- WITTENS QUANTUM SINGULARITY THEORY
- ____________________________________________________________________________
Annals of Mathematics 178 (2013), 1–106
http://dx.doi.org/10.4007/annals.2013.178.1.1
The Witten equation, mirror symmetry,
and quantum singularity theory
By Huijun Fan, Tyler Jarvis, and Yongbin Ruan
Abstract
For any nondegenerate, quasi-homogeneous hypersurface singularity, we
describe a family of moduli spaces, a virtual cycle, and a corresponding
cohomological field theory associated to the singularity. This theory is
analogous to Gromov-Witten theory and generalizes the theory of r-spin
curves, which corresponds to the simple singularity Ar−1.
We also resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual, and the second conjecture is
that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are
also discussed.
Contents
1. Introduction 2
1.1. Organization of the paper 9
1.2. Acknowledgments 9
2. W-curves and their moduli 10
2.1. W-structures on orbicurves 10
2.2. Moduli of stable W-orbicurves 20
2.3. Admissible groups G and W g,k,G 30
2.4. The tautological ring of W g,k 32
3. The state space associated to a singularity 35
3.1. Lefschetz thimble 35
3.2. Orbifolding and state space 37
H. F. was partially Supported by NSFC 10401001, NSFC 10321001, and NSFC 10631050.
T. J. was partially supported by National Science Foundation grants DMS-0605155 and
DMS-0105788 and the Institut Mittag-Leffler (Djursholm, Sweden).
Y. R. was partially supported by the National Science Foundation and the Yangtze Center
of Mathematics at Sichuan University.
c 2013 Department of Mathematics, Princeton University.
1
2 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN
4. Virtual cycles and axioms 39
4.1. î
W (Γ)óvir and its axioms 39
4.2. Cohomological field theory 49
5. ADE-singularities and mirror symmetry 55
5.1. Relation between QW and HN (C
N , W ∞, C) 56
5.2. Self-mirror cases 58
5.3. Simple singularities that are not self-mirror 66
6. ADE-hierarchies and the generalized Witten conjecture 70
6.1. Overview of the results on integrable hierarchies 70
6.2. Reconstruction theorem 73
6.3. Computation of the basic four-point correlators in the A-model 82
6.4. Computation of the basic four-point correlators in the B-model 91
6.5. Proof of Theorem 6.1.3 97
References 101
1. Introduction
The study of singularities has a long history in mathematics. For example,
in algebraic geometry it is often necessary to study algebraic varieties with singularities even if the initial goal was to work only with smooth varieties. Many
important surgery operations, such as flops and flips, are closely associated
with singularities. In lower-dimensional topology, links of singularities give
rise to many important examples of 3-manifolds. Singularity theory is also an
important subject in its own right. In fact, singularity theory has been well
established for many decades (see [AGZV85]). One of the most famous examples is the ADE-classification of hypersurface singularities of zero modality.
We will refer to this part of singularity theory as classical singularity theory
and review some aspects of the classical theory later. Even though we are
primarily interested in the quantum aspects of singularity theory, the classical
theory always serves as a source of inspiration.
Singularity theory also appears in physics. Given a polynomial W :
C
n ✲ C with only isolated critical (singular) points, one can associate to it
the so-called Landau-Ginzburg model. In the early days of quantum cohomology, the Landau-Ginzburg model and singularity theory gave some of the first
examples of Frobenius manifolds. It is surprising that although the LandauGinzburg model is one of the best understood models in physics, there has been
no construction of Gromov-Witten type invariants for it until now. However,
our initial motivation was not about singularities and the Landau-Ginzburg
model. Instead, we wanted to solve the Witten equation
∂u¯
i +
∂W
∂ui
= 0,
QUANTUM SINGULARITY THEORY 3
where W is a quasi-homogeneous polynomial and ui
is interpreted as a section
of an appropriate orbifold line bundle on a Riemann surface C .
The simplest Witten equation is the Ar−1 case. This is of the form
∂u¯ + ru¯
r−1 = 0.
It was introduced by Witten [Wit93a] more than fifteen years ago as a generalization of topological gravity. Somehow, it was buried in the literature
without attracting much attention. Several years ago, Witten generalized his
equation for an arbitrary quasi-homogeneous polynomial [Wit] and coined it
the “Landau-Ginzburg A-model.” Let us briefly recall the motivation behind
Witten’s equation. Around 1990, Witten proposed a remarkable conjecture relating the intersection numbers of the Deligne-Mumford moduli space of stable
curves with the KdV hierarchy [Wit91]. His conjecture was soon proved by
Kontsevich [Kon92]. About the same time, Witten also proposed a generalization of his conjecture. In his generalization, the stable curve is replaced by a
curve with a root of the canonical bundle (r-spin curve), and the KdV-hierarchy
was replaced by more general KP-hierarchies called nKdV, or Gelfand-Dikii,
hierarchies. The r-spin curve can be thought of as the background data to be
used to set up the Witten equation in the Ar−1-case. Since then, the moduli
space of r-spin curves has been rigorously constructed by Abramovich, Kimura,
Vaintrob and the second author [AJ03], [Jar00], [Jar98], [JKV01]. The more
general Witten conjecture was proved in genus zero several years ago [JKV01],
in genus one and two by Y.-P. Lee [Lee06], and recently in higher genus by
Faber, Shadrin, and Zvonkine [FSZ10].
The theory of r-spin curves (corresponding to the Ar−1-case of our theory)
does not need the Witten equation at all. This partially explains the fact that
the Witten equation has been neglected in the literature for more than ten
years. In the r-spin case, the algebro-geometric data is an orbifold line bundle
L satisfying the equation L r = Klog. Assume that all the orbifold points
are marked points. A marked point with trivial orbifold structure is called
a broad (or Ramond in our old notation) marked point, and a marked point
with nontrivial orbifold structure is called a narrow (or Neveu-Schwarz in our
old notation) marked point. Contrary to intuition, broad marked points are
much harder to study than narrow marked points. If there is no broad marked
point, a simple lemma of Witten’s shows that the Witten equation has only the
zero solution. Therefore, our moduli problem becomes an algebraic geometry
problem. In the r-spin case the contribution from the broad marked point
to the corresponding field theory is zero (the decoupling of the broad sector).
This was conjectured by Witten and proved true for genus zero in [JKV01]
and for higher genus in [Pol04]. This means that in the r-spin case, there is no
need for the Witten equation, which partly explains why the moduli space of
4 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN
higher spin curves has been around for a long time while the Witten equation
seems to have been lost in the literature.
In the course of our investigation, we discovered that in the Dn-case the
broad sector gives a nonzero contribution. Hence, we had to develop a theory
that accounts for the contribution of the solution of the Witten equation in
the presence of broad marked points.
It has taken us a while to understand the general picture, as well as various
technical issues surrounding our current theory. In fact, an announcement was
made in 2001 by the last two authors for some special cases coupled with an
orbifold target. We apologize for the long delay because we realized later that
(1) the theory admits a vast generalization to an arbitrary quasi-homogeneous
singularity and (2) the broad sector has to be investigated. We would like
to mention that the need to investigate the broad sector led us to the space
of Lefschetz thimbles and other interesting aspects of the Landau-Ginzburg
model, including Seidel’s work on the Landau-Ginzburg A-model derived category [Sei08]. In many ways, we are happy to have waited for several years to
arrive at a much more complete and more interesting theory!
To describe our theory, let us first review some classical singularity theory.
Let W : C
N ✲ C be a quasi-homogeneous polynomial. Recall that W
is a quasi-homogeneous polynomial if there are positive integers d, n1, . . . , nn
such that W(λ
n1 x1, . . . , λnN xn) = λ
dw(x1, . . . , xN ). We define the weight (or
charge), of xi to be qi
:= ni
d
. We say W is nondegenerate if (1) the choices of
weights qi are unique and (2) W has a singularity only at zero. There are many
examples of nondegenerate quasi-homogeneous singularities, including all the
nondegenerate homogeneous polynomials and the famous ADE-examples.
Example 1.0.1.
An: W = x
n+1, n ≥ 1;
Dn: W = x
n−1 + xy2
, n ≥ 4;
E6: W = x
3 + y
4
;
E7: W = x
3 + xy3
;
E8: W = x
3 + y
5
.
The simple singularities (A, D, and E) are the only examples with so-called
central charge cˆW < 1. There are many more examples with ˆcW ≥ 1.
In addition to the choice of a nondegenerate singularity W, our theory also
depends on a choice of subgroup G of the group Aut(W) of diagonal matrices
γ such that W(γx) = W(x). We often use the notation GW := Aut(W), and
we call this group the maximal diagonal symmetry group of W. The group
GW always contains the exponential grading (or total monodromy) element
J = diag(e
2πiq1
, . . . , e2πiqN ), and hence it is always nontrivial.
QUANTUM SINGULARITY THEORY 5
Given a choice of nondegenerate W and a choice of admissible (see Section 2.3) subgroup G ≤ GW with hJi ≤ G ≤ Aut(W), we construct a cohomological field theory whose state space is defined as follows. For each γ ∈ G, let
C
Nγ be the fixed point set of γ and Wγ = W|C
Nγ . Let Hγ,G be the G-invariants
of the middle-dimensional relative cohomology
Hγ,G = Hmid(C
Nγ
,(ReW)
−1
(M, ∞), C)
G
of C
Nγ
for M >> 0, as described in Section 3. The state space of our theory is
the sum
HW,G =
M
γ∈G
Hγ,G.
The state space HW,G admits a grading and a natural nondegenerate pairing.
For α1, . . . , αk ∈ HW,G and a sequence of nonnegative integers l1, . . . , lk,
we define (see Definition 4.2.6) the genus-g correlator
hτl1
(α1), . . . , τlk
(αk)i
W,G
g
by integrating over a certain virtual fundamental cycle. In this paper we
describe the axioms that this cycle satisfies and the consequences of those
axioms. In a separate paper [FJR] we construct the cycle and prove that it
satisfies the axioms.
Theorem 1.0.2. The correlators hτl1
(α1), . . . , τlk
(αk)iW,G
g
satisfy the usual
axioms of Gromov-Witten theory (see Section 4.2), but where the divisor axiom
is replaced with another axiom that facilitates computation.
In particular, the three-point correlator together with the pairing defines
a Frobenius algebra structure on HW,G by the formula
hα ? β, γi = hτ0(α), τ0(β), τ0(γ)i
W,G
0
.
One important point is the fact that our construction depends crucially on
the Abelian automorphism group G. Although there are at least two choices
of group that might be considered canonical (the group generated by the exponential grading operator J or the maximal diagonal symmetry group GW ),
we do not know how to construct a Landau-Ginzburg A-model defined by W
alone. In this sense, the orbifold LG-model W/G is more natural than the
LG-model for W itself.
We also remark that our theory is also new in physics. Until now there has
been no description of the closed-string sector of the Landau-Ginzburg model.
Let us come back to the Witten-Kontsevich theorem regarding the KdV
hierarchy in geometry. Roughly speaking, an integrable hierarchy is a system of differential equations for a function of infinitely many time variables
F(x, t1, t2, . . .) where x is a spatial variable and t1, t2, . . . , are time variables.
6 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN
The PDE is a system of evolution equations of the form
∂F
∂tn
= Rn(x, Fx, Fxx, . . .),
where Rn is a polynomial. Usually, Rn is constructed recursively. There is
an alternative formulation in terms of the so-called Hirota bilinear equation
which e
F will satisfy. We often say that e
F is a τ -function of hierarchy. It
is well known that KdV is the A1-case of more general ADE-hierarchies. As
far as we know, there are two versions of ADE-integrable hierarchies: the first
constructed by Drinfeld-Sokolov [DS84] and the second constructed by KacWakimoto [KW89]. Both of them are constructed from integrable representations of affine Kac-Moody algebras. These two constructions are equivalent by
the work of Hollowood and Miramontes [HM93].
Witten’s original motivation was to generalize the geometry of DeligneMumford space to realize ADE-integrable hierarchies. Now, we can state his
integrable hierarchy conjecture rigorously. Choose a basis αi (i ≤ s) of HW,G.
Define the genus-g generating function
Fg,W,G =
X
k≥0
hτl1
(αi1
), . . . , τln
(αin
)i
W,G
g
t
l1
i1
· · ·t
ln
is
n!
.
Define the total potential function
DW,G = exp ÑX
g≥0
h
2g−2Fg,W,Gé
.
Conjecture 1.0.3 (Witten’s ADE-integrable hierarchy conjecture). The
total potential functions of the A, D, and E singularities with the symmetry
group hJi generated by the exponential grading operator, are τ -functions of the
corresponding A, D, and E integrable hierarchies.
In the An case, this conjecture is often referred as the generalized Witten
conjecture, as compared to the original Witten conjecture proved by Kontsevich [Kon92]. As mentioned earlier, the conjecture for the An-case has been
established recently by Faber, Shadrin, and Zvonkine [FSZ10]. The original
Witten conjecture also inspired a great deal of activity related to GromovWitten theory of more general spaces. Those cases are 2-Toda for CP1 by
Okounkov-Pandharipande [OP06a] and the Virasoro constraints for toric manifolds by Givental [Giv01], and Riemann surfaces by Okounkov-Pandharipande
[OP06b]. In some sense, the ADE-integrable hierarchy conjecture is analogous
to these lines of research but where the targets are singularities.
The main application of our theory is the resolution of the ADE-integrable
hierarchy conjecture, as manifested by the following two theorems.
QUANTUM SINGULARITY THEORY 7
Theorem 1.0.4. The total potential functions of the singularities Dn with
even n ≥ 6, and E6, E7, and E8, with the group hJi are τ -functions of the
corresponding Kac-Wakimoto/Drinfeld-Sokolov hierarchies.
We expect the conjecture for D4 to be true as well. However, our calculational tools are not strong enough to prove it at this moment. We hope to
come back to it at another occasion.
Surprisingly, the Witten conjecture for Dn with n odd is false. Note that
in the case of n even, the subgroup hJi has index two in the maximal group
GDn of diagonal symmetries, but in the case that n is odd, hJi is equal to GDn
.
In this paper we prove
Theorem 1.0.5. (1) For all n > 4, the total potential function of the
Dn-singularity with the maximal diagonal symmetry group GDn
is a
τ -function of the A2n−3-Kac-Wakimoto/Drinfeld-Sokolov hierarchies.
(2) For all n > 4, the total potential function of W = x
n−1y+y
2
(n ≥ 4) with
the maximal diagonal symmetry group is a τ -function of the Dn-KacWakimoto/Drinfeld-Sokolov hierarchy.
The above two theorems realize the ADE-hierarchies completely in our
theory. Moreover, it illustrates the important role that the group of symmetries
plays in our constructions: When the symmetry group is GDn
, we have the
A2n−3-hierarchy, but when the symmetry group is hJi, and when hJi is a
proper subgroup of GDn
, we have the Dn-hierarchy.
Readers may wonder about the singularity W = x
n−1y + y
2
(which is
isomorphic to A2n−3). Its appearance reveals a deep connection between integrable hierarchies and mirror symmetry. (See more in Section 6.)
Although the simple singularities are the only singularities with central
charge ˆcW < 1, there are many more examples of singularities. It would be an
extremely interesting problem to find other integrable hierarchies corresponding to singularities with ˆcW ≥ 1.
Witten’s second conjecture is the following ADE self-mirror conjecture
which interchanges the A-model with the B-model.
Conjecture 1.0.6 (ADE self-mirror conjecture). If W is a simple singularity, then for the symmetry group hJi, generated by the exponential grading
operator, the ring HW,hJi
is isomorphic to the Milnor ring of W.
The second main theorem of this paper is the following.
Theorem 1.0.7. (1) Except for Dn with n odd, the ring HW,hJi of any
simple (ADE) singularity W with group hJi is isomorphic to the Milnor
ring QW of the same singularity.
8 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN
(2) The ring HDn,GDn
of Dn with the maximal diagonal symmetry group
GDn
is isomorphic to the Milnor ring QA2n−3
of W = x
n−1y + y
2
.
(3) The ring HW,GW of W = x
n−1y + y
2
(n ≥ 4) with the maximal diagonal
symmetry group GW is isomorphic to the Milnor ring QDn of Dn.
The readers may note the similarities between the statements of the above
mirror symmetry theorem and our integrable hierarchies theorems. In fact,
the mirror symmetry theorem is the first step towards the proof of integrable
hierarchies theorems.
Of course we cannot expect that most singularities will be self-mirror,
but we can hope for mirror symmetry beyond just the simple singularities.
Since the initial draft of this paper, much progress has been made [FJJS12],
[Kra10], [KPA+10] for invertible singularities. An invertible singularity has the
property that the number of monomials is equal to the number of variables.
This is a large class of quasi-homogeneous singularities.
In general, it is a very difficult problem to compute Gromov-Witten invariants of compact Calabi-Yau manifolds. While there are many results for
low genus cases [Giv98], [LLY97], [Zin08], there are only a very few compact
examples [MP06], [OP06b] where one knows how to compute Gromov-Witten
invariants in all genera by either mathematical or physical methods. (For some
recent advances, see [HKQ09].)
Note that a Calabi-Yau hypersurface of weighted projective space defines
a quasi-homogenenous singularity and hence an LG-theory. This type of singularity has P
i qi = 1. In the early 1990s, Martinec-Vafa-Warner-Witten proposed a famous conjecture [Mar90], [VW89], [Wit93b] connecting these two
points of view.
Conjecture 1.0.8 (Landau-Ginzburg/Calabi-Yau correspondence). The
LG-theory of a generic quasi-homogeneous singularity W/hJi and the corresponding Calabi-Yau theory are isomorphic in a certain sense.
This is certainly one of the most important conjectures in the subject.
The importance of the conjecture comes from the physical indication that the
LG theory and singularity theory is much easier to compute than the CalabiYau geometry. The precise mathematical statement of the above conjecture is
still lacking at this moment (see [CR10] also). We hope to come back to it on
another occasion.
We conclude by noting that it would be a very interesting problem to
explore how to extend our results to a setting like that treated by Guffin and
Sharpe in [GS09a], [GS09b]. They have considered twisted Landau-Ginzburg
models without coupling to topological gravity, but over more general orbifolds,
QUANTUM SINGULARITY THEORY 9
whereas our model couples to topological gravity, but we work exclusively with
orbifold vector bundles.
1.1. Organization of the paper. A complete construction of our theory
will be carried out in a series of papers. In this paper, we give a complete
description of the algebro-geometric aspects of our theory. The information
missing is the analytic construction of the moduli space of solutions of the
Witten equation and its virtual fundamental cycle, which is done in a separate
paper [FJR]. Here, we summarize the main properties or axioms of the cycle
and their consequences. The main application is the proof of Witten’s selfmirror conjecture and integrable hierarchies conjecture for ADE-singularities.
The paper is organized as follows. In Section 2, we will set up the theory
of W-structures. This is the background data for the Witten equation and
a generalization of the well-known theory of r-spin curves. The analog of
quantum cohomology groups and the state space of the theory will be described
in Section 3. In Section 4, we formulate a list of axioms of our theory. The
proof of Witten’s mirror symmetry conjecture is in Section 5. The proof of his
integrable hierarchies conjecture is in Section 6.
1.2. Acknowledgments. The third author would like to express his special thanks to E. Witten for explaining to him his equation in 2002 and for his
support over these years. Thanks also go to K. Hori and A. Klemm for many
stimulating discussions about Landau-Ginzburg models.
The last two authors would like to thank R. Kaufmann for explaining his
work, for many helpful discussions, and for sharing a common interest and
support on this subject for these years. We also thank Marc Krawitz for showing us the Berglund-H¨ubsch mirror construction, Eric Sharpe for explaining
to us some aspects of his work in [GS09a], [GS09b], and Alessandro Chiodo
for his insights. The second author would also like to thank T. Kimura for
helpful discussions and insights, and he thanks the Institut Mittag-Leffler for
providing a stimulating environment for research.
The first author would like to thank K. C. Chang, Weiyue Ding, and
J. Jost for their long-term encouragement and support, and especially he wants
to thank Weiyue Ding for fruitful discussions and warm help for many years.
He also thanks Bohui Chen for many useful suggestions and comments. Partial
work was done when the first author visited MPI in Leipzig, MSRI in Berkeley,
and the University of Wisconsin-Madison respectively; he appreciates their
hospitality.
The first and second authors thank H. Tracy Hall for many helpful discussions and for his ideas and insights related to Proposition 2.3 and equation (99).
10 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN
Finally, the third author would like to thank the University of WisconsinMadison, where much of the current work has taken place, for warm support
and fond memories.
2. W-curves and their moduli
2.1. W-structures on orbicurves.
2.1.1. Orbicurves and line bundles. Recall that an orbicurve C with marked
points p1, . . . , pk is a (possibly nodal) Riemann surface C with orbifold structure at each pi and each node. That is to say, for each marked point pi
,
there is a local group Gpi
and (since we are working over C) a canonical isomorphism Gpi
∼= Z/mi for some positive integer mi
. A neighborhood of pi
is uniformized by the branched covering map z ✲ z
mi
. For each node p,
there is again a local group Gp
∼= Z/nj whose action is complementary on
the two different branches. That is to say, a neighborhood of a nodal point
(viewed as a neighborhood of the origin of {zw = 0} ⊂ C
2
) is uniformized by
a branched covering map (z, w) ✲ (z
nj
, wnj ), with nj ≥ 1, and with group
action e
2πi/nj (z, w) = (e
2πi/nj z, e−2πi/njw).
Definition 2.1.1. We will call the orbicurve C smooth if the underlying
curve C is smooth, and we will call the orbicurve nodal if the underlying curve
C is nodal.
Note that this definition agrees with that of algebraic geometers for smooth
Deligne-Mumford stacks, but it differs from that of many topologists (e.g.,
[CR04]) since orbicurves with nontrivial orbifold structure at a point will still
be called smooth when the underlying curve is smooth.
We denote by % : C ✲ C the natural projection to the underlying
(coarse, or nonorbifold) Riemann surface C. If L is a line bundle on C, it can
be uniquely lifted to an orbifold line bundle %
∗L over C . When there is no
danger of confusion, we use the same symbol L to denote its lifting.
Definition 2.1.2. Let KC be the canonical bundle of C. We define the
log-canonical bundle of C to be the line bundle
KC,log := K ⊗ O(p1) ⊗ · · · ⊗ O(pk),
where O(p) is the holomorphic line bundle of degree one whose sections may
have a simple pole at p. This bundle KC,log can be thought of as the canonical
bundle of the punctured Riemann surface C − {p1, . . . , pk}.
The log-canonical bundle of C is defined to be the pullback to C of the
log-canonical bundle of C:
(1) KC ,log := %
∗KC,log.
QUANTUM SINGULARITY THEORY 11
Near a marked point p of C with local coordinate x, the bundle KC,log is
locally generated by the meromorphic one-form dx/x. If the local coordinate
near p on C is z, with z
m = x, then the lift KC ,log := %
∗
(KC,log) is still locally
generated by m dz/z = dx/x. When there is no risk of confusion, we will
denote both KC,log and KC ,log by Klog. Near a node with coordinates z and
w satisfying zw = 0, both K and Klog are locally generated by the one-form
dz/z = −dw/w.
Note that although %
∗KC,log = KC ,log, the usual canonical bundle does
not pull back to itself:
(2) %
∗KC = KC ⊗ O
−
X
k
i=1
(mi − 1)pi
!
6= KC ,
where mi
is the order of the local group at pi
. This can be seen from the fact
that when x = z
m, we have
(3) dx = mzm−1
dz.
2.1.2. Pushforward to the underlying curve. If L is an orbifold line bundle
on a smooth orbicurve C , then the sheaf of locally invariant sections of L is
locally free of rank one and hence dual to a unique line bundle |L | on C .
We also denote |L | by %∗L , and it is called the “desingularization” of L in
[CR04, Prop. 4.1.2]. It can be constructed explicitly as follows.
We keep the local trivialization at nonorbifold points and change it at
each orbifold point p. If L has a local chart ∆ ×C with coordinates (z, s) and
if the generator 1 ∈ Z/m ∼= Gp acts locally on L by
(z, s) 7→ (exp(2πi/m)z, exp(2πiv/m)s),
then we use the Z/m-equivariant map Ψ : (∆ − {0}) × C ✲ ∆ × C given by
(4) (z, s) ✲ (z
m, z−v
s),
where Z/m acts trivially on the second ∆ × C. Since Z/m acts trivially, this
gives a line bundle over C, which is |L |.
If the orbicurve C is nodal, then the pushforward %∗L of a line bundle L
may not be a line bundle on C. In fact, if the local group Gp at a node acts
nontrivially on L , then the invariant sections of L form a rank-one torsionfree sheaf on C (see [AJ03]). However, we may take the normalizations C
‹
and
C‹ to get (possibly disconnected) smooth curves, and the pushforward of L
from C
‹
will give a line bundle on C‹
. Thus |L | is a line bundle away from the
nodes of C, but its fiber at a node is two-dimensional; that is, there is (usually)
no gluing condition on |L | at the nodal points. The situation is slightly more
subtle than this (see [AJ03]), but for our purposes, it will be enough to consider
the pushforward |L | as a line bundle on the normalization C‹
where the local
group acts trivially on L .
12 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN
It is also important to understand more about the sections of the pushforward %∗L . Suppose that s is a section of |L | having local representative
g(u). Then (z, zv
g(z
m)) is a local section of L . Therefore, we obtain a section
%
∗
(s) ∈ Ω
0
(L ) which equals s away from orbifold points under the identification given by equation 4. It is clear that if s is holomorphic, so is %
∗
(s). If
we start from an analytic section of L , we can reverse the above process to
obtain a section of |L |. In particular, L and |L | have isomorphic spaces of
holomorphic sections:
%
∗
: H0
(C, |L |)−→fiH0
(C , L ).
In the same way, there is a map %
∗
: Ω0,1
(|L |) ✲ Ω
0,1
(L ), where Ω0,1
(L )
is the space of orbifold (0, 1)-forms with values in L . Suppose that g(u)du¯
is a local representative of a section of t ∈ Ω
0,1
(|L |). Then %
∗
(t) has a local
representative z
v
g(z
m)mz¯
m−1dz¯. Moreover, % induces an isomorphism
%
∗
: H1
(C, |L |)−→fiH1
(C , L ).
Example 2.1.3. The pushforward |KC | of the log-canonical bundle of any
orbicurve C is again the log-canonical bundle of C, because at a point p with
local group Gp
∼= Z/m, the one-form m dz/z = dx/x is invariant under the
local group action.
Similarly, the pushforward |KC | of the canonical bundle of C is just the
canonical bundle of C:
(5) |KC | = %∗KC = KC,
because the local group Z/m acts on the one-form dz by exp(2πi/m)dz, and
the invariant holomorphic one-forms are precisely those generated by mzm−1dz
= dx.
2.1.3. Quasi-homogeneous polynomials and their Abelian automorphisms.
Definition 2.1.4. A quasi-homogeneous (or weighted homogeneous) polynomial W ∈ C[x1, . . . , xN ] is a polynomial for which there exist positive rational numbers q1, . . . , qN ∈ Q>0
, such that for any λ ∈ C
∗
,
(6) W(λ
q1 x1, . . . , λqN xN ) = λW(x1, . . . , xN ).
We will call qj the weight of xj . We define d and ni for i ∈ {1, . . . , N} to
be the unique positive integers such that (q1, . . . , qN ) = (n1/d, . . . , nN /d) with
gcd(d, n1, . . . , nN ) = 1.
Throughout this paper we will need a certain nondegeneracy condition
on W.
Definition 2.1.5. We call W nondegenerate if
(1) W contains no monomial of the form xixj for i 6= j;
QUANTUM SINGULARITY THEORY 13
(2) the hypersurface defined by W in weighted projective space is nonsingular or, equivalently, the affine hypersurface defined by W has an
isolated singularity at the origin.
The following proposition was pointed out to us by N. Priddis and follows
from [HK, Thm. 3.7(b)].
Proposition 2.1.6. If W is a nondegenerate, quasi-homogeneous polynomial, then the weights qi are bounded by qi ≤
1
2
and are unique.
From now on, when we speak of a quasi-homogeneous polynomial W, we
will assume it to be nondegenerate.
Definition 2.1.7. Write the polynomial W =
Ps
j=1 Wj as a sum of monomials Wj = cj
QN
`=1 x
bj`
`
, with bj` ∈ Z
≥0 and with cj 6= 0. Define the s × N
matrix
(7) BW := (bj`),
and let BW = V T Q be the Smith normal form of BW [Art91, §12, Thm. 4.3].
That is, V is an s × s invertible integer matrix and Q is an N × N invertible
integer matrix. The matrix T = (tj`) is an s × N integer matrix with tj` = 0
unless ` = j, and t`,` divides t`+1,`+1 for each ` ∈ {1, . . . , N − 1}.
Lemma 2.1.8. If W is nondegenerate, then the group
GW := {(α1, . . . , αN ) ∈ (C
∗
)
N | W(α1x1, . . . , αN xN ) = W(x1, . . . , xN )}
of diagonal symmetries of W is finite.
Proof. The uniqueness of the weights qi
is equivalent to saying that the
matrix BW has rank N. We may as well assume that BW is invertible. Now
write γ = (α1, . . . , αN ) ∈ GW , as αj = exp(uj + vj i) for uj ∈ R uniquely
determined, and vj ∈ R determined up to integral multiple of 2πi. The equation W(α1x1, . . . , αN xN ) = W(x1, . . . , xN ) can be written as BW (u + vi) ≡ 0
(mod 2πi), where u + vi = (u1 + v1i, . . . , uN + vN i) and 0 is the zero vector.
Invertibility of BW shows that u` = 0 for all `. Thus GW is a subgroup of
U(1)N , and a straightforward argument shows that the number of solutions
(modulo 2πi) to the equation BW (vi) ≡ 0 (mod 2πi) is also finite.
Definition 2.1.9. We write each element γ ∈ GW (uniquely) as
γ = (exp(2πiΘ
γ
1
), . . . , exp(2πiΘ
γ
N )),
with Θγ
i ∈ [0, 1) ∩ Q.
There is a special element J of the group GW , which is defined to be
J := (exp(2πiq1), . . . , exp(2πiqN )),
14 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN
where the qi are the weights defined in Definition 2.1.5. Since qi 6= 0 for all i,
we have ΘJ
i = qi
. By definition, the order of the element J is d. The element
J will play an important role in the remainder of the paper.
For any γ ∈ GW , let C
Nγ
:= (C
N )
γ be the set of fixed points of γ in
C
N , let Nγ denote its complex dimension, and let Wγ := W|C
Nγ be the quasihomogeneous singularity restricted to the fixed point locus of γ. The polynomial Wγ defines a quasi-homogeneous singularity of its own in C
Nγ
γ , and
Wγ has its own Abelian automorphism group. However, we prefer to think of
the original group GW acting on C
Nγ
. Note that GW preserves the subspace
C
Nγ ⊆ C
N .
Lemma 2.1.10. If W is a nondegenerate, quasi-homogeneous polynomial,
then for any γ ∈ GW , the polynomial Wγ has no nontrivial critical points.
Therefore, Wγ is itself a nondegenerate, quasi-homogeneous polynomial in the
variables fixed by γ.
Proof. Let m ⊂ C[x1, . . . , xN ] be the ideal generated by the variables not
fixed by γ, and write W as W = Wγ + Wmoved, where Wmoved ∈ m. In fact, we
have Wmoved ∈ m2 because if any monomial in Wmoved does not lie in m2
, it
can be written as xmM, where M is a monomial fixed by γ. However, γ ∈ GW
acts diagonally, and it must fix W, and hence it must fix every monomial
of W, including xmM. Since it fixes M and xmM, it must also fix xm—a
contradiction. This shows that Wmoved ∈ m2
.
Now we can show that there are no nontrivial critical points of Wγ. For
simplicity, re-order the variables so that x1, . . . , x` are the fixed variables and
x`+1, . . . , xN are the remaining variables. If there were a nontrivial critical
point of Wγ, say (α1, . . . , α`) ∈ C
`
, then the point (α1, . . . , α`
, 0, . . . , 0) ∈ C
N
would be a nontrivial critical point of W. To see this, note that for any
i ∈ {1, . . . , N}, we have
∂Wmoved
∂xi
(α1,...,α`,0,...,0)
= 0
since Wmoved ∈ m2
. This gives
∂W
∂xi
(α1,...,α`,0,...,0)
=
∂Wγ
∂xi
(α1,...,α`)
+
∂Wmoved
∂xi
(α1,...,α`,0,...,0)
= 0,
which shows that (α1, . . . , α`
, 0, . . . , 0) is a nontrivial critical point of W—a
contradiction.
2.1.4. W-structures on an orbicurve. A W-structure on an orbicurve C is
essentially a choice of N line bundles L1, . . . , LN so that for each monomial
QUANTUM SINGULARITY THEORY 15
Wj = x
bj,1
1
· · · x
bj,N
N , we have an isomorphism of line bundles
ϕj : L
⊗bj,1
1
· · · L
⊗bj,N
N
✲ Klog.
However, the isomorphisms ϕj need to be compatible, in the sense that at any
point p there exists a trivialization Li
|
p
∼= C for every i and Klog|
p
∼= C · dz/z
such that for all j ∈ {1, . . . , s}, we have ϕj (1, . . . , 1) = 1 · dz/z ∈ C. If s = N,
we can choose such trivializations for any choice of maps {ϕj}, but if s > N,
then the choices of {ϕj} need to be related. To do this we use the Smith
normal form to give us a sort of minimal generating set of isomorphisms that
will determine all the maps {ϕj}.
Definition 2.1.11. For any nondegenerate, quasi-homogeneous polynomial
W ∈ C[x1, . . . , xN ], with matrix of exponents BW = (b`j ) and Smith normal form BW = V T Q, let A := (aj`) := V
−1B = T Q, and let u` be the
sum of the entries in the `-th row of V
−1
(i.e., the `-th term in the product
V
−1
(1, 1, . . . , 1)T
).
For any ` ∈ {1, . . . , N}, denote by A`(L1, . . . , LN ) the tensor product
A`(L1, . . . , LN ) := L
⊗a`1
1 ⊗ · · · ⊗ L
⊗a`N
N .
We define a W-structure on an orbicurve C to be the data of an N-tuple
(L1, . . . , LN ) of orbifold line bundles on C and isomorphisms
ϕ˜`
: A`(L1, . . . , LN )−→fiK
u`
C ,log
for every ` ∈ {1, . . . , N}.
Note that for each point p ∈ C , an orbifold line bundle L on C induces a
representation Gp ✲ Aut(L ). Moreover, a W-structure on C will induce a
representation rp : Gp ✲ U(1)N . For all our W-structures, we require that
this representation rp be faithful at every point.
The next two propositions follow immediately from the definitions.
Proposition 2.1.12. The Smith normal form is not necessarily unique,
but for any two choices of Smith normal form B = V T Q = V
0T
0Q0
, a
W-structure (L1, . . . , LN , ϕ˜1, . . . , ϕ˜N ) with respect to V T Q induces a canonical W-structure (L1, . . . , LN , ϕ˜
0
1
, . . . , ϕ˜
0
N ) with respect to V
0T
0Q0
, where the
isomorphism ϕ˜
0
i
is given by
ϕ˜
0
i = ˜ϕ
zi1
1 ⊗ · · · ⊗ ϕ˜
ziN
N
and where Z = (zij ) := (V
0
)
−1V .
16 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN
Proposition 2.1.13. For each j ∈ {1, . . . , s}, the maps {ϕ˜`} induce an
isomorphism
ϕj := ˜ϕ
vj1
1 ⊗ · · · ⊗ ϕ˜
vjN
N : Wj (L1, . . . , LN )(8)
= L
⊗bj1
1 ⊗ · · · ⊗ L
⊗bj,N
N =L
P
`
vj`a`1
1 ⊗ · · · ⊗ L
P
`
vj`a`N
N
✲ KC ,log,
where V = (vj`).
Moreover, if B is square (and hence invertible)
______________________________________________________________________________________
- 6. ADE-hierarchies and the generalized Witten conjecture The main motivation for Witten to introduce his equation was the following conjecture. Conjecture 6.0.1 (ADE-integrable hierarchy conjecture). The total potential functions of the A, D, and E singularities with group hJi are τ -functions of the corresponding A, D, and E integrable hierarchies. The An-case was established recently by Faber-Shadrin-Zvonkin [FSZ10]. One of our main results is the resolution of Witten’s integrable hierarchies conjecture for the D and E series. It turns out that Witten’s conjecture needs a modification in the Dn case for n odd. This modification is extremely interesting because it reveals a surprising role that mirror symmetry plays in integrable hierarchies. 6.1. Overview of the results on integrable hierarchies. Let us start from the ADE-hierarchies. As we mentioned in the introduction, there are two equivalent versions of ADE-integrable hierarchies—that of Drinfeld-Sokolov [DS84] and that of Kac-Wakimoto [KW89]. The version directly relevant to us is the Kac-Wakimoto ADE-hierarchies because the following beautiful work of Frenkel-Givental-Milanov reduces the problem to an explicit problem in Gromov-Witten theory. Let us describe their work. Let W be a nondegenerate quasi-homogeneous singularity, and let φi (i ≤ µ) be the monomial basis of the Milnor ring with φ1 = 1. Consider the miniversal deformation space C µ where a point λ = (t1, . . . , tµ) parametrizes the polynomial W +t1φ1+t2φ2 · · ·+tµφµ. We can assign a degree to ti such that the above perturbed polynomial has the degree one; i.e., deg(ti) = 1−deg(φi). The tangent space Tλ carries an associative multiplication ◦ and a Euler vector field E = P i deg(ti)∂ti with the unit e = ∂Wλ ∂t1 . It is more subtle to construct a metric. We can consider residue pairing hf, giλ = Resx=0 fgω ∂Wλ ∂x1 · · · ∂Wλ ∂xN using a holomorphic n-form ω. A deep theorem of Saito [Sai81] states that one can choose a primitive form ω such that the induced metric is flat. Together, it defines a Frobenius manifold structure on a neighborhood of zero of C µ . We should mention that there is no explicit formula for the primitive form in general. However, it is known that for ADE-singularities, the primitive form can be chosen to be a constant multiple of standard volume form; i.e., c dx for An and c dx ∧ dy for DE series. Furthermore, one can define a potential function F, playing the role of genus-zero Gromov-Witten theory with only primary fields. It is constructed QUANTUM SINGULARITY THEORY 71 as follows. We want to work in flat coordinates si with the property that degC(si) = degC(ti) and hsi , sj i are constant. The flat coordinates depend on the flat connection of the metric and hence the primitive form. Its calculation is important and yet a difficult problem. Nevertheless, we know that the flat coordinates exist thanks to the work of Saito [Sai81]. Then, consider the threepoint correlator Cijk = h∂si , ∂sj , ∂sk i as a function near zero in C µ . We can integrate Cijk to obtain F. Here, we normalize F such that F has leading term of degree three. We can differentiate F by the Euler vector field. It has the property LEF = (ˆcW −3)F. Namely, F is homogeneous of degree ˆcW −3. The last condition means that, in the Taylor expansion F = Xa(n1, . . . , nµ) s n1 1 · · · s nµ µ n1! · · · nµ! , we have a(n1, . . . , nµ) 6= 0 only when P ni− P ni(1−degC(si)) = P degC(si) = cˆW − 3. Note that the degree in the Frobenius manifold is different from that of the A-model. For example, the unit e has degree 1 instead of zero. The A-model degree is 1 minus the B-model degree. With this relation in mind, we will treat the insertion si with degree 1 − degC(si). Then, the above formula is precisely the selection rule of quantum singularity theory. It is known that the Frobenius manifold of a singularity is semisimple in the sense that the Frobenius algebra on Tλ at a generic point λ is semisimple. On any semisimple Frobenius manifold, Givental constructed a formal Gromov-Witten potential function. We will only be interested in the case that the Frobenius manifold is the one corresponding to the miniversal deformation space of a quasi-homogeneous singularity W. We denote it by DW,formal = exp ÑX g≥0 h 2g−2F g formalé . The construction of DW,formal is complicated, but we only need its following formal properties: (1) F0 formal agrees with F for primitive fields, i.e., with no descendants. (2) F g formal satisfies the same selection rule as a Gromov-Witten theory with C1 = 0 and dimension ˆcW . (3) DW,formal satisfies all the formal axioms of Gromov-Witten theory. The first property is obvious from the construction. The second property is a consequence of the fact that DW,formal satisfies the dilaton equation and Virasoro constraints. A fundamental theorem of Frenkel-Givental-Milanov [GM05], [FGM10] is Theorem 6.1.1. For ADE-singularities, DW,formal is a τ -function of the corresponding Kac-Wakimoto ADE-hierarchies. 72 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN Remark 6.1.2. Givental-Milanov first constructed a Hirota-type equation for Givental’s formal total potential function. Later, Frenkel-Givental-Milanov proved that Givental-Milanov’s Hirota equation is indeed the same as that of Kac-Wakimoto. Our main theorem is Theorem 6.1.3. (1) Except for Dn with n odd and D4, the total potential functions of DE-singularities with the group hJi are equal to the corresponding Givental formal Gromov-Witten potential functions up to a linear change of variables. (2) DDn,Gmax = DA2n−3,formal, up to a linear change of variables. (3) For DT n = x n−1y + y 2 (n > 4), DDT n ,Gmax = DDn,formal, up to a linear change of variables. Using the theorem of Frenkel-Givental-Milanov, we obtain Corollary 6.1.4. (1) Except for Dn with n odd and D4, the total potential function of DE-singularities with the group hJi is a τ -function of the corresponding Kac-Wakimoto hierarchies (and hence Drinfeld-Sokolov hierarchies ). (2) The total potential function of all Dn-singularities for n > 4 with the maximal diagonal symmetry group is a τ -function of the A2n−3 Kac-Wakimoto hierarchies (and hence Drinfeld-Sokolov hierarchies ). (3) The total potential function of DT n = x n−1y + y 2 (n > 4) with the maximal diagonal symmetry group is a τ -function of the Dn Kac-Wakimoto hierarchies (and hence Drinfeld-Sokolov hierarchies ). Remark 6.1.5. There is a technical issue in Givental’s formal theory, as follows. For any semisimple point t of Saito’s Frobenius manifold, he defined an ancestor potential At . From this he obtains a descendant potential function D = Sˆ tAt , where Sˆ t is certain quantization of a symplectic transformation St determined by the Frobenius manifold. Then, he showed D is independent of t. However, to compare with our A-model calculation, we need to expand D as formal power series at t = 0. Although D is expected to have a power series expansion at t = 0, we have been informed that a proof is not yet in the literature. Our strategy to avoid this problem is to show that (i) the A- and B-models have isomorphic Frobenius manifolds, and (ii) in the ADE cases the ancestor functions of both models are completely determined by their respective Frobenius manifolds. Therefore, the A- and B-model have the same ancestor potentials and hence the same descendant potentials. Definition 6.1.6. An ancestor correlator is defined as hτl1 (α1), . . . , τln (αn)i W,G g (t) = X k hτl1 (α1), . . . , τln (αn), k copies z }| { t, . . . , ti W,G g . QUANTUM SINGULARITY THEORY 73 Then, we define ancestor generating function F W,G g (t) of our theory with these correlators similarly. Givental ancestor potential is defined for semisimple points t 6= 0. In the above definition, t is only a formal variable. To be able to choose an actual value t 6= 0, we need to show that the ancestor correlator is convergent for that choice of t. This is done in the following lemma. Lemma 6.1.7. Choose a basis T i of HW,G, and write t = P i tiT i . For the simple (ADE) singularities, the ancestor correlator hτl1 (α1), . . . , τln (αn)iW,G g (t) is a polynomial in the variables ti . Furthermore, if l1 = · · · = ln = 0, (i.e., if there are no ψ-classes ), the ancestor correlator is also a polynomial in the variables α1, . . . , αn. Proof. Consider correlator hτ1(α1), . . . , τn(αn), Ti1 , . . . , Tik iW,G g . The dimension condition is 2((ˆcw − 3)(1 − g) + n + k) = X i (2li + degW αi) + X j degW Tij . This implies that X j (2 − degW Tij ) = X i (2li + degW αi) − 2((ˆcw − 3)(1 − g) + n). Therefore, if we redefine deg0 W Tij := 2 − degW Tij , the ancestor correlator is homogeneous of a fixed degree. When W is an ADE-singularity, it is straightforward to check that 2 − degW Tij > 0. Hence, it must be a polynomial. The same argument implies the second case. This lemma shows that we can consider F W,G t and A W,G t for a semisimple point t 6= 0. The proof of the main theorem depends on four key ingredients. The first ingredient is the reconstruction theorem for the ADE-theory, which shows that the two ancestor potentials are both determined by their corresponding Frobenius manifolds. The second step is to show that the Frobenius manifolds are completely determined by genus-zero, three-point correlators and certain explicit four-point correlators. The third ingredient is the Topological Euler class axiom for narrow sectors, which enables us to compute all the three-point and required four-point correlators. The last ingredient is the mirror symmetry of ADE-singularities we proved in last section. The required modification in the Dn case when n is odd is transparent from mirror symmetry. 6.2. Reconstruction theorem. In this subsection, we will establish the reconstruction theorem simultaneously for ADE-quantum singularity theory and Givental’s formal Gromov-Witten theory in the ADE-case. We use the facts that 74 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN (i) both theories satisfy the formal axioms of Gromov-Witten theories, (ii) they both have the same selection rules, (iii) they both have isomorphic quantum rings up to a mirror transformation. The last fact has been established in the previous section. To simplify the notation, we state the theorem for the quantum singularity theory of the A-model. We start with the higher genus reconstruction using an idea of FaberShadrin-Zvonkine [FSZ10]. Theorem 6.2.1. If c <ˆ 1, then the ancestor potential function is uniquely determined by the genus-zero primary potential (i.e., without gravitational descendants ). If cˆ = 1, then the ancestor potential function is uniquely determined by its genus-zero and genus-one primary potentials. The proof of Theorem 6.2.1 is a direct consequence of the following two lemmas, using the Faber-Shadrin-Zvonkine reduction technique. For this argument, we always assume that ˆc ≤ 1. Lemma 6.2.2. Let αi ∈ Hγi,G for all i ∈ {1, . . . , n}, and let β be any product of ψ classes. If c <ˆ 1, then the integral Z Mg,n+k β · Λ W g,n+k (α1, . . . , αn, Ti1 , . . . , Tik ) vanishes if deg β < g for g ≥ 1. If cˆ = 1, then the above integral vanishes when deg β < g for g ≥ 2. Proof. The integral R Mg,n+k β · ΛW g,n+k (α1, . . . , αn, Ti1 , . . . , Tik ) does not vanish only if deg β = 3g − 3 + n + k − D − nX +k τ=1 Nγτ /2, where D = ˆc(g − 1) + P τ ιγτ . Recall that ιγ = PN i=1(Θγ i − qi). Now we have the inequality deg β = (3 − cˆ)(g − 1) + nX +k τ=1 (1 − ιγτ )(90) = (3 − cˆ)(g − 1) + nX +k τ=1 (1 − cˆ+ ˆc − ιγτ − Nγτ /2) ≥ (3 − cˆ)(g − 1) + (n + k)(1 − cˆ), where we used the fact (easily verified for the simple singularities AD and E) that the complex degree degC αγ = ιγ + Nγ/2 of a class αγ ∈ Hγ always satisfies degC αγ = ιγ + Nγ/2 ≤ c. ˆ QUANTUM SINGULARITY THEORY 75 Hence if g ≥ 2, we have deg β ≥ g. If g = 1, then deg β > 0 for ˆc < 1 and deg β ≥ 0 for ˆc = 1, where the equality holds if and only if degC αγτ = ˆc for all τ . The following lemma treats the integral for higher-degree ψ classes. It was proved in [FSZ10], where it was called g-reduction. Lemma 6.2.3. Let P be a monomial in the ψ and κ-classes in Mg,k of degree at least g for g ≥ 1 or at least 1 for g = 0. Then the class P can be represented by a linear combination of dual graphs, each of which has at least one edge. Proof of Theorem 6.2.1. Take any correlators hτd1 (α1)· · · τdk (αn), Ti1 , . . . , Tik ig,n+k = Z Mg,n+k ψ d1 1 · · · ψ dn k Λ W g,n+k (α1, . . . , αn, Ti1 , . . . , Tik ). The total degree of the ψ-classes must either match the hypothesis of Lemma 6.2.2 or match the hypothesis of Lemma 6.2.3. If the total degree is small, then it vanishes by Lemma 6.2.2. If it is large, then the integral is changed to the integral over the boundary classes while decreasing the degree of the total integrated ψ or κ classes. Applying the degeneration and composition laws, the genus of the moduli spaces involved will also decrease. It is easy to see that one can continue this process until the original integral is represented by a linear combination of integrals over moduli spaces of genus zero and genus one, without gravitational descendants. Remark 6.2.4. There is an alternative higher-genus reconstruction, using Teleman’s recent announcement [Tel12] of a proof of Givental’s conjecture [Giv01]. However, in the ADE-case the above argument is much simpler and achieves the same goal. The above theorem implies that all the ancestor correlators are determined by genus-zero ancestor correlators without ψ classes. On the B-model side, Givental’s genus-zero generating function is equal to Saito’s genus-zero generating function. Hence, it is well defined at t = 0. Furthermore, Lemma 6.1.7 shows that both the A- and B-model genus-zero functions without descendants are polynomials and are defined over the entire Frobenius manifold. Finally, we observe that the genus-zero ancestor generating function is determined by the ordinary genus-zero generating function (i.e., at t = 0). Therefore, it is enough to compare the ordinary genus zero generating functions. Next, we consider the reconstruction of genus-zero correlators using the Witten-Dijkgraaf-Verlinde-Verlinde Equation (WDVV). 76 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN Definition 6.2.5. We call a class γ primitive if it cannot be written as γ = γ1 ? γ2 for 0 < degC(γi) < degC(γ) (or, in the case of our A-model singularity theory, 0 < degW (γi) < degW (γ)). We have the following lemma. Lemma 6.2.6 (Reconstruction Lemma). Any genus-zero k-point correlator of the form hγ1, . . . , γk−3, α, β, ε ? φi0 can be rewritten as hγ1, . . . , γk−3, α, β, ε ? φi = S + hγ1, . . . , γk−3, α, ε, β ? φi0 (91) + hγ1, . . . , γk−3, α ? ε, β, φi0 − hγ1, . . . , γk−3, α ? β, ε, φi0 , where S is a linear combination of genus-zero correlators with fewer than k insertions. Moreover, all the genus-zero k-point correlators hγ1, . . . , γki0 are uniquely determined by the pairing, by the three-point correlators, and by correlators of the form hα1, . . . , αk 0−2, αk 0−1, αk 0i0 for k 0 ≤ k and such that αi primitive for all i ≤ k 0 − 2. Proof. Choose a basis {δi} such that δ0 = ε ? φ, and let δ 0 i be the dual basis with respect to the pairing (i.e., hδi , δ0 j i = δij ). Using WDVV and the definition of the multiplication ?, we have the formula hγ1, . . . , γk−3, α, β, ε ? φi0 = hγ1, . . . , γk−3, α, β, ε ? φi0 hδ 0 0 , ε, φi0 (92) = X k−3=I∪J X ` hγi∈I , α, ε, δ`i0 hδ 0 ` , φ, β, γj∈J i0 − X k−3=I∪J J6=∅ X ` hγi∈I , α, β, δ`i0 hδ 0 ` , φ, ε, γj∈J i0 . All of the terms on the right-hand side are k 0 -point correlators with k 0 < k except X ` hγi≤k−3, α, ε, δ`i0 hδ 0 ` , φ, βi0 + X ` hα, ε, δ`i0 hδ 0 ` , φ, β, γj≤k−3i0 − X ` hα, β, δ`i0 hδ 0 ` , φ, ε, γj≤k−3i0 = hγj≤k−3, α, ε, φ ? βi0 + hα ? ε, φ, β, γj≤k−3i0 − hα ? β, ε, φ, γj≤k−3i0 , as desired. This proves equation (91). Now, suppose that hγ1, . . . , γki0 is such that γk is not primitive, so γk = ε ? φ with ε primitive. Applying equation (91) shows that hγ1, . . . , γki0 can be rewritten as a linear combination of correlators S with fewer insertions plus QUANTUM SINGULARITY THEORY 77 three more terms: hγ1, . . . , γki0 = S + hγj≤k−3, γk−2, ε, γk−1 ? φi0 + hγj≤k−3, γk−2 ? ε, γk−1, φi0 − hγj≤k−3, γk−2 ? γk−1, ε, φi0 . Note that we have replaced γk−2, γk−1, γk in the original correlator by γk−2, ε, φ in the first and third terms and by γk−1, φ, γk−2 ? ε in the second term. So the first and third terms now have a primitive class ε where there was originally γk−1. The second term has replaced γk by φ, which has lower degree. We repeat the above argument on the second term hγj≤k−3, γk−2 ? ε, γk−1, φi0 to show that the original correlator hγ1, . . . , γk−3, γk−2, γk−1, γki0 can be rewritten in terms of correlators that are either shorter (k 0 < k) or that have replaced one of the three classes γk−2,γk−1, or γk by a primitive class. Now move the primitive class into the set γi≤k−3. Pick another nonprimitive class and continue the induction. In this way, we can replace all the insertions by primitive classes except the last two. Definition 6.2.7. We call a correlator a basic correlator if it is of the form described in the previous lemma; that is, if all insertions are primitive but the last two. For a basic correlator, we still have the dimension formula (93) X i degC(ai) = ˆc + k − 3. This gives the following lemmas. Lemma 6.2.8. If degC(a) ≤ cˆ for all classes a and if P is the maximum complex degree of any primitive class, then all the genus-zero correlators are uniquely determined by the pairing and k-point correlators with (94) k ≤ 2 + 1 + ˆc 1 − P . Proof. Let ha1, . . . , ak−2, ak−1, aki0 be a basic correlator, so ai≤k−2’s are primitive. Then, degC(ai) ≤ P for i ≤ k − 2 and degC(ak−1), degC(ak) ≤ cˆ. By the dimension formula, we have cˆ+ k − 3 ≤ (k − 2)P + 2ˆc. Lemma 6.2.9. All the genus-zero correlators for the An, Dn+1, E6, E7, E8, and DT n+1 singularities, in either the A-model or the B-model, are uniquely determined by the pairing, the three-point correlators, and the four-point correlators. 78 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN Proof. Since the pairing, the three-point correlators and the selection rules in the A-model and the B-model have been shown to be mirror, it suffices to prove the conclusion in the A-model side. Let P be the maximum complex degree of any primitive class. It is easy to obtain the data for these singularities: An : P = 1 n+1 , cˆ = n−1 n+1 , E6 : P = 1 3 , cˆ = 5 6 , E7 : P = 1 3 , cˆ = 8 9 , E8 : P = 1 3 , cˆ = 14 15 , Dn+1(n even) : P = 1 n , cˆ = n−1 n , Dn+1(n odd) : P = n−1 2n , cˆ = n−1 n , DT n+1 : P = n−1 2n , cˆ = n−1 n . By formula (94), we know that (1) k ≤ 4 for An, E6, E7, E8, and Dn+1(n even) singularities, (2) k ≤ 5 for Dn+1(n odd) and DT n+1 singularities. For the singularities Dn+1(n odd) and DT n+1, we need a more refined estimate. For the singularity Dn+1 (n odd), we have the isomorphism (HDn+1,hJi , ?) ∼= QDn+1 . Here QDn+1 is generated by {1, X, . . . , Xn−1 , Y } and satisfies the relations nXn−1 + Y 2 ≡ 0 and XY ≡ 0. X and Y are the only primitive forms, and they have complex degrees as follows: degC X = 1 n , degC Y = n − 1 2n . The basic genus-zero, five-point correlators may have the form hX, Y, Y, α, βi0. By the dimension formula (93) for k = 5, we have degC α + degC β = ˆc + 2 − n − 1 n − 1 n = 2n − 1 n > 2n − 2 n = 2ˆc. This is impossible, since for any element a, we have degC(a) ≤ cˆ. Similarly we can rule out the existence of the basic five-point functions of the form hX, X, X, α, βi0 and hX, X, Y, α, βi0. Therefore the only possible basic fivepoint functions have the form hY, Y, Y, α, βi0. In this case, we have the degree formula degC α + degC β = 3n + 1 2n . Because of the fact that X ? Y ≡ 0, and for dimension reasons, α or β cannot contain Y . So the only possible form of the basic five-point correlators are hY, Y, Y, Xi , X 3n+1 2 −i i0, i > 0. QUANTUM SINGULARITY THEORY 79 Using formula (91) with α = Y, β = Xi , ε = X, and φ = X 3n−1 2 −i , we have hY, Y, Y, Xi , X 3n+1 2 −i i0 = S + hY, Y, Y, X, X 3n−1 2 i0 + hY, Y, X ? Y, Xi , X 3n−1 2 −i i0 − hY, Y, Y ? Xi , X 3n−1 2 −i , Xi0 = S. This shows that any basic, genus-zero, five-point correlators can be uniquely determined by two-, three-, and four-point correlators. For the DT n+1 singularity, we have the isomorphism (HDT n+1 , ?) ∼= QDn+1 = C[X, Y ]/hnXn−1 + Y 2 , XY i and the degrees for the primitive classes X and Y degC X = 1 n , degC Y = n − 1 2n . Hence the reduction from basic five-point correlators to the fewer-point correlators is exactly the same as for the singularity Dn+1 with n odd. The Reconstruction Lemma yields more detailed information for the basic correlators as well. Theorem 6.2.10. (1) All genus-zero correlators in the An−1 case for both our (A-model) and the Saito (B-model) theory are uniquely determined by the pairing, the three-point correlators, and a single four-point correlator of the form hX, X, Xn−2 , Xn−2 i0 , where X denotes the primitive class that is the image of x via the Frobenius algebra isomorphism from QAn = C[x]/(x n−1 ). (2) All genus-zero correlators in the Dn+1 case of our (A-model) theory with maximal symmetry group and in the DT n+1 case of the Saito (B-model) are uniquely determined by the pairing, the three-point correlators, and a single four-point correlator of the form hX, X, X2n−2 , X2n−2 i0 . Again, X denotes the primitive class that is the image of x via the Frobenius algebra isomorphism from QDn+1 = C[x, y]/(nxn−1 + y 2 , xy). (3) All genus-zero correlators in the DT n+1 case of our (A-model) theory, in the Dn+1 case of our theory with n odd and symmetry group hJi, and in the Dn+1 case of the Saito (B-model) theory, are uniquely determined by the pairing, the three-point correlators, and four-point correlators of the form hX, X, Xn−1 , Xn−2 i0 and hX, X, Y, X2 i0 . The second of these occurs only in the case that n = 3. Here X and Y denote the primitive classes that are the 80 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN images of x and y, respectively, via the Frobenius algebra isomorphism from QDT n+1 = C[x, y]/(x n−1y, xn + 2y). (4) In the E6 case of our theory (A-model) with maximal symmetry group and in the E6 case of the Saito (B-model) theory, all genus-zero correlators are uniquely determined by the pairing, the three-point correlators, and the correlators hY, Y, Y 2 , XY 2 i0 and hX, X, XY, XY i0 . Here X and Y denote the primitive classes that are the images of x and y, respectively, via the Frobenius algebra isomorphism from QE6 = C[x, y]/(x 2 , y3 ). (5) In the E7-case of our theory (A-model) with maximal symmetry group and in the E7 case of the Saito (B-model) theory, all genus-zero correlators are uniquely determined by the pairing, the three-point correlators, and the correlators hX, X, X2 , XY i0 , hX, Y, X2 , X2 i0 , and hY, Y, XY, X2Y i0 . Here X and Y denote the primitive classes that are the images of x and y, respectively, via the Frobenius algebra isomorphism from QE7 = C[x, y]/(3x 2 + y 3 , xy2 ). (6) In the E8-case of our theory (A-model) with maximal symmetry group and in the E8 Saito (B-model) theory, all genus-zero correlators are uniquely determined by the pairing, the three-point correlators, and by the correlators hY, Y, Y 3 , XY 3 i0 , and hX, X, X, XY 3 i0 . Here X and Y denote the primitive classes that are the images of x and y, respectively, via the Frobenius algebra isomorphism from QE8 = C[x, y]/(x 2 , y4 ). Proof. Applying Lemma 6.2.9, all genus zero correlators are uniquely determined by the pairing, three- or four-point correlators. Let us study the genus zero four-point correlators in more detail. In the An−1 case, X is the only ring generator and hence the only primitive class. It has degC X = 1/(n+ 1). A dimension count shows that the only fourpoint correlator of the form hX, X, α, βi0 is hX, X, Xn−2 , Xn−2 i0 . A similar argument shows that in the Dn+1 A-model with the maximal symmetry group and DT n+1 B-model cases the only basic four-point correlator is hX, X, X2n−2 , X2n−2 i0 . In the case of the DT n+1 A-model, and for the Dn+1 A-model for n odd with symmetry group J, and for the Dn+1 B-model, the central charges are the same, ˆc = n−1 n , and all have only two primitive classes X and Y with the same degrees degC X = 1 n , degC Y = n − 1 2n . Hence the basic four-point correlators are the same for the three cases. Let us consider the case Dn+1 A-model for n odd with symmetry group J. There are several cases for the form of the basic four-point correlators. Case A: form hX, X, α, βi0 . The dimension formula shows that degC α+degC β = 2n−3 n . So the only possibility is hX, X, Xn−1 , Xn−2 i0 . QUANTUM SINGULARITY THEORY 81 Case B: form hX, Y, α, βi0 . By the dimension formula, we have degC α + degC β = 3n−3 2n . There are two cases: Case B1: α, β do not contain Y . Then for j > 1, the correlator has the form hX, Y, Xi , Xj i0 . Setting α = Y, β = Xi , ε = Xj−1 , φ = X in formula (91), we have hX, Y, Xi , Xj i0 = S + hX, Y, Xj−1 , Xi+1i0 + hX, Y ? Xj−1 , Xi , Xi0 − hX, Y ? Xi , Xj−1 , Xi0 = S + hX, Y, Xj−1 , Xi+1i0 = · · · = S + hX, Y, X, Xi0 i0 . The dimension formula shows that the only four-point correlator hX, Y, X, Xi i0 does not vanish only if n = 3 and in this case i = 2. Case B2: α, β contain Y. In this case hX, Y, α, βi0 has the form hY, Y, X, βi0 which can be included in the following Case C. Case C: form hY, Y, α, βi0 . We have the degree formula degC α + degC β = 1. There are two cases: Case C1: α, β do not contain Y . We have the form hY, Y, Xi , Xj i0 with j > 1. Let α = Y, β = Xi , ε = Xj−1 , φ = X in formula (91); we obtain hY, Y, Xi , Xj i0 = S + hY, Y, Xj−1 , Xi+1i0 + hY, Y ? Xj−1 , Xi , Xi0 − hY, Y ? Xi , Xj−1 , Xi0 = S + hY, Y, Xj−1 , Xi+1i0 = · · · = S + hY, Y, X, Xn−1 i0 . Now hY, Y, X, Xn−1 i0 = hX, Y, Y, Xn−1 i0 = S + hX, Y, X, Y ? Xn−2 i0 + hX, X ? Y, Y, Xn−2 i0 − hX, Y 2 , Xn−2 , Xi0 = S. Case C2: α, β contain Y . The basic correlator has the form hY, Y, Y, X n+1 2 i0 . Similarly, we have hY, Y, Y, X n+1 2 i0 = S + hY, Y, X, Y ? X n−1 2 i0 + hY, Y ? X, Y, X n−1 2 i0 − hY, Y 2 , X n−1 2 , Xi0 = S. In summary, if n > 3, then the basic four-point correlator is only hX, X, Xn−1 , Xn−2 i0 ; if n = 3, then the basic four-point correlators are hX, X, X2 , Xi0 , hX, Y, X, X2 i0 . 82 HUIJUN FAN, TYLER JARVIS, and YONGBIN RUAN In the E6 case, the primitive classes are X, Y . The dimension condition shows that the only four-point correlators with two primitive insertions are hY, Y, Y 2 , XY 2 i0 , hX, X, X, XY 2 i0 , hX, X, XY, XY i0 . Applying equation (91) and the fact that X2 = 0, we can reduce hX,X,X,XY 2 i0 to hX, X, XY, XY i0 . In the E7 case, the primitive classes are X and Y with degC X = 1/3, degC Y = 2/9, and ˆc = 8/9. The dimension condition shows that the only basic four-point correlators are hX, X, X2 , XY i0 , hX, X, X, X2Y i0 , hX, Y, X2 , X2 i0 , hX, Y, Y 2 , X2Y i0 , hY, Y, XY, X2Y i0 . We can use equation (91) to further reduce hX, X, X, X2Y i0 to the remaining four and to reduce hX, Y, Y 2 , X2Y i0 = hY, X, Y 2 , X2Y i0 to hY, Y, XY, X2Y i0 . Finally, in the E8 case, a dimension count shows that the only basic fourpoint correlators are hX, X, X, XY 3 i0 , hX, X, XY, XY 2 i0 , hY, Y, Y 3 , XY 3 i0 . Again equation (91) shows that hX, X, XY, XY 2 i0 can be expressed in terms of hX, X, X, XY 3 i0 . 6.3. Computation of the basic four-point correlators in the Amodel. 6.3.1. Computing classes in complex codimension one. Definition 6.3.1. Let Γg,k,W denote the set of all connected single-edged W-graphs of genus g with k tails decorated by elements of HW . Further, denote by Γg,k,W,cut the set of all W-graphs with no edges (possibly disconnected), but with one pair of tails labeled + and −, respectively, such that gluing the tail + to the tail − gives an element of Γg,k,W . Furthermore, we require that the decorations γ+ and γ− satisfy γ+γ− = 1. Similarly, let Γg,k,W (γ1, . . . , γk) and Γg,k,W,cut(γ1, . . . , γk) denote the subset of Γg,k,W and of Γg,k,W , respectively, consisting of decorated W-graphs with the i-th tail decorated by γi for each i ∈ {1, . . . , k}. For any graph Γcut ∈ Γg,k,W,cut, we denote by Γ ∈ Γg,k,W the uniquely determined graph in Γg,k,W obtained by gluing the two tails + and −. We further denote the underlying undecorated graph by |Γ|, and we denote the closure in Mg,k of the locus of stable curves with dual graph |Γ| by M(|Γ|). Finally, denote the Poincar´e dual of this locus by î M(|Γ|) ó ∈ H∗ (Mg,k, C). QUANTUM SINGULARITY THEORY 83 Remark 6.3.2. In genus zero, the local group at an edge (or at the tails labelled + and −) is completely determined by the local group at each of the tails. Theorem 6.3.3. Assume the W-structure is concave (i.e., π∗
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