# Relations on $\overline{M}_{g,n}$ via equivariant Gromov-Witten theory of
$\mathbb{P}^{1}$

Felix Janda

###### Abstract

We give a proof of Pixton’s generalized Faber–Zagier relations in the
tautological Chow ring of $\overline{M}_{g,n}$. The strategy is very similar
to the work of Pandharipande–Pixton–Zvonkine, who have given a proof of the
same result in cohomology. The main tool is the Givental–Teleman
classification of semisimple cohomological field theories. While in general
only known in cohomology, the Givental–Teleman classification for the
equivariant Gromov–Witten theory of the projective line is also valid in the
Chow ring, as can be shown via virtual localization. We obtain the relations
just from the latter theory.

##  1 Introduction

The study of the Chow ring of the moduli space $M_{g}$ of genus $g$ algebraic
curves was initiated by Mumford in [1]. For this he introduced tautological
classes, which reflect the geometry of the objects parametrized by the moduli
space. The tautological ring $R^{*}(M_{g})$ is the ring generated by
tautological classes and the definitions were later extended to the
Deligne–Mumford compactification $\overline{M}_{g,n}$ of stable curves of
genus $g$ with $n$ markings, where $R^{*}(\overline{M}_{g,n})$ is compactly
defined [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, that is
the maps obtained from forgetting marked points or gluing curves along
markings. See the recent survey article [3] for a discussion of the
tautological ring and the topics surrounding this paper.

There is an explicit set of generators of the ring $R^{*}(\overline{M}_{g,n})$
of the form

$\xi_{\Gamma*}\left(\prod_{v\in\Gamma}P_{v}\right),$ (1)

which we briefly describe now. $\Gamma$ is a dual graph describing a stratum
of $\overline{M}_{g,n}$ by the topological type of its generic elements; that
is the vertices $v$ of $\Gamma$ correspond to irreducible components and are
labeled by a genus $g(v)$, edges correspond to nodes and there are numbered
legs corresponding to the markings. Each edge is thought of as glued from two
half-edges and the legs also count as half-edges. Corresponding to $\Gamma$,
there is a gluing map

$\xi_{\Gamma}\colon\prod_{v\in\Gamma}\overline{M}_{g(v),n(v)}\to\overline{M}_{g,n},$

finite of degree $|\operatorname{Aut}(\Gamma)|$, where $n(v)$ is the number of
half-edges at vertex $v$. The $P_{v}$ in (1) are arbitrarily chosen monomials
in the first Chern classes $\psi_{1},\dotsc,\psi_{n(v)}$ of the cotangent line
bundles at the markings and the $\kappa$-classes

$\kappa_{i}=\pi_{*}(\psi_{n(v)+1}^{i+1}),$

where $\pi$ is the forgetful map

$\pi:\overline{M}_{g(v),n(v)+1}\to\overline{M}_{g(v),n(v)}.$

See [4, Appendix A] for a review of these generators of the tautological ring.
There, it is shown how multiplication, push-forward and pull-back under the
tautological maps can be defined on the level of generators. In [5], the
strata algebra $\mathcal{S}_{g,n}$ is defined as the formal algebra generated
by formal classes of the form (1). The discussion of [4, Appendix A] defines
the ring structure on $\mathcal{S}_{g,n}$ and lifts of the tautological maps
under the natural forgetful homomorphism $q\colon\mathcal{S}_{g,n}\to
R^{*}(\overline{M}_{g,n})$.

While there is an explicit set of generators, the set of relations between the
generators is not known despite much study. The conjectures of Faber [6, 7]
would give a description of the relations. However, recently, Petersen and
Tommasi [8] gave counter-examples to the Gorenstein conjecture, which is the
only one of Faber’s conjectures that had remained unproven. On the other hand,
Pixton’s set [5] of generalized Faber–Zagier relations gives a well-tested
conjectural description for the set of tautological relations.

We now give a brief description of Pixton’s relations. As for the original
relations of Faber and Zagier in $R^{*}(M_{g})$, the main input to the
formulation of the relations of Pixton are the hypergeometric series

$A(z)=\sum_{i=0}^{\infty}\frac{(6i)!}{(3i)!(2i)!}\frac{(-z)^{i}}{1728^{i}}=1-\frac{5}{144}z\pm\dotsb,\\\
B(z)=\sum_{i=0}^{\infty}\frac{(6i)!}{(3i)!(2i)!}\frac{1+6i}{1-6i}\frac{(-z)^{i}}{1728^{i}}=1+\frac{7}{144}z\mp\dotsb.$
(2)

The functions $A$ and $B$ have appeared in many different ways in the study of
the moduli space of curves and are strongly related to the asymptotic
expansion of the Airy function. See [9] for a review of some of these
occurrences.

Given a collection $(a_{1},\dotsc,a_{n})\in\\{0,1\\}^{n}$, Pixton constructs
from these series certain elements

$\sum_{\Gamma}\frac{1}{|\operatorname{Aut}(\Gamma)|}\frac{1}{2^{h_{1}(\Gamma)}}\xi_{\Gamma*}\left(\left[\prod_{v}\kappa_{v}\prod_{i}B_{i}\prod_{e}\Delta_{e}\right]_{\prod_{v}\zeta_{v}^{g(v)-1}}\right)$
(3)

in the strata algebra $\mathcal{S}_{g,n}$, where the sum is over all dual
graphs $\Gamma$ and the products are over all vertices $v$, markings $i$ and
edges $e$ of $\Gamma$. At each vertex $v$ a parity condition is singled out
using a variable $\zeta_{v}$ satisfying $\zeta_{v}^{2}=1$. The objects
$\kappa_{v}$, $B_{i}$ and $\Delta_{e}$ in (3) depend on the $\zeta$-variables
and the square brackets single out a specific coefficient in these variables.
Define

$B_{i}=\begin{cases}A(\zeta_{i}\psi_{i}),&\text{if $a_{i}=0$,}\\\
B(\zeta_{i}\psi_{i}),&\text{if $a_{i}=1$,}\end{cases}$

where $\zeta_{i}$ is $\zeta_{v}$ if the marking $i$ is at vertex $v$, and
define

$\Delta_{e}=\frac{\zeta^{\prime}+\zeta^{\prime\prime}-A(\zeta^{\prime}\psi^{\prime})\zeta^{\prime\prime}B(\zeta^{\prime\prime}\psi^{\prime\prime})-\zeta^{\prime}B(\zeta^{\prime}\psi^{\prime})A(\zeta^{\prime\prime}\psi^{\prime\prime})}{\psi^{\prime}+\psi^{\prime\prime}},$

where $\psi^{\prime}$ and $\psi^{\prime\prime}$ are the cotangent line classes
at both sides of the node corresponding to $e$, and $\zeta^{\prime}$ and
$\zeta^{\prime\prime}$ are the parity variables at the vertices connected by
$e$. Finally, $\kappa_{v}$ is the polynomial in formal $\kappa$-classes of
$\overline{M}_{g(v),n(v)}$ defined by

$\kappa_{v}=\sum_{k=0}^{\infty}\pi_{*}\left(\prod_{i=n(v)+1}^{n(v)+k}(\psi_{i}-\psi_{i}A(\zeta_{v}\psi_{i}))\right),$

where $\pi$ forgets the last $k$ markings, via Faber’s formula [10].

In this paper we give a proof of the following result, which has been
conjectured by Pixton in [5].

######  Theorem 1.

Let $(g,n)$ be in the stable range $2g-2+n>0$, and let $d\geq 0$ satisfy the
condition

$3d>g-1+\sum_{i=1}^{n}a_{i}.$ (4)

Then the degree $d$ part of (3) is a tautological relation, that is, lies in
the kernel of the forgetful map $q\colon\mathcal{S}_{g,n}\to
R^{*}(\overline{M}_{g,n})$.

Pixton furthermore conjectures that the smallest ideal in $\mathcal{S}_{g,n}$
including the relations of Theorem 1 and stable under push-forward under the
tautological maps coincides with the set of all tautological relations.

In [11], Theorem 1 is proven on the level of cohomology; in other words, it is
proven that (3) is mapped to zero under the composition of $q$ and the cycle
class map. The main ingredient of the proof of [11] is the Givental–Teleman
classification of semisimple cohomological field theories (CohFTs) applied in
the example of Witten’s 3-spin class. All arguments of [11] also apply to Chow
rings except for the fact that the classification cannot be used since its
proof by Teleman uses topological arguments in an essential way.

For the proof of Theorem 1, we will use the same arguments as in [11] except
that we use another CohFT, namely the equivariant Gromov–Witten theory of the
projective line. In this example, the classification is well known to be valid
also in the Chow setting by a localization calculation [12] of Givental. The
CohFT is slightly more complicated than the CohFT of Witten’s 3-spin class in
that it depends on one additional parameter. We will, however, see that under
a particular specialization of the parameters, it agrees with the CohFT of
Witten’s 3-spin class.

The CohFT that we use can also alternatively be defined using the moduli space
of stable quotients to $\mathbb{P}^{1}$ [13]. Virtual localization for this
moduli space has been used in [14] to give the first proof of the Faber–Zagier
relations of $R^{*}(M_{g})$. The current work can be viewed as a
simplification of the author’s first proof [15] of Theorem 1 generalizing the
arguments of [14]. We prefer to use Gromov–Witten theory over stable quotients
here because the reconstruction of the CohFT in the Chow ring via localization
had already essentially been proven by Givental [12].

Instead of $\mathbb{P}^{1}$, we could also have studied the Gromov–Witten
theory of higher-dimensional projective spaces in order to find relations in
the tautological ring. The resulting relations become increasingly complicated
with rising dimension. Still, in [16] it is shown that all of them can be
expressed in terms of Pixton’s relations.

### Plan of the paper

We start by reviewing standard material on CohFTs including the
Givental–Teleman reconstruction in Section 2. In Section 3 we restrict
ourselves to the example of the equivariant Gromov–Witten theory of
$\mathbb{P}^{1}$, studying first the Frobenius algebra underlying the theory
in Section 3.2 and the $R$-matrix in Section 3.3. In Section 3.4, we observe
that under a particular specialization of parameters this data coincides with
the 3-spin theory. We use this in Section 3.5 to conclude Theorem 1.

In the appendix, we make explicit Givental’s proof of the reconstruction of
the Gromov–Witten theory of projective spaces and we recall his mirror
symmetry description of the small theories.

### Acknowledgments

I thank my PhD advisor R. Pandharipande for all his support while I worked on
this project, especially for the frequent meetings at the beginning of this
project.

I am also very grateful for comments from him and the anonymous referee, which
led to a considerable improvement of the current text.

Special thanks are due to Y. P. Lee for discussions at the conference
_Cohomology of the moduli space of curves_ organized by the
_Forschungsinstitut für Mathematik_ at ETH Zürich, which changed my way of
thinking about the stable quotient relations. I am also grateful for relevant
discussions with A. Marian, D. Oprea, A. Pixton and D. Zvonkine at various
points of time.

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
[17] to formalize the structure of classes from Gromov–Witten theory. Let $V$
be an $N$-dimensional $\mathbb{C}$-vector space and $\eta$ a nonsingular
bilinear form on $V$.

######  Definition 1.

A cohomological field theory (CohFT) $\Omega$ on $(V,\eta)$ (on the level of
the Chow ring) is a system

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

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

$S_{n}$-equivariance

The multilinear form $\Omega_{g,n}$ is invariant with respect to the
$S_{n}$-action permuting the factors of $(V^{*})^{\otimes n}$ and the markings
of $\overline{M}_{g,n}$ simultaneously.

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
points glued together. 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 points
glued together.

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

$\eta(uv,w)=\Omega_{0,3}(u,v,w).$ (5)

######  Definition 3.

A CohFT is called semisimple if the algebra $V$ is semisimple, that is, if it
admits a basis of orthogonal idempotent elements.

######  Remark 1.

Many CohFTs come in families. The definition of a CohFT directly generalizes
to the setting where $V$ is a module over a commutative ring $T$ by demanding
$T$-linearity instead of $\mathbb{C}$-linearity.

###  2.2 Examples

######  Example 1.

For each Frobenius algebra there is the _trivial CohFT_ (also called
_topological field theory_ or TQFT) $\Omega_{g,n}$ characterized by (5) and
the condition

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

Let us record an explicit formula for the appendix: 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 $\Delta_{i}=\eta(\epsilon_{i},\epsilon_{i})$, is a corresponding
orthonormal basis of _normalized idempotents_. It is not difficult to show

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

since, in general, $\Omega_{0,3}$ is uniquely determined by $\eta$ and the
quantum product, and the other $\Omega_{g,n}$ are determined by considering
the restriction to the locus of curves glued together from $2g-2+n$ rational
curves with three special points.

######  Example 2.

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

######  Example 3.

Let $X$ be a smooth, projective variety such that the cycle class map gives an
isomorphism between Chow and cohomology rings. Then the Gromov–Witten theory
of $X$ defines a CohFT based on the module $A^{*}(X)\otimes N$ over the
Novikov ring $N:=\mathbb{C}[\\![q^{\beta}]\\!]$, which is generated by formal
variables $q^{\beta}$ indexed by effective, integral curve classes $\beta$.
The CohFT is defined by setting, for $v_{1},\dotsc,v_{n}\in A^{*}(X)$,

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

where the sum ranges over effective, integral curve classes,
$\operatorname{ev}_{i}$ is the $i$th evaluation map and $p$ is the forgetful
map $p\colon\overline{M}_{g,n}(X,\beta)\to\overline{M}_{g,n}$. 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 Gromov–Witten theory.

For a torus action on $X$, this example can be enhanced to give a CohFT from
the $T$-equivariant Gromov–Witten theory of $X$. In this case, the CohFT is
defined on the module $A_{T}^{*}(X)\otimes N$. The case that
$X=\mathbb{P}^{1}$ with a $\mathbb{C}^{*}$-action will be the main example we
will study. We will make the definition of the CohFT more concrete in Section
3.1.

###  2.3 Reconstruction

The (upper half of the) _symplectic loop group_ corresponding to a vector
space $V$ with nonsingular bilinear form $\eta$ is the group of endomorphism-
valued power series $V[\\![z]\\!]$ such that the _symplectic condition_

$R(z)R^{t}(-z)=1$ (6)

holds. Here $R^{t}$ is the adjoint of $R$ with respect to $\eta$. There is an
action of this group on the space of all CohFTs based on a fixed semisimple
Frobenius algebra structure of $V$. The action is often named after Givental
because he introduced it on the level of arbitrary genus Gromov–Witten
potentials.

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

$R\Omega_{g,n}(v_{1},\dotsc,v_{n})=\sum_{\Gamma}\frac{1}{\operatorname{Aut}(\Gamma)}\xi_{*}\left(\prod_{v}\sum_{k=0}^{\infty}\frac{1}{k!}\pi_{*}\Omega_{g(v),n(v)+k}(\dots)\right),$
(7)

where $\xi\colon\prod_{v}\overline{M}_{g(v),n(v)}\to\overline{M}_{g,n}$ is the
gluing map of curves of topological type $\Gamma$ from their irreducible
components, $\pi\colon\overline{M}_{g(v),n(v)+k}\to\overline{M}_{g(v),n(v)}$
forgets the last $k$ markings and we still need to specify what is put into
the arguments of $\prod_{v}\Omega_{g(v),n(v)+k}$. Instead of allowing only
vectors in $V$ to be put into $\Omega_{g,n}$, we will allow elements of
$V[\\![\psi_{1},\dotsc,\psi_{n}]\\!]$, where $\psi_{i}$ acts on the cohomology
of the relevant moduli space of curves by multiplication with the $i$th
cotangent line class.

  * •

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

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

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.

  * •

At each of the additional arguments for each vertex put

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

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

The following reconstruction result (on the level of potentials) was first
proposed by Givental [18].

######  Theorem 2 ([19]).

In cohomology, the $R$-matrix action is free and transitive on the space of
semisimple CohFTs based on a given Frobenius algebra. In particular, for any
CohFT $\Omega$ with underlying TQFT $\omega$ there exists an $R$-matrix $R$
such that $\Omega=R\omega$.

######  Remark 2.

Teleman’s proof relies heavily on topological results (Mumford’s
conjecture/Madsen-Weiss theorem) and it is therefore not known if the same
classification result also holds in general when we work with Chow rings
instead of cohomology.

######  Remark 3.

In the appendix, we review a virtual localization computation of Givental,
which implies that Theorem 2 is true in the Chow ring for the equivariant
Gromov–Witten theory of projective spaces. Furthermore, mirror symmetry gives
an explicit description of the $R$-matrix.

######  Example 4.

By Mumford’s Grothendieck–Riemann–Roch calculation [1], 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 the $B_{2i}$ 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.

##  3 Equivariant Gromov–Witten theory of $\mathbb{P}^{1}$

In this section we consider the projective line $\mathbb{P}^{1}$ together with
a $\mathbb{C}^{*}$-action with weights $(0,1)$ and the tautological relations
resulting from it.

###  3.1 Definition of the CohFT

We make the CohFT described in Example 3 more concrete in our example.

The only effective curve classes on $\mathbb{P}^{1}$ are nonnegative multiples
of $[\mathbb{P}^{1}]$. Hence the Novikov ring is the ring
$\mathbb{C}[\\![q]\\!]$, where $q$ corresponds to $[\mathbb{P}^{1}]$.

Let $\lambda$ be the first Chern class of the dual of the one dimensional
vector space with a $\mathbb{C}^{*}$-action of weight 1. Then the equivariant
Chow ring of $\mathbb{P}^{1}$ is isomorphic to

$A_{\mathbb{C}^{*}}^{*}(\mathbb{P}^{1})\cong\mathbb{Q}[\lambda,H]/H(H-\lambda);$

this expresses the fact that the equivariant classes $H$ and $H-\lambda$ of
the two fixed points 0 and $\infty$ intersect trivially. The module
$A_{\mathbb{C}^{*}}^{*}(\mathbb{P}^{1})\otimes N$ over
$\mathbb{C}[\lambda]\otimes N$ is the state space of the CohFT.

The CohFT $\Omega_{g,n}^{\mathbb{P}^{1}}$ corresponding to the equivariant
Gromov–Witten theory of $\mathbb{P}^{1}$ is defined by setting for
$v_{1},\dotsc,v_{n}\in A_{\mathbb{C}^{*}}^{*}(\mathbb{P}^{1})$,

$\Omega_{g,n}(v_{1},\dotsc,v_{n})=\sum_{d=0}^{\infty}p_{*}\left(\prod_{i=1}^{n}\operatorname{ev}_{i}^{*}(v_{i})\cap[\overline{M}_{g,n}(\mathbb{P}^{1},d)]^{\mathrm{vir}}\right)q^{d},$
(8)

where $\overline{M}_{g,n}(\mathbb{P}^{1},d)$ is the moduli space of stable
maps of degree $d$ to $\mathbb{P}^{1}$ and $p$ is the forgetful map to
$\overline{M}_{g,n}$. The dimension of the virtual class
$[\overline{M}_{g,n}(\mathbb{P}^{1},d)]^{\mathrm{vir}}$ is

$(3-\dim\mathbb{P}^{1})(g-1)+n+\langle
dH,c_{1}(T_{\mathbb{P}^{1}})\rangle=2g-2+n+2d.$

This means that for any $a_{1},\dotsc,a_{n}\in\\{0,1\\}$ the coefficient of
$q^{d}$ in $\Omega_{g,n}(H^{a_{1}},\dotsc,H^{a_{n}})$ has degree

$3g-3+n-(2g-2+n+2d)+\sum_{i=1}^{n}a_{i}=g-1-2d+\sum_{i=1}^{n}a_{i}.$

Since this integer is negative for large enough $d$, the sum over $d$ in (8)
is finite and defines a CohFT on the module
$A_{\mathbb{C}^{*}}^{*}(\mathbb{P}^{1})[q]$ over the ring
$\mathbb{C}[\lambda,q]$.

###  3.2 The underlying TQFT

The nonsingular bilinear form on $A_{T}^{*}(\mathbb{P}^{1})$ is given by the
equivariant Poincaré pairing. In the basis $\\{1,H\\}$ it takes the form

$\begin{pmatrix}0&1\\\ 1&\lambda\end{pmatrix}.$

Define

$\widetilde{H}:=H-\frac{\lambda}{2},$

so that in the basis $\\{1,\widetilde{H}\\}$ the Poincaré pairing becomes just

$\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$

The equivariant quantum product of $\mathbb{P}^{1}$ is defined using three-
pointed genus zero equivariant Gromov–Witten invariants as in (5). Explicitly,
the relation defining the classical equivariant cup product gets deformed to

$H(H-\lambda)=q.$

Equivalently, we can write

$\widetilde{H}^{2}=\phi,$

where we set

$\phi=\frac{\lambda^{2}}{4}+q.$

Notice that the product is semisimple if and only if $\phi\neq 0$.

By choosing a root $\sqrt{\phi}$ of $\phi$ (working in an extension of the
base ring), we can easily write down idempotents

$\frac{1}{\Delta_{\pm}}\widetilde{H}+\frac{1}{2},$

where $\Delta_{\pm}=\pm 2\sqrt{\phi}$. By choosing further roots
$\sqrt{\Delta_{\pm}}$, we can define normalized idempotents such that the
change of basis $\Psi$ from the basis of normalized idempotents to the basis
$\\{1,\widetilde{H}\\}$ is given by

$\Psi=\begin{pmatrix}\frac{\sqrt{\phi}}{\sqrt{\Delta_{+}}}&\frac{-\sqrt{\phi}}{\sqrt{\Delta_{-}}}\\\
\frac{1}{\sqrt{\Delta_{+}}}&\frac{1}{\sqrt{\Delta_{-}}}\end{pmatrix}.$

For convenience, we also note that

$\Psi^{-1}=\begin{pmatrix}\frac{1}{\sqrt{\Delta_{+}}}&\frac{\sqrt{\phi}}{\sqrt{\Delta_{+}}}\\\
\frac{1}{\sqrt{\Delta_{-}}}&\frac{-\sqrt{\phi}}{\sqrt{\Delta_{-}}}\end{pmatrix}.$

By induction, we can show that the underlying TQFT $\omega_{g,n}$ of
$\Omega_{g,n}^{\mathbb{P}^{1}}$ is explicitly given by

$\omega_{g,n}(1^{\otimes a},\widetilde{H}^{\otimes
b})=\begin{cases}2^{g}\phi^{\frac{g-1+b}{2}},&\text{if $g-1+b$ is even,}\\\
0,&\text{else,}\end{cases}$ (9)

where we have used a shorthand notation for the arguments of $\omega_{g,n}$,
meaning $a$ insertions of 1 and $b$ insertions of $\widetilde{H}$. Notice that
the TQFT applied to basis vectors takes values in $\mathbb{Q}[\phi]$.

###  3.3 The $R$-matrix

As reviewed in the appendix, the $R$-matrix necessary for reconstructing the
equivariant Gromov–Witten theory of $\mathbb{P}^{1}$ can be found by using
virtual localization for a nontrivial $\mathbb{C}^{*}$-action and made
explicit using stationary phase asymptotics of certain oscillating integrals
on the so-called mirror of $\mathbb{P}^{1}$. The coefficients of the
$R$-matrix obtained from virtual localization are valued in
$\mathbb{Q}[\lambda^{\pm}][\\![q,z]\\!]$, but the mirror description shows
that the coefficients are obtained by expanding elements of
$\mathbb{Q}[\lambda^{\pm},\phi^{\pm}][\\![z]\\!]$ around $q=0$. In any case,
the $R$-matrix is defined over a nontrivial ring extension of
$\mathbb{Q}[\lambda,q]$, and it is, at first sight, not clear how this is
compatible with the fact that the CohFT is defined over
$\mathbb{C}[\lambda,q]$. As we will see in Section 3.5, such a contrast
implies the existence of tautological relations.

As reviewed in Section A.6 of the appendix (see also [9]), the $R$-matrix can
be computed from the stationary phase asymptotics of the oscillating integrals

$\frac{1}{\sqrt{-2\pi
z}}\int\limits_{\Gamma_{\pm}}e^{F(x)/z}\left(e^{x}-\frac{\lambda}{2}\right)^{1-i}\mathrm{d}x,$
(10)

where $\Gamma_{\pm}$ are suitable real, one-dimensional cycles, and

$F(x)=e^{x}+qe^{-x}-\lambda x$

is the superpotential. Let $u_{\pm}$ be the critical values at the critical
points

$e^{x}=\frac{\lambda}{2}\pm\sqrt{\phi}.$

of $F$. The $\Gamma_{\pm}$ are the corresponding Lefschetz thimbles, each
going through exactly one of the critical points.

We start with the stationary phase asymptotics for the integral (10) when
$i=1$. Shifting each critical point to the origin, rescaling $x$, expanding
the exponential and using

$\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{\infty}e^{-\frac{x^{2}}{2}}x^{i}\mathrm{d}x=\begin{cases}(i-1)!!,&\text{if
$i$ is even,}\\\ 0,&\text{else,}\end{cases}$

gives

$\displaystyle\frac{1}{\sqrt{-2\pi
z}}\int\limits_{\Gamma_{\pm}}e^{\left(u_{\pm}+\Delta_{\pm}\frac{x^{2}}{2}+\lambda\frac{x^{3}}{6}+\Delta_{\pm}\frac{x^{4}}{24}+\lambda\frac{x^{5}}{120}+\dotsb\right)/z}\mathrm{d}x$
$\displaystyle\asymp$
$\displaystyle\frac{e^{\frac{u_{\pm}}{z}}}{\sqrt{\Delta_{\pm}}\sqrt{2\pi}}\int\limits_{-\infty}^{\infty}e^{-\frac{x^{2}}{2}}e^{-\frac{x^{3}}{6}\frac{\lambda}{\Delta_{\pm}^{3/2}}\sqrt{-z}-\frac{x^{4}}{24}\frac{1}{\Delta_{\pm}}(-z)-\frac{x^{5}}{120}\frac{\lambda}{\Delta_{\pm}^{5/2}}(-z)^{3/2}-\dotsb}\mathrm{d}x$
$\displaystyle\asymp$
$\displaystyle\frac{e^{\frac{u_{\pm}}{z}}}{\sqrt{\Delta_{\pm}}}F_{0}\left(z\frac{\lambda^{2}}{\Delta_{\pm}^{3}},\frac{\Delta_{\pm}^{2}}{\lambda^{2}}\right)$

for some power series

$F_{0}(x,y)=1-\left(\frac{5}{24}-\frac{1}{8}y\right)x+\left(\frac{385}{1152}-\frac{77}{192}y+\frac{9}{128}y^{2}\right)x^{2}+\dotsb\in\mathbb{Q}[y][\\![x]\\!].$

When $i=0$, the additional factor

$e^{x}-\frac{\lambda}{2}$

in (10) becomes

$\left(\frac{\lambda}{2}\pm\sqrt{\phi}\right)e^{x}-\frac{\lambda}{2}$

after translation and

$\pm\sqrt{\phi}\left(\left(1+\frac{\lambda}{\Delta_{\pm}}\right)e^{x\Delta_{\pm}^{-1/2}\sqrt{-z}}-\frac{\lambda}{\Delta_{\pm}}\right).$

after scaling. So for $i=0$, the asymptotic expansion is given by

$\frac{\pm\sqrt{\phi}e^{\frac{u_{\pm}}{z}}}{\sqrt{\Delta_{\pm}}}F_{1}\left(z\frac{\lambda^{2}}{\Delta_{\pm}^{3}},\frac{\Delta_{\pm}^{2}}{\lambda^{2}}\right),$

where

$F_{1}(x,y)=1+\left(\frac{7}{24}-\frac{3}{8}y\right)x-\left(\frac{455}{1152}-\frac{33}{64}y+\frac{15}{128}y^{2}\right)x^{2}+\dotsb\in\mathbb{Q}[y][\\![x]\\!].$

In total, the $S$-matrix from the basis of normalized idempotents to the basis
$\\{1,\widetilde{H}\\}$ is given by

$\begin{pmatrix}\frac{\sqrt{\phi}}{\sqrt{\Delta_{+}}}F_{1}\left(z\frac{\lambda^{2}}{\Delta_{+}^{3}},\frac{\Delta_{+}^{2}}{\lambda^{2}}\right)&\frac{-\sqrt{\phi}}{\sqrt{\Delta_{-}}}F_{1}\left(z\frac{\lambda^{2}}{\Delta_{-}^{3}},\frac{\Delta_{-}^{2}}{\lambda^{2}}\right)\\\
\frac{1}{\sqrt{\Delta_{+}}}F_{0}\left(z\frac{\lambda^{2}}{\Delta_{+}^{3}},\frac{\Delta_{+}^{2}}{\lambda^{2}}\right)&\frac{1}{\sqrt{\Delta_{-}}}F_{0}\left(z\frac{\lambda^{2}}{\Delta_{-}^{3}},\frac{\Delta_{-}^{2}}{\lambda^{2}}\right)\end{pmatrix}\begin{pmatrix}e^{\frac{u_{+}}{z}}&0\\\
0&e^{\frac{u_{-}}{z}}\end{pmatrix}.$

Multiplying the first factor from the right by $\Psi^{-1}$ gives the
$R$-matrix written in the basis $\\{1,\widetilde{H}\\}$:

$R(z)=\begin{pmatrix}F_{1}^{e}&\sqrt{\phi}F_{1}^{o}\\\
\frac{1}{\sqrt{\phi}}F_{0}^{o}&F_{0}^{e}\end{pmatrix}\left(z\frac{\lambda^{2}}{\Delta_{+}^{3}},\frac{\Delta_{+}^{2}}{\lambda^{2}}\right)=\begin{pmatrix}F_{1}^{e}&\sqrt{\phi}F_{1}^{o}\\\
\frac{1}{\sqrt{\phi}}F_{0}^{o}&F_{0}^{e}\end{pmatrix}\left(z\frac{\lambda^{2}}{8\phi^{3/2}},4\frac{\phi}{\lambda^{2}}\right),$
(11)

where

$F_{i}^{e}(x,y)=\frac{F_{i}(x,y)+F_{i}(-x,y)}{2},\qquad
F_{i}^{o}(x,y)=\frac{F_{i}(x,y)-F_{i}(-x,y)}{2}.$

The symplectic condition (6) implies

$R^{-1}(z)=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}R^{t}(-z)\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$ (12)

So we can easily compute

$R^{-1}(z)=\begin{pmatrix}F_{0}^{e}&\sqrt{\phi}F_{1}^{o}\\\
\frac{1}{\sqrt{\phi}}F_{0}^{o}&F_{1}^{e}\end{pmatrix}\left(-z\frac{\lambda^{2}}{8\phi^{3/2}},4\frac{\phi}{\lambda^{2}}\right).$

######  Remark 4.

The power series $F_{i}^{e}(x,y)$ are even in $x$; the power series
$F_{i}^{o}(x,y)$ are odd. In particular, the coefficients of $R(z)$ in the
basis $\\{1,\widetilde{H}\\}$ are power series in $z$ whose coefficients are
Laurent polynomials in $\lambda$ and $\phi$.

###  3.4 Limit

In this section, we show how, out of the CohFT
$\Omega_{g,n}^{\mathbb{P}^{1}}$, a CohFT can be extracted that is very similar
to the theory of the shifted Witten’s 3-spin class as studied in [11].

We use the isomorphisms

$\mathbb{Q}[\lambda,q]\cong\mathbb{Q}[\lambda,\phi],\qquad
A_{\mathbb{C}^{*}}^{*}(\mathbb{P}^{1})\otimes\mathbb{Q}[q]\cong\mathbb{Q}[\lambda,\phi]\langle
1,\widetilde{H}\rangle,$

defined by

$q\mapsto\phi-\frac{\lambda^{2}}{4},\qquad
H\mapsto\widetilde{H}+\frac{\lambda}{2},$

once and for all so that the CohFT $\Omega_{g,n}^{\mathbb{P}^{1}}$ is defined
over the ring $\mathbb{Q}[\lambda,\phi]$ and based on the module
$\mathbb{Q}[\lambda,\phi]\langle 1,\widetilde{H}\rangle$. (It follows from the
discussion of Section 3.3 that we can use $\mathbb{Q}$-coefficients (instead
of $\mathbb{C}$-coefficients), but it is not of much importance which field we
use.)

We can define a new CohFT $\widetilde{\Omega}_{g,n}^{\mathbb{P}^{1}}$ on the
module $\mathbb{Q}[\lambda^{\pm},\phi]\langle 1,\widetilde{H}\rangle$ over the
ring $\mathbb{Q}[\lambda^{\pm 1},\phi]$ as the composition

$\widetilde{\Omega}_{g,n}^{\mathbb{P}^{1}}=\varphi\circ\Omega_{g,n}^{\mathbb{P}^{1}},$

where

$\varphi\colon A^{*}(\overline{M}_{g,n})\otimes\mathbb{Q}[\lambda,\phi]\to
A^{*}(\overline{M}_{g,n})\otimes\mathbb{Q}[\lambda^{\pm},\phi]$

is the ring homomorphism induced by

$C\mapsto\left(-\frac{4}{3\lambda^{2}}\right)^{i}C$

for any $C\in A^{i}(\overline{M}_{g,n})$. By definition, $\varphi$ leaves
elements of $A^{0}(\overline{M}_{g,n})\otimes\mathbb{Q}[\lambda,\phi]$
invariant and therefore the TQFTs of $\Omega_{g,n}^{\mathbb{P}^{1}}$ and
$\widetilde{\Omega}_{g,n}^{\mathbb{P}^{1}}$ are the same (up to the change of
base ring). Since the formal variable $z$ of an $R$-matrix measures degree in
the Chow ring, we have

$\widetilde{R}(z)=R\left(-\frac{4}{3}z\lambda^{-2}\right)=\begin{pmatrix}F_{1}^{e}&\sqrt{\phi}F_{1}^{o}\\\
\frac{1}{\sqrt{\phi}}F_{0}^{o}&F_{0}^{e}\end{pmatrix}\left(-\frac{z}{6\phi^{3/2}},4\frac{\phi}{\lambda^{2}}\right),$
(13)

where $R(z)$ and $\widetilde{R}(z)$ denote the $R$-matrices of
$\Omega_{g,n}^{\mathbb{P}^{1}}$ and
$\widetilde{\Omega}_{g,n}^{\mathbb{P}^{1}}$, respectively.

Notice that no positive power of $\lambda$ appears in the TQFT (9) and in
$\widetilde{R}$. So by the reconstruction, which also holds for
$\widetilde{\Omega}_{g,n}^{\mathbb{P}^{1}}$ in the Chow ring, the base ring
for the CohFT $\widetilde{\Omega}_{g,n}^{\mathbb{P}^{1}}$ can be taken to be
the intersection

$\mathbb{Q}[\lambda^{\pm},\phi]\cap\mathbb{Q}[\lambda^{-1},\phi^{\pm}]=\mathbb{Q}[\lambda^{-1},\phi].$

We define a new CohFT $\Omega_{g,n}$ on the module $\mathbb{Q}[\phi]\langle
1,\widetilde{H}\rangle$ over the base ring $\mathbb{Q}[\phi]$ by setting
$\lambda^{-1}=0$ in $\widetilde{\Omega}_{g,n}^{\mathbb{P}^{1}}$.
Reconstruction in the Chow ring for
$\widetilde{\Omega}_{g,n}^{\mathbb{P}^{1}}$ implies reconstruction in the Chow
ring for $\Omega_{g,n}$.

The CohFT $\Omega_{g,n}$ has the same underlying TQFT (9) as before but the
simpler $R$-matrix

$\begin{pmatrix}F_{1}^{e}&\sqrt{\phi}F_{1}^{o}\\\
\frac{1}{\sqrt{\phi}}F_{0}^{o}&F_{0}^{e}\end{pmatrix}\left(-\frac{z}{6\phi^{3/2}},0\right)=\begin{pmatrix}F_{1}^{e}&-\sqrt{\phi}F_{1}^{o}\\\
-\frac{1}{\sqrt{\phi}}F_{0}^{o}&F_{0}^{e}\end{pmatrix}\left(\frac{z}{6\phi^{3/2}},0\right).$

From their definitions, we can make the $F_{i}(z,0)$ explicit:

$\displaystyle F_{0}(z,0)$
$\displaystyle\asymp\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{\infty}e^{-\frac{x^{2}}{2}}e^{-\frac{x^{3}}{6}\sqrt{-z}}\mathrm{d}x\asymp\sum_{i=0}^{\infty}\frac{(6i-1)!!}{(2i)!}\frac{(-z)^{i}}{36^{i}}$
$\displaystyle=A(6z)$ $\displaystyle F_{1}(z,0)$
$\displaystyle\asymp\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{\infty}e^{-\frac{x^{2}}{2}}e^{-\frac{x^{3}}{6}\sqrt{-z}}(1+x\sqrt{-z})\mathrm{d}x\asymp\dotsb$
$\displaystyle=B(6z)$

Here, the Faber–Zagier $A$\- and $B$-series (2) appear. Using the symplectic
condition (12), we can also compute the inverse of the $R$-matrix to be equal
to

$\begin{pmatrix}A^{e}&\sqrt{\phi}B^{o}\\\
\frac{1}{\sqrt{\phi}}A^{o}&B^{e}\end{pmatrix}\left(\frac{z}{\phi^{3/2}}\right),$
(14)

where $A^{e}$ and $B^{e}$ are the even parts of $A$ and $B$, respectively, and
similarly $A^{o}$ and $B^{o}$ denote the odd parts. The coefficients of the
inverse $R$-matrix are elements of $\mathbb{Q}[\phi^{\pm}][\\![q]\\!]$.

In cohomology, the CohFT of the shifted Witten’s 3-spin class as described in
[11] coincides with $\Omega_{g,n}$ if we identify the parameters $\phi$ here
and in [11] as well as the bases $\\{1,\widetilde{H}\\}$ and
$\\{\partial_{x},\partial_{y}\\}$. This is because the TQFT (9) (compare to
[11, Lemma 3.3]) and inverse $R$-matrix (14) (compare to [11, Equation (14)])
completely agree.111Note that the TQFT and inverse $R$-matrix in [11] are
written in the rescaled basis $\\{\hat{\partial}_{x},\hat{\partial}_{y}\\}$,
where $\hat{\partial}_{x}=\phi^{1/4}\partial_{x}$ and
$\hat{\partial}_{y}=\phi^{-1/4}\partial_{y}$. It is not clear whether
$\Omega_{g,n}$ agrees with Witten’s 3-spin class in the Chow ring since it is
not known whether reconstruction in the Chow ring holds for the 3-spin theory.

###  3.5 Relations

Recall from the previous section that there exists a CohFT $\Omega_{g,n}$
defined on the $\mathbb{Q}[\phi]$-module $\mathbb{Q}[\phi]\langle
1,\widetilde{H}\rangle$, which coincides with the CohFT of shifted Witten’s
3-spin class in cohomology, but for which reconstruction holds in the Chow
ring. Let us fix a choice of $a_{1},\dotsc,a_{n}\in\\{0,1\\}$ for this
section.

The fact that the inverse $R$-matrix (14) has poles in $\phi$ implies the
existence of tautological relations: The reconstruction formula (7) for the
CohFT $\Omega_{g,n}$ defines an element

$\mathcal{R}_{g,n}(\widetilde{H}^{a_{1}},\dotsc,\widetilde{H}^{a_{n}})\in\mathcal{S}_{g,n}\otimes\mathbb{Q}[\phi^{\pm}],$

projecting to
$\Omega_{g,n}(\widetilde{H}^{a_{1}},\dotsc,\widetilde{H}^{a_{n}})$ under
$q\colon\mathcal{S}_{g,n}\to R^{*}(\overline{M}_{g,n})$. So all coefficients
of $\phi^{c}$ in $\mathcal{R}_{g,n}$ for $c<0$ have to project to zero and
thus are tautological relations. As we will see in this section these
relations turn out to be nontrivial and are in fact enough to prove Theorem 1.

| $\lambda$ | $q$ | $\phi$ | $C\in A^{i}(\overline{M}_{g,n})$
---|---|---|---|---
$\Omega_{g,n}^{\mathbb{P}^{1}}(\widetilde{H}^{a_{1}},\dotsc,\widetilde{H}^{a_{n}})$ | 1 | 2 | 2 | i
$\widetilde{\Omega}_{g,n}^{\mathbb{P}^{1}}(\widetilde{H}^{a_{1}},\dotsc,\widetilde{H}^{a_{n}})$ | 1 | 2 | 2 | 3i
$\Omega_{g,n}(\widetilde{H}^{a_{1}},\dotsc,\widetilde{H}^{a_{n}})$ |  |  | 2 | 3i
Table 1: Degree rules

We can characterize the part of $\mathcal{R}_{g,n}$ with poles in $\phi$ in a
different way: By the dimension analysis from Section 3.1, the coefficient of
$\lambda^{j}q^{d}$ in
$\Omega_{g,n}^{\mathbb{P}^{1}}(\widetilde{H}^{a_{1}},\dotsc,\widetilde{H}^{a_{n}})$
is an element of $A^{k}(\overline{M}_{g,n})$ for

$k=g-1-2d-j+\sum_{i=1}^{n}a_{i}.$

Hence,
$\Omega_{g,n}^{\mathbb{P}^{1}}(\widetilde{H}^{a_{1}},\dotsc,\widetilde{H}^{a_{n}})$
is homogeneous of degree

$g-1+\sum_{i=1}^{n}a_{i}$ (15)

if we assign degrees according to the first row of Table 1.

Recall from Section 3.4 that
$\widetilde{\Omega}_{g,n}^{\mathbb{P}^{1}}=\varphi\circ\Omega_{g,n}^{\mathbb{P}^{1}}$,
where $\varphi$ replaces each $C\in A^{i}(\overline{M}_{g,n})$ by a constant
multiple of $C\lambda^{-2i}$. Therefore
$\widetilde{\Omega}_{g,n}^{\mathbb{P}^{1}}(\widetilde{H}^{a_{1}},\dotsc,\widetilde{H}^{a_{n}})$
is also homogeneous of degree (15), if we assign degrees as in the second row
of Table 1. Finally,
$\Omega_{g,n}(\widetilde{H}^{a_{1}},\dotsc,\widetilde{H}^{a_{n}})$ is
homogeneous of the same degree (15) if we assign degrees as in the third row
of Table 1.

The conclusion is that the part of

$\mathcal{R}_{g,n}(\widetilde{H}^{a_{1}},\dotsc,\widetilde{H}^{a_{n}})$ (16)

with poles in $\phi$ corresponds to the part whose degree in the Chow ring is
greater than

$\frac{g-1+\sum_{i=1}^{n}a_{i}}{3}.$ (17)

This is exactly the degree of Witten’s 3-spin class. In fact, as discussed
before, in cohomology the CohFT $\Omega_{g,n}$ is the same as the CohFT of
Witten’s 3-spin class and, as in [11], the considered relations are those of
degree greater than the degree (17) of Witten’s class. So the relations we
consider are the same elements of the strata algebra as in [11]. Starting from
the fact that the Givental–Teleman classification of semisimple CohFTs holds
in the Chow ring for the CohFT $\Omega_{g,n}^{\mathbb{P}^{1}}$ we have shown
that the relations considered in [11] hold not only in cohomology but also in
the Chow ring.

In [11], the resulting relations are formally simplified to give the relations
of Theorem 1. Thus, we can conclude the proof of Theorem 1 here. For the
convenience of the reader we provide a summary of the arguments below.

###  3.6 Computing the relations

Following [11], we indicate how the relations from the vanishing of (16) in
degree greater than (17) imply the relations of Theorem 1. For simplicity, let
us set $\phi=1$. We expand the reconstruction formula (7) defining (16) using
the explicit inverse $R$-matrix (14) and the formula (9) for the TQFT. The
terms which give a relation are those of degree greater than (17), which
explains inequality (4). For a dual graph $\Gamma$, the powers of $2$ in (9)
yield

$\prod_{v}2^{g(v)}=2^{g}2^{-h^{1}(\Gamma)}.$

The edge term in the reconstruction formula is

$\frac{R^{-1}(\psi^{\prime})\otimes
R^{-1}(\psi^{\prime\prime})-\operatorname{Id}\otimes\operatorname{Id}}{-\psi^{\prime}-\psi^{\prime\prime}}\eta^{-1}\\\
=\frac{1\otimes\widetilde{H}+\widetilde{H}\otimes
1-(A^{e}(\psi^{\prime})1+A^{o}(\psi^{\prime})\widetilde{H})\otimes(B^{o}(\psi^{\prime\prime})1+B^{e}(\psi^{\prime\prime})\widetilde{H})}{\psi^{\prime}+\psi^{\prime\prime}}\\\
-\frac{(B^{o}(\psi^{\prime})1+B^{e}(\psi^{\prime})\widetilde{H})\otimes(A^{e}(\psi^{\prime\prime})1+A^{o}(\psi^{\prime\prime})\widetilde{H})}{\psi^{\prime}+\psi^{\prime\prime}}\\\
=[\Delta_{e}]_{\zeta^{\prime 0}\zeta^{\prime\prime 0}}(1\otimes
1)+[\Delta_{e}]_{\zeta^{\prime 1}\zeta^{\prime\prime 0}}(\widetilde{H}\otimes
1)+[\Delta_{e}]_{\zeta^{\prime 0}\zeta^{\prime\prime
1}}(1\otimes\widetilde{H})+[\Delta_{e}]_{\zeta^{\prime 1}\zeta^{\prime\prime
1}}(\widetilde{H}\otimes\widetilde{H}),$

where $[\Delta_{e}]_{\zeta^{\prime i}\zeta^{\prime\prime j}}$ is the
coefficient of $\zeta^{\prime i}\zeta^{\prime\prime j}$ in the edge series
$\Delta_{e}$ from the introduction. Similarly, in the reconstruction formula,
the contribution of the $i$th marking and of an extra marking can be expressed
in terms of $B_{i}$ and $\kappa_{v}$, respectively. As necessitated by the
case distinction in (9), we can keep track of the parity of the number of
$\widetilde{H}$-insertions into the TQFT at vertex $v$ using an additional
variable $\zeta_{v}$. By careful inspection, we see that (16) at $\phi=1$
coincides with (3) times $2^{g}$. We have thus arrived at the statement of
Theorem 1.

##  Appendix A Virtual localization for projective spaces (after Givental)

In this appendix, we recall Givental’s localization calculation [12] (see also
[20] for a more leisurely treatment), 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. In
Section A.5, we collect properties of the $R$-matrix. Finally, in Section A.6,
we give Givental’s mirror description [18] of the $R$-matrix.

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

Let $T=(\mathbb{C}^{*})^{m+1}$ act diagonally on $\mathbb{P}^{m}$. The
equivariant Chow rings of a point $\mathrm{pt}$ 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

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

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 [21] implies that the virtual fundamental
class can be split into a sum

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

of contributions of fixed loci $X$. Here, $N^{\mathrm{vir}}_{X,T}$ 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 defined only after localizing by the elements
$\lambda_{0},\dotsc,\lambda_{m}$. By studying the $\mathbb{C}^{*}$-action on
deformations and obstructions of stable maps, $e_{T}(N^{\mathrm{vir}}_{X,T})$
can be computed explicitly.

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

  * •

a graph $(V,E)$,

  * •

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

  * •

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

  * •

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

  * •

an assignment $p\colon\\{1,\dotsc n\\}\to V$ 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 conditions
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. An unstable vertex $v$ with $(g(v),n(v))=(0,2)$ corresponds to either
a node connecting two multiple covers of a line or a marking at the end of one
of the multiple covers. When $(g(v),n(v))=(0,2)$, the vertex $v$ corresponds
to an unmarked point at the end of a multiple cover. 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
$e_{T}(N^{\mathrm{vir}}_{X,T})$ is a product of factors corresponding to the
geometry of the graph:

$e_{T}(N^{\mathrm{vir}}_{X,T})=\prod_{v\text{,
stable}}\frac{e(\mathbb{E}^{*}\otimes
T_{\mathbb{P}^{m},\zeta(v)})}{e_{\zeta(v)}}\prod_{\text{nodes}}\frac{e_{\zeta}}{-\psi_{1}-\psi_{2}}\\\
\prod_{\begin{subarray}{c}g(v)=0\\\
n(v)=1\end{subarray}}(-\psi_{v})\prod_{e}\mathrm{Contr}_{e}$ (18)

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 $n(v)=2$. By $\psi_{1}$ 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

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

where $p_{j}$ is the other fixed point on the fixed line. The explicit
expressions for the last two factors of (18) can be found in [21], but will
play no role for us. It is only important that they depend only 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

$\Omega_{g,n}^{t}(v_{1},\dotsc,v_{n})\\\
=\sum_{d,k=0}^{\infty}\frac{q^{d}}{k!}\pi_{*}p_{*}\left(\prod_{i=1}^{n}\operatorname{ev}_{i}^{*}(v_{i})\prod_{i=n+1}^{n+k}\operatorname{ev}_{i}^{*}(t)\cap[\overline{M}_{g,n+k}(\mathbb{P}^{m},d)]^{\mathrm{vir}}\right),$
(19)

where

$t=t_{0}\phi_{0}+\dotsb+t_{m}\phi_{m}$

is a formal point on $W$, the map $\pi$ forgets the last $k$ markings and $p$
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 for an endomorphism-valued power series
$R_{\mathbb{P}^{m}}(z)$. In the following we will systematically suppress the
dependence on $t$ in the notation. In the main part of this paper, we are
interested in the case where $t$ is set to zero.

For the localization computation, we start by remarking that for each
localization graph for (19) there exists a dual graph of $\overline{M}_{g,n}$
corresponding to the topological type of the stabilization under $\pi p$ 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 of rational curves:

  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

  3. 3.

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

Here, a stable component is a component of the source curve not contracted by
the stabilization for $\pi p$. The first two types of trees correspond,
respectively, to the preimage under the stabilization of

  1. 1.

one of the $n$ markings,

  2. 2.

one of the nodes.

Each type of tree gives rise to a series of localization contributions, and we
want to record it using the fact that the same contributions already occur in
genus zero.

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 $\mathbb{P}^{m}$. We will later identify $W^{\prime}$ with $W$
in a nontrivial way (see Equation (28)). The type (1) contributions are
recorded by

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

the homomorphism-valued power series such that

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

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

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

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

The type (2) contributions are recorded by the bivector

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

which is defined by

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

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

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

Finally, the type (3) contribution is a vector

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

which is defined by

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

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

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

With these contributions we can write the CohFT already in a form quite
similar to the $R$-matrix action formula (7). 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 for $n=0$ is defined similarly as in Example 1. We have

$\Omega_{g,n}^{t}(v_{1},\dotsc,v_{n})=\sum_{\Gamma}\frac{1}{\operatorname{Aut}(\Gamma)}\xi_{*}\left(\prod_{v}\sum_{k=0}^{\infty}\frac{1}{k!}\pi_{*}\omega_{g_{v},n_{v}+k}(\dots)\right),$
(20)

where we put

  1. 1.

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

  2. 2.

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

  3. 3.

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

We will still need to apply the string and dilaton equation in order to make
$\widetilde{T}(\psi)$ a multiple of $\psi^{2}$, like the corresponding series
in the reconstruction, express the Hodge classes via Mumford’s formula 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

$\sum_{k=0}^{\infty}\frac{1}{k!}\pi_{*}\left(\prod_{i=1}^{n}\frac{1}{-x_{i}-\psi_{i}}\prod_{i=n+1}^{n+k}Q(\psi_{i})\right),$
(21)

where $\pi\colon\overline{M}_{g,n+k}\to\overline{M}_{g,n}$ 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, (21) 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 $-u$, that is applying
$e^{t\mathcal{L}^{\prime}}|_{t=-u}$ to (21), gives

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

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 $Q^{\prime}$ satisfies
$Q^{\prime}_{0}=0$. This is because the string equation implies
$\mathcal{L}u=1$ and therefore applying $e^{t\mathcal{L}}|_{t=-u}$ to $u$
gives, on the one hand, $u-t|_{t=-u}=0$ and, on the other hand, the definition
of $u$ with $Q$ replaced by $Q^{\prime}$, which for dimension reasons is a
nonzero multiple of $Q^{\prime}_{0}$.

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

$\sum_{k=0}^{\infty}\frac{1}{k!}\pi_{*}\left(\prod_{i=1}^{n}\frac{1}{-x_{i}-\psi_{i}}\prod_{i=n+1}^{n+k}Q(\psi_{i})\right)\\\
=\sum_{k=0}^{\infty}\frac{\Delta^{\frac{2g-2+n+k}{2}}}{k!}\pi_{*}\left(\prod_{i=1}^{n}\frac{e^{-\frac{u}{x_{i}}}}{-x_{i}-\psi_{i}}\prod_{i=n+1}^{n+k}Q^{\prime\prime}(\psi_{i})\right),$
(22)

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

$\frac{1}{-z-w}+\sum_{k=1}^{\infty}\frac{1}{k!}\int\limits_{\overline{M}_{0,2+k}}\frac{1}{-z-\psi_{1}}\frac{1}{-w-\psi_{2}}\prod_{i=3}^{2+k}Q(\psi_{i})=\frac{e^{-u/z+-u/w}}{-z-w}.$
(23)

In order to see that (23) is true, we use that the left-hand side is
annihilated by $\mathcal{L}+1/z+1/w$ in order to move from $Q$ to $Q^{\prime}$
via the string flow and notice that there all the integrals vanish for
dimension reasons. Similarly, there is the identity

$1-\frac{Q(z)}{z}-\frac{1}{z}\sum_{k=2}^{\infty}\frac{1}{k!}\int\limits_{\overline{M}_{0,1+k}}\prod_{i=2}^{1+k}Q(\psi_{i})\\\
=e^{-u/z}\left(1-\frac{Q^{\prime}(z)}{z}\right)=e^{-u/z}\left(\Delta^{-\frac{1}{2}}-\frac{Q^{\prime\prime}(z)}{z}\right),$
(24)

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

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

###  A.4 Computation of the localization series

We apply (22) to (20) and obtain

$\Omega_{g,n}^{t}(v_{1},\dotsc,v_{n})=\sum_{\Gamma}\frac{1}{\operatorname{Aut}(\Gamma)}\xi_{*}\left(\prod_{v}\sum_{k=0}^{\infty}\frac{1}{k!}\pi_{*}\omega^{\prime}_{g_{v},n_{v}+k}(\dots)\right),$
(25)

where we put

  1. 1.

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

  2. 2.

one 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 $\widetilde{R}^{-1}$,
$\widetilde{V}$ and $\widetilde{T}$ 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

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

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 (23), where the first summand stands
for the case that the vertex is unstable and the second summand stands for the
case that the vertex is stable with $k$ trees of type 3 and one tree of type 1
attached to it. Applying (23), we obtain

$S_{i}^{-1}(z)=e^{-\frac{u_{i}}{z}}R_{i}^{-1}(z).$ (26)

Using the shorthand notation

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

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

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

By the identity axiom and Witten-Dijgraaf-Verlinde-Verlinde equation, we have

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

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

$\frac{S_{i}^{-1}(z)\otimes S_{j}^{-1}(w)}{-z-w}\eta^{-1}\\\
=\frac{\eta(e_{i}\phi_{i},e_{j}\phi_{j})}{-z-w}+\sum_{d,k=0}^{\infty}\frac{q^{d}}{k!}\int\limits_{\overline{M}_{0,2+k}(\mathbb{P}^{m},d)}\frac{\operatorname{ev}_{1}^{*}(e_{i}\phi_{i})}{-z-\psi_{1}}\frac{\operatorname{ev}_{2}^{*}(e_{j}\phi_{j})}{-w-\psi_{2}}\prod_{l=3}^{k+2}\operatorname{ev}_{l}^{*}(t).$
(27)

We compute the right-hand side via localization. There are two cases in the
localization depending on whether the first and second markings are at the
same vertex or at different ones. In the first case, we apply (23) 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 (23) at the
two vertices and obtain

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

So all together

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

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

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

via localization. Applying (24) 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 (25), the underlying TQFT of $\Omega^{t}_{g,0}$ 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 $\widetilde{T}$ vanishes at $(t,q)=0$, we
have that $\Delta_{i}$ agrees with $e_{i}$ at $(t,q)=0$. Therefore, we can
identify $W^{\prime}$ with $W$ via

$W^{\prime}\to W,\qquad w_{i}\mapsto\sqrt{\Delta_{i}/e_{i}}\ \epsilon_{i},$
(28)

where $\epsilon_{i}$ is the idempotent element which coincides with $\phi_{i}$
at $(t,q)=0$. The previous results then say exactly that $\Omega^{t}$ 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 $R_{\text{Mumford}}$ 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 define the endomorphism $R_{\mathbb{P}^{m}}$ to be the composition

$R_{\mathbb{P}^{m}}=R\cdot R_{\text{Mumford}}.$

Therefore, the $R$-matrix action of $R_{\mathbb{P}^{m}}$ takes the TQFT to
$\Omega^{t}$.

###  A.5 Characterization of $R$-matrix

We study the $R$-matrix $R_{\mathbb{P}^{m}}$ defined in the previous section
as the composition of $R$ and $R_{\text{Mumford}}$.

As we have seen, there is a power series $S^{-1}(z)$ in $z^{-1}$ strongly
related to $R$. By considering (27) as $w+z\to 0$, we see that $S^{-1}(z)$
satisfies the symplectic condition; that is, its inverse $S(z)$ is the adjoint
of $S^{-1}(-z)$ with respect to $\eta$. Using this, we can say, more
explicitly, that the evaluation of $S(z)$ at the $i$th normalized idempotent
is the vector

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

In terms of $S(z)$, the relation between $R$ and $S$, when written in the
basis of normalized idempotents, is

$R(z)=S(z)e^{-\operatorname{diag}(u_{0},\dotsc,u_{m})/z}.$ (29)

The series $S(z)$ satisfies the _quantum differential equation_

$z\frac{\partial}{\partial t_{\mu}}S(z)=\phi_{\mu}\star S(z)$ (30)

for any $\mu$. This follows from the genus zero topological recursion
relations

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

where, as previously, $\eta^{-1}$ should be inserted at the two $\bullet$.

The divisor axiom in Gromov–Witten theory determines the $q$-dependence of
$S$. We have

$\left\langle\frac{\sqrt{e_{i}}\phi_{i}}{z-\psi},\eta^{-1},H\right\rangle=\left\langle\frac{\sqrt{e_{i}}H\phi_{i}}{z(z-\psi)},\eta^{-1}\right\rangle+d\left\langle\frac{\sqrt{e_{i}}\phi_{i}}{z-\psi},\eta^{-1}\right\rangle\\\
=\left(\frac{\lambda_{i}}{z}+d\right)\left\langle\frac{\sqrt{e_{i}}\phi_{i}}{z-\psi},\eta^{-1}\right\rangle$

and hence

$H\star S_{i}(z)=DS_{i}(z)+\lambda_{i}S_{i}(z),$ (31)

where

$D=zq\frac{\partial}{\partial q}.$

The analog of equation (29) defines a new series $S_{\mathbb{P}^{1}}$. It
satisfies the same symplectic condition, quantum differential equation and
$q$-dependence (31) since $R_{\text{Mumford}}$ is diagonal and does not depend
on the $t_{i}$.

Let us consider the classical limit $(t,q)\to 0$. In this case,
$\Delta_{i}=e_{i}$, $w_{i}=\phi_{i}$ and $\widetilde{R}=R=\operatorname{Id}$.
Therefore,

$R_{\mathbb{P}^{1}}|_{(t,q)=0}=R_{\text{Mumford}}=\exp(\operatorname{diag}(b_{0},\dotsc,b_{m})),$
(32)

where

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

######  Proposition 1.

The series $R_{\mathbb{P}^{1}}(z)$ written in the basis of normalized
idempotents is the unique matrix-valued power series in $z$ depending on
parameters $\lambda_{0},\dotsc,\lambda_{m}$, $q$ and $t_{0},\dotsc,t_{m}$ such
that there exist functions $u_{0},\dotsc,u_{m}$ independent of $z$ and a
matrix-valued power series $S_{\mathbb{P}^{1}}$ in $z^{-1}$ such that

  1. 1.

the series $R_{\mathbb{P}^{1}}$ and $S_{\mathbb{P}^{1}}$ are related by (29),

  2. 2.

$S_{\mathbb{P}^{1}}$ satisfies the quantum differential equation (30),

  3. 3.

$S_{\mathbb{P}^{1}}$ satisfies (31) and

  4. 4.

the classical limit $(t,q)\to 0$ of $R_{\mathbb{P}^{1}}$ is given by (32).

###### Proof.

In this Section we have checked that $R_{\mathbb{P}^{1}}$ indeed satisfies
these properties.

By a general analysis [18, Proposition 6.7] of the quantum differential
equation we see that the first two properties determine $R_{\mathbb{P}^{1}}$
up to functions constant in the $t_{i}$. Via (29), equation (31) also
determines the dependence of $R_{\mathbb{P}^{1}}$ on $q$. Hence
$R_{\mathbb{P}^{1}}$ is uniquely determined from its classical limit. ∎

###  A.6 Small equivariant mirror

Following [18], we give a concrete description of the $R$-matrix for the
equivariant Gromov–Witten theory of $\mathbb{P}^{m}$. In contrast to the
previous sections of the appendix, we will soon restrict ourselves to the
small quantum cohomology case $t=0$. To simplify the notation, we will leave
out the subscript $\mathbb{P}^{m}$ in $R_{\mathbb{P}^{m}}$ and
$S_{\mathbb{P}^{m}}$.

If we use alternative coordinates

$t=\widetilde{t}_{0}+\widetilde{t}_{1}H+\dotsb+\widetilde{t}_{m}H^{m},$

the quantum differential equation implies

$z\frac{\partial}{\partial\widetilde{t}_{1}}S_{i}(z)=H\star
S_{i}(z)=DS_{i}(z)+\lambda_{i}S_{i}(z).$ (33)

So we can rewrite the differential equation (33) satisfied by $S_{i}$ to

$\left(zq\frac{\partial}{\partial q}+\lambda_{i}\right)S_{i}(z)=H\star
S_{i}(z).$ (34)

Now it makes sense to restrict to $t=0$ and we will do so from now on (without
changing the notation).

If we write $S_{\mu i}=\eta(S_{i},\phi_{\mu})$, we can rewrite (34) to

$D(S_{\mu i}e^{\ln(q)\lambda_{i}/z})=S_{(\mu+1)i}e^{\ln(q)\lambda_{i}/z}$ (35)

for $\mu<m$ and

$\prod_{j=0}^{m}(D-\lambda_{j})(S_{0i}e^{\ln(q)\lambda_{i}/z})=qS_{0i}e^{\ln(q)\lambda_{i}/z}.$
(36)

######  Proposition 2.

The series $R$ is the unique matrix-valued power series in $z$ depending on
parameters $\lambda_{0},\dotsc,\lambda_{m}$, $q$ such that there exist
functions $u_{0},\dotsc,u_{m}$ independent of $z$ and a matrix-valued power
series $S$ in $z^{-1}$ such that

  1. 1.

the series $R$ and $S$ are related by (29),

  2. 2.

$S$ satisfies the quantum differential equations (35) and (36),

  3. 3.

the classical limit $q\to 0$ of $R$ is given by (32).

###### Proof.

Equation (35) determines all of $S$ from $S_{0i}$, which is determined from
(36) up to additive integration constants depending on
$\lambda_{0},\dotsc,\lambda_{m}$. Choosing the integration constants such that
the classical limit condition holds determines $S$ up to constants (not
depending on $\lambda_{0},\dotsc,\lambda_{m}$), which is enough to determine
$R$ uniquely via (29). ∎

Givental constructs asymptotic solutions $S_{0i}$ to (36) via oscillating
integrals on the mirror manifold

$\\{(T_{0},\dotsc,T_{m}):e^{T_{0}}\cdot\dots\cdot
e^{T_{m}}=q\\}\subset\mathbb{C}^{m+1}$

with superpotential

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

The integrals are given by

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

along $m$-cycles $\Gamma_{i}$ through a specific critical point of the
superpotential $F$ constructed using the Morse theory of the real part of
$F/z$. 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 $m+1$ critical points at which it is possible to do a stationary
phase expansion of $S_{0i}$. Let us write $P_{i}$ 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 factor222The
function $u_{i}$ of this appendix does not completely agree with the function
$u_{i}$ of Section A.3. They differ by the constant $c=\sum_{j\neq
i}(\lambda_{i}-\lambda_{j})(-1+\ln(\lambda_{i}-\lambda_{j}))$.
Correspondingly, the $S_{i}$ defined via the oscillating integral also
coincides with the series from localization only up to a factor $e^{c/z}$.
These differences are irrelevant for the computation of the $R$-matrix.

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

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

$S_{0i}=e^{u_{i}/z}\int\exp\left(-\sum_{j}(P_{i}-\lambda_{j})\sum_{k=3}^{\infty}\frac{T_{j}^{k}(-z)^{(k-2)/2}}{k!}\right)\mathrm{d}\mu_{i}$
(37)

for the conditional Gaussian distribution

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

For the asymptotic expansion, we formally expand the exponential in (37) and
integrate over the real part of the image of the mirror under the
transformations we have performed. The integrals are moments of $\mu_{i}$
which can be computed using the covariance matrix

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

Since odd moments of Gaussian distributions, vanish we find that the
asymptotic expansion of $e^{-u_{i}/z}S_{0i}$ is a power series in $z$. So,
using (26) as a definition, by (35) the entries of the $R$-matrix in the basis
of normalized idempotents are given by similar asymptotic expansion of

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

We need to check that the $R$-matrix given in terms of asymptotics of
oscillating integrals behaves correctly in the limit as $q\to 0$. By
definition, in this limit, $P_{i}\to\lambda_{i}$. By symmetry, it is enough to
consider the zeroth 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 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 as $q\to 0$, 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 (32) of $R_{00}$ for $q\to 0$.

## References

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Departement Mathematik
ETH Zürich
<EMAIL_ADDRESS>
Institut de Mathématiques de Jussieu
Paris
<EMAIL_ADDRESS>


# Nonreciprocal electromagnetic wave manipulation via a single reflection

###### Abstract

Electric field manipulation plays a key role in applications such as electron
acceleration, nonlinear light-matter interaction, and radiation engineering.
Nonreciprocal materials, such as Weyl semimetals, enable the manipulation of
the electric field in a full photonic manner, owing to their intrinsic time-
reversal symmetry breaking, leading to asymmetric material response for
photons with $+\mathbf{k}$ and $-\mathbf{k}$ momenta. Here, the results
suggest that a simple planar interface between semi-infinite air and a
nonreciprocal material can achieve spatio-temporal manipulation of the
electric field. In particular, this work presents three compelling scenarios
for electromagnetic wave manipulation: radiation pattern redistribution (with
closed-form expressions provided), carrier-envelope-phase control, and spatial
profile control. The presented results pave the way for electric field
manipulation using pattern-free nonreciprocal materials.

###### keywords:

nonreciprocity, electric field manipulation, photonics, far-field

Lu Wang*

Dr. Lu Wang
Address: State Key Laboratory of Magnetic Resonance and Atomic and Molecular
Physics, Wuhan Institute of Physics and Mathematics, Innovation Academy for
Precision Measurement Science and Technology, Chinese Academy of Sciences,
Wuhan 430071, China
Email Address<EMAIL_ADDRESS>
##  1 Introduction

Electric field manipulation plays a key role in applications such as electron
acceleration [1], nonlinear light-matter interaction [2], and radiation
engineering [3]. Though with crucial impacts, the manipulation of the electric
field is commonly achieved by complicated optical elements such as
metamaterial [4], grating [5, 6], and wave plate [7]. These complicated
structures bring challenges in manufacturing and thus limit related
applications. In addition, it is worth noting that the majority of optical
elements exhibit limited responsiveness within specific frequency ranges.
Consequently, even if these elements are accessible in the optical regime,
acquiring them for long wavelengths, proves to be exceedingly difficult due to
significant loss during propagation through the optical elements and the
pronounced diffraction effects experienced by long wavelengths.

In light of these obstacles, I propose an alternative method that utilizes
reflection on planar surfaces to manipulate electric fields. The potential
applicability of this approach spans a broad frequency range, encompassing
terahertz to optical frequencies. It enables effortless engineering in a
purely photonics-based manner, opening up exciting opportunities in this
field. This can be realized via nonreciprocal materials such as Weyl
semimetals.

To comprehensively understand the spatio-temporal behavior of the electric
field when influenced by nonreciprocal materials, I investigate three
different scenarios. In the first scenario, I explore how the presence of the
nonreciprocal material influences the radiation pattern of an ideal dipole,
where closed-form expressions are presented. Furthermore, I examine how the
nonreciprocal material affects the carrier-envelope phase and phase-front
distortions of the electric field, which are relevant for practical
applications. The presented results pave the way for electric field
manipulation using pattern-free nonreciprocal materials.

##  2 Theoretical model

To achieve the proposed aim i.e. electromagnetic wave manipulation using a
planar structure, the nonreciprocal material is the key. Weyl semimetals, a
newly discovered class of quantum materials, would serve the purpose. Weyl
semimetals possess unique intrinsic time-reversal symmetry breaking, which
leads to nonreciprocity. In the momentum space, Weyl nodes appear in pairs
with opposite chirality, where the Weyl points are the sink or source of the
Berry curvature [8, 9]. Several Weyl semimetals have been experimentally
discovered such as $\text{WP}_{\text{2}}$, $\text{HgCr}_{2}\text{Se}_{4}$,
TaAs and $\text{Co}_{3}\text{Sn}_{2}\text{S}_{2}$ [10, 11, 12, 13], which are
ready to be explored and implemented. The material property of the
nonreciprocal material can be characterized via the permittivity tensor [14]

$\bm{\epsilon}=\begin{pmatrix}\epsilon_{d}&0&\epsilon_{xz}\\\
0&\epsilon_{d}&0\\\ -\epsilon_{xz}&0&\epsilon_{d}\end{pmatrix},$ (1)

where without loss of generalization, the tensor elements are chosen to be
$\epsilon_{d}=1.5+0.3\,i$ and $\epsilon_{xz}=1.5\,i$, which are standard
values. Note that Equation (1) is written in the Cartesian coordinates with
azimuthal angle $\phi=0$ (see Figure 1).

The reflection of the nonreciprocal material is calculated by the transfer-
matrix method [3, 15, 16]. The propagation of the fields is derived using the
dyadic Green function. Detailed analyses related to the dyadic Green function
can be found in the supplementary material (referred to as SM in the following
context).

##  3 Results

###  3.1 Radiation distribution manipulation

Figure 1: Illustration of the dipole emitter $\bm{\mu}$ in the presence of the
nonreciprocal material. The dipole is located at $(0,0,z_{0})$ and dipole
moment is aligned along the $z$ axis $\bm{\mu}=(0,0,\mu_{z})$. The observation
point is denoted by a black dot $r=(x,y,z)$. In the far-field limit, there are
only two values of momenta responding to two paths $p_{0}$ and $p_{r}$ that
are relevant. The path directly links the source and the observation point is
represented by $p_{0}$. The path that reaches the observation point via
reflection off from the nonreciprocal material surface is represented by
$p_{r}$. The variables $\theta_{0}$ and $\theta_{r}$ are polar angles of
$p_{0}$ and $p_{r}$, respectively.

I focus on the ideal dipole as the radiation source since in principle, the
results can be extended to the emission of arbitrary objects using the
discrete dipole approximation [17]. The propagation of the radiation is
governed by the dyadic Green function (see SM Sections S2 and S3).

With the ideal dipole (point source) located at $r_{0}=(0,0,z_{0})$, and the
dipole moment $\bm{\mu}=(0,0,\mu_{z})$, one can write the total electric field
as

$\mathbf{E}=\omega_{0}^{2}\mu_{0}\mathbf{G}^{E}\bm{\mu}=\omega_{0}^{2}\mu_{0}\left(\mathbf{G}_{0}^{E}+\mathbf{G}_{r}^{E}\right)\bm{\mu}=\mathbf{E}_{0}+\mathbf{E}_{r},$
(2)

where $\mathbf{G}^{E}$ is a rank two tensor and represents the dyadic Green
function, $\omega_{0}=2\pi c/\lambda_{0}$ is the angular frequency, $c$ is the
speed of light in vacuum, $\lambda_{0}$ is the wavelength of the radiation in
the vacuum, and $\mu_{0}$ is the vacuum permeability. The subscripts "0" and
"r" represent the initial electric field radiated by the dipole and field
after the interaction (reflection) with the nonreciprocal material,
respectively.

Commonly the Weyl identity is used to describe the dyadic Green function,
which requires integrating over the entire $k_{x},\,k_{y}$ momentum space [18,
19]. This double integral can be cumbersome to analyze the underlying physics
and needs high computational demand for implementations. Here, with the far-
field limit and the stationary phase approximation [20], I present closed-form
expressions (SM Section S3.3 ).

Figure 2: Radiation patterns of the ideal dipole in the presence of vacuum
(a), reciprocal material (b), and nonreciprocal material (c). The observation
plane is chosen to be $z=0.1\lambda_{0}$. The dipole location
$z_{0}=25\lambda_{0}$. The amplitude of the time-averaged Poynting vector is
represented by $|{S}|$, where
$\bm{S}=(S_{x},S_{y},S_{z})=\frac{1}{2}\text{Re}[\bm{E}\times\bm{H}^{*}]$. The
white arrows on each panel represent vector $(S_{x},S_{y})$.

The results obtained indicate that the final radiation pattern can be
attributed to two distinct propagation paths, denoted as $p_{0}$ and $p_{r}$
(see Figure 1). The $p_{0}$ path corresponds to $\mathbf{G}_{0}^{E}$, where
the radiation travels directly from the dipole to the observation point,
covering a distance denoted as $r_{0}=\sqrt{x^{2}+y^{2}+(z-z_{0})^{2}}$. On
the other hand, the $p_{r}$ path corresponds to $\mathbf{G}_{r}^{E}$, where
the radiation from the dipole reaches the surface of the nonreciprocal
material and is subsequently reflected towards the observation point. The
distance covered along this path is denoted as
$r_{r}=\sqrt{x^{2}+y^{2}+(z+z_{0})^{2}}$. To facilitate further analysis, one
can define
$f_{n}={\omega_{0}^{2}\mu_{0}\mu_{z}\sin{(\theta_{n})}\exp{(ik_{0}r_{n})}}/{(4\pi
r_{n})}$ with $n\in\\{0,r\\}$, where "0" and "r" again represent $p_{0}$ and
$p_{r}$, respectively. The analytical expressions of the electric fields can
be written as:

$\displaystyle\mathbf{E}_{0}=$ $\displaystyle
f_{0}\left\\{-\cos{(\theta_{0})}\left[\cos{(\phi)}\hat{x}+\sin{(\phi)}\hat{y}\right]+\sin{(\theta_{0})}\hat{z}\right\\}=-f_{0}\hat{\theta}_{0}$
(3) $\displaystyle\mathbf{E}_{r}=$ $\displaystyle
f_{r}\left\\{\left[-r_{pp}\cos{(\theta_{r})}\cos{(\phi)}+r_{ps}\sin{(\phi)}\right]\hat{x}-\left[r_{pp}\cos{(\theta_{r})}\sin{(\phi)}+r_{ps}\cos{(\phi)}\right]\hat{y}+r_{pp}\sin{(\theta_{r})}\hat{z}\right\\}$
$\displaystyle=$ $\displaystyle-
f_{r}\left(r_{pp}\hat{\theta}_{r}+r_{ps}\hat{\phi}\right).$ (4)

The unit vector in the Cartesian coordinates is written as
$(\hat{x},\hat{y},\hat{z})$ and the unit vector in the Spherical coordinates
is written as $(\hat{r},\hat{\theta}_{n},\hat{\phi})$, $n\in\\{0,\text{r}\\}$.
The relation between the unit vectors in Cartesian and spherical coordinates
can be found in SM Equation S26-S30. In this context, $\theta_{0}$ corresponds
to the polar angle associated with $p_{0}$, whereas $\theta_{r}$ represents
the polar angle of $p_{r}$ and also serves as the angle of incidence at the
surface of the nonreciprocal material. Additionally, $\phi$ denotes the
azimuthal angle. The reflection coefficient is denoted by $r_{ij}$ with
$i,\,j\,\in\\{s,p\\}$ and the $r_{sp}$, for example, represents the reflection
from s- to p-polarized light.

Since the dipole radiation in the far-field only contains p-polarisation [21],
$r_{ss}$ and $r_{sp}$ do not show up in Equation (4). A more general
expression can be seen in SM Section 3. The results for radiation patterns in
the presence of different materials are shown in Figure 2. The radiation
pattern is defined as the amplitude of the time-averaged Poynting vector
$|\bm{S}|$, where $\bm{S}=\frac{1}{2}\text{Re}[\bm{E}\times\bm{H}^{*}]$,
$\bm{H}$ is the magnetic field strength, and Re$[\,]$ represents taking the
real part. Without loss of generalization, the parameters for plotting are
chosen to be $z_{0}=25\lambda_{0}$, and $z=0.1\lambda_{0}$.

Now it has been proven that a simple air-nonreciprocal material planar surface
can lead to asymmetric radiation distribution. In the following context, I
focus specifically on the case presented in Figure 2c, where the material is
nonreciprocal with $\epsilon_{d}=1.5+0.3\,i$ and $\epsilon_{xz}=1.5\,i$.

###  3.2 Carrier-Envelope-Phase (CEP) manipulation

Here we investigate how nonreciprocal material can be leveraged to manipulate
the carrier-envelope-phase (CEP) of the pulse. Without loss of generalization,
we look into one example with $\phi=45$ degree and the pulse duration related
parameter $\tau=5/\omega_{0}$. By defining the amplitude of the s- and p-
polarized light as

$\displaystyle E_{s}=\exp{(-t^{2}/\tau^{2})}\exp{(iw_{0}t)},$ (5)
$\displaystyle E_{p}=\exp{(-t^{2}/\tau^{2})}\exp{(iw_{0}t+i\pi/2)}$ (6)

one can write the input circularly polarized electric field as
$\bm{E}_{\text{in}}(t)=\rm Re$$[E_{s}\hat{s}+E_{p}\hat{p}]$ (see Figure 3b),
where $\hat{s}$ ($\hat{p}$) is the unit vector for s-polarisation
(p-polarisation). Since $\bm{E}_{\text{in}}(t)$ is circularly polarized, the
CEP of the $E_{s}$ and $E_{p}$ are 0 and $pi/2$ (90 degrees), respectively.
The electric field after reflection from the nonreciprocal material surface is
written as

$E_{\text{out}}(t)=\text{Re}[r_{sp}E_{s}\hat{p}+r_{ps}E_{p}\hat{s}+r_{pp}E_{p}\hat{p}+r_{ss}E_{s}\hat{s}].$
(7)

Figure 3: Panel a displays the CEP of the s- and p- polarized light plotted
against the incidence angle. The vertical dashed lines represent incidence
angles 20 and 60 degrees, corresponding to the reflected electric field
presented in c and d, respectively. Panel b shows the circularly polarized
incidence field.

Figure 3a shows how the CEP of s- and p- polarized light change as a function
of incidence angle. In particular, two incidence angles 20 degrees (Figure 3c)
and 60 degrees (Figure 3d) corresponding to elliptically and linearly
polarized light after reflection, are presented as examples. These two angles
are marked by the vertical dashed lines in Figure 3a.

The results suggest that the electric field transforms from a circularly
polarized pulse into an elliptically polarized or linearly polarized pulse by
simply reflecting from the nonreciprocal material surface. By simply changing
the pulse incidence angle, continuous tuning of CEP is achieved. This brings
alternatives for manipulating electromagnetic waveform, particularly in
frequency ranges where traditional methods result in significant loss and beam
distortions, such as the terahertz regime.

###  3.3 Phase-Front Induced Spatial Profile Manipulation

Here, I present how the nonreciprocal material can potentially be used to
achieve spatial profile manipulation via phase-front manipulation. In this
example, $\phi=0$ is chosen for simplicity, because in this configuration,
$r_{sp}=r_{ps}=0$. With the p-polarized incident light, $r_{pp}$ is the only
parameter of interest. The beam size related parameter is normalized to the
wavelength ($\sigma=2\lambda_{0}$). The input electric field is at normal
incidence and is defined as

$E_{\text{in}}(x)=\exp{[-(x/\sigma)^{2}]}\exp{(\psi)},$ (8)

where the purely real part represents the spatial Gaussian envelope and the
phase-front is noted by $\exp{(\psi)}$. Flat phase-front and quadratic phase-
front are represented by $\psi=0$ and $\psi=\pm i(2x/\sigma)^{2}$,
respectively.

Figure 4 shows that with a symmetric distribution of both the envelope and the
phase-front, after reflection, an asymmetric spatial profile is obtained owing
to the asymmetric response of the nonreciprocal material for $+\bm{k}$ and
$-\bm{k}$. Since the space $x$ and momentum $k_{x}$ are a Fourier pair, owing
to the spatial distribution in $x$, though the input field is at normal
incidence, it still contains photons with different incidence angles i.e. a
distribution in $k_{x}$. Remember that the reflection coefficient is an
incidence angle dependent variable too, i.e. $r_{pp}(k_{x})$. For more
information on the calculations performed, please refer to Section S5 in the
SM document. The results suggest that one can achieve spatial profile
manipulation using phase-front manipulation.

Figure 4: Influence of the nonreciprocal material on the spatial profile of a
Gaussian pulse upon reflection. The black curve represents the input pulse
spatial profile. The red curve represents the reflected pulse without any
additional phase. The yellow and blue curves represent the reflected pulse
with quadratic phases of "+" and "-" signs, respectively.

##  4 Conclusions

These findings demonstrate that a simple planar interface between semi-
infinite air and a nonreciprocal material enables effective spatio-temporal
electromagnetic wave manipulation. The utilization of a single reflection on a
planar surface presents a method with the potential for wide-frequency-range
application. This characteristic is especially advantageous in frequency
ranges like terahertz, where the avoidance of propagation through optical
elements alleviates potential absorption, distortion, and diffraction. In
addition, the closed-form expressions for a single dipole can be extended to
the emission of arbitrary objects using the discrete dipole approximation.

These findings lay the foundation for pattern-free electric field manipulation
using nonreciprocal materials, offering promising avenues for future
advancements in this field.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the
author.

Acknowledgements:

Lu Wang thanks Prof. Xiaojun Liu for helpful discussions on this manuscript.

Conflict of Interest:

The author declares no conflict of interest.

Data Availability Statement:

The data and code are available upon request.

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