# Comparing tautological relations from the equivariant Gromov-Witten theory
of projective spaces and spin structures

Felix Janda

###### Abstract

Pandharipande-Pixton-Zvonkine’s proof of Pixton’s generalized Faber-Zagier
relations in the tautological ring of $\overline{M}_{g,n}$ has started the
study of tautological relations from semisimple cohomological field theories.
In this article we compare the relations obtained in the examples of the
equivariant Gromov-Witten theory of projective spaces and of spin structures.
We prove an equivalence between the $\mathbb{P}^{1}$\- and $3$-spin relations,
and more generally between restricted $\mathbb{P}^{m}$-relations and similarly
restricted $(m+2)$-spin relations. We also show that the general
$\mathbb{P}^{m}$-relations imply the $(m+2)$-spin relations.

##  1 Introduction

The study of the Chow ring of the moduli space of curves was initiated Mumford
in [11]. Because it is difficult to understand the whole Chow ring in general,
the tautological subrings of classes reflecting the geometry of the objects
parametrized by the moduli space were introduced. The tautological ring
$R^{*}(\overline{M}_{g,n})$ is compactly described [2] as the smallest system

$R^{*}(\overline{M}_{g,n})\subseteq A^{*}(\overline{M}_{g,n})$

of subrings compatible with push-forward under the tautological maps, i.e. the
maps obtained from forgetting marked points or gluing curves along common
markings.

There is a canonical set of generators parametrized by decorated graphs [5].
The formal vector space $\mathcal{S}_{g,n}$ generated by them, the _strata
algebra_ , therefore admits a surjective map to $R^{*}(\overline{M}_{g,n})$
and the structure of the tautological ring is determined by the kernel of this
surjection. Elements of the kernel are called _tautological relations_.

In [15] A. Pixton proposed a set of (at the time conjectural) relations
generalizing the relations of Faber-Zagier in $R^{*}(M_{g})$. Furthermore, he
conjectured that these give all tautological relations. The first proof [13]
of the fact that the conjectural relations are actual relations (in
cohomology) brought cohomological field theories (CohFTs) into the picture.

A CohFT on a free module $V$ of finite rank over a base ring $A$ is a system
of classes $\Omega_{g,n}$ behaving nicely under pull-back via the tautological
maps. A CohFT can also be used to give $V$ the structure of a Frobenius
algebra. The CohFT is called semisimple if, after possible base extension, the
algebra $V$ has a basis of orthogonal idempotents.

For semisimple CohFTs there is a conjecture by Givental [3] proven in some
cases by himself and in full generality in cohomology by Teleman [16], giving
a reconstruction of the CohFT from its genus $0$, codimension $0$ part and the
data of a power series $R(z)$ of endomorphisms of $V$. The formula naturally
lifts to the strata algebra.

To get relations from a semisimple cohomological field theory we can use that
the reconstructed CohFT of elements in the strata algebra is in general only
defined over an extension $B\leftarrow A$. However since we have started out
with a CohFT over $A$, this implies that certain linear combinations of
elements in the strata algebra have to vanish under the projection to the
tautological ring.

This procedure was essentially used in the proof [13] in the special example
of the CohFT defined from Witten’s $3$-spin class. There the base ring is a
polynomial ring in one variable but the reconstructed CohFT seems to have
poles in this variable.

In [14] (in preparation) the authors construct tautological relations using
Witten’s $r$-spin class for any $r\geq 3$. Given a list of integers
$a_{1},\dots,a_{n}\in\\{0,\dots,r-2\\}$, Witten’s class
$W_{g,n}(a_{1},\dots,a_{n})$ is a cohomology class on $\overline{M}_{g,n}$ of
pure degree

$D_{g,n}(a_{1},\dots,a_{n})=\frac{(r-2)(g-1)+\sum_{i=1}^{n}a_{i}}{r}.$

Witten’s class can be “shifted” by any vector in the vector space $\langle
e_{0},\dots,\linebreak[1]e_{r-2}\rangle$ to obtain a semisimple CohFT. In
practice, the authors use two particular shifts for which the answer can be
explicitly computed. Shifted Witten’s class is of mixed degree: more
precisely, the degrees of its components go from 0 to
$D_{g,n}(a_{1},\dots,a_{n})$. On the other hand, the Givental-Teleman
classificiation of semisimple CohFTs gives an expression of the shifted Witten
class in terms of tautological classes. The authors conclude that the
components of this expression beyond degree $D_{g,n}(a_{1},\dots,a_{n})$ are
tautological relations.

This article studies how relations from spin structures are related to the
relations obtained from the CohFT defined from the equivariant Gromov-Witten
theory of projective spaces. The following two theorems are our main results.

######  Theorem 1 (rough version).

The relations obtained from the equivariant Gromov-Witten theory of
$\mathbb{P}^{m}$ imply the $(m+2)$-spin relations.

######  Theorem 2 (rough version).

A special restricted set of relations from equivariant $\mathbb{P}^{m}$ is
equivalent to a corresponding restricted set of $(m+2)$-spin relations. For
$\mathbb{P}^{1}$ and 3-spin no restriction is necessary.

Since for equivariant $\mathbb{P}^{m}$ the reconstruction holds in Chow,
Theorem 1 implies that the higher spin relations also hold in Chow.

We will give strong evidence that the method of proof for Theorem 2 cannot
directly be extended to an equivalence between the full $\mathbb{P}^{m}$\- and
$(m+2)$-spin relations for $m>2$. Possibly, there are more $\mathbb{P}^{m}$\-
than $(m+2)$-spin relations.

Any of the theorems gives another proof of the fact that Pixton’s relations
hold in Chow. In fact, the proof of Theorem 1 in the case $m=1$ is essentially
a simplified version of the author’s previous proof in [8].

This article does not give a comparison between relations from CohFTs of
different dimensions, nor does it consider all relations from equivariant
$\mathbb{P}^{m}$. On the other hand, if indeed Pixton’s relations are all
tautological relations, the $3$-spin relations have to imply the relations
from any other semisimple CohFT. Yet, for example it not clear how the
$4$-spin relations can be written in terms of $3$-spin relations.

The article is structured as follows. In Section 2 we give definitions of
CohFTs, discuss the $R$-matrix action on CohFTs and the reconstruction result.
We then in Section 2.5 turn to the two examples of equivariant
$\mathbb{P}^{m}$ and the CohFT from the $A_{m+1}$-singularity. In Section 2.6
we describe the general procedure of obtaining relations from semisimple
CohFTs and general methods of proving that the relations from one CohFT imply
the relations from another. We then state precise versions of Theorem 1 and 2.
Section 3 discusses explicit expression of the $R$-matrices in both theories
in terms of asymptotics of oscillating integrals. The constraints following
from these expressions will be used in the next sections. We also note a
connection to Airy functions. Section 4 and Section 5 give proofs of Theorem 1
and 2. Finally, Section 6 gives evidence why, with the methods used in the
proofs of the theorems, an equivalence between $\mathbb{P}^{m}$\- and
$(m+2)$-spin relations cannot be established. Since the reconstruction result
of Givental we use to get relations in Chow has never appeared explicitly in
the literature, we recall its proof in Appendix A.

### Acknowledgments

This work started out of discussions at the conference _Cohomology of the
moduli space of curves_ organized by the _Forschungsinstitut für Mathematik_
at ETH Zürich. The author is especially grateful for encouragement and
explanations from Y.P. Lee. The author thanks the _Mathematics Department of
the University of Utah_ for the hospitality during his visit in January 2014\.
The author is grateful for comments of A. Pixton and D. Zvonkine on earlier
versions of the paper, a copy of a draft of [14] from R. Pandharipande and for
helpful discussions with S. Keel.

The author was supported by the Swiss National Science Foundation grant SNF
200021_143274.

##  2 Cohomological field theories

###  2.1 Definitions

Cohomological field theories were first introduced by Kontsevich and Manin in
[10] to formalize the structure of classes from GW-theory. Let $A$ be an
integral, commutative $\mathbb{Q}$-algebra, $V$ a free $A$-module of finite
rank and $\eta$ a non-singular bilinear form on $V$.

######  Definition 1.

A cohomological field theory (CohFT) $\Omega$ on $(V,\eta)$ is a system

$\Omega_{g,n}\in A^{*}(\overline{M}_{g,n})\otimes_{\mathbb{Q}}(V^{*})^{\otimes
n}$

of multilinear forms with values in the Chow ring of $\overline{M}_{g,n}$
satisfying the following properties:

Symmetry

$\Omega_{g,n}$ is symmetric in its $n$ arguments

Gluing

The pull-back of $\Omega_{g,n}$ via the gluing map

$\overline{M}_{g_{1},n_{1}+1}\times\overline{M}_{g_{2},n_{2}+1}\to\overline{M}_{g,n}$

is given by the direct product of $\Omega_{g_{1},n_{2}+1}$ and
$\Omega_{g_{2},n_{2}+1}$ with the bivector $\eta^{-1}$ inserted at the two
gluing points. Similarly for the gluing map
$\overline{M}_{g-1,n+2}\to\overline{M}_{g,n}$ the pull-back of $\Omega_{g,n}$
is given by $\Omega_{g-1,n+2}$ with $\eta^{-1}$ inserted at the two gluing
points.

Unit

There is a special element $\mathbf{1}\in V$ called the _unit_ such that

$\Omega_{g,n+1}(v_{1},\dotsc,v_{n},\mathbf{1})$

is the pull-back of $\Omega_{g,n}(v_{1},\dotsc,v_{n})$ under the forgetful map
and

$\Omega_{0,3}(v,w,\mathbf{1})=\eta(v,w).$

######  Definition 2.

The quantum product $(u,v)\mapsto uv$ on $V$ with unit $\mathbf{1}$ is defined
by the condition

$\eta(uv,w)=\Omega_{0,3}(u,v,w).$ (1)

######  Definition 3.

A CohFT is called semisimple if there is a base extension $A\to B$ such that
the algebra $V\otimes_{A}B$ is semisimple.

###  2.2 First Examples

######  Example 1.

For each Frobenius algebra there is the _trivial CohFT_ (also called
_topological field theory_ or TQFT) $\Omega_{g,n}$ characterized by (1) and
that

$\Omega_{g,n}\in A^{0}(\overline{M}_{g,n})\otimes(V^{*})^{\otimes n}.$

Let us record an explicit formula for Appendix A: In the case that the
Frobenius algebra is semisimple, there is a basis $\epsilon_{i}$ of orthogonal
idempotents of $V$ and

$\tilde{\epsilon}_{i}=\frac{\epsilon_{i}}{\sqrt{\Delta_{i}}},$

where $\Delta_{i}^{-1}=\eta(\epsilon_{i},\epsilon_{i})$, is a corresponding
orthonormal basis of _normalized idempotents_. We have

$\Omega_{g,n}(\tilde{\epsilon}_{i_{1}},\dotsc,\tilde{\epsilon}_{i_{n}})=\begin{cases}\sum_{j}\Delta_{i_{j}}^{g-1},&\text{if
}n=0,\\\ \Delta_{i_{1}}^{\frac{2g-2+n}{2}},&\text{if }i_{1}=\dotsb=i_{n},\\\
0,&\text{else.}\end{cases}$

######  Example 2.

The Chern polynomial $c_{t}(\mathbb{E})$ of the Hodge bundle $\mathbb{E}$
gives a $1$-dimensional CohFT over $\mathbb{Q}[t]$.

######  Example 3.

Let $X$ be a smooth, projective variety such that the cycle class map gives an
isomorphism between Chow and cohomology rings. Let
$A=\mathbb{Q}[\\![q^{\beta}]\\!]$ be its Novikov ring. Then the Gromov-Witten
theory of $X$ defines a CohFT based on the $A$-module $A^{*}(X)\otimes A$ by
the definition

$\Omega_{g,n}(v_{1},\dotsc,v_{n})=\sum_{\beta}\pi_{*}\left(\prod_{i=1}^{n}\operatorname{ev}_{i}^{*}(v_{i})\cap[\overline{M}_{g,n}(X,\beta)]^{vir}\right)q^{\beta},$

where the sum ranges over effective, integral curve classes,
$\operatorname{ev}_{i}$ is the $i$-th evaluation map and $\pi$ is the
forgetful map $\pi:\overline{M}_{g,n}(X,\beta)\to\overline{M}_{g,n}$. The
gluing property follows from the splitting axiom of virtual fundamental
classes. The fundamental class of $X$ is the unit of the CohFT and the unit
axioms follow from the identity axiom in GW-theory.

For a torus action on $X$, this example can be enhanced to give a CohFT from
the equivariant GW-theory of $X$.

###  2.3 The $R$-matrix action

######  Definition 4.

The (upper part of the) symplectic loop group is defined as the subgroup of
the group of endomorphism valued power series $R=1+O(z)$ in $z$ satisfying the
symplectic condition

$\eta(R(z)v,R(-z)w)=\eta(v,w)$

for all vectors $v$ and $w$.

An action of this group on the space of CohFTs makes it interesting for us. In
its definition the endomorphism valued power series $R$ is evaluated at
cotangent line classes and applied to vectors.

Given a CohFT $\Omega_{g,n}$ the new CohFT $R\Omega_{g,n}$ takes the form of a
sum over dual graphs $\Gamma$

$R\Omega_{g,n}(v_{1},\dotsc,v_{n})=\sum_{\Gamma}\frac{1}{\operatorname{Aut}(\Gamma)}\xi_{*}\left(\prod_{v}\sum_{k=0}^{\infty}\frac{1}{k!}\varepsilon_{*}\Omega_{g_{v},n_{v}+k}(\dots)\right),$

where $\xi:\prod_{v}\overline{M}_{g_{v},n_{v}}\to\overline{M}_{g,n}$ is the
gluing map of curves of topological type $\Gamma$ from their irreducible
components,
$\varepsilon:\overline{M}_{g_{v},n_{v}+k}\to\overline{M}_{g_{v},n_{v}}$
forgets the last $k$ markings and we still need to specify what is put into
the arguments of $\prod_{v}\Omega_{g_{v},n_{v}+k_{v}}$.

  * •

Into each argument corresponding to a marking of the curve, put $R^{-1}(\psi)$
applied to the corresponding vector.

  * •

Into each pair of arguments corresponding to an edge put the bivector

$\frac{R^{-1}(\psi_{1})\eta^{-1}R^{-1}(\psi_{2})^{t}-\eta^{-1}}{-\psi_{1}-\psi_{2}}\in\operatorname{Hom}(V^{*},V)[\\![\psi_{1},\psi_{2}]\\!]\cong
V^{\otimes 2}[\\![\psi_{1},\psi_{2}]\\!],$

where one has to substitute the $\psi$-classes at each side of the
normalization of the node for $\psi_{1}$ and $\psi_{2}$. By the symplectic
condition this is well-defined.

  * •

Into each of the additional arguments for each vertex put

$T(\psi):=\psi(1-R^{-1}(\psi))\mathbf{1},$

where $\psi$ is the cotangent line class corresponding to that vertex. Since
$T(z)=O(z^{2})$ the above $k$-sum is finite.

######  Reconstruction Conjecture (Givental).

The $R$-matrix action is free and transitive on the space of semisimple CohFTs
based on a given Frobenius algebra.

######  Theorem 3 (Givental[3]).

Reconstruction for the equivariant GW-theory of toric targets holds in Chow.

######  Theorem 4 (Teleman[16]).

Reconstruction holds in cohomology.

######  Remark 1.

Givental’s original conjecture was only stated in terms of the descendent
integrals of the CohFT and there is no explicit proof of Theorem 3 in the
literature. Therefore in Appendix A we recall the well-known lift of
Givental’s proof to CohFTs.

######  Example 4.

By Mumford’s Grothendieck-Riemann-Roch calculation [11] the single entry of
the $R$-matrix taking the trivial one-dimensional CohFT to the CohFT from
Example 2 is given by

$\exp\left(\sum_{i=1}^{\infty}\frac{B_{2i}}{2i(2i-1)}(tz)^{2i-1}\right),$

where $B_{2i}$ are the Bernoulli numbers, defined by

$\sum_{i=0}^{\infty}B_{i}\frac{x^{i}}{i!}=\frac{x}{e^{x}-1}.$

More generally, if we consider a more general CohFT given by a product of
Chern polynomials (in different variables) of the Hodge bundle, the $R$-matrix
from the trivial CohFT is the product of the $R$-matrices of the factors.

###  2.4 Frobenius manifolds and the quantum differential equation

There is a natural way to deform a CohFT $\Omega_{g,n}$ on $V$ over $A$ to a
CohFT over $A[\\![V]\\!]$. For a basis $\\{e_{\mu}\\}$ of $V$ let

$p=\sum t^{\mu}e_{\mu}$

be a formal point on $V$. Then the deformed CohFT is given by

$\Omega_{g,n}^{p}(v_{1},\dotsc,v_{n})=\sum_{k=0}^{\infty}\frac{1}{k!}\pi_{*}\Omega_{g,n+k}(v_{1},\dotsc,v_{n},p,\dotsc,p).$

Notice that the deformation is constant in the direction of the unit.

The quantum product on the deformed CohFT gives $V$ the structure of a
(formal) _Frobenius manifold_ [1]. The $e_{\mu}$ induce flat vector fields on
$V$ corresponding to the _flat coordinates_ $t^{\mu}$. Greek indices will
stand for flat coordinates with an exception stated in Section 2.5.

A Frobenius manifold is called _conformal_ if it admits an _Euler vector
field_ , i.e. a vector field $E$ of the form

$E=\sum_{\mu}(\alpha_{\mu}t^{\mu}+\beta_{\mu})\frac{\partial}{\partial
t^{\mu}},$

such that the quantum product, the unit and the metric are eigenfunctions of
the Lie derivative $L_{E}$ with eigenvalues $1$, $-1$ and $2-\delta$
respectively. Here $\delta$ is a rational number called conformal dimension.
Assuming that $A$ itself is the ring of (formal) functions of a variety $X$ we
say that the Frobenius manifold is _quasi-conformal_ if there is vector field
$E$ on $X\times V$ satisfying the axioms of an Euler vector field.

A CohFT $\Omega_{g,n}$ is called _homogeneous_ (_quasi-homogeneous_) if its
Frobenius manifold is conformal (quasi-conformal) and the extended CohFT is an
eigenvector of of $L_{E}$ of eigenvalue $(g-1)\delta+n$. As the name suggests
CohFTs are homogeneous if they carry a grading such that all natural
structures are homogeneous with respect to the grading.

We say that the Frobenius manifold $V$ is _semisimple_ if there is a basis of
idempotent vector fields $\epsilon_{i}$ defined after possible base extension
of $A$. The idempotents can be formally integrated to _canonical coordinates_
$u_{i}$. We will use roman indices for them. Let $\mathbf{u}$ be the diagonal
matrix with entries $u_{i}$ and $\Psi$ be the transition matrix from the basis
of normalized idempotents corresponding to the $u_{i}$ to the flat basis
$e_{i}$.

The $R$-matrix from the trivial theory to $\Omega^{p}$ satisfies a
differential equation which is related to the _quantum differential equation_

$z\frac{\partial}{\partial t^{\alpha}}S_{j}=e_{\alpha}\star S_{j}$

for vectors $S_{j}$. We assemble the $S_{j}$ into a matrix $S$.

######  Proposition 1 (see [4]).

If $V$ is semisimple and after a choice of canonical coordinates $u_{i}$ has
been made, there exists a fundamental solution $S$ to the quantum differential
equation of the form

$S=\Psi Re^{\mathbf{u}/z},$ (2)

such that $R$ satisfies the symplectic condition $R(z)R^{t}(-z)=1$. The matrix
$R$ is unique up to right multiplication by a diagonal matrix of the form

$\exp(a_{1}z+a_{3}z^{3}+a_{5}z^{5}+\dotsb)$

for constant diagonal matrices $a_{i}$.

In the case that there exists an Euler vector field $E$, there is a unique
matrix $R$ defined from a fundamental solution $S$ by (2) which satisfies the
homogeneity

$z\frac{\mathrm{d}}{\mathrm{d}z}R+L_{E}R=0.$

Such an $R$ automatically satisfies the symplectic condition.

######  Remark 2.

The matrix $R$ should be thought as the matrix representation of an
endomorphism in the basis of normalized idempotents. The symplectic condition
in Proposition 1 is then the same as in Definition 4.

######  Remark 3.

The exponential in (2) has to be thought as a formal expression. All the
quantities in Proposition 1 are only defined after base change of $A$
necessary to define the canonical coordinates.

######  Remark 4.

The quantum differential equation is equivalent to the differential equation

$[R,\mathrm{d}\mathbf{u}]+z\Psi^{-1}\mathrm{d}(\Psi R)=0$ (3)

for $R$.

In the conformal case Teleman showed that the uniquely determined homogeneous
$R$-matrix of Proposition 1 is the one appearing in the reconstruction, taking
the trivial theory to the given one.

Equivariant projective spaces $\mathbb{P}^{m}$ only give a quasi-conformal
Frobenius manifold. However Givental showed, and we will recall the proof in
Appendix A, that in this case in the reconstruction one should take $R$ such
that in the classical limit $q\to 0$ it assumes the diagonal form

$R|_{q=0}=\exp(\operatorname{diag}(b_{0},\dotsc,b_{m})),$ (4)

where, using the notation from Section 2.5,

$b_{j}=\sum_{i=1}^{\infty}\frac{B_{2i}}{2i(2i-1)}\sum_{l\neq
j}\left(\frac{z}{\lambda_{l}-\lambda_{j}}\right)^{2i-1}.$

The $R$-matrix is uniquely determined by this additional property and the
homogeneity.

###  2.5 The two CohFTs

The cohomological field theory corresponding to the $A_{m+1}$-singularity
$f(X)=X^{m+2}/(m+2)$ is defined using Witten’s $(m+2)$-spin class on the
moduli of curves with $(m+2)$-spin structures. See [13] for a discussion of
different constructions of Witten’s class. In comparison to [13] we use a
different normalization for Witten’s class and a different basis for the free
module in order to have a more direct comparison to the
$\mathbb{P}^{m}$-theory.

The CohFT is based on the rank $(m+1)$ free module of versal deformations

$f_{t}(X)=\frac{X^{m+2}}{m+2}+t^{m}X^{m}+\dotsb+t^{1}X+t^{0}$

of $f$. In this article, using the deformation from Section 2.4, we will view
the CohFT as being based on

$k_{A_{m+1}}=\mathbb{Q}[t^{1},\dotsc,t^{m}],$

the space of regular functions on the Frobenius manifold where the
$t^{0}$-coordinate vanishes. Because of dimension constraints we do not need
to look at formal functions, and because the CohFT stays constant along the
$t^{0}$ direction we can restrict to the $(t^{0}=0)$-subspace.

The algebra structure is given by $k_{A_{m+1}}[X]/(f^{\prime}_{t})$, where
$X^{\mu}$ corresponds to $\frac{\partial}{\partial t^{\mu}}$. The metric is
given by the residue pairing

$\eta(a,b)=\frac{1}{2\pi\imath}\oint\frac{ab}{f^{\prime}_{t}(X)}\mathrm{d}X.$

Written as a matrix in the basis $1,\dotsc,X^{m}$, the metric $\eta$ has
therefore zeros above the antidiagonal, ones at the antidiagonal and again
zeros in the first antidiagonal below it. Notice also that $\eta$ has no
dependence on $t^{1}$. Therefore, while the $t^{\mu}$ do not give a basis of
flat vector fields on the Frobenius manifold, there is a triangular matrix
independent of $t^{1}$, sending the $1,\dotsc,X^{m}$ to a basis of flat vector
fields such that $X$ is mapped to itself. With this we can pretend that the
$t^{\mu}$ were flat coordinates if we consider in the quantum differential
equation only differentiation by $t^{1}$.

For $(\mathbb{C}^{*})^{m+1}$-equivariant $\mathbb{P}^{m}$ the CohFT is based
on the equivariant Chow ring

$A_{(\mathbb{C}^{*})^{m+1}}^{*}(\mathbb{P}^{m})[\\![q]\\!]\cong
k_{\mathbb{P}^{m}}[H]/\prod_{i=0}^{m}(H-\lambda_{i}),$

of $\mathbb{P}^{m}$, an $(m+1)$-dimensional free module over

$k_{\mathbb{P}^{m}}=\mathbb{Q}[\lambda_{0},\dotsc,\lambda_{m}][\\![q]\\!],$

and depends on the Novikov variable $q$ and the torus parameters
$\lambda_{i}$. We will not consider the deformation from Section 2.4. The
algebra structure is given by the small quantum equivariant Chow ring

$QA_{(\mathbb{C}^{*})^{m+1}}^{*}(\mathbb{P}^{m})\cong
k_{\mathbb{P}^{m}}[H]/\left(\prod_{i=0}^{m}(H-\lambda_{i})-q\right)$

and the pairing is the Poincaré pairing

$\eta(a,b)=\frac{1}{2\pi\imath}\oint\frac{ab}{\prod_{i=0}^{m}(H-\lambda_{i})}\mathrm{d}H$

in the equivariant Chow ring.

To match up this data we set

$\displaystyle X=$ $\displaystyle H-\bar{\lambda},$ $\displaystyle
X^{m+1}+\sum_{\mu=0}^{m-1}(\mu+1)t^{\mu+1}X^{\mu}=$
$\displaystyle\prod_{i=0}^{m}(X+\bar{\lambda}-\lambda_{i})-q,$

where

$\bar{\lambda}=\sum_{i=0}^{m}\frac{\lambda_{i}}{m+1}.$

So in particular

$t^{1}=-q+\prod_{i=0}^{m}(\bar{\lambda}-\lambda_{i})=:-q-\lambda$

and we have described a ring map

$\Phi:k_{A_{m+1}}[\lambda]\to k_{\mathbb{P}^{m}},$

whose image are the polynomials, symmetric in the torus parameters and
vanishing if all torus parameters coincide. Therefore, after base extension,
the Frobenius algebras from the $A_{m+1}$-singularity and equivariant
$\mathbb{P}^{m}$ match completely up.

On the $\mathbb{P}^{m}$-side, let $Q_{i}$ be the power series solution to

$\prod_{i=0}^{m}(Y+\bar{\lambda}-\lambda_{i})=q$

with limit $\lambda_{i}-\bar{\lambda}$ as $q\to 0$. In particular, the $Q_{i}$
are solutions to

$Y^{m+1}+\sum_{\mu=0}^{m-1}(\mu+1)t^{\mu+1}Y^{\mu}.$

On the $A_{m+1}$-side, let the $Q_{i}$ be the solutions to this equation in
any order. On both sides we can then define

$\Delta_{i}=\prod_{j\neq
i}(Q_{i}-Q_{j})=(m+1)Q_{i}^{m}-\sum_{\mu=1}^{m-1}(\mu+1)\mu
t^{\mu+1}Q_{i}^{\mu-1}$

and the _discriminant_

$\operatorname{disc}=\prod_{i}\Delta_{i}\in k_{A_{m+1}}.$

The choice of the $Q_{i}$ gives a bijection between the idempotents

$\epsilon_{i}=\frac{\prod_{j\neq i}(X-Q_{j})}{\Delta_{i}}.$

We will also need to make a choice of square roots of the $\Delta_{i}$ to be
able to define the normalized idempotents

$\tilde{\epsilon}_{i}=\frac{\prod_{j\neq i}(X-Q_{j})}{\sqrt{\Delta_{i}}}.$

The $A_{m+1}$-theory is conformal with Euler vector field

$E=\sum_{i=1}^{m}\frac{m+2-i}{m+2}t^{\mu}\frac{\partial}{\partial t^{\mu}},$

while the equivariant $\mathbb{P}^{m}$-theory is semi-conformal with Euler
vector field

$E=(m+1)q\frac{\partial}{\partial
q}+\sum_{i=0}^{m}\lambda_{i}\frac{\partial}{\partial\lambda_{i}}.$

###  2.6 Relations from CohFTs

Let $\Omega$ be a semisimple CohFT defined on $V$ over $A$. Formal properties
of the reconstruction theorem will imply tautological relations. The main
point is that the $R$-matrix from the trivial theory written in flat
coordinates lives only in

$\operatorname{End}(V\otimes_{A}B)[\\![z]\\!],$

for some $\mathbb{Q}$-algebra extension $B$111In our examples
$B=A[\operatorname{disc}^{-1}]$. of $A$. Let $C$ be the $A$-module quotient
fitting into the exact sequence

$0\to A\to B\xrightarrow{p}C\to 0.$ (5)

The reconstruction gives elements

$\overline{\Omega}_{g,n}\in\mathcal{S}_{g,n}\otimes(V^{*})^{\otimes n}\otimes
B.$

However since we have started out with a CohFT defined over $A$, we know that
the projection of

$p(\overline{\Omega}_{g,n})\in\mathcal{S}_{g,n}\otimes(V^{*})^{\otimes
n}\otimes C$

to $R^{*}(\overline{M}_{g,n})\otimes(V^{*})^{\otimes n}\otimes C$ has to
vanish222Assuming that reconstruction holds in this case.. Since $C$ is a
$\mathbb{Q}$-vector space, we obtain a system of vector spaces
$T_{g,n}^{\Omega}$ of relations. The complete system $\bar{T}_{g,n}^{\Omega}$
of _tautological relations obtained from the CohFT $\Omega$_ is the vector
space generated by

$\xi_{*}(\pi_{*}(T_{g_{1},n_{1}+m}^{\Omega}P)\times\mathcal{S}_{g_{2},n_{2}}\times\dotsb\times\mathcal{S}_{g_{k},n_{k}}),$

where $P$ is the vector space of polynomials in $\psi$-classes, and $\xi_{*}$
and $\pi_{*}$ are the formal analogues of the push-forwards along gluing and
forgetful maps.

We say that a vector space of tautological relations $T_{g,n}$ implies another
$T^{\prime}_{g,n}$ if the vector space, obtained from $T_{g,n}$ by the
completion process as described right above, is contained in
$T^{\prime}_{g,n}$. Using this definition we can also define an equivalence
relation between vector spaces of tautological relations.

Let us describe two relation preserving actions on the space of all CohFTs on
$V$ over $A$. The first is an action of the multiplicative monoid of $A$. The
action of $\varphi\in A$ is given by multiplication by $\varphi^{d}$ in
codimension $d$. This replaces the $R$-matrix $R(z)$ of the theory by
$R(\varphi z)$. Since multiplication by $\varphi$ is well-defined in $C$,
relations are preserved. The second action is the action of an $R$-matrix
defined over $A$.

The second action automatically proves equivalence of relations since
$R$-matrices are always invertible. Similarly, the first action proves
equivalence if $\varphi$ is invertible.

_Extending scalars_ also preserves relations. By this we mean tensoring
$\Omega$ with $A\to A^{\prime}$ under the condition that this preserves the
exactness of the sequence (5). We call the special case when $A^{\prime}=A/I$
for some ideal $I$ of $A$ a _limit_. If $C\to C\otimes_{A}A^{\prime}$ is
injective, extending scalars proves an equivalence of relations.

Let us again state our now well-defined results.

######  Theorem 1.

The relations from the equivariant Gromov-Witten theory of $\mathbb{P}^{m}$
imply the $(m+2)$-spin relations, both CohFTs as defined in Section 2.5.

The main statement necessary to be proven here is that the $R$-matrix for
$\mathbb{P}^{m}$ after replacing $z\mapsto z\lambda^{-1}$ admits the limit
$\lambda^{-1}\to 0$ and that this limit is the $R$-matrix for the
$A_{m+1}$-theory. In order for this to make sense, one uses the matchup from
Section 2.5 and views both as being defined over

$\mathbb{Q}[\\![\lambda_{0},\dotsc,\lambda_{m},q]\\!][\lambda^{-1}]$

In Section 3 we will see that for both original theories to define the
$R$-matrix it is enough to localize by $\operatorname{disc}$. So the extension
of scalars does not lose relations.

Motivated from Section 3.1 let us call the limit $t^{2},\dotsc,t^{m}=0$ the
_Airy limit_. For $\mathbb{P}^{m}$ the Airy limit concretely means, assuming
the sum of all torus weights is zero, that we restrict ourselves to the case
that up to a factor the torus weights are the $(m+1)$-th roots of unity.

######  Theorem 2.

In the Airy limit the $\mathbb{P}^{m}$\- and $(m+2)$-spin relations are
equivalent.

The main point for the proof is to show there is a series

$\varphi\in\lambda\mathbb{Q}[\\![t^{1}\lambda^{-1}]\\!],$

and an $R$-matrix $R$ without poles in $\operatorname{disc}$ such that the
Airy limit $\mathbb{P}^{m}$-$R$-matrix is obtained from the Airy limit
$A_{m+1}$-$R$-matrix by applying the transformation $z\mapsto z\varphi$,
followed by the action of $R$. We will show in the proof that there is only
one possible choice for $\varphi$. For Theorem 2 both theories can be viewed
as living over

$\mathbb{Q}[\\![\lambda_{0},\dotsc,\lambda_{m},q,t^{1}\lambda^{-1}]\\!][\lambda^{-1}]/(t^{1}+q+\lambda,t^{2},\dotsc,t^{m}).$

In Section 6 we will give evidence that the method of proof of Theorem 2 does
not work outside the Airy limit. What we will show is that assuming a
procedure as in the proof of Theorem 2 exists and is well-defined in the Airy
limit, the information that $\varphi$ was unique in the limit implies that the
$R$-matrix in the $R$-matrix action cannot be defined over the base ring.

#### Relations from degree vanishing

The more classical way of [13] and [14] to obtain tautological relations works
by considering cohomological degrees: Assume that $\Omega$ is in addition
quasi-homogenous for an Euler vector field $E$ and that all $\beta_{i}$ vanish
and all $\alpha_{i}$ are positive. Then the quasi-homogeneity implies that the
cohomological degree of $\Omega_{g,n}(\frac{\partial}{\partial
t^{i_{1}}},\dotsc,\frac{\partial}{\partial t^{i_{n}}})$ is bounded by

$(g-1)\delta+n-\sum_{j}\alpha_{i_{j}}.$

However the reconstructed theory might also contain terms of higher
cohomological degree. These thus have to vanish, giving tautological
relations.

Notice that these relations coming from degree considerations are implied from
the relations we have described previously: With respect to the grading on $B$
induced by the Euler vector field, no element of $A$ has negative degree.
Therefore the negative degree parts of $B$ and $C$ are isomorphic. Thus, the
homogeneity of the CohFT implies that the degree vanishing relations are
obtained from the previous relations by restricting to the negative degree
part of $C$.

The way of obtaining tautological relations by looking at poles in the
discriminant has already previously been studied by D. Zvonkine.

##  3 Oscillating integrals

###  3.1 For the $A_{m+1}$-singularity

We want to describe the $A_{m+1}$-$R$-matrix in terms of asymptotics of
oscillating integrals. For the purposes of this article the integrals can be
treated as purely formal objects.

The quantum differential equation with one index lowered says that

$\displaystyle z\frac{\partial}{\partial t^{1}}S_{\mu k}=$ $\displaystyle
S_{(\mu+1)k},$ $\displaystyle\text{for }\mu<m,$ $\displaystyle
z\frac{\partial}{\partial t^{1}}S_{mk}=$
$\displaystyle-\sum_{\mu=0}^{m-1}(\mu+1)t^{\mu+1}S_{\mu k},$

where the Greek indices stand for components in the basis of the $X^{\mu}$. It
is not difficult to see that the oscillating integrals

$\frac{1}{\sqrt{-2\pi z}}\int_{\Gamma_{k}}x^{\mu}\exp(f_{t}(x)/z)\mathrm{d}x,$

where $f_{t}$ as before is the deformed singularity, for varying cycles
$\Gamma_{k}$, if convergent, provide solutions to this system of differential
equations, and also satisfy homogeneity with respect to the Euler vector
field.

For generic choices of parameters, to each critical point $Q_{k}$ there
corresponds a cycle $\Gamma_{k}$ constructed via the Morse theory of
$\Re(f_{t}(x)/z)$, which moves through that critical point in the direction of
steepest descent and avoids all other critical points. By moving to the
critical point and scaling coordinates we obtain

$S_{\mu
k}=\frac{e^{u_{k}/z}}{\sqrt{2\pi\Delta_{k}}}\int\left(\frac{x(-z)^{1/2}}{\sqrt{\Delta_{k}}}+Q_{k}\right)^{\mu}\\\
\exp\left(-\sum_{l=2}^{m+2}\frac{x^{l}(-z)^{(l-2)/2}}{l!}\frac{f_{t}^{(l)}(Q_{k})}{\Delta_{k}^{l/2}}\right)\mathrm{d}x,$

where $u_{k}=f_{t}(Q_{k})$. By the method of steepest descend, we obtain the
asymptotics as $z\to 0$ by expanding the integrand as a formal power series in
$z$ and integrating from $-\infty$ to $\infty$. Since the $(l=2)$-term in the
sum is $-x^{2}/2$, we can use the formula for the moments of the Gaussian
distribution to write the asymptotics of $\sqrt{\Delta_{k}}e^{-u_{k}/z}S_{\mu
k}$ as a formal power series in $z$ with values in
$k_{A_{m+1}}[Q_{k},\Delta_{k}^{-1}]$.

The entries of the $R$-matrix are then given by

$R_{ik}\asymp\frac{1}{\sqrt{\Delta_{i}}}e^{-u_{k}/z}\prod_{j\neq
i}\left(\frac{\partial}{\partial t^{1}}-Q_{j}\right)S_{0k}.$

Noticing that the change of basis from normalized idempotents to the basis
$1,X,\dotsc,X^{m}$ can be defined over $k_{A_{m+1}}[Q_{k},\Delta_{k}^{-1/2}]$,
recalling that $\operatorname{disc}=\prod\Delta_{i}$ and applying Galois
theory, we see that the endomorphism $R$ is defined over
$k_{A_{m+1}}[\operatorname{disc}^{-1}]$.

In the Airy limit $t^{2},\dotsc,t^{m}\to 0$ the quantum differential equation
becomes the slightly modified higher Airy differential equation [9]

$\left(z\frac{\partial}{\partial t^{1}}\right)^{m+1}S_{0k}=-t^{1}S_{0k}.$

The entries of the $R$-matrix in this case are therefore related to the
asymptotic expansions of the higher Airy functions and their derivatives when
their complex argument approaches $\infty$.

In the case of the $A_{2}$-singularity we do not need to take any limit and
discover the hypergeometric series $A$ and $B$ of Faber-Zagier in the
expansions of the (slightly modified) usual Airy function

$\frac{e^{\frac{2}{3}(t^{1})^{3/2}/z}}{\sqrt{-2\pi
z}}\int_{\Gamma_{k}}e^{\left(\frac{x^{3}}{3}+t^{1}x\right)/z}\mathrm{d}x\asymp\frac{1}{\sqrt{2\pi\Delta}}\int\limits_{-\infty}^{\infty}\exp\left(-\frac{x^{2}}{2}-\frac{x^{3}}{3}\frac{\sqrt{-z}}{\Delta^{3/2}}\right)\mathrm{d}x\\\
\asymp\frac{1}{\sqrt{\Delta}}\sum_{i=0}^{\infty}\frac{(6i-1)!!}{(2i)!}\left(\frac{-z}{9\Delta^{3}}\right)^{i}=\frac{1}{\sqrt{\Delta}}\sum_{i=0}^{\infty}\frac{(6i)!}{(3i)!(2i)!}\left(\frac{-z}{72\Delta^{3}}\right)^{i}$

and a derivative of it

$\frac{e^{\frac{2}{3}(t^{1})^{3/2}/z}}{\sqrt{-2\pi
z}}\int_{\Gamma_{k}}xe^{\left(\frac{x^{3}}{3}+t^{1}x\right)/z}\mathrm{d}x\asymp\frac{\sqrt{-t^{1}}}{\sqrt{\Delta}}\sum_{i=0}^{\infty}\frac{(6i)!}{(3i)!(2i)!}\frac{1+6i}{1-6i}\left(\frac{-z}{72\Delta^{3}}\right)^{i}.$

Here $\Delta=2\sqrt{-t^{1}}$. The cycle $\Gamma_{k}$ determines which square-
root of $(-t^{1})$ we take.

###  3.2 For equivariant $\mathbb{P}^{m}$

Givental [3] has given explicit solutions to the quantum differential equation
for projective spaces in the form of complex oscillating integrals. Let us
recall their definition and see how they behave in the match up with the
$(m+2)$-spin theory.

Using the divisor axiom of Gromov-Witten invariants, the quantum differential
equation implies the differential equations

$(D+\lambda_{i})S_{i}=H\star S_{i}.$

for the fundamental solutions $S_{i}$ at the origin. Here we have written
$D=zq\frac{\partial}{\partial q}$. Equivalently, the equation says

$D(S_{i}e^{\ln(q)\lambda_{i}/z})=H\star S_{i}e^{\ln(q)\lambda_{i}/z}.$

Therefore the entries of $S$ with one index lowered satisfy

$(D-\bar{\lambda})(S_{\mu
i}e^{\ln(q)\lambda_{i}/z})=S_{(\mu+1)i}e^{\ln(q)\lambda_{i}/z},$

where $S_{(m+1)i}$ is defined such that

$\prod_{j=0}^{m}(D-\lambda_{j})(S_{0i}e^{\ln(q)\lambda_{i}/z})=qS_{0i}e^{\ln(q)\lambda_{i}/z}.$

The Greek indices stand for the basis of flat vector fields corresponding to
$1,H-\bar{\lambda},\dotsc,(H-\bar{\lambda})^{m}$.

Givental’s oscillating integral solutions for $S_{0i}$ are stationary phase
expansions of the integrals

$S_{0i}e^{\ln(q)\lambda_{i}/z}=(-2\pi
z)^{-m/2}\int\limits_{\Gamma_{i}\subset\\{\sum T_{j}=\ln
q\\}}e^{F_{i}(T)/z}\omega$

along $m$-cycles $\Gamma_{i}$ through a specific critical point of $F_{i}(T)$
inside a $m$-dimensional $\mathbb{C}$-subspace of $\mathbb{C}^{m+1}$, where

$F_{i}(T)=\sum_{j=0}^{m}(e^{T_{j}}+\lambda_{j}T_{j}).$

The form $\omega$ is the restriction of
$\mathrm{d}T_{0}\wedge\dotsb\wedge\mathrm{d}T_{m}$. To see that the integrals
are actual solutions, notice that applying $D-\lambda_{j}$ to the integral has
the same effect as multiplying the integrand by $e^{T_{j}}$.

There are $m+1$ possible critical points at which one can do a stationary
phase expansion of $S_{0i}$. Let us write $P_{i}=Q_{i}+\bar{\lambda}$ for the
solution to

$\prod_{i=0}^{m}(X-\lambda_{i})=q$

with limit $\lambda_{i}$ as $q\to 0$. For each $i$ we need to choose the
critical point $e^{T_{j}}=P_{i}-\lambda_{j}$ in order for the factor

$\exp(u_{i}/z):=\exp\left(\left(\sum_{j=0}^{m}(P_{i}-\lambda_{j}+\lambda_{j}\ln(P_{i}-\lambda_{j}))-\lambda_{i}\ln(q)\right)/z\right)$

of $S_{0i}$ to be well-defined in the limit $q\to 0$. Shifting the integral to
the critical point and scaling coordinates by $\sqrt{-z}$ we find

$S_{0i}=e^{u_{i}/z}\int\exp\left(-\sum_{j}(Q_{i}-\bar{\lambda}_{j})\sum_{k=3}^{\infty}\frac{T_{j}^{k}(-z)^{(k-2)/2}}{k!}\right)\mathrm{d}\mu_{i}$

for the conditional Gaussian distribution

$\mathrm{d}\mu_{i}=(2\pi)^{-m/2}\exp\left(-\sum_{j}(Q_{i}-\bar{\lambda}_{j})\frac{T_{j}^{2}}{2}\right)\omega.$

The covariance matrices are given by

$\sigma_{i}(T_{k},T_{l})=\frac{1}{\Delta_{i}}\begin{cases}-\prod_{j\notin\\{k,l\\}}(Q_{i}-\bar{\lambda}_{j}),&\text{for
}k\neq l,\\\ \sum_{m\neq
k}\prod_{j\notin\\{k,m\\}}(Q_{i}-\bar{\lambda}_{j}),&\text{for
}k=l.\end{cases}$

From here we can see that the integral is symmetric in the $\bar{\lambda}_{j}$
and therefore we can write its asymptotics as $z\to 0$ completely in terms of
data from $A_{m+1}$. Since odd moments of Gaussian distributions vanish we
find that $e^{-u_{i}/z}S_{0i}$ is a power series in $z$ with values in
$\Delta_{i}^{-1/2}k_{A_{m+1}}[Q_{i},\Delta_{i}^{-1},\lambda]$.

So the entries of the $R$-matrix in the basis of normalized idempotents are
given by

$\Delta_{k}^{-1/2}\prod_{j\neq k}(D+\lambda_{i}-P_{j})\ (e^{-u_{i}/z}S_{0i}).$

Since $\frac{\mathrm{d}P_{i}}{\mathrm{d}q}=\frac{1}{\Delta_{i}}$ these entries
are in

$k_{A_{m+1}}[Q_{0},\dotsc,Q_{m},\Delta_{0}^{-1/2},\dotsc,\Delta_{m}^{-1/2},\lambda].$

So, with the arguments from Section 3.1, the endomorphism $R$ can be defined
over $k_{\mathbb{P}^{m}}[\operatorname{disc}^{-1}]$.

We need to check that the $R$-matrix given in terms of oscillating integrals
behaves correctly in the limit $q\to 0$. By definition, in this limit
$P_{i}\to\lambda_{i}$. By symmetry it is enough to consider the $0$-th column.
Set $x_{i}=e^{T_{i}}$. Then

$\lim\limits_{q\to 0}R_{j0}\asymp\lim\limits_{q\to
0}e^{-u_{0}/z}\Delta_{0}^{-1/2}\prod_{k\neq
j}(zq\frac{\mathrm{d}}{\mathrm{d}q}+\lambda_{j}-\lambda_{k})\ S_{00}\\\
=\lim\limits_{q\to 0}\frac{e^{-u_{0}/z}}{\sqrt{\Delta_{0}}(-2\pi z)^{m/2}}\int
e^{(\sum_{k}(e^{T_{k}}-(\lambda_{0}-\lambda_{k})T_{k}))/z+\sum_{k\neq
j}T_{k}}\omega\\\ =\lim\limits_{q\to
0}\frac{e^{-u_{0}/z}}{\sqrt{\Delta_{0}}(-2\pi z)^{m/2}}\int
e^{(\sum\limits_{k\neq
0}(x_{k}-(\lambda_{0}-\lambda_{k})T_{k})+\frac{q}{\prod\limits_{k\neq
0}x_{k}})/z}\prod_{k\neq j}x_{j}\bigwedge_{k=1}^{m}\mathrm{d}T_{k}.$

In the last step we have moved to the chart

$x_{0}=\frac{q}{\prod_{j\neq 0}x_{j}}.$

Since in this chart $\lim_{q\to 0}x_{0}=0$, we have that $R_{j0}$ vanishes
unless $j=0$. On the other hand in the limit $q\to 0$ the integral for
$R_{00}$ splits into one-dimensional integrals

$\lim\limits_{q\to 0}R_{00}\asymp\lim\limits_{q\to
0}\frac{e^{-u_{0}/z}}{\sqrt{\Delta_{0}}(-2\pi z)^{m/2}}\prod_{k\neq
0}\int\limits_{0}^{\infty}e^{(x-(\lambda_{0}-\lambda_{k})\ln(x))/z}\mathrm{d}x.$

Let us temporarily set $z_{k}=-z/(\lambda_{0}-\lambda_{k})$. The prefactors
also split into pieces in the limit and we calculate the factor corresponding
to $k$ to be

$\frac{e^{(1-\ln(\lambda_{0}-\lambda_{k}))/z_{k}}}{\sqrt{-2\pi
z(\lambda_{0}-\lambda_{k})}}\int\limits_{0}^{\infty}e^{(x-(\lambda_{0}-\lambda_{k})\ln(x))/z}\mathrm{d}x=\frac{e^{(1-\ln(1/z_{k}))/z_{k}}}{\sqrt{2\pi/z_{k}}}\Gamma\left(1+\frac{1}{z_{k}}\right)\\\
=\frac{e^{(1-\ln(1/z_{k}))/z_{k}}}{\sqrt{2\pi
z_{k}}}\Gamma\left(\frac{1}{z_{k}}\right)\asymp\exp\left(\sum_{l=1}^{\infty}\frac{B_{2l}}{2l(2l-1)}\left(\frac{z}{\lambda_{k}-\lambda_{0}}\right)^{2l-1}\right),$

using Stirling’s approximation of the gamma function in the last step. So the
product of the factors gives the expected limit (4) of $R_{00}$ for $q\to 0$.
This calculation gives a proof for the results [7] of Ionel on the main
generating function used in [12] and [8] without having to use Harer’s
stability results.

##  4 $\mathbb{P}^{m}$ relations imply $(m+2)$-spin relations

We prove Theorem 1 in this section. As already mentioned, for this it is
enough to show that, after the change $z\mapsto z\lambda^{-1}$, the
$\mathbb{P}^{m}$-$R$-matrix converges to the $A_{m+1}$-$R$-matrix in the limit
$\lambda\to\infty$. For this we have to compare the differential equations
satisfied by the $R$-matrices.

Inserting the vector field corresponding to the hyperplane into (3) and using
the divisor equation as in Section 3.2 gives the equation

$[R_{\mathbb{P}^{m}},\xi]+zq\frac{\mathrm{d}R_{\mathbb{P}^{m}}}{\mathrm{d}q}+zq\Psi^{-1}\frac{\mathrm{d}\Psi}{\mathrm{d}q}R_{\mathbb{P}^{m}}=0,$
(6)

where $\xi$ denotes the diagonal matrix of quantum multiplication by
$H-\bar{\lambda}$.

######  Lemma 1.

$R_{\mathbb{P}^{m}}(z/\lambda)$ admits a limit $R$ for $\lambda\to\infty$. The
matrix $R$ satisfies

$\displaystyle[R,\xi]+z\frac{\mathrm{d}R}{\mathrm{d}t^{1}}+z\Psi^{-1}\frac{\mathrm{d}\Psi}{\mathrm{d}t^{1}}R=0$
$\displaystyle
z\frac{\mathrm{d}R}{\mathrm{d}z}+\sum_{\mu=1}^{m}\frac{m+2-\mu}{m+2}t^{\mu}\frac{\mathrm{d}R}{\mathrm{d}t^{\mu}}=0$

###### Proof.

The $\mathbb{P}^{m}$-$R$-matrix satisfies the homogeneity property

$z\frac{\mathrm{d}R_{\mathbb{P}^{m}}}{\mathrm{d}z}+(m+1)q\frac{\mathrm{d}R_{\mathbb{P}^{m}}}{\mathrm{d}q}+\sum_{i=0}^{m}\lambda_{i}\frac{\mathrm{d}R_{\mathbb{P}^{m}}}{\mathrm{d}\lambda_{i}}=0.$

So $R^{\prime}(z):=R_{\mathbb{P}^{m}}(z/\lambda)$ written with the
$A_{m+1}$-variables satisfies

$(m+2)z\frac{\mathrm{d}R^{\prime}}{\mathrm{d}z}+(m+1)\lambda\frac{\mathrm{d}R^{\prime}}{\mathrm{d}\lambda}+\sum_{\mu=1}^{m}(m+2-\mu)t^{\mu}\frac{\mathrm{d}R^{\prime}}{\mathrm{d}t^{\mu}}=0.$

The main differential equation satisfied by $R^{\prime}$ is

$[R^{\prime},\xi]+z\left(1+\frac{t^{1}}{\lambda}\right)\frac{\mathrm{d}R^{\prime}}{\mathrm{d}t^{1}}+z\left(1+\frac{t^{1}}{\lambda}\right)\Psi^{-1}\frac{\mathrm{d}\Psi}{\mathrm{d}t^{1}}R^{\prime}=0.$

From the expression of $R_{\mathbb{P}^{m}}$ in terms of oscillating integrals
we know that the entries of the $z^{i}$-part $R^{\prime}_{i}$ of $R^{\prime}$
live in

$\lambda^{-i}k_{A_{m+1}}[Q_{0},\dotsc,Q_{m},\Delta_{0}^{-1/2},\dotsc,\Delta_{m}^{-1/2},\lambda].$

To show that the limit exists we need to show that $\lambda$ occurs in no
positive power. We will show this by induction by $i$. It certainly holds for
$R^{\prime}_{0}=1$. Since $\xi$ is diagonal with pairwise distinct entries
$Q_{j}$, the $z^{i}$-part of the differential equation determines the off-
diagonal coefficients of $R^{\prime}_{i}$ in terms of $R^{\prime}_{i-1}$.
Because $\Psi^{-1}\frac{\mathrm{d}\Psi}{\mathrm{d}t^{1}}$ does not depend on
$\lambda$, the off-diagonal coefficients of $R^{\prime}_{i}$ will admit the
limit $\lambda\to\infty$. Since
$\Psi^{-1}\frac{\mathrm{d}\Psi}{\mathrm{d}t^{1}}$ in general vanishes on the
diagonal the diagonal coefficient of the $z^{i+1}$-part of the differential
equation determines the diagonal of
$\frac{\mathrm{d}R^{\prime}_{i}}{\mathrm{d}t^{1}}$ from the off-diagonal
entries of $R^{\prime}_{i}$. Apart from a possible term constant in $t^{1}$ we
therefore know that also the diagonal entries of $R^{\prime}_{i}$ admit the
limit.

Let us consider such a possible ambiguity $a_{i}$. Since all products of
$\Delta_{j}$ have dependence in $t^{1}$, the “denominator” of $a_{i}$ can only
be a power of $\lambda$ less than $i$. However then $a_{i}$ cannot possibly
satisfy the homogeneity. By induction therefore the limit $R$ exists. The
properties of $R$ easily follow from the corresponding ones of $R^{\prime}$. ∎

By inserting the vector field $\frac{\partial}{\partial t^{1}}$ into (3) and
similar arguments as in the proof of Lemma 1 one can show the following lemma.

######  Lemma 2.

The $A_{m+1}$-$R$-matrix is uniquely determined from the differential equation

$[R_{A_{m+1}},\xi]+z\frac{\mathrm{d}R_{A_{m+1}}}{\mathrm{d}t^{1}}+z\Psi^{-1}\frac{\mathrm{d}\Psi}{\mathrm{d}t^{1}}R_{A_{m+1}}=0,$

the homogeneity

$z\frac{\mathrm{d}R_{A_{m+1}}}{\mathrm{d}z}+\sum_{\mu=1}^{m}\frac{m+2-\mu}{m+2}t^{\mu}\frac{\mathrm{d}R_{A_{m+1}}}{\mathrm{d}t^{\mu}}=0$

and that the entries of the $z$-series coefficients of $R_{A_{m+1}}$ should
lie in

$k_{A_{m+1}}[Q_{0},\dotsc,Q_{m},\Delta_{0}^{-1/2},\dotsc,\Delta_{m}^{-1/2}].$

The lemmas imply that the modified $\mathbb{P}^{m}$-$R$-matrix contains only
non-positive powers of $\lambda$ and the part constant in $\lambda$ equals the
$A_{m+1}$-$R$-matrix. Therefore the $A_{m+1}$-relations are contained in the
modified $\mathbb{P}^{m}$-relations as the $\lambda^{0}$-part, and we have
completed the proof of Theorem 1.

##  5 Equivalence of relations

We want to give a proof of Theorem 2 in this section. So we will consider the
CohFTs in the Airy limit, i.e. with all $t^{\mu}$ but $t:=t^{1}$ set to zero.
In this limit the metric becomes $\eta(X^{i},X^{j})=\delta_{i+j,m}$, the
quantum product stays semisimple and the Euler vector field for the
$A_{m}$-singularity

$E=\frac{m+1}{m+2}t\frac{\partial}{\partial t}$

is a multiple of $X$.

Rewriting (6) for the $\mathbb{P}^{m}$-$R$-matrix
$\tilde{R}_{\mathbb{P}^{m}}=\Psi R_{\mathbb{P}^{m}}\Psi^{-1}$ written in flat
coordinates gives

$\displaystyle[\tilde{R}_{\mathbb{P}^{m}},\xi]-zqL_{E}\tilde{R}_{\mathbb{P}^{m}}+zq\tilde{R}_{\mathbb{P}^{m}}\mu=0,$

where $\xi$ is multiplication by $E$ in flat coordinates and
$\mu=-(L_{E}\Psi)\Psi^{-1}$.

We need to find a series $\varphi$ in $t$ and an $R$-matrix $R$ sending the
modified $A_{m+1}$-theory to equivariant $\mathbb{P}^{m}$:

$\tilde{R}_{\mathbb{P}^{m}}(z)=R(z)\tilde{R}_{A_{m+1}}(z\varphi).$

We know that $\tilde{R}_{A_{m+1}}$ satisfies

$[\tilde{R}_{A_{m+1}},\xi]+zL_{E}\tilde{R}_{A_{m+1}}-z\tilde{R}_{A_{m+1}}\mu=0$

and the weighted homogeneity condition

$\left(z\frac{\mathrm{d}}{\mathrm{d}z}+L_{E}\right)\tilde{R}_{A_{m+1}}+[\mu,\tilde{R}_{A_{m+1}}]=0.$

Putting these together we find that $R$ must satisfy

$0=[R,\xi]-zqL_{E}R+zq\frac{L_{E}\varphi}{\varphi}R\mu\\\
+\frac{1}{\varphi}\left(q+\varphi-q\frac{L_{E}\varphi}{\varphi}\right)R[\tilde{R}_{A_{m+1}}(z\varphi),\xi]\tilde{R}_{A_{m+1}}^{-1}(z\varphi).$

######  Lemma 3.

The series $\tilde{R}_{A_{m+1}}\xi\tilde{R}_{A_{m+1}}^{-1}$ is not a
polynomial in $z$.

Because of the lemma and the homogeneity of $\tilde{R}_{A_{m+1}}$ we see that
in order for $R$ to exist in the limit $\operatorname{disc}\to 0$ the function
$\varphi$ has to satisfy

$q+\varphi-q\frac{L_{E}\varphi}{\varphi}=0$

or equivalently

$-q^{-1}=\varphi^{-1}+L_{E}\varphi^{-1}.$

There is a unique solution $\varphi^{-1}$ to this differential equation.
Concretely, we have

$\varphi^{-1}=\lambda^{-1}\sum_{i=0}^{\infty}\frac{m+2}{m+2+i(m+1)}\left(-\frac{t}{\lambda}\right)^{i}.$

Since it is not necessary for the proof of Theorem 2, we will prove Lemma 3 in
Section 6.

Let us from now on assume that $\varphi$ is this solution. Then the
differential equation for $R$ spells

$[R,\xi]-zqL_{E}R+zq\frac{L_{E}\varphi}{\varphi}R\mu=0.$ (7)

The following lemma implies that the matrix
$\tilde{R}_{\mathbb{P}^{m}}(z)\tilde{R}_{A_{m+1}}^{-1}(z\varphi)$ does not
have any poles in $t$ and this concludes the proof of Theorem 2.

######  Lemma 4.

For any solution $R(z)$ of (7) of the form

$R(z)=\sum_{i=0}^{\infty}(R_{jk}^{i})z^{i}=1+O(z),$

for Laurent series $R_{jk}^{i}$ in $t$, actually all the $R_{jk}^{i}$ have to
be polynomials.

###### Proof.

The matrices $\xi$ and $\mu$ can be explicitly calculated

$\xi_{jk}=t\frac{m+1}{m+2}\delta_{j,k+1}(-t)^{\delta_{0,j}},\qquad\mu_{jk}=\frac{2j-m}{2(m+2)}\delta_{j,k},$

where all indices are understood modulo $(m+1)$.

Assume that we have already constructed $R^{i-1}$ and its entries have no
negative powers in $t$. Looking at the $z^{i}$-part of (7) gives expressions
for $R_{j(k+1)}^{i}\xi_{(k+1)k}-\xi_{j(j-1)}R_{(j-1)k}^{i}$ as power series
with no poles in $t$. From here we see that if we can determine the
$R_{j0}^{i}$ as power series with no poles, then the other entries are given
by

$R_{jk}^{i}\equiv(-t)^{\delta_{k>j}}R_{(j-k)0}^{i}$

modulo terms with no poles in $t$, determined from $R^{i-1}$. The exponent
$\delta_{k>j}$ is $1$ for $k>j$ and $0$ otherwise.

From the $z^{i+1}$-part of (7) we then get expressions with no poles in $t$
for

$(m+1)t\frac{\mathrm{d}R_{j0}^{i}}{\mathrm{d}t}+jR_{j0}^{i},$

thus determining all $R_{j0}^{i}$ but $R_{00}^{i}$ up to a constant. Therefore
all the $R_{jk}^{i}$ are polynomials in $t$. ∎

######  Remark 5.

The derivation in this section would have worked the same if $q$ was any other
invertible power series in $t$.

##  6 Higher dimensions

We would like to show that for $m>1$ there is no pair of function $\varphi$
and matrix power series $R(z)$, both well-defined in the limit
$\operatorname{disc}\to 0$, such that

$\tilde{R}_{\mathbb{P}^{m}}(z)=R(z)\tilde{R}_{A_{m+1}}(z\varphi),$ (8)

where again $\tilde{R}_{*}=\Psi R_{*}\Psi^{-1}$. We will need to assume that
that $\varphi$ is well-defined in the Airy limit. Then we can use the
discussion from Section 5 to derive that $\varphi$ is of the form

$\varphi=\lambda+c_{0}\lambda^{0}+c_{-1}\lambda^{-1}+\dotsb,$

where the $c_{i}$ are independent of $\lambda$ and $c_{-1}$ in the Airy limit
becomes a constant multiple of $(t^{1})^{2}$. For the uniqueness of $\varphi$
we needed Lemma 3.

###### Proof of Lemma 3.

Recall that we have to show that
$P:=\tilde{R}_{A_{m+1}}\xi\tilde{R}_{A_{m+1}}^{-1}$ is not a polynomial in
$z$. From the differential equation for $\tilde{R}_{A_{m+1}}$ we obtain a
differential equation for $P$.

$[P,\xi]=z^{2}\frac{\mathrm{d}P}{\mathrm{d}z}-z[P,\mu]$

By definition we also have the initial condition $P|_{z=0}=\xi$. Write
$P=\xi+zP_{1}+z^{2}P_{2}+\dotsb$. The homogeneity condition for
$\tilde{R}_{A_{m+1}}$ implies that the only nonzero entries of $P_{i}$ are at
the $(i-1)$-th diagonal, where by this we mean the entries on $j$-th row,
$k$-th column such that $k-j\equiv i-1\pmod{m+1}$.

Assume we have shown that $P_{i}\neq 0$ has a nonzero entry on the $(i-1)$-th
diagonal row. Recalling the proof of Lemma 4 we see that essentially the
differences of two subsequent entries in the $i$-th diagonal of $P_{i+1}$ are
a multiple of an entry on the $(i-1)$-th diagonal of $P_{i}$. Since the
absolute value of any entry of $\mu$ is less than $\frac{1}{2}$, all of these
multiples are nonzero. Therefore it is impossible for all entries on the
$i$-th diagonal of $P_{i+1}$ to be zero. The lemma follows by induction. ∎

To show that there is no suitable intermediate $R$-matrix $R$ it will be
enough to consider the $z^{1}$-term of (8). It says

$\tilde{r}_{\mathbb{P}^{m}}=r+\varphi\tilde{r}_{A_{m+1}},$

where $r_{*}$ stands for the $z^{1}$-term of $R_{*}$. Since
$\tilde{r}_{\mathbb{P}^{m}}$ has no negative powers in $\lambda$, the
$\lambda^{-1}$-terms on the right hand side have to cancel. However the
bottom-left coefficient of $\tilde{r}_{A_{m+1}}$ has a pole in the
discriminant. Since for $m>2$ the coefficient $c_{-1}$ cannot be a multiple of
the discriminant for degree reasons, in this case $r$ has to have a pole in
the discriminant. Contradiction.

It remains to look at the case $m=2$. Here it is similarly enough to show that
there is one coefficient in the $R$-matrix with a second order pole in the
discriminant in order to derive a contradiction. We look at the coefficient
$r_{20}$ calculated from the oscillating integral of Section 3.1. We need to
calculate the $z^{1}$-coefficient of the asymptotic expansion of

$\sum_{Q}\frac{1}{\sqrt{2\pi}\Delta}\int\limits_{-\infty}^{\infty}\exp\left(-\frac{x^{2}}{2}-x^{3}\sqrt{-z}\frac{Q}{\Delta^{3/2}}-\frac{x^{4}}{4}(-z)\frac{1}{\Delta^{2}}\right)\mathrm{d}x,$

where we sum over roots $Q$ of the polynomial defining the singularity and
here $\Delta=3Q^{2}+2t^{2}$. Expanding the Gaussian integral we find the
coefficient to be equal to

$-\sum_{Q}\frac{15}{2}\frac{Q^{2}}{\Delta^{4}}+\sum_{Q}\frac{3}{\Delta^{3}}.$

It is straightforward to check that the first summand equals

$-\frac{15}{2}\frac{-2(2t^{2})^{3}+27(t^{1})^{2}}{(-4(2t^{2})^{3}-27(t^{1})^{2})^{2}},$

whereas the second term has only a first order pole in the discriminant.

##  Appendix A Givental’s localization calculation

We want to recall Givental’s localization calculation [4], which proves that
the CohFT from equivariant $\mathbb{P}^{m}$ can be obtained from the trivial
theory via a specific $R$-matrix action. We first recall localization in the
space of stable maps to $\mathbb{P}^{m}$ in Section A.1. Next, in Section A.2
we group the localization contributions according to the dual graph of the
source curve. We collect identities following from the string and dilaton
equation in Section A.3 before applying them to finish the computation in
Section A.4.

###  A.1 Localization in the space of stable maps

Let $T=(\mathbb{C}^{*})^{m+1}$ act diagonally on $\mathbb{P}^{m}$. The
equivariant Chow ring of a point and $\mathbb{P}^{m}$ are given by

$\displaystyle A^{*}_{T}(\mathrm{pt})\cong$
$\displaystyle\mathbb{Q}[\lambda_{0},\dotsc,\lambda_{m}]$ $\displaystyle
A^{*}_{T}(\mathbb{P}^{m})\cong$
$\displaystyle\mathbb{Q}[H,\lambda_{0},\dotsc,\lambda_{m}]/\prod_{i=0}^{m}(H-\lambda_{i}),$

where $H$ is a lift of the hyperplane class. Furthermore, let $\eta$ be the
equivariant Poincaré pairing.

There are $m+1$ fixed points $p_{0},\dotsc,p_{m}$ for the $T$-action on
$\mathbb{P}^{m}$. The characters of the action of $T$ on the tangent space
$T_{p_{i}}\mathbb{P}^{m}$ are given by $\lambda_{i}-\lambda_{j}$ for
$j\not=i$. Hence the corresponding equivariant Euler class $e_{i}$ is given by

$e_{i}=\prod_{j\neq i}(\lambda_{i}-\lambda_{j}).$

The equivariant class $e_{i}$ also serves as the inverse of the norms of the
equivariant (classical) idempotents

$\phi_{i}=e_{i}^{-1}\prod_{j\neq i}(H-\lambda_{j}).$

The virtual localization formula [6] implies that the virtual fundamental
class can be split into a sum

$\displaystyle[\overline{M}_{g,n}(\mathbb{P}^{m},d)]^{vir}_{T}=\sum_{X}\iota_{X,*}\frac{[X]^{vir}_{T}}{e_{T}(N^{vir}_{X,T})}$

of contributions of fixed loci $X$. Here $N^{vir}_{X,T}$ denotes the virtual
normal bundle of $X$ in $\overline{M}_{g,n}(\mathbb{P}^{m},d)$ and $e_{T}$ the
equivariant Euler class. Because of the denominator, the fixed point
contributions are only defined after localizing by the elements
$\lambda_{0},\dotsc,\lambda_{m}$. By studying the $\mathbb{C}^{*}$-action on
deformations and obstructions of stable maps, $e_{T}(N^{vir}_{X,T})$ can be
computed explicitly.

The fixed loci can be labeled by certain decorated graphs. These consist of

  * •

a graph $(V,E)$,

  * •

an assignment $\zeta:V\to\\{p_{0},\dotsc,p_{m}\\}$ of fixed points,

  * •

a genus assignment $g:V\to\mathbb{Z}_{\geq 0}$,

  * •

a degree assignment $d:E\to\mathbb{Z}_{>0}$,

  * •

an assignment $p:\\{1,\dotsc n\\}\to V$ of marked points,

such that the graph is connected and contains no self-edges, two adjacent
vertices are not assigned to the same fixed point and we have

$g=h^{1}(\Gamma)+\sum_{v\in V}g(v),\qquad d=\sum_{e\in E}d(e).$

A vertex $v\in V$ is called stable if $2g(v)-2+n(v)>0$, where $n(v)$ is the
number of outgoing edges at $v$.

The fixed locus corresponding to a graph is characterized by the condition
that stable vertices $v\in V$ of the graph correspond to contracted genus
$g(v)$ components of the domain curve, and that edges $e\in E$ correspond to
multiple covers of degree $d(e)$ of the torus fixed line between two fixed
points. Such a fixed locus is isomorphic to a product of moduli spaces of
curves

$\prod_{v\in V}\overline{M}_{g(v),n(v)}$

up to a finite map.

For a fixed locus $X$ corresponding to a given graph the Euler class
$e_{T}(N^{vir}_{X,T})$ is a product of factors corresponding to the geometry
of the graph

$e_{T}(N^{vir}_{X,T})=\prod_{v\text{, stable}}\frac{e(\mathbb{E}^{*}\otimes
T_{\mathbb{P}^{m},\zeta(v)})}{e_{\zeta(v)}}\prod_{\text{nodes}}\frac{e_{\zeta}}{-\psi_{1}-\psi_{2}}\\\
\prod_{\begin{subarray}{c}g(v)=0\\\
n(v)=1\end{subarray}}(-\psi_{v})\prod_{e}\mathrm{Contr}_{e}.$ (9)

In the first product $\mathbb{E}^{*}$ denotes the dual of the Hodge bundle,
$T_{\mathbb{P}^{m},\zeta(v)}$ is the tangent space of $\mathbb{P}^{m}$ at
$\zeta(v)$, and all bundles and Euler classes should be considered
equivariantly. The second product is over nodes forced onto the domain curve
by the graph. They correspond to stable vertices together with an outgoing
edge, or vertices $v$ of genus $0$ with $n(v)=2$. With $\psi_{1}$ and
$\psi_{2}$ we denote the (equivariant) cotangent line classes at the two sides
of the node. For example, the equivariant cotangent line class $\psi$ at a
fixed point $p_{i}$ on a line mapped with degree $d$ to a fixed line is more
explicitly given by

$-\psi=\frac{\lambda_{j}-\lambda_{i}}{d},$

where $p_{j}$ is the other fixed point on the fixed line. The explicit
expressions for the terms in the second line of (9) can be found in [6], but
will play no role for us. It is only important that they only depend on local
data.

###  A.2 General procedure

We set $W$ to be $A^{*}_{T}(\mathbb{P}^{m})$ with all equivariant parameters
localized. For $v_{1},\dotsc,v_{n}\in W$ the (full) CohFT $\Omega_{g,n}$ from
equivariant $\mathbb{P}^{m}$ is defined by

$\Omega_{g,n}^{p}(v_{1},\dotsc,v_{n})\\\
=\sum_{d,k=0}^{\infty}\frac{q^{d}}{k!}\varepsilon_{*}\pi_{*}\left(\prod_{i=1}^{n}\operatorname{ev}_{i}^{*}(v_{i})\prod_{i=n+1}^{n+k}\operatorname{ev}_{i}^{*}(p)\cap[\overline{M}_{g,n+k}(\mathbb{P}^{m},d)]^{vir}\right),$
(10)

where $p$ is a point on the formal Frobenius manifold, $\varepsilon$ forgets
the last $k$ markings and $\pi$ forgets the map. We want to calculate the
push-forward via virtual localization. In the end we will arrive at the
formula of the $R$-matrix action as described in Section 2.3. In the following
we will systematically suppress the dependence on $p$ in the notation.

We start by remarking that for each localization graph for (10) there exists a
dual graph of $\overline{M}_{g,n}$ corresponding to the topological type of
the stabilization of a generic source curve of that locus. What gets
contracted under the stabilization maps are trees of rational curves. There
are three types of these unstable trees:

  1. 1.

those which contain one of the $n$ markings and are connected to a stable
component,

  2. 2.

those which are connected to two stable components and contain none of the $n$
markings and

  3. 3.

those which are connected to one stable component but contain none of the $n$
markings.

These give rise to series of localization contributions and we want to record
those, using the fact that they already occur in genus 0.

Let $W^{\prime}$ be an abstract free module over the same base ring as $W$
with a basis $w_{0},\dotsc,w_{m}$ labeled by the fixed points of the
$T$-action on $\mathbb{P}^{m}$. The type 1 contributions are recorded by

$\tilde{R}^{-1}=\sum_{i}\tilde{R}_{i}^{-1}w_{i}\in\operatorname{Hom}(W,W^{\prime})[\\![z]\\!],$

the homomorphism valued power series such that

$\tilde{R}_{i}^{-1}(v)=\eta(e_{i}\phi_{i},v)+\sum_{d,k=0}^{\infty}\frac{q^{d}}{k!}\sum_{\Gamma\in
G_{d,k,i}^{1}}\frac{1}{\operatorname{Aut}(\Gamma)}\operatorname{Contr}_{\Gamma}(v)$

where $G_{d,k,i}^{1}$ is the set of localization graphs for
$\overline{M}_{0,2+k}(\mathbb{P}^{m},d)$ such that the first marking is at a
valence 2 vertex at fixed point $i$ and $\operatorname{Contr}_{\Gamma}(v)$ is
the contribution for graph $\Gamma$ for the integral

$\int_{\overline{M}_{0,2+k}(\mathbb{P}^{m},d)}\frac{e_{i}}{-z-\psi_{1}}\operatorname{ev}_{2}^{*}(v)\prod_{l=3}^{2+k}\operatorname{ev}_{l}^{*}(p).$

We define the integral in the case $(d,k)=(0,0)$ to be zero and will do
likewise for other integrals over non-existing moduli spaces.

The type 2 contributions are recorded by the bivector

$\tilde{V}=\sum_{i}\tilde{V}^{ij}w_{i}\otimes w_{j}\in W^{\prime\otimes
2}[\\![z,w]\\!]$

which is defined by

$V^{ij}=\sum_{d,k=0}^{\infty}\frac{q^{d}}{k!}\sum_{\Gamma\in
G_{d,k,i,j}^{2}}\frac{1}{\operatorname{Aut}(\Gamma)}\operatorname{Contr}_{\Gamma},$

where $G_{d,k,i,j}^{2}$ is the set of localization graphs for
$\overline{M}_{0,2+k}(\mathbb{P}^{m},d)$ such that the first and second
marking are at valence 2 vertices at fixed points $i$ and $j$, respectively,
and $\operatorname{Contr}_{\Gamma}$ is the contribution for graph $\Gamma$ for
the integral

$\int_{\overline{M}_{0,2+k}(\mathbb{P}^{m},d)}\frac{e_{i}e_{j}}{(-z-\psi_{1})(-w-\psi_{2})}\prod_{l=3}^{2+k}\operatorname{ev}_{l}^{*}(p).$

Finally, the type 3 contribution is a vector

$\tilde{T}=\sum_{i}\tilde{T}_{i}w_{i}\in W^{\prime}[\\![z]\\!]$

which is defined by

$\tilde{T}_{i}=p+\sum_{d,k=0}^{\infty}\frac{q^{d}}{k!}\sum_{\Gamma\in
G_{d,k,i}^{3}}\frac{1}{\operatorname{Aut}(\Gamma)}\operatorname{Contr}_{\Gamma}$

where $G_{d,k,i}^{3}$ is the set of localization graphs for
$\overline{M}_{0,1+k}(\mathbb{P}^{m},d)$ such that the first marking is at a
valence 2 vertex at fixed point $i$ and $\operatorname{Contr}_{\Gamma}(v)$ is
the contribution for graph $\Gamma$ for the integral

$\int_{\overline{M}_{0,1+k}(\mathbb{P}^{m},d)}\frac{e_{i}}{-z-\psi}\prod_{l=2}^{1+k}\operatorname{ev}_{l}^{*}(p).$

With these contributions we can write the CohFT already in a form quite
similar to the reconstruction formula. Let $\omega_{g,n}$ be the $n$-form on
$W^{\prime}$ which vanishes if $w_{i}$ and $w_{j}$ for $i\neq j$ are inputs,
which satisfies

$\omega_{g,n}(w_{i},\dotsc,w_{i})=\frac{e(\mathbb{E}^{*}\otimes
T_{\mathbb{P}^{m},p_{i}})}{e_{i}}=e_{i}^{g-1}\prod_{j\neq
i}c_{\lambda_{j}-\lambda_{i}}(\mathbb{E})$

and which is for $n=0$ defined similarly as in Example 1. We have

$\Omega_{g,n}^{p}(v_{1},\dotsc,v_{n})=\sum_{\Gamma}\frac{1}{\operatorname{Aut}(\Gamma)}\xi_{*}\left(\prod_{v}\sum_{k=0}^{\infty}\frac{1}{k!}\varepsilon_{*}\omega_{g_{v},n_{v}+k}(\dots)\right),$
(11)

where we put

  1. 1.

$\tilde{R}^{-1}(\psi)(v_{i})$ into the argument corresponding to marking $i$,

  2. 2.

a half of $\tilde{V}(\psi_{1},\psi_{2})$ into an argument corresponding to a
node and

  3. 3.

$\tilde{T}(\psi)$ into all additional arguments.

We will still need to apply the string and dilaton equation in order to make
$\tilde{T}(\psi)$ to be a multiple of $\psi^{2}$, like the corresponding
series in the reconstruction, and then relate the series to the $R$-matrix.

###  A.3 String and Dilaton Equation

We want to use the string and dilaton equation to bring a series

$\sum_{k=0}^{\infty}\frac{1}{k!}\varepsilon_{*}\left(\prod_{i=1}^{n}\frac{1}{-x_{i}-\psi_{i}}\prod_{i=n+1}^{n+k}Q(\psi_{i})\right),$
(12)

where $\varepsilon:\overline{M}_{g,n+k}\to\overline{M}_{g,n}$ is the forgetful
map and $Q=Q_{0}+zQ_{1}+z^{2}Q_{2}+\dotsb$ is a formal series, into a
canonical form.

By the string equation, (12) is annihilated by

$\mathcal{L}^{\prime}=\mathcal{L}+\sum_{i=1}^{n}\frac{1}{x_{i}},$

where $\mathcal{L}$ is the string operator

$\mathcal{L}=\frac{\partial}{\partial Q_{0}}-Q_{1}\frac{\partial}{\partial
Q_{0}}-Q_{2}\frac{\partial}{\partial Q_{1}}-Q_{3}\frac{\partial}{\partial
Q_{2}}-\dotsb.$

Moving along the string flow for some time $-u$, i.e. applying
$e^{t\mathcal{L}^{\prime}}|_{t=-u}$, to (12) gives

$\sum_{k=0}^{\infty}\frac{1}{k!}\varepsilon_{*}\left(\prod_{i=1}^{n}\frac{e^{-\frac{u}{x_{i}}}}{-x_{i}-\psi_{i}}\prod_{i=n+1}^{n+k}Q^{\prime}(\psi_{i})\right),$

for a new formal series
$Q^{\prime}=Q^{\prime}_{0}+zQ^{\prime}_{1}+z^{2}Q^{\prime}_{2}+\dotsb$. In the
case that

$u=\sum_{k=1}^{\infty}\frac{1}{k!}\int\limits_{\overline{M}_{0,2+k}}\prod_{i=3}^{2+k}Q(\psi_{i}),$

which we will assume from now on, the new series $Q^{\prime}$ will satisfy
$Q^{\prime}_{0}=0$ since by the string equation $\mathcal{L}u=1$ and therefore
applying $e^{t\mathcal{L}}|_{t=-u}$ to $u$ gives on the one hand zero and on
the other hand the definition of $u$ with $Q$ replaced by $Q^{\prime}$, and
for dimension reasons this is a nonzero multiple of $Q^{\prime}_{0}$.

Next, by applying the dilaton equation we can remove the linear part from the
series $Q^{\prime}_{0}$

$\sum_{k=0}^{\infty}\frac{1}{k!}\varepsilon_{*}\left(\prod_{i=1}^{n}\frac{1}{-x_{i}-\psi_{i}}\prod_{i=n+1}^{n+k}Q(\psi_{i})\right)\\\
=\sum_{k=0}^{\infty}\frac{\Delta^{\frac{2g-2+n+k}{2}}}{k!}\varepsilon_{*}\left(\prod_{i=1}^{n}\frac{e^{-\frac{u}{x_{i}}}}{-x_{i}-\psi_{i}}\prod_{i=n+1}^{n+k}Q^{\prime\prime}(\psi_{i})\right),$
(13)

where $Q^{\prime\prime}=Q^{\prime}-Q^{\prime}_{1}z$ and

$\Delta^{\frac{1}{2}}=(1-Q^{\prime}_{1})^{-1}=\sum_{k=0}^{\infty}\frac{1}{k!}\int\limits_{\overline{M}_{0,3+k}}\prod_{i=4}^{3+k}Q(\psi_{i}).$

We will also need identities in the degenerate cases $(g,n)=(0,2)$ and
$(g,n)=(0,1)$. In the first case, there is the identity

$\frac{1}{-z-w}+\sum_{k=1}^{\infty}\frac{1}{k!}\int\limits_{\overline{M}_{0,2+k}}\frac{1}{-z-\psi_{1}}\frac{1}{-w-\psi_{2}}\prod_{i=3}^{2+k}Q(\psi_{i})=\frac{e^{-u/z+-u/w}}{-z-w}.$
(14)

In order to see that (14) is true, we use that the left hand side is
annihilated by $\mathcal{L}+\frac{1}{z}+\frac{1}{w}$ in order to move from $Q$
to $Q^{\prime}$ via the string flow and notice that there all of the integrals
vanish for dimension reasons. Similarly, there is the identity

$1-\frac{Q(z)}{z}-\frac{1}{z}\sum_{k=2}^{\infty}\frac{1}{k!}\int\limits_{\overline{M}_{0,1+k}}\prod_{i=2}^{1+k}Q(\psi_{i})\\\
=e^{-u/z}\left(1-\frac{Q^{\prime}(z)}{z}\right)=e^{-u/z}\left(\Delta^{-\frac{1}{2}}-\frac{Q^{\prime\prime}(z)}{z}\right),$
(15)

which can be proven like the previous identity by using that the left hand
side is annihilated by $\mathcal{L}+\frac{1}{z}$.

We define the functions $u_{i}$ and $(\Delta_{i}/e_{i})^{\frac{1}{2}}$ for
$i\in\\{0,\dotsc,m\\}$ to be the $u$ and $\Delta^{\frac{1}{2}}$ at the points
$Q=\tilde{T}_{i}$ from the previous section.

###  A.4 Expressing localization series in terms of Frobenius structures

We apply (13) to (11) and obtain

$\Omega_{g,n}^{p}(v_{1},\dotsc,v_{n})=\sum_{\Gamma}\frac{1}{\operatorname{Aut}(\Gamma)}\xi_{*}\left(\prod_{v}\sum_{k=0}^{\infty}\frac{1}{k!}\varepsilon_{*}\omega^{\prime}_{g_{v},n_{v}+k}(\dots)\right),$
(16)

where we put

  1. 1.

$R^{-1}(\psi)(v_{i})$ into the argument corresponding to marking $i$,

  2. 2.

a half of $V(\psi_{1},\psi_{2})$ into an argument corresponding to a node and

  3. 3.

$T(\psi)$ into all additional arguments.

Here $R^{-1}$, $V$ and $T$ are defined exactly as $\tilde{R}^{-1}$,
$\tilde{V}$ and $\tilde{T}$ but with the replacement

$\frac{e_{i}}{-x-\psi}\rightsquigarrow\frac{e_{i}e^{-\frac{u_{i}}{x}}}{-x-\psi}$

made at the factors we put at the ends of the trees. The form
$\omega^{\prime}_{g,n}$ satisfies

$\omega^{\prime}_{g,n}(w_{i},\dotsc,w_{i})=\Delta_{i}^{\frac{2g-2+n}{2}}e_{i}^{-\frac{n}{2}}\prod_{j\neq
i}c_{\lambda_{j}-\lambda_{i}}(\mathbb{E}).$

We now want to compute $R^{-1}$, $V$ and $T$ in terms of the homomorphism
valued series $S^{-1}(z)\in\operatorname{Hom}(W,W^{\prime})[\\![z]\\!]$ with
$w_{i}$-component

$S_{i}^{-1}(z)=\langle\frac{e_{i}\phi_{i}}{-z-\psi},-\rangle\\\
:=\eta(e_{i}\phi_{i},-)+\sum_{d,k=0}^{\infty}\frac{q^{d}}{k!}\int\limits_{\overline{M}_{0,2+k}(\mathbb{P}^{m},d)}\frac{\operatorname{ev}_{1}^{*}(e_{i}\phi_{i})}{-z-\psi_{1}}\operatorname{ev}_{2}^{*}(-)\prod_{j=3}^{2+k}\operatorname{ev}_{j}^{*}(p).$

We start by computing $S^{-1}$ via localization. Using that in genus zero the
Hodge bundle is trivial we find that at the vertex with the first marking we
need to compute integrals exactly as in (14), where the first summand stands
for the case that the vertex is unstable and the $k$-summand stands for the
case that the vertex is stable with $k$ trees of type 3 and one tree of type
1\. Applying (14) we obtain

$S_{i}^{-1}(z)=e^{-\frac{u_{i}}{z}}R_{i}^{-1}(z).$

Using the short-hand notation

$\left\langle\frac{v_{1}}{x_{1}-\psi},\frac{v_{2}}{x_{2}-\psi},\frac{v_{3}}{x_{3}-\psi}\right\rangle\\\
:=\sum_{d,k=0}^{\infty}\frac{q^{d}}{k!}\int\limits_{\overline{M}_{0,3+k}(\mathbb{P}^{m},d)}\frac{\operatorname{ev}_{1}^{*}v_{1}}{x_{1}-\psi_{1}}\frac{\operatorname{ev}_{2}^{*}v_{2}}{x_{2}-\psi_{2}}\frac{\operatorname{ev}_{3}^{*}v_{3}}{x_{3}-\psi_{3}}\prod_{i=4}^{3+k}\operatorname{ev}_{i}^{*}(p)$

for genus zero Gromov-Witten invariants and applying the string equation, we
can also write

$S_{i}^{-1}(z)=-\frac{1}{z}\langle\frac{e_{i}\phi_{i}}{-z-\psi},\mathbf{1},-\rangle.$

We have by the identity axiom and WDVV equation

$\langle\frac{e_{i}\phi_{i}}{-z-\psi},\frac{e_{j}\phi_{j}}{-w-\psi},\mathbf{1}\rangle=\langle\frac{e_{i}\phi_{i}}{-z-\psi},\frac{e_{j}\phi_{j}}{-w-\psi},\bullet\rangle\langle\bullet,\mathbf{1},\mathbf{1}\rangle\\\
=\langle\frac{e_{i}\phi_{i}}{-z-\psi},\mathbf{1},\bullet\rangle\langle\bullet,\mathbf{1},\frac{e_{j}\phi_{j}}{-w-\psi}\rangle,$

where in the latter two expressions the $\bullet$ should be filled with
$\eta^{-1}$, so that

$\frac{S_{i}^{-1}(z)\eta^{-1}S_{j}^{-1}(w)^{t}}{-z-w}\\\
=\frac{\eta(e_{i}\phi_{i},e_{j}\phi_{j})}{-z-w}+\sum_{d,k=0}^{\infty}\frac{q^{d}}{k!}\int\limits_{\overline{M}_{0,2+k}(\mathbb{P}^{m},d)}\frac{\operatorname{ev}_{1}^{*}(e_{i}\phi_{i})}{-z-\psi_{1}}\frac{\operatorname{ev}_{2}^{*}(e_{j}\phi_{j})}{-w-\psi_{2}}\prod_{l=3}^{2+k}\operatorname{ev}_{l}^{*}(p).$
(17)

We compute the right hand side via localization. There are two cases in the
localization depending on whether the first and second marking are at the same
or a different vertex. In the first case we apply (14) at this common vertex
and obtain the total contribution

$\frac{e_{i}\delta_{ij}e^{-\frac{u_{i}}{z}-\frac{u_{j}}{w}}}{-z-w},$

which includes the unstable summand. In the other case, we apply (14) at the
two vertices and obtain

$e^{-\frac{u_{i}}{z}-\frac{u_{j}}{w}}V^{ij}(z,w).$

So all together

$V^{ij}(z,w)=\frac{R_{i}^{-1}(z)\eta^{-1}R_{j}^{-1}(w)^{t}-e_{i}\delta_{ij}}{-z-w}.$

Finally we express $T$ in terms of $R$ by computing

$S_{i}^{-1}(z)\mathbf{1}=e_{i}-\frac{1}{z}\sum_{d,k=0}^{\infty}\frac{q^{d}}{k!}\int\limits_{\overline{M}_{0,1+k}(\mathbb{P}^{m},d)}\frac{\operatorname{ev}_{1}^{*}(e_{i}\phi_{i})}{-z-\psi_{1}}\prod_{j=2}^{1+k}\operatorname{ev}_{j}^{*}(p)$

via localization. Applying (15) at the first marking we find that

$S_{i}^{-1}(z)\mathbf{1}=e^{-\frac{u_{i}}{z}}\left(\Delta_{i}^{-\frac{1}{2}}e_{i}^{\frac{1}{2}}-\frac{T_{i}(z)}{z}\right).$

So

$T(z)=z\left(\sum_{i}\Delta_{i}^{-\frac{1}{2}}e_{i}^{\frac{1}{2}}w_{i}-R^{-1}(z)\mathbf{1}\right).$

By (16) the underlying TQFT of $\Omega^{p}_{g,0}$ is given by

$\sum_{i}\Delta_{i}^{g-1}.$

This implies that the $\Delta_{i}$ need to be the inverses of the norms of the
idempotents for the quantum product of equivariant $\mathbb{P}^{m}$ (because
these are pairwise different). Since $\tilde{T}$ vanishes at $(p,q)=0$,
$\Delta_{i}$ is $e_{i}^{-1}$ at $(p,q)\to 0$. Therefore we can identify
$W^{\prime}$ with $W$ by mapping $w_{i}$ to $\sqrt{\Delta_{i}/e_{i}}$ times
the idempotent element which coincides with $\phi_{i}$ at $(p,q)=0$. The
previous results then say exactly that $\Omega^{p}$ is obtained from the CohFT
$\omega^{\prime}$ by the action of the $R$-matrix $R$. In turn, Example 4
implies that $\omega^{\prime}$ is obtained from the TQFT by the action of an
$R$-matrix which is diagonal in the basis of idempotents and has entries

$\exp\left(\sum_{i=1}^{\infty}\frac{B_{2i}}{2i(2i-1)}\sum_{j\neq
i}\left(\frac{z}{\lambda_{j}-\lambda_{i}}\right)^{2i-1}\right).$

We still need to check that $R$ satisfies the quantum differential equation
and has the correct limit (4) as $q\to 0$. By considering (17) as $w+z\to 0$
we see that $S^{-1}(z)$ satisfies the symplectic condition, i.e. its inverse
$S(z)$ is the adjoint with respect to $\eta$ of $S^{-1}(-z)$. More explicitly
the evaluation of $S(z)$ at the $i$th normalized idempotent is the vector

$\left\langle\frac{\sqrt{e_{i}}\phi_{i}}{z-\psi},\eta^{-1}\right\rangle.$

By the genus 0 topological recursion relations for any flat vector field $X$

$z\left\langle
X,\frac{\sqrt{e_{i}}\phi_{i}}{z-\psi},\eta^{-1}\right\rangle=\langle
X,\eta^{-1},\bullet\rangle\left\langle\bullet,\frac{\sqrt{e_{i}}\phi_{i}}{z-\psi}\right\rangle,$

where again $\eta^{-1}$ should be inserted at the $\bullet$. Therefore $S$
satisfies the quantum differential equation

$zXS(z)=X\star S(z),$

where on the left hand side the action of vector fields and the right hand
side quantum multiplication is used.

At $q=0$, we can check that $R$ becomes the identity matrix and therefore the
$R$-matrix of $\Omega^{p}$ becomes the $R$-matrix of the CohFT
$\omega^{\prime}$, which has the correct limit (4).

## References

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Departement Mathematik
ETH Zürich
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