12 i i 2 q t an + q t bn ,
(10)
and n ∞ 1 1 λ − 1 n − 12 −i i 2 −i − S− ( i ) = q t a q t b exp −∑ n −n − n . n=1 n 1 − q
(11)
We propose operator product formula DOI: 10.4236/am.2018.94033
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S− ( n ) S+ ( n ) S− ( n − 1) S+ ( n − 1) S− ( 2 ) S+ ( 2 ) S− (1) S+ (1) .
(12)
After some careful computation via the commutative relation (6) and (9), the Formula (12) is equal to
∏
1≤ j
