q-Deformed Oscillators and D-branes on Conifold

We study the q-deformed oscillator algebra acting on the wavefunctions of non-compact D-branes in the topological string on conifold. We find that the mirror B-model curve of conifold appears from the commutation relation of the q-deformed oscillators.


Introduction
The topological string is an interesting playground to study the gauge/string duality via the geometric transition [1,2,3]. It is also interesting to study how the target space geometry is quantized in this context. Recently, it is realized that the A-model side is described by a statistical model of crystal melting [4,5], while the B-model side is reformulated as matrix models [6,7]. In both cases, a spectral curve appears either as the limit shape of molten crystal or from the loop equation of matrix model. It is expected that the spectral curve should be viewed as a "quantum Riemann surface" in the sense that the coordinates of this curve become non-commutative at finite string coupling g s . It is argued that the natural language to deal with this phenomenon is the D-module [8,9].
In this paper, we study the non-commutative algebraic structure in the mirror B-model side of the topological string on the resolved conifold O(−1) ⊕ O(−1) → P 1 . As noticed in [10], there is an underlying q-deformed oscillator (or, q-oscillator for short) structure in the wavefunction of non-compact D-branes on conifold. We study the representation of q-oscillators in terms of non-commutative coordinates and show that the mirror curve of conifold appears from the commutation relation of the q-oscillators. This paper is organized as follows. In section 2, we construct the q-oscillators A ± acting on the D-brane wavefunctions in terms of variables obeying the commutation relation [p, x] = g s . In section 3, we show that the commutation relation of q-oscillators A ± is nothing but the mirror curve of resolved conifold. In section 4, we revisit the computation of the partition function of Chern-Simons theory using the q-oscillators. We conclude in section 5 with discussion.

Our Notations
Here we summarize our notations and elementary formulas used in the text. We denote the string coupling as g s , and the Kähler parameter of the base P 1 of resolved conifold as t = g s N . Then we introduce the parameters q and Q by (1.1) We also introduce the canonical pair of coordinates x and p satisfying We use the representation such that x acts as a multiplication and p acts as a derivative p = g s ∂ x . In particular, when acting on a constant function "1", we get x · 1 = x, p · 1 = 0, e ax · 1 = e ax , e bp · 1 = 1,

Operator Representation of D-brane Wavefunctions
In this section, we consider an operator representation of the wavefunction of D-branes on conifold. The wavefunction of a D-brane on conifold in the standard framing is given by [11][12][13][14][15][16][17] where the q-binomial is defined as In order to rewrite the wavefunction Z N (x) in the operator language, let us recall the q-binomial formula for the variables z and w obeying the relation wz = qzw In the last step, we used the commutation relation (1.5). By comparing (2.1) and (2.4), we see that the D-brane wavefunction is written as the operator (e p − e −x+p ) N acting on the constant function "1", according to the rule in (1.3). Namely, the D-brane wavefunction has a simple expression where A + is given by Using the commutation relation (1.5), A + is also written as One can see that the product form of Z N (x) in (2.1) easily follows from our simple expression Z N (x) = A N + · 1. By repeatedly using the relation (1.5), we can change the ordering so that e p comes to the rightmost position (2.8) When acting on "1", the last expression of A N + gives the product form of wavefunction in (2.1).

q-Oscillators and D-brane Wavefunctions
In this section, we consider the q-oscillator structure of the wavefunction Z N (x). The q-oscillator structure for the Rogers-Szegö polynomials and the Stieltjes-Wigert polynomials, which are related to our wavefunction Z N (x) by a change of framing [15,16,18], was studied in [19,20].
From the definition Z N (x) = A N + · 1, it follows that A + acts as the raising operator Next consider the operator lowering the index of Z N (x). From the relation and one can see that the operator A − defined by lowers the index of Z N (x) as desired: Note that A − annihilates the constant function "1" This implies that the constant function "1" can be identified as the vacuum of q-oscillator Now let us see that A + and A − obey the q-oscillator algebra. From (3.1) and (3.5), we find (3.8) It follows that A + and A − satisfy the q-oscillator algebra Here the operator q N is defined as We can directly compute the algebra of A ± using the commutation relations (1.5) without acting them on the wavefunction as above. From the expression of A ± in terms of variables x, p in (2.6) and (3.4), we can show that A ± satisfy (3.11) Therefore, we arrive at the expression of q N in terms of x and p as (3.12) One can check that the RHS of (3.12) satisfies the defining properties of q N (3.10). When acting on Z N (x), the relation (3.12) leads to the following constraint on the wavefunction This agrees with the known mirror B-model curve for the conifold [21,22]. In other words, we find an interesting interpretation of the mirror curve (3.13): it represents the q-oscillator relation [A − , A + ] = q N written in the canonical variables x, p.

Wavefunction of anti-D-Brane
In contrast to the ordinary oscillator, in the case of q-oscillator we can consider the (3.14) Repeating the similar calculation as in (2.8), we find (3.15) From this expression, we see that . (3.16) As argued in [12], this is interpreted as the wavefunction of anti-D-brane, up to a shift of x. Z −N (x) has another interpretation as the wavefunction of D-brane ending on a different leg of the toric diagram of conifold [15,16,17]. The constraint equation which can be rewritten as

Closed String Partition Function and the q-Oscillators
In this section, we consider the partition function of closed topological string on conifold from the viewpoint of q-oscillators. Although our computation is essentially the same as [24], we emphasize that the q-oscillator structure makes the computation more transparent. There is essentially no new result in this section, but we include this for completeness.
As shown in [25,2,1], the closed string partition function of conifold is given by the U (N ) Chern-Simons theory on S 3 . Later, it was noticed in [24] that the same partition function is written as the log-normal matrix model (4.1) The orthogonal polynomial associated with this log-normal measure is known as the Stieltjes-Wigert polynomial S N (x) [26], which is given by In the following we will show that S N (x) is related to Z N (x) by the following change of framing x → x − p + 1 2 g s , p → p.
Note that the operator U preserves the vacuum In terms of the conjugated q-oscillators the Stieltjes-Wigert polynomial is written in the same form as Z N (x) = A N + · 1 Using the commutation relation (1.5), one can see that (4.7) agrees with the expression (4.2), as promised In order to calculate the partition function (4.1), we need the norm of the function which is known as the FZZT wavefunction [23,27]. We see that the q-oscillator representation (4.7) simplify the computation of the norm.
The log-normal measure associated with (4.1) becomes Gaussian under the change of (4.10) The inner product with respect to this measure is defined as . (4.11) Let us consider the adjoint of B + with respect to this measure f, B + g = (B + ) † f, g . (4.12) Using the representation of B + in terms of x, p (4.6), the action of B + on a function g(x) reads Then the inner product f, B + g is written as (4.14) From this equation, we find that the adjoint of B + is proportional to B − (4.6) Now it is straightforward to calculate the norm of S N (x). In order to do that, it is convenient to use the bra-ket notation (4.16) Noticing that B − satisfies the same relation as A − (3.5) when acting on the state |n 17) and using the relation between (B + ) † and B − (4.15), we find Now let us compute the norm n|n . First, the norm of unit function 1 = |0 is given The norm of higher state |n is determined by the following recursion relation Finally, the norm of |n is found to be (1 − q k ). (4.21) This agrees with the known result of the norm of S n (x) with respect to the measure (4.10) [26].
The partition function of matrix model (4.1) is given by the product of the norm of Ψ n (x) in (4.9) Then the partition function of matrix model (4.1) is given by where η( q) denotes the η-function On the other hand, the partition function of the U (N ) Chern-Simons theory on S 3 is which agrees with Z log up to terms in the free energy which are polynomial in t.

Discussion
In this paper, we find that the mirror B-model curve of resolved conifold has an interesting interpretation as the q-oscillator relation [A − , A + ] = q N itself. It would be interesting to find the physical origin of this algebraic structure.
Recently, the partition function of the Donaldson-Thomas theory of the noncommutative version of conifold is calculated by Szendröi [29] Z NC = The last factor of Z NC in (5.1) is the same as the Chern-Simons partition function in (4.26), but the first factor is different This difference is discussed in the context of the wall-crossing phenomena [30,31,32]. It is tempting to identify the extra factor in Z NC as the effect of anti-D-branes [33]