On fermionic representation of the framed topological vertex

The Gromov-Witten invariants of \mathbb{C}^3 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 Govomov-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 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 §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.
In the rest of this paper, after reviewing some preliminaries in §2, we will first propose in §3 a generalization of the ADKMV Conjecture to the framed topological vertex. For the precise statement see §3.3. We will refer to this conjecture as the Framed ADKMV Conjecture. Secondly, we will prove the one-legged and twolegged cases of the Framed ADKMV Conjecture in §4 and §5 respectively. In the final §6 we will derive a determinatal formula for the framed topological vertex in the three-legged case based on the Framed ADKMV Conjecture.
Acknowledgements. The first author is partially supported by NSFC grants (11001148 and 10901152) and the President Fund of GUCAS. The second author is partially supported by two NSFC grants (10425101 and 10631050) and a 973 project grant NKBRPC (2006cB805905).

Partitions.
A partition µ of a positive integral number n is a decreasing finite sequence of integers µ 1 ≥ · · · ≥ µ l > 0, such that |µ| := µ 1 + · · · + µ l = n. The following number associated to µ will be useful in this paper: It is very useful to graphically represent a partition by its Young diagram. This leads to many natural definitions. First of all, by transposing the Young diagram one can define the conjugate µ t of µ. Secondly assume the Young diagram of µ has k boxes in the diagonal. Define m i = µ i − i and n i = µ t i − i for i = 1, · · · , k, then it is clear that m 1 > · · · > m k ≥ 0 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 (2) κ µ = 2 e∈µ c(e).

2.2.
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) .
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: 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.
The anti-commutation relations for these operators are (11) [ψ r , ψ * s ] := ψ r ψ * s + ψ * s ψ r = δ r,s id 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 So the operators {ψ r , ψ * −r } r>0 are called the fermionic creators. The normally ordered product is defined as In other words, an annihilator is always put on the right of a creator.
2.5. The boson-fermion correspondence. For any integer n, define an operator α n on the fermionic Fock space F as follows: Then the boson-fermion correspondence is a linear isomorphism Φ : F → B given by 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 (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
where c 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 Z (a1,a2,a3) (q; . Closed and open Gromov-Witten invariants of local toric Calabi-Yau 3-folds can be obtained from the topological vertex by suitable gluing process.
3.2. 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 andψ * 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 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.
3.3. 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, §5.11]. For details, see §5.2. It is surprising that there is only little difference between them.
A straightforward application of the ADKMV Conjecture and the Framed AD-KMV Conjecture is that they establish a connection between the topological vertex and integrable hierarchies as pointed out in [1].

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.

4.2.
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).
By the boson-fermion correspondence (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.

5.2.
The determination of A ij mn (q; a 1 , a 2 ). Note that the charge 0 subspace (F ⊗ F ) (0) of F ⊗ F has a decomposition The Framed ADKMV Conjecture predicts the existence of an operator T of the form . In this subsection we modify the method in [1, §5.11] to the framed case to derive explicit expressions for A ij mn (q; a 1 , a 2 ). Because the operators {ψ ij mn } commute with each other and square to zero, we have Take µ = (m|n) and ν = ∅ or take ν = (m|n) and µ = ∅, as in the one-legged case we get for i = 1, 2: Take µ = (m|n) and ν = (m ′ |n ′ ), then it is clear that the coefficient of |(m|n) ⊗ |(m ′ |n ′ ) in T (q; a 1 , a 2 )(|0 ⊗ |0 ) is (−1) n+n ′ (A 11 mn (q; a 1 , a 2 )A 22 m ′ n ′ (q; a 1 , a 2 ) − A 12 mn ′ (q, a 1 , a 2 )A 21 m ′ n (q, a 1 , a 2 )). Assuming the Framed ADKMV Conjecture, one should have: a 1 , a 2 )).
The left-hand side can be rewritten as follows: Therefore, by (47) we have A 12 mn ′ (q; a 1 , a 2 ) · A 21 m ′ n (q; a 1 , a 2 )
Proof. Expanding the determinants one has where ǫ(στ ) is the sign of the permutation στ , and in the second equality we have used (49). By Proposition 2.3, The proof is complete.
Let M be a set of nonnegative integers {m 1 > m 2 > · · · > m k } written in decreasing order. For a subset A = {m i1 > · · · > m ir } of M , also written in decreasing order, denote by ε(M/A) the sign of the permutation (m i1 , · · · , m ir , m j1 , · · · , m j k−r ) → (m 1 , · · · , m k ), where (m j1 , · · · 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 (11), the following Lemma is easy to prove. Lemma 5.4. Let µ = (M |N ) and γ = (A|B) be two partitions such that γ < µ, then Now it is straightforward to get the following Lemma 5.5. Let C (a1,a2) µν (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 Proof. For r ≥ 0, let where the condition ( * r) in the summation is given by (60). Then we have For r > 0, we use Lemma 5.2 , Lemma 5.3 and Lemma 5.5 to get: where the condition ( * r) in the summation is given by (60).
This matches with (43), so the proof of Theorem 5.1 is completed.
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

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. 6.1. From fermionic representation to determinantal representation. If one assumes the Framed ADKMV Conjecture, one can determine A ij mn (i, j = 1, 2, 3) by modifying the method of [1] as in §5.2. They are indeed given by (27), (28), (29). By (11), we can expand T as follows: i,j=1,2,3 m,n≥0 where the summation is over all partitions µ 11 , µ 12 , . . . , µ 33 . Now let µ i = (M i |N i ) = (m i 1 , m i 2 , · · · , m i ki |n i 1 , n i 2 , · · · , n i k i ) (when k i = 0, µ i is the empty partition). Denote by C (a) µ 1 ,µ 2 ,µ 3 the right-hand side of (26). It is clear that The ± signs can be tracked off using the Koszul sign convention. More precisely we have the following Lemma 6.1. Let C (a) µ 1 ,µ 2 ,µ 3 be the right-hand side of (26). Then one has C (a) µ 1 ,µ 2 ,µ 3 = (−1) ||N λ ||+||N µ ||+||N ν || (−1) r 32 r 12 +r 31 r 32 +r 21 r 21 +r 32 r 13 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: 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.
Similar to the proof of Proposition 5.6, one can prove the following Proposition, which gives the determinantal form of C Here the summation is taken over all r ij ≥ 0, i = j, i, j = 1, 2, 3 satisfying the conditions r ic(i) + r ic 2 (i) = r c(i)i + r c 2 (i)i ≤ k i , i =