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Acta Polytechnica Vol. 51 No. 1/2011

Toda Tau Functions with Quantum Torus Symmetries

K. Takasaki

Abstract

The quantumtorus algebra plays an important role in a special class of solutions of theTodahierarchy. Typical examples
are the solutions related to the melting crystal model of topological strings and 5D SUSY gauge theories. The quantum
torus algebra is realized by a 2D complex free fermion system that underlies the Toda hierarchy, and exhibits mysterious
“shift symmetries”. This article is based on collaboration with Toshio Nakatsu.

Keywords: Toda hierarchy, melting crystal model, quantum torus algebra.

1 Introduction
This paper is a review of our recentwork [1, 2] on an
integrable structure of the melting crystal model of
topological strings [3] and 5Dgauge theories [4]. It is
shown here that the partition function of this model,
onbeing suitablydeformedby special externalpoten-
tials, is essentially a tau function of the Toda hierar-
chy [5]. A technical clue to this observation is a kind
of symmetries (referred to as “shift symmetries”) in
the underlying quantum torus algebra. These sym-
metries 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 extended to some
otherToda tau functions that are related to the topo-
logical vertex [6] and the double Hurwitz numbers of
the Riemann sphere [7].

2 Quantum torus algebra

Throughout this paper, q denotes a constant with
|q| < 1, and Λ and Δ denote the Z×Z matrices

Λ=
∑
i∈Z

Ei,i+1 =(δi+1,j), Δ=
∑
i∈Z

iEii =(iδij).

Their combinations

v(k)m = q
−km/2ΛmqkΔ (k, m ∈ Z) (1)

satisfy the commutation relations

[v(k)m , v
(l)
n ] = (q

(lm−kn)/2 − q(kn−lm)/2)v(k+l)m+n (2)
of the quantum torus algebra. This Lie algebra can
thus be embedded into theLie algebragl(∞) ofZ×Z
matrices A = (aij) for which ∃N such that aij = 0
for |i − j| > N.
To formulate a fermionic realization of this Lie al-

gebra, we introduce the creation/annihilation opera-
tors ψi, ψ

∗
i (i ∈ Z) with anti-commutation relations

ψiψ
∗
j + ψ

∗
j ψi = δi+j,0,

ψiψj + ψj ψi = 0,

ψ∗i ψ
∗
j + ψ

∗
j ψ

∗
i = 0

and the 2D free fermion fields

ψ(z)=
∑
i∈Z

ψiz
−i−1, ψ∗(z)=

∑
i∈Z

ψ∗i z
−i.

The vacuum states 〈0|, |0〉 of the Fock space and its
dual space are characterized by the vacuum condi-
tions

ψi|0〉 =0 (i ≥ 0), ψ∗i |0〉 =0 (i ≥ 1),
〈0|ψi =0 (i ≤ −1), 〈0|ψ∗i =0 (i ≤ 0).

To any element A =(aij) of gl(∞), one can asso-
ciate the fermion bilinear

 =
∑

i,j∈Z
aij:ψ−iψ

∗
j :,

:ψ−iψ
∗
j : = ψ−iψ

∗
j − 〈0|ψ−iψ

∗
j |0〉.

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

V (k)m = v̂
(k)
m =

qk/2
∮
dz
2πi

zm:ψ(qk/2z)ψ∗(q−k/2z): (3)

satisfy the commutation relations

[V (k)m , V
(l)
n ] = (q

(lm−kn)/2 − q(kn−lm)/2) ·(
V
(k+l)
m+n −

qk+l

1− qk+l
δm+n,0

)
(4)

for k and l with k + l �=0 and

[V (k)m , V
(−k)
n ] = (q

−k(m+n)/2 − qk(m+n)/2) ·
V
(0)
m+n + mδm+n,0. (5)

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Acta Polytechnica Vol. 51 No. 1/2011

Thus ̂gl(∞) contains a central extension of the quan-
tum torus algebra, in which the û(1) algebra is real-
ized by

Jm = V
(0)
m = Λ̂

m (m ∈ Z). (6)

3 Shift symmetries

Let us introduce the operators

G± = exp

(
∞∑

k=1

qk/2

k(1 − qk)
J±k

)
,

W0 =
∑
n∈Z

n2:ψ−nψ
∗
n:. (7)

G±’s play the role of “transfermatrices” in themelt-
ing crystalmodel [3, 4]. W0 is a fermionic formof the
so called “cut-and-join” operator for Hurwitz num-
bers [8].

G± and q
W0/2 induce the following two types of

“shift symmetries” [1, 2] in the (centrally extended)
quantum torus algebra.
• First shift symmetry

G−G+

(
V (k)m − δm,0

qk

1− qk

)
(G−G+)

−1 =

(−1)k
(

V
(k)
m+k − δm+k,0

qk

1− qk

)
(8)

• Second shift symmetry

qW0/2V (k)m q
−W0/2 = V (k−m)m (9)

4 Toda tau function in
melting crystal model

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

τ(s, T , T̄)=

〈s|exp
(

∞∑
k=1

TkJk

)
gexp

(
−

∞∑
k=1

T̄kJ−k

)
|s〉, (10)

where T =(T1, T2, · · ·) and T̄ =(T̄1, T̄2, · · ·) are time
variables of the Toda hierarchy, 〈s| and |s〉 are the
ground states

〈s| = 〈−∞| · · · ψ∗s−1ψ
∗
s , |s〉 = ψ−sψ−s+1 · · · | − ∞〉

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 thedeformedmelting crystalmodel [1, 2]
can be cast into the apparently similar (but essen-
tially different) form

Z(s, t)= 〈s|G+eH(t)QL0G−|s〉, (11)

where Q and t =(t1, t2, · · ·) are coupling constants of
the model, and H(t) and L0 the following operators:

H(t) =
∞∑

k=1

tkHk, Hk = V
(k)
0 ,

L0 =
∑
n∈Z

n:ψ−nψ
∗
n:. (12)

The shift symmetries (8) and (9) imply the oper-
ator identity

G+e
H(t)G−1+ = exp

(
∞∑

k=1

tkq
k

1− qk

)
G−1− q

−W0/2 ·

exp

(
∞∑

k=1

(−1)ktkJk

)
qW0/2G−.

Inserting this identity and using the fact that

〈s|G−1− q
−W0/2 = q−s(s+1)(2s+1)/12〈s|,

q−W0/2G−1+ |s〉 = q
−s(s+1)(2s+1)/12|s〉,

we can rewrite Z(s, t) as

Z(Q, s, t) = exp
( ∞∑

k=1

tkq
k

1− qk
)
·

q−s(s+1)(2s+1)/6τ(s, T ,0), (13)

Tk = (−1)ktk,

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

g = qW0/2G−G+Q
L0G−G+q

W0/2. (14)

Actually, the shift symmetries imply the operator
identity

G−1− e
H(t)G− = exp

(
∞∑

k=1

tkq
k

1− qk

)
G+q

W0/2 ·

exp

(
∞∑

k=1

(−1)ktkJ−k

)
q−W0/2G−1+

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 ex-

plained by the intertwining relations

Jkg = gJ−k (k =1,2, . . .), (15)

which, too, area consequenceof the shift symmetries.
These intertwining relations imply the constraints(

∂Tk + ∂T̄k
)
τ(s, T , T̄)= 0 (k =1,2, . . .) (16)

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Acta Polytechnica Vol. 51 No. 1/2011

on the tau function. The tau function τ(s, T , T̄)
thereby becomes a function τ(s, T − T̄) of the differ-
ence T − T̄ . 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 hier-
archy.
(15) are a special case of the more general inter-

twining relations

(V (k)m − δm,0
qk

1− qk
)g =

Q−kg(V
(−k)
−2k−m − δ2k+m,0

q−k

1− q−k
). (17)

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

5 Other models
The followingToda tau functions canbe treatedmore
or less in the sameway as the foregoing tau function.
We shall discuss this issue elsewhere.
1. The generating function of the two-leg ampli-
tude Wλμ in the topological vertex [6] is aToda
tau function determined by

g = qW0/2G+G−q
W0/2. (18)

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

g = e−βW0QL0. (19)

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

Acknowledgement

This work has been partly supported by the JSPS
Grants-in-Aid for Scientific Research No. 19104002,
No. 21540218 and No. 22540186 from the Japan So-
ciety for the Promotion of Science.

References

[1] Nakatsu,T., Takasaki,K.: Melting crystal, quan-
tum torus and Toda hierarchy, Comm. Math.
Phys. 285 (2009), 445–468.

[2] Nakatsu, T., Takasaki, K.: Integrable structure
of melting crystal model with external potential,
Advanced Studies in Pure. Math. vol. 59 (Math.
Soc. Japan, 2010), pp. 201–223.

[3] Okounkov, A., Reshetikhin, N., Vafa, C.: Quan-
tum Calabi-Yau and classical crystals, In:
P. Etingof, V. Retakh, I. M. Singer (eds.),
The unity of mathematics, Progr. Math. 244,
Birkhäuser, 2006, pp. 597–618.

[4] Maeda, T., Nakatsu, T., Takasaki, K., Tamako-
shi, T.: Five-dimensional supersymmetric Yang-
Mills theoriesandrandomplanepartitions,JHEP
0503 (2005), 056.

[5] Takasaki, K., Takebe, T.: Integrable hierarchies
and dispersionless limit, Rev. Math. Phys. 7
(1995), 743–808.

[6] Zhou, J.: Hodge integrals and integrable hierar-
chies, arXiv:math.AG/0310408.

[7] Okounkov, A.: Toda equations for Hurwitz num-
bers, Math. Res. Lett. 7 (2000), 447–453.

[8] Kazarian, M.: KP hierarchy for Hodge integrals,
Adv. Math. 221 (2009), 1–21.

Kanehisa Takasaki
E-mail: takasaki@math.h.kyoto-u.ac.jp
Graduate School of Human
and Environmental Studies
Kyoto University
Yoshida, Sakyo, Kyoto, 606-8501, Japan

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