We show how the smooth geometry of Calabi-Yau manifolds emerges from the
thermodynamic limit of the statistical mechanical model of crystal melting
defined in our previous paper arXiv:0811.2801. In particular, the thermodynamic
partition function of molten crystals is shown to be equal to the classical
limit of the partition function of the topological string theory by relating
the Ronkin function of the characteristic polynomial of the crystal melting
model to the holomorphic 3-form on the corresponding Calabi-Yau manifold.Comment: 4 pages; v2: revised discussion on wall crossing; v3: typos
  corrected, published versio

B. Feng

Hirosi Ooguri

L. I. Ronkin

Masahito Yamazaki

R. P. Thomas

S. K. Donaldson

English

arXiv

Crossref

Emergent Calabi-Yau Geometry

We show how the smooth geometry of Calabi-Yau manifolds emerges from the thermodynamic limit of the statistical mechanical model of crystal melting defined in our previous paper. In particular, the thermodynamic partition function of molten crystals is shown to be equal to the classical limit of the partition function of the topological string theory by relating the Ronkin function of the characteristic polynomial of the crystal melting model to the holomorphic 3-form on the corresponding Calabi-Yau manifold

Ooguri, Hirosi

Yamazaki, Masahito

Caltech Authors - Main

Emergent Calabi-Yau Geometry
Hirosi Ooguri1,2 and Masahito Yamazaki1,2,3
1California Institute of Technology, Pasadena, California 91125, USA
2Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8586, Japan
3Department of Physics, University of Tokyo, Hongo 7-3-1, Tokyo 113-0033, Japan
(Received 27 February 2009; published 21 April 2009)
We show how the smooth geometry of Calabi-Yau manifolds emerges from the thermodynamic limit of
the statistical mechanical model of crystal melting defined in our previous paper. In particular, the
thermodynamic partition function of molten crystals is shown to be equal to the classical limit of the
partition function of the topological string theory by relating the Ronkin function of the characteristic
polynomial of the crystal melting model to the holomorphic 3-form on the corresponding Calabi-Yau
manifold.
DOI: 10.1103/PhysRevLett.102.161601 PACS numbers: 11.25.Uv, 11.25.Mj
Introduction.—Stringy effects and quantum gravity ef-
fects are expected to modify our concept of space and time,
and understanding such effects quantitatively is an impor-
tant step in the exploration of physics at the Planck scale.
In the perturbative string theory, where the string coupling
constant gs is assumed to be small, the string length is
longer than the Planck length. In this regime, stringy
corrections to spacetime geometry kicks in first and quan-
tum gravity effects are obscured. In the context of the
topological string theory on C3, it was pointed out in
Ref. [1] that the partition function can be resummed, and
the result can be expressed in terms of a statistical model of
crystal melting for large gs. In our previous paper [2], we
generalized this construction to an arbitrary toric Calabi-
Yau threefold. In the crystal melting model, the Calabi-Yau
geometry is discretized, and each atom of the crystal can be
regarded as a fundamental unit of the geometry. In this
Letter, we will show how the smooth Calabi-Yau geometry
emerges from the discrete structure of the crystal melting
model in the thermodynamic limit, where gs ! 0. The
topological string theory is relevant for counting micro-
states of black holes in the superstring theory [3], and we
expect that our result sheds some light on the quantum
nature of spacetime in the superstring theory also.
A toric Calabi-Yau 3-fold M is a Ka¨hler quotient of
CFþ3 by Uð1ÞF, and its mirror manifold ~M is defined by
the polynomial equation [4,5],
uvþ Pðz; wÞ ¼ 0; ðu; vÞ 2 C; ðz; wÞ 2 C:
(1)
Here Pðz; wÞ is a Newton polynomial of the form,
Pðz; wÞ ¼ XFþ3
i¼1
ciðtÞzniwmi ; (2)
and ciðtÞ’s are functions of the Ka¨hler moduli t of the
original toric 3-fold M. The exponents ðni; miÞ 2 Z2 cor-
respond to lattice points of the toric diagram. For example,
for the mirror of Oð1Þ þOð1Þ bundle over CP1,
Pðz; wÞ is given by
Pðz; wÞ ¼ 1þ zþ wþ etzw: (3)
In this Letter, we will show that the Newton polynomial
Pðz; wÞ for the mirror of a Calabi-Yau manifold is identical
to the characteristic polynomial of the corresponding dimer
model, which is the partition function of the model on a
torus. The relation between Pðz; wÞ and the characteristic
polynomial had been discussed earlier in [6]. Here, we will
prove their precise equality including the dependence on
the moduli t.
The dimer model enters into our discussion because of
its relevance to counting of D-brane bound states. Consider
D0 and D2 branes with a single D6 brane on M. The low
energy effective theory is a supersymmetric quiver quan-
tum mechanics characterized by a quiver diagram on T2
[6,7]. In our previous paper [2], we showed that the dimer
model defined in Refs. [8,9] gives a generating function of
the Witten indices for bound states in quiver quantum
mechanics [10].
We also constructed a statistical model of crystal melt-
ing equivalent to the dimer model. The crystal consists of
atoms of different types, each of which corresponds to a
node of the quiver diagram for the quantummechanics, and
the chemical bonds between the atoms are dictated by
edges of the diagram. The quiver diagram is drawn on
T2. To construct the initial configuration of the crystal,
we start with the universal covering of the diagram over the
plane and pile up atoms on nodes following the rules
prescribed in Ref. [2]. This initial configuration corre-
sponds to a single D6 brane with no D0 and D2 charges.
Removing atoms from the crystal generates bound states
with nonzero D0 and D2 charges. Such molten crystal
configurations are in one-to-one correspondence with per-
fect matchings of the dimer model defined in Ref. [9], as
shown in the Appendix of Ref. [2].
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It is reasonable to expect that a classical geometric
picture emerges in the limit of large D0 and D2 charges
since it can represent a large black hole in the superstring
theory. The corresponding thermodynamic limit in the
dimer model was studied in Ref. [11]. In this Letter, we
will show that the partition function of the dimer model
evaluated in the thermodynamic limit is equal to the
genus-0 limit of the partition function of the topological
string theory onM. The dimer model has been formulated
to describe the noncommutative Donaldson-Thomas
theory [8,9], while the topological string theory for a
general toric Calabi-Yau manifold is equivalent to the
commutative Donaldson-Thomas theory [12], as shown
by [13]. A relation of our results to the wall-crossing
formula [14–17] between the noncommutative and com-
mutative Donaldson-Thomas theories is currently under
investigation.
The emergence of Calabi-Yau geometry from the ther-
modynamic limit has been observed in Ref. [1] in the case
of C3. In this Letter, we make the connection sharper and
more explicit by showing the direct connection between
the partition functions of the crystal melting model and the
topological theory for a general toric Calabi-Yau 3-fold.
Thermodynamic limit of the crystal melting model.—The
main object of study in this Letter is the partition function,
Z ¼ X
q0;qA
ðq0; qAÞegsq0tAqA ; (4)
where ðq0; qAÞ is the Witten index for bound states of q0
D0 branes and qA D2 branes on the Ath 2-cycle (A ¼
1; . . . ; dimH2ðMÞ) with a single D6 brane on a toric
Calabi-Yau manifold. According to the dictionary in
Ref. [2], the Witten index is equal (up to a sign) to the
number of molten crystal configurations where q0 is the
total number of atoms removed, whereas the relative num-
bers of different types of atoms removed from the crystal
are specified by qA’s. Later we will identify gs as the
topological string coupling constant and tA as the Ka¨hler
moduli of the toric Calabi-Yau manifold M.
The behavior of Z for gs ! 0 can be evaluated by using
the result of Ref. [11]. Consider a finite covering the
original quiver diagram, N times in one direction and N
times in another direction on T2. N is introduced as an
infrared regulator, and we will take N ! 1 at the end of
the computation so that we have the dimer model on the
plane R2. The surface of the crystal is determined by the
height function h over the plane. To define h, we start with
the canonical perfect matching m0 of the dimer model
corresponding to the initial crystal configuration with no
D0 or D2 charges. For any other perfect matching m, the
superposition ofm0 andm gives a set of closed loops on the
dimer graph. If m corresponds to a bound state with finite
D0 and D2 charges, m and m0 differ only in a finite region
on the graph. The function hi (i: node of the N  N cover
of the quiver) is defined so that it is 0 far away from the
region where m and m0 differ, and it increases by 1 every
time we cross a closed loop as we move inside of the
region. The corresponding molten crystal configuration is
obtained by removing hi atoms of the initial crystal over
the node i. In particular,X
i
hi ¼ q0: (5)
To take the thermodynamic limit, it is useful to introduce
the Cartesian coordinates (x, y), 0  x, y  1, on the N 
N covering of T2. In the limit where gs  1 and 1 N,
the height hi becomes a smooth function hðx; yÞ. We re-
scale the height function by the factor of 1=N so that
N2
Z 1
0
dxdyhðx; yÞ ¼ q0
N
; (6)
to take into account the large N scaling of the partition
function discussed in Ref. [18] and quoted as Theorem 2.1
in Ref. [11]. The statistical weight in the thermodynamic
limit is given by an integral of a surface tension ð@hÞ,
which is a function of the gradient of h, as [19]
Z exp

N2max
h
Z 1
0
dxdy½ð@hÞ  gsNhðx; yÞ

: (7)
The integral of gsNhðx; yÞ in the exponent comes from the
weight factor egsq0 in (4), and we used (6). In the thermo-
dynamic limit, we look for a height function hðx; yÞ which
maximize the exponent.
To derive the macroscopic surface tension from the
microscopic crystal melting model, we first define a char-
acteristic polynomial ~Pðz; wÞ as the sum of perfect match-
ings with weights assigned to edges of the dimer model on
T2 [11]. We then define its Ronkin function Rðx; yÞ by
Rðx; yÞ ¼
Z 2
0
ln ~Pðexþi; eyþiÞ ddð2Þ2 : (8)
According to Theorem 3.6 of Ref. [11], the surface tension
ð@xh; @yhÞ is the Legendre transform of the Ronkin func-
tion with respect to ðs; tÞ ¼ ð@xh; @yhÞ as
Rðx; yÞ ¼ max
s;t
½ðs; tÞ þ xsþ yt: (9)
The first step in relating the dimer model to the topo-
logical string theory is to show that the characteristic
polynomial ~Pðz; wÞ of the dimer model is equal to the
Newton polynomial Pðz; wÞ for the mirror Calabi-Yau
manifold (1),
~Pðz; wÞ ¼ Pðz; wÞ: (10)
According to Ref. [20], there is a one-to-one correspon-
dence between perfect matchings of the dimer model on T2
and bifundamental fields of the gauged linear sigma model
appearing in the Ka¨hler quotient construction of the toric
Calabi-Yau manifold M. They are then related, by change
of variables described in Ref. [20], to lattice points of the
toric diagram and to terms zniwmi in the Newton polyno-
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mial (1). This shows that there is a one-to-one correspon-
dence between terms in Pðz; wÞ and ~Pðz; wÞ, as pointed out
by [6]. Furthermore we can show that their coefficients
agree. The Ka¨hler moduli of M are the Fayet-Iliopoulos
(FI) parameters of the quiver quantum mechanics.
According to the dictionary, between quiver gauge theories
and dimer models [7], FI parameters are associated with
nodes of the quiver diagram, or equivalently to faces of the
dimer model. Thus, we can identify the FI parameters with
the magnetic fluxes through faces of the dimer model,
which parametrize the energy of each perfect matching
of the dimer model [11]. Each perfect matching appears in
~Pðz; wÞ with the weight given by an exponential of the
fluxes. On the other hand, the Newton polynomial is a sum
of zniwmi , each of which corresponds to a lattice point of
the toric diagram and is weighted by an exponential of the
Ka¨hler moduli t [4]. We have verified that the weight for
perfect matchings and the weight for lattice points of the
toric diagram agree, and this proves the identity (10).
Combining (7) and (9) and discarding a term in total
derivative in (x, y), which is justified by the subtraction of
the linear piece in Rðx; yÞ discussed in the next paragraph,
we find
Z exp

N2
Z
dxdyR

gsN
2
x;
gsN
2
y

: (11)
By rescaling (x, y) by the factor of gN=2, this becomes
Z exp

4
g2s
Z
dxdyRðx; yÞ

: (12)
Note that theN dependence has disappeared except that the
range of the (x, y) integral has been rescaled by the factor
of gN=2. For N ! 1 with small but fixed gs, we have an
integral over the whole (x, y) plane.
The integral (12) in the large N limit is divergent. To
identify and subtract the divergent part, it is convenient to
introduce the concept of the amoeba [21] which is a subset
of R2 defined by,
Amoeba ¼ fðx; yÞ 2 R2 Pðexþi; eyþiÞ ¼ 0
for some ð;Þg: (13)
In the thermodynamic limit, the amoeba corresponds to the
liquid phase of the crystal [11]. If there are no interior
points in the toric diagram, the complement of the amoeba
is the solid phase, where the crystal retains its original
shape. There, the Ronkin function Rðx; yÞ is linear [24]. If
there are interior lattice points in the toric diagram, the
amoeba acquires holes, inside of which are in the gas
phase, where the Ronkin function is again linear but the
slope of the crystal surface is different from the original
one. The integral (12) becomes finite if we subtract the
linear piece of the Ronkin function in the solid phase so
that the partition function is normalized to be 1 for the
initial crystal configuration.
Topological string at genus 0.—Our next task is to
compute the genus-0 topological string partition function
F 0 of the toric Calabi-Yau manifold M and compare it
with the thermodynamic limit of the partition function of
the crystal melting model (12). For this purpose, it is
convenient to use the mirror Calabi-Yau manifold ~M de-
fined by the Eq. (1) since F 0 can be evaluated by the
classical period integral as,
F 0 ¼
Z
0
; (14)
where  is the holomorphic 3-form on the mirror, and 0
is the Lagrangian 3-cycle which is the mirror of the 6-cycle
filling the entire toric Calabi-Yau manifold.
According to the microscopic derivation of the mirror
symmetry by Hori and Vafa [4], the sigma model on the
toric Calabi-Yau manifold is equivalent to the Landau-
Ginzburg model with the superpotential
Wðu; x; yÞ ¼ euPðex; eyÞ: (15)
It was shown in Ref. [5] that an integral of eW in a three-
dimensional subspace of the (u, x, y) plane can be trans-
formed into a period integral of the holomorphic 3-form
on the mirror Calabi-Yau manifold. Thus, we should be
able to evaluate (14) as an integral of eW . To do so, we need
to identify the contour of the integral.
Since gs and t
A’s in (4) are taken to be real in the dimer
model, the Newton polynomial Pðx; yÞ in our case is with
real coefficients. The mirror manifold (1) has the complex
conjugation involution, and thus the fixed point set is a
natural candidate for 0. In fact, following the mirror
symmetry transformation as described in Ref. [4], we
find that the 6-cycle of the original toric Calabi-Yau mani-
foldM corresponds to the real section in the (u, x, y) space
in the mirror ~M. Thus, we find
F 0 ¼
Z 1
0
du
Z 1
1
dxdyee
uPðex;eyÞ
¼ 
Z
dxdy lnPðex; eyÞ: (16)
The divergent part of the integral in the (x, y) plane can be
removed by subtracting a linear term in lnPðex; eyÞ for x,
y! 1 as we did for (12). The integral (16) is almost equal
to the exponent of the partition function (12) of the crystal
melting model, except that we do not have the averaging
over the phases (, ) to define the Ronkin function as in
(8). It turns out that the integral over (x, y) in (16) removes
the dependence on ð;Þ, and thus the averaging process is
not necessary.
To see this, let us define a generalization of F 0 for an
integral of ðx; yÞ with arbitrary phases as
F ð;Þ ¼
Z
dxdy lnPðexþi; eyþiÞ: (17)
Taking derivatives of F with respect to ð;Þ, we find the
PRL 102, 161601 (2009) P HY S I CA L R EV I EW LE T T E R S
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integrand becomes a total derivative in ðx; yÞ as in


@
@
þ  @
@

F ¼
Z
dxdyi


@
@x
þ  @
@y

 lnPðexþi; eyþiÞ: (18)
If we choose ð;Þ so that it is not in the direction of the
tentacles of the amoeba (13), the boundary term is removed
by the regularization and we find ð@ þ @ÞF ¼ 0.
Since ,  is arbitrary except in the direction of the
tentacles of the amoeba, F is independent of (, ) and
agrees with its average. Namely,
F 0 ¼ 
Z
dxdy lnPðx; yÞ ¼ 
Z
dxdyRðx; yÞ: (19)
Thus, we found that the thermodynamic limit of the
partition function of the crystal melting model given
by (12) is equal to expð 4
g2s
F 0Þ, which is the genus-0
partition function of the topological string theory. This is
what we wanted to show.
We would like to thank Alexei Borodin, Kentaro Hori,
Kentaro Nagao, and Andrei Okounkov for stimulating
discussions. We thank Mina Aganagic for her comment
on the earlier version of this Letter. This work is supported
in part by DOE Grant No. DE-FG03-92-ER40701 and by
the World Premier International Research Center Initiative
of MEXT of Japan. H.O. is also supported in part by a
Grant-in-Aid for Scientific Research (C) 20540256 of JSPS
and by the Kavli Foundation. M.Y. is also supported in part
by JSPS and GCOE for Phys. Sci. Frontier of MEXT of
Japan.
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[2] H. Ooguri and M. Yamazaki, arXiv:0811.2801 [Comm.
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[3] H. Ooguri, A. Strominger, and C. Vafa, Phys. Rev. D 70,
106007 (2004).
[4] K. Hori and C. Vafa, arXiv:hep-th/0002222.
[5] K. Hori, A. Iqbal, and C. Vafa, arXiv:hep-th/0005247.
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ph/0311005.
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[18] S. Sheffield, Ph.D. thesis, Stanford University, 2003.
[19] Here we only show the gs dependence explicitly and the
dependence on the Ka¨hler moduli tA is in .
[20] S. Franco and D. Vegh, J. High Energy Phys. 11 (2006)
054.
[21] See Ref. [22] for a survey of aspects of amoebae, and
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Emergent Calabi-Yau geometry

Emergent Calabi-Yau Geometry

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