On fermionic representation of the framed topological vertex

The Gromov-Witten invariants of ℂ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb{C}}^3 $$\end{document} with branes is encoded in the topological vertex which has a very complicated combinatorial expression. A simple formula for the topological vertex was proposed by Aganagic et al. in the fermionic picture. We will propose a similar formula for the framed topological vertex and prove it in the case when there are one or two branes.


Introduction
The topological vertex [2,4] is the basic building block for the theory of open and closed Gromov-Witten invariants of toric Calabi-Yau threefolds. It encodes open Gromov-Witten invariants of C 3 with three special D-branes, and Gromov-Witten invariants of any toric Calabi-Yau threefold, both open and closed, can be computed from it by certain explicit gluing process. One way to understand the gluing is through taking inner products or vacuum expectation values on the bosonic Fock space [8]. More precisely, the boundary condition on each of the three D-branes is indexed by a partition µ i , i = 1, 2, 3. It is well-known that the space Λ of symmetric functions has some natural basis indexed by JHEP12(2015)019 partitions (e.g. the Newton functions). One then understands the topological vertex as an element in the tensor product Λ ⊗3 , and the gluing is achieved by taking inner products on the corresponding copies of Λ. In this picture the topological vertex has an extremely complicated combinatorial expression in terms of skew Schur functions. Suggested by the boson-fermion correspondence, a deep conjecture was made in [2] and [1] that the topological vertex has a surprisingly simple expression in the fermionic picture: it is a Bogoliubov transform of the fermionic vacuum, i.e. the fermionic vacuum acted upon by an exponential of a quadratic expression of fermionic operators. We will refer to this as the ADKMV Conjecture. See section 3.2 for a precise statement.
A straightforward application as pointed out in [1] is related to integrable hierarchies: the one-legged case is related to the KP hierarchy, the two-legged case to the 2-dimensional Toda hierarchy, and the three-legged case to the 3-component KP hierarchy. The one-legged and the two-legged cases can also be seen directly from the bosonic picture [9], but the three-legged case can only be seen through the fermionic picture.
The ADKMV conjecture was only checked for the case of hook partitions in [1]. In the present paper, we propose a generalization of the ADKMV conjecture to the framed topological vertex, and give a proof of it in the one-legged and two legged cases.
In the rest of this paper, after reviewing some preliminaries in section 2, we will first propose in section 3 a generalization of the ADKMV Conjecture to the framed topological vertex. For the precise statement see section 3.3. We will refer to this conjecture as the Framed ADKMV Conjecture. Secondly, we will prove the one-legged and two-legged cases of the Framed ADKMV Conjecture in section 4 and section 5 respectively. In the final section 6 we will derive a determinatal formula for the framed topological vertex in the three-legged case based on the Framed ADKMV Conjecture.

JHEP12(2015)019
and n 1 > · · · > n k ≥ 0. The partition µ is completely determined by the numbers m i , n i . We often denote the partition µ by (m 1 , . . . , m k |n 1 , . . . , n k ), this is called the Frobenius notation. A partition of the form (m|n) in Frobenius notation is called a hook partition.
For a box e at the position (i, j) in the Young diagram of µ, define its content by c(e) = j − i. Then it is easy to see that A straightforward application of (2.2) is the following: . . , m k |n 1 , n 2 , . . . , n k ) be a partition written in the Frobenius notation. Then we have In particular, Proof. It is clear that:

Schur functions and skew Schur functions
Let Λ be the space of symmetric functions in x = (x 1 , x 2 , . . . ). For a partition µ, let s µ := s µ (x) be the Schur function in Λ. If we write µ = (m 1 , · · · , m k |n 1 , · · · , n k ) in Frobenius notation, then there is a determinantal formula that expresses s µ in terms of s (m|n) .
The inner product on the space Λ is defined by setting the set of Schur functions as an orthonormal basis. Given two partitions µ and ν, the skew Schur functions s µ/ν is defined by the condition (s µ/ν , s λ ) = (s µ , s ν s λ )

Specialization of symmetric functions
Let q ρ := (q −1/2 , q −3/2 , . . . ). It is easy to see that where h(e) is the hook number of e.

Fermionic Fock space
We say a set of half-integers A = {a 1 , a 2 , . . . } ⊂ Z + 1 2 , a 1 > a 2 > · · · , is admissible if it satisfies the following two conditions: where Z − is the set of negative integers.

JHEP12(2015)019
Consider the linear space W spanned by a basis {a|a ∈ Z+ 1 2 }, indexed by half-integers. For an admissible set A = {a 1 > a 2 > . . . }, we associate an element A ∈ ∧ ∞ W as follows: Then the free fermionic Fock space F is defined as One can define an inner product on F by taking {A : A ⊂ Z + 1 2 is admissible} as an orthonormal basis.

JHEP12(2015)019
and other anti-commutation relations are zero. It is clear that for r > 0, so the operators {ψ −r , ψ * r } r>0 are called the fermionic annihilators. For a partition µ = (m 1 , m 2 , . . . , m k |n 1 , n 2 , . . . , n k ), it is clear that (2.13) So the operators {ψ r , ψ * −r } r>0 are called the fermionic creators. The normally ordered product is defined as −ψ * r ψ r , r < 0. In other words, an annihilator is always put on the right of a creator.

The boson-fermion correspondence
For any integer n, define an operator α n on the fermionic Fock space F as follows: where |0 m = − 1 2 + m ∧ − 3 2 + m ∧ · · · . It is clear that Φ induces an isomorphism between F (0) and Λ. Explicitly, this isomorphism is given by The boson-fermionic correspondence plays an important role in Kyoto school's theory of integrable hierarchies. For example, Proposition 2.5. If τ ∈ Λ corresponds to |v ∈ F (0) under the boson-fermion correspondence, then τ is a tau-function of the KP hierarchy in the Miwa variable t n = pn n if and only if |v satisfies the bilinear relation A state |v ∈ F (0) satisfies the bilinear relation (2.16) if and only if it lies in the orbit GL ∞ |0 . There is also a multi-component generalization of the boson-fermion correspondence which can be used to study multi-component KP hierarchies [3].

The topological vertex
The topological vertex introduced in [2] is defined by It can also be rewritten as follows (see e.g. [10]): The framed topological vertex in framing (a 1 , a 2 , a 3 ) is given by: Even though the topological vertex is presented here in its combinatorial expression, its significance lies in its geometric origin as open Gromov-Witten invariants. In the mathematical theory of the topological vertex [4], the open Gromov-Witten invariants are defined by localizations on relative moduli spaces. This leads to some special Hodge integrals on the Deligne-Mumford moduli spaces, whose generating series can be shown to be . Closed and open Gromov-Witten invariants of local toric Calabi-Yau 3-folds can be obtained from the topological vertex by suitable gluing process.

The ADMKV Conjecture
It is conjectured in [2] and [1] that the topological vertex has a simple expression in the fermionic picture as follows. On the three-component femionic Fock space F ⊗ F ⊗ F, define for i = 1, 2, 3 operators ψ i r and ψ i * r , r ∈ Z + 1 2 . They act on the i-th factor of the tensor product as the operators ψ r and ψ * r respectively, and we use the Koszul sign convention for the anti-commutation relations of these operators, i.e., we set . Then the ADKMV Conjecture states that where for i = 1, 2, 3, Here it is understood that A 34 mn = A 31 mn and A 10 mn = A 13 mn . This is very surprising because in the bosonic picture the expression for the topological vertex is very complicated.

The Framed ADMKV Conjecture
We make the following generalization of the above ADKMV Conjecture to the framed topological vertex: for A ij mn (q; a) similar to A ij mn (q) above: Here a = a 1 , a 2 , a 3 . We refer to this conjecture as the Framed ADKMV Conjecture. We derive A ij m,n (q; a) by the same method as for the derivation of A ij mn (q) in [1, section 5.11]. For details, see section 5.2. It is surprising that there is only little difference between them.
A straightforward application of the ADKMV Conjecture and the Framed ADKMV Conjecture is that they establish a connection between the topological vertex and integrable hierarchies as pointed out in [1].

JHEP12(2015)019
4 Proof of the one-legged case In this section, as a warm up exercise we will derive a fermionic representation of the framed one-legged topological vertex, hence establishing the one-legged case of the Framed ADKMV Conjecture.

The framed one-legged topological vertex in terms of Schur functions
The generating functional of the Gromov-Witten invariants of C 3 with one brane is encoded in W (a,0,0) µ,(0),(0) . It is also the generating function of certain Hodge integrals on the moduli spaces of pointed stable curves. Let (4.1) By (3.2) one then has: By (2.15), this corresponds to an element V (a) (q) in the charge 0 ferminonic Fock subspace F (0) :

Proof of the one-legged case of the Framed ADKMV Conjecture
By the Framed ADKMV Conjecture we should have for some A mn (q; a).   Theorem 4.2. In the case of one-legged topological vertex, the Framed ADKMV Conjecture holds for the above A mn (q; a).
For later reference, note we have proved the following identity:  The framed two-legged topological vertex encodes the open Gromov-Witten invariants of C 3 with two branes: Recall the following identity proved in [10]: The following identity proved in [7] will play a key role below: Based on this formula, the following formula is proved in [10]: Therefore, (5.1) can be rewritten as follows: By the boson-fermion correspondence (2.15), this corresponds to the following element in the femionic picture: Using this we will prove the two-legged case of the Framed ADKMV Conjecture.

(5.11)
The left-hand side can be rewritten as follows:
Proof. One can use Lemma 2.1 to get:

JHEP12(2015)019
Proof. Expanding the determinants one has where ǫ(στ) is the sign of the permutation στ , and in the second equality we have used (5.12).
By Proposition 2.3, Now note det(h b j −s i ) 1≤i,j≤r = 0 if s i = s j for some 1 ≤ i < j ≤ r, and det(e b ′ j −t i ) 1≤i,j≤r = 0

JHEP12(2015)019
if t i = t j for some 1 ≤ i < j ≤ r. Therefore, The proof is complete.

JHEP12(2015)019
Let M be a set of nonnegative integers {m 1 > m 2 > · · · > m k } written in decreasing order. For a subset A = {m i 1 > · · · > m ir } of M , also written in decreasing order, denote by ε(M/A) the sign of the permutation (m i 1 , · · · , m ir , m j 1 , · · · , m j k−r ) → (m 1 , · · · , m k ), where (m j 1 , · · · m j k−r ) is the set M \A written in decreasing order. Let µ = (M |N ) and γ = (A|B) be two partitions, we define γ < µ if A ⊂ M and B ⊂ N . If γ < µ holds, then µ\γ := (M \A|N \B) is naturally defined as a partition. By (2.11), the following Lemma is easy to prove. (q) = µ, ν|T (q; a 1 , a 2 )|0 ⊗ |0 . Then one has Proposition 5.6. We have where the condition ( * r) in the summation is given by where the condition ( * r) in the summation is given by (5.23). Then we have

Proof of the Framed ADKMV Conjecture in the two-legged case
In this subsection we finish the proof of Theorem 5.1. We now simplify the summation in (5.22). Let η = (s 1 , . . . , s r |t 1 , . . . , t r ). We first take γ=(A|B), r(γ)=r :
Remark 5.1. Note the charge 0 subspace (F ⊗ F) (0) has a direct sum decomposition The two-legged topological vertex corresponds to only the component of T (|0 ⊗ |0 ) in F (0) ⊗ F (0) . It is interesting to find the geometric meaning of other components.
From the above proof one can also see that

JHEP12(2015)019
6 Towards a proof of the three-legged case In this section we present an intermediate result which should be useful for a proof of the three-legged case of the Framed ADKMV Conjecture.

(6.2)
Here c ∈ S 3 is the 3-cycle translation that transforms 1 to 2, 2 to 3 and 3 to 1, the summation is over all partitions γ ij = (M ij |N ij ) satisfying the following conditions:

3)
and ǫ(M ii , M ic(i) , M ic 2 (i) ) is the sign of the transformation that rearranges the ordered set of numbers (M ii , M ic(i) , M ic 2 (i) ) in a decreasing order.