Toda tau functions with quantum torus symmetries

This paper is a review of our recent work [1,2] on an integrable structure of the melting crystal model of topological strings [3] and 5D gauge theories [4]. It is shown here that the partition function of this model, on being suitably deformed by special external potentials, is essentially a tau function of the Toda hierarchy [5]. A technical clue to this observation is a kind of symmetries (referred to as "shift symmetries") in the underlying quantum torus algebra. These symmetries enable us, firstly, to convert the deformed partition function to the tau function and, secondly, to show the existence of hidden symmetries of the tau function. These results can be exntended to some other Toda tau functions that are related to the topological vertex [6] and the double Hurwitz numbers of the Riemann sphere [7].

Throughout this paper, q denotes a constant with |q| < 1, and Λ and ∆ denote the

Their combinations

satisfy the commutation relations

of the quantum torus algebra. This Lie algebra can thus be embedded into the Lie algebra gl(∞) of Z × Z matrices A = (a ij ) for which ∃N such that a ij = 0 for |i -j| > N.

To formulate a fermionic realization of this Lie algebra, we introduce the creation/annihilation operators ψ i , ψ * i (i ∈ Z) with anti-commutation relations

and the 2D free fermion fields

The vacuum states 0|, |0 of the Fock space and its dual space are characterized by the vacuum conditions

To any element A = (a ij ) of gl(∞), one can associate the fermion bilinear

These fermion bilinears form a one-dimensional central extension gl(∞) of gl(∞). The special fermion bilinears [1, 2]

satisfy the commutation relations

for k and l with k + l = 0 and

Thus gl(∞) contains a central extension of the quantum torus algebra, in which the u(1) algebra is realized by

Let us introduce the operators

G ± ’s play the role of “transfer matrices” in the melting crystal model [3,4]. W 0 is a fermionic form of the so called “cut-and-join” operator for Hurwitz numbers [8]. G ± and q W 0 /2 induce the following two types of “shift symmetries” [1,2] in the (centrally extended) quantum torus algebra.

• First shift symmetry

• Second shift symmetry

A general tau function of the 2D Toda hierarchy [5] is given by

where T = (T 1 , T 2 , • • • ) and T = ( T1 , T2 , • • • ) are time variables of the Toda hierarchy, s| and |s are the ground states

in the charge-s sector of the Fock space, and g is an element of GL(∞) = exp gl(∞) .

On the other hand, the partition function Z(Q, s, t) of the deformed melting crystal model [1,2] can be cast into the apparently similar (but essentially different) form

where Q and t = (t 1 , t 2 , • • • ) are coupling constants of the model, and H(t) and L 0 the following operators:

The shift symmetries ( 8) and (9) imply the operator identity

Inserting this identity and using the fact that

we can rewrite Z(s, t) as

where the GL(∞) element g defining the tau function is given by

Actually, the shift symmetries imply the operator identity

as well. This leads to another expression of Z(Q, s, t, ) in which τ (s, T , 0) is replaced with τ (s, 0, -T ).

The existence of different expressions can be explained by the intertwining relations

which, too, are a consequence of the shift symmetries. These intertwining relations imply the constraints

on the tau function. The tau function τ (s, T , T ) thereby becomes a function τ (s, T -T ) of the difference T -T . In particular, τ (s, T , 0) and τ (s, 0, -T ) coincide. The reduced function τ (T , s) may be thought of as a tau function of the 1D Toda hierarchy.

(15) are a special case of the more general intertwining relations

We can translate these relations to the language of the Lax formalism of the Toda hierarchy. A study on this issue is now in progress.

The following Toda tau functions can be treated more or less in the same way as the foregoing tau function. We shall discuss this issue elsewhere.

1. The generating function of the two-leg amplitude W λµ in the topological vertex [6] is a Toda tau function determined by

2. The generating function of double Hurwitz numbers of the Riemann sphere [7] is a Toda tau function determined by

The parameter q is interpreted as q = e -β .

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