MnLargeSymbols’164 MnLargeSymbols’171

# Permutation-equivariant
quantum K-theory X.
Quantum Hirzebruch-Riemann-Roch in genus 0

Alexander GIVENTAL

###### Abstract.

We extract genus $0$ consequences of the all genera Quantum HRR formula proved
in Part IX. This includes re-proving and generalizing the adelic
characterization of genus $0$ quantum K-theory found in [5]. Extending some
results of Part VIII and of [5], we derive the invariance of a certain variety
(the “big J-function”), constructed from the genus $0$ descendant potential of
permutation-equivariant quantum K-theory, under the action of certain finite
difference operators in Novikov’s variables, apply this to reconstructing the
whole variety from one point on it, and give an explicit description of it in
the case of the point target space.

This material is based upon work supported by the National Science Foundation
under Grant DMS-1611839, by the IBS Center for Geometry and Physics, POSTECH,
Korea, and by IHES, France

##  1\. Adelic characterization

Let $\mathcal{F}_{X}$ denote the genus-0 descendant potential of permutation-
equivariant quantum K-theory on a compact Kähler manifold $X$, or more
precisely, its dilaton-shifted version. By definition (see Part IX)

$\mathcal{F}_{X}({\mathbf{v}}+{\mathbf{t}}):=\sum_{{\mathbf{l}},d}\frac{Q^{d}}{\prod_{r}l_{r}!}\langle\dots,{\mathbf{t}}_{1},\dots;\dots;\dots,{\mathbf{t}}_{k},\dots;\dots\rangle_{0,{\mathbf{l}},d}.$

Here ${\mathbf{t}}=({\mathbf{t}}_{1},\dots,{\mathbf{t}}_{k},\dots)$ is a
sequence of Laurent polynomials in $q$ with vector coefficients in
$K:=K^{0}(X)\otimes\Lambda$,
${\mathbf{v}}:=(1-q)(\mathbf{1},\mathbf{1},\dots)$ is the dilaton vector, and
$\mathbf{1}$ is the unit in $K^{0}(X)$. The ground ring $\Lambda$, containing
Novikov’s variables $Q$, is a $\lambda$-algebra, i.e. is equipped with Adams’
operations $\Psi^{m}$, $m=1,2,\dots$, $\Psi^{1}=\operatorname{id}$, which act
on Novikov’s variables by $\Psi^{m}(Q^{d})=Q^{md}$. We assume that $\Psi^{m}$
with $m>1$ increase the descending filtration of $\Lambda$ by the powers
$\Lambda_{+}^{d}$ of a certain ideal $\Lambda_{+}$ containing all Novikov’s
monomials $Q^{d}$ with $d\neq 0$. The correlators
$\langle\dots\rangle_{g,{\mathbf{l}},d}$ are defined in terms of K-theory on
the moduli space $X_{g,n,d}$ of genus-$g$ degree-$d$ stable maps to $X$ of
compact complex nodal curves with $n$ marked points. Let $h$ denote the
automorphism of $X_{g,n,d}$ induced by the renumbering of the $n$ marked
points with the cycle structure determined by a partition
${\mathbf{l}}=(l_{1},\dots,l_{r},\dots)$, where $l_{r}$ stands for the number
of cycles of length $r=1,2,\dots$. The correlator

$\langle{\mathbf{u}}_{1},\dots,{\mathbf{u}}_{l_{1}};{\mathbf{v}}_{1},\dots,{\mathbf{v}}_{l_{2}};\dots;{\mathbf{w}}_{1},\dots,{\mathbf{w}}_{l_{r}};\dots\rangle_{g,{\mathbf{l}},d},$

where ${\mathbf{u}}_{i},{\mathbf{v}}_{j},{\mathbf{w}}_{k}$ are K-valued
Laurent polynomials of $q$, is defined as follows. To such a Laurent
polynomial, say, ${\mathbf{w}}(q)=\sum_{m}\phi_{m}q^{m}$, $\phi_{m}\in
K^{0}(X)\otimes\Lambda$, we associate an $h$-equivariant vector bundle $W$ on
$X_{g,n,d}$:

$W=\bigotimes_{\alpha=1}^{r}\sum_{m}\operatorname{ev}_{\sigma_{\alpha}}^{*}(\phi_{m})L_{\sigma_{\alpha}}^{\otimes
m},$

where $\sigma_{1},\dots,\sigma_{r}$ are indices of marked points cyclically
permuted by $h$, $\operatorname{ev}_{\sigma}:X_{g,n,d}\to X$ is the evaluation
map at the marked point $\sigma$, and $L_{\sigma}$ is the line orbibundle over
$X_{g,n,d}$ formed by the cotangent lines to the curves at the $\sigma$th
marked point. With this notation, the above correlator equals

$\left(\prod_{r=1}^{\infty}\frac{1}{r^{l_{r}}}\right)\
\operatorname{str}_{h}H^{*}\left(X_{g,n,d};\mathcal{O}_{g,n,d}\bigotimes_{i=1}^{l_{1}}U_{i}\bigotimes_{j=1}^{l_{2}}V_{j}\cdots\bigotimes_{k=1}^{l_{r}}W_{k}\cdots\right),$

where $\mathcal{O}_{g,n,d}$ is the virtual structure sheaf introduced by Y.-P.
Lee [6].

With the function $\mathcal{F}_{X}$ we associate the “big J-function” which to
a sequence ${\mathbf{t}}=({\mathbf{t}}_{1},{\mathbf{t}}_{2},\dots)$ of
$K$-valued Laurent polynomials in $q$ associates the following sequence
${\mathbf{f}}=({\mathbf{f}}_{1},{\mathbf{f}}_{2},\dots)$ of $K$-valued
rational functions111We remind that by a rational function, Laurent
polynomial, etc. in $q$ we mean a $Q$-series whose coefficients are such
rational functions, Laurent polynomials, etc. of $q$:

$\displaystyle{\mathbf{f}}_{1}=$
$\displaystyle(1-q)\mathbf{1}+{\mathbf{t}}_{1}(q)+\sum_{{\mathbf{l}},d,\alpha}\frac{Q^{d}\phi_{\alpha}}{\prod_{s}{l_{s}!}}\langle\frac{\phi^{\alpha}}{1-qL},{\mathbf{t}}_{1}(L),\dots;{\mathbf{t}}_{2}(L),\dots\rangle_{d,{\mathbf{l}}+\mathbf{1}_{1}}$
$\displaystyle\dots$ $\displaystyle{\mathbf{f}}_{r}=$
$\displaystyle(1-q)\mathbf{1}+{\mathbf{t}}_{r}(q)+\sum_{{\mathbf{l}},d,\alpha}\frac{Q^{d}\phi_{\alpha}}{\prod_{s}{l_{s}!}}\langle\frac{\phi^{\alpha}}{1-qL},{\mathbf{t}}_{k}(L),\dots;{\mathbf{t}}_{2k}(L),\dots\rangle_{d,{\mathbf{l}}+\mathbf{1}_{1}}$
$\displaystyle\dots$

Here the cycle structure ${\mathbf{l}}+\mathbf{1}_{1}$ is obtained from
${\mathbf{l}}$ by adding one cycle of length $1$.

The range of the J-function is a (formal, infinite dimensional) subvariety in
the space $\mathcal{K}^{\infty}$ of sequenses
${\mathbf{f}}=({\mathbf{f}}_{1},{\mathbf{f}}_{2},\dots)$ of $K$-valued
rational functions in $q$ allowed to have poles at $q=0$,$\infty$, or roots of
unity. We denote this range by $\mathcal{L}_{X}$.

Another way to describe $\mathcal{L}_{X}$ is to consider $\mathcal{F}_{X}$ as
a family of functions of the first input ${\mathbf{t}}_{1}$ depending on
${\mathbf{t}}_{2},{\mathbf{t}}_{3},\dots$ as parameters. Then the map
${\mathbf{t}}_{1}\mapsto{\mathbf{f}}_{1}({\mathbf{t}}_{1},{\mathbf{t}}_{2},{\mathbf{t}}_{3},\dots)$
parameterizes the graph of the differential $d_{t_{1}}\mathcal{F}_{X}$ for
given values of the parameters ${\mathbf{t}}_{2},{\mathbf{t}}_{3},\dots$, and
${\mathbf{f}}_{r}:={\mathbf{f}}_{1}({\mathbf{t}}_{r},{\mathbf{t}}_{2r},{\mathbf{t}}_{3r},\dots)$.
The sequences ${\mathbf{f}}=({\mathbf{f}}_{1},{\mathbf{f}}_{2},\dots)$ of
differentials lie in the cotangent bundle of $T^{*}\mathcal{K}_{+}^{\infty}$
of the subpace $\mathcal{K}_{+}^{\infty}\subset\mathcal{K}^{(\infty)}$,
consisting of sequences of $K$-valued Laurent polynomaials in $q$. The
cotangent bundle space is identified with the whole of $\mathcal{K}^{\infty}$
by means of the polarization
$\mathcal{K}^{\infty}=\mathcal{K}^{\infty}_{+}\oplus\mathcal{K}^{\infty}_{-}$,
Lagrangian with respect to the symplectic form

$\displaystyle\Omega^{\infty}({\mathbf{f}},{\mathbf{g}})$
$\displaystyle=\sum_{r=1}^{\infty}\frac{\Psi^{r}}{r}\Omega({\mathbf{f}}_{r},{\mathbf{g}}_{r}),\
$ $\displaystyle\Omega(f,g)=$
$\displaystyle-[\operatorname{Res}_{q=0}+\operatorname{Res}_{q=\infty}](f(q^{-1}),g(q))\,\frac{dq}{q}.$

Here $(a,b):=\chi(X;a\otimes
b)=\int_{X}\operatorname{td}(T_{X})\operatorname{ch}(a)\operatorname{ch}(b)$
is the K-theoretic Poincaré pairing on $K^{0}(X)$. By definition, the negative
space $\mathcal{K}_{-}^{\infty}$ of the polarization consists of sequences
${\mathbf{f}}=({\mathbf{f}}_{1},{\mathbf{f}}_{2},\dots)$ satisfying
${\mathbf{f}}_{r}(\infty)=0$, ${\mathbf{f}}_{r}(0)\neq\infty$.

Remark. Note that the graph of each ${\mathbf{f}}_{r}$ for fixed values of the
parameters ${\mathbf{t}}_{2r},{\mathbf{t}}_{3r},\dots$ is a Lagrangian variety
in $(\mathcal{K},\Omega)$ (where $\mathcal{K}$ is the space of $K$-valued
Laurent polynomials in $q$), and the whole family of such Lagrangian varieties
for, say, $r=1$ determines $\mathcal{L}_{X}$. However, a value of
${\mathbf{f}}_{1}$, though “knows” its input ${\mathbf{t}}_{1}$, does not
“remember” the values of the parameters
${\mathbf{t}}_{2},{\mathbf{t}}_{2},\dots$. One reason to introduce
$\mathcal{L}_{X}$ is that from a point on it, the corresponding argument
${\mathbf{t}}\in\mathcal{K}^{\infty}_{+}$ is reconstructed by projection along
$\mathcal{K}^{\infty}_{-}$. On the other hand, there is no reason for
$\mathcal{L}_{X}$ to be Lagrangian in
$(\mathcal{K}^{\infty},\Omega^{\infty})$.

We will give an adelic characterization of $\mathcal{L}_{X}$ in terms of fake
genus-0 K-theoretic GW-invariant of $X$. The fake holomorphic Euler
characteristic of a bundle $V$ over $X_{g,n,d}$ is defined by the right hand
side of the Hirzebruch-RR formula:

$\chi^{fake}(X_{g,n,d};V):=\int_{[X_{g,n,d}]}\operatorname{td}(T_{X_{g,n,d}})\operatorname{ch}(V),$

where $[X_{g,n,d}]$ is the virtual fundamental class, and $T_{X_{g,n,d}}$ is
the virtual tangent bundle. The genus-0 descendant potential
$\mathcal{F}_{X}^{fake}$ is defined as a function of $t(q)=\sum_{m\geq
0}\phi_{m}(q-1)^{m}\in\mathcal{K}_{+}^{fake}$ by

$\mathcal{F}_{X}^{fake}(t)=\sum_{n,d}\frac{Q^{d}}{n!}\langle
t(L),\dots,t(L)\rangle_{0,n,d}^{fake},$

where
$\langle\phi_{\alpha_{1}}L^{m_{1}},\dots,\phi_{\alpha_{n}}L^{m_{n}}\rangle_{0,n,d}:=\chi^{fake}\left(X_{0,n,d};\otimes_{i=1}^{n}\operatorname{ev}_{i}^{*}(\phi_{\alpha_{i}})L_{i}^{m_{i}}\right)$.

The graph of the differential $d\mathcal{F}_{X}^{fake}$ is identified with a
Lagrangian variety
$\mathcal{L}_{X}^{fake}\subset(\mathcal{K}^{fake},\Omega^{fake})$. By
$\mathcal{K}^{fake}$ we denote the space of $K$-valued Laurent series in
$q-1$. It is equipped with the symplectic form

$\Omega^{fake}(f,g):=\operatorname{Res}_{q=1}(f(q^{-1}),g(q))\,\frac{dq}{q},$

Lagrangian polarization $\mathcal{K}^{fake}_{\pm}$, where
$\mathcal{K}^{fake}_{+}$ consists of power series, and
$\mathcal{K}^{fake}_{-}$ consists of the principal parts of the Laurent
series, as well as the dilaton vector $(1-q)\mathbf{1}$. Explicitly,
$\mathcal{L}_{X}^{fake}$ consists of vector-valued Laurent series in $q-1$ of
the form

$1-q+t(q)+\sum_{\alpha,n,d}\frac{Q^{d}\phi_{\alpha}}{n!}\langle\frac{\phi^{\alpha}}{1-qL},t(L),\dots,t(L)\rangle^{fake}_{0,n+1,d}.$

In fact $\mathcal{L}_{X}^{fake}$ is an overruled Lagrangian cone, i.e. each
tangent space $T$ of $\mathcal{L}_{X}^{fake}$ contain
$(1-q)T\subset\mathcal{L}_{X}^{fake}$ and is tangent to
$\mathcal{L}_{X}^{fake}$ everywhere along $(1-q)T$. The cone
$\mathcal{L}_{X}^{fake}$ can be explicitly described in terms of cohomological
GW-theory of $X$ (see [1, 3, 5] or Part IX).

Theorem 1. Let
${\mathbf{f}}=({\mathbf{f}}_{1},{\mathbf{f}}_{2},\dots,{\mathbf{f}}_{r},\dots)$
be a sequence of rational functions of $q$ with values in
$K=K^{0}(X)\otimes\Lambda$ considered as an element of the symplectic loop
space $(\mathcal{K}^{\infty},\Omega^{\infty})$. This sequence represents a
point in the range $\mathcal{L}_{X}\subset\mathcal{K}^{\infty}$ of the “big
J-function” of permutation-equivariant quantum K-theory of $X$ if and only if
it satisfies the following criteria:

(i) For each $r=1,2,\dots$, the Laurent series expansion
${\mathbf{f}}_{r}^{(1)}$ of ${\mathbf{f}}_{r}(q)$ near $q=1$ lies in the
Lagrangian submanifold
$\mathcal{L}_{X}^{fake}\subset(\mathcal{K}^{fake},\Omega^{\operatorname{fake}})$
representing the graph of the differential of of the genus 0 descendant
potential $\mathcal{F}_{X}^{fake}$ of fake quantum K-theory of $X$;

(ii) For each $r$, and each root of unity $\zeta\neq 1$, the Laurent series
expansion ${\mathbf{f}}_{r}^{(\zeta)}$ of ${\mathbf{f}}_{r}(q^{1/m}/\zeta)$
near $q=1$, where $m$ is the primitive order of $\zeta$, lies in the
Lagrangian subspace

$\triangle_{\zeta}\Psi^{m}(T_{{\mathbf{f}}_{rm}^{(1)}}\mathcal{L}_{X}^{fake})\otimes_{\Psi^{m}(\Lambda)}\Lambda,$

where the operator of multiplication $\triangle_{\zeta}$ is given by the
formula

$\triangle_{\zeta}:=e^{\textstyle\sum_{k>0}\left(\frac{\Psi^{k}(T^{*}_{X}-1)}{k(1-\zeta^{-k}q^{k/m})}-\frac{\Psi^{km}(T^{*}_{X}-1)}{k(1-q^{km})}\right)}.$

Theorem 1 generalizes several previously known results. Setting
${\mathbf{t}}_{k}=0$ for all $k>1$ we obtain the result of [5] which
characterizes the version of genus-0 quantum K-theory which does not involve
permutations of the marked points. Setting all ${\mathbf{t}}_{k}$ with $k>1$
equal to each other, we obtain an adelic characterization of permutation-
equivariant J-function studied in Part VII and Part VIII.

##  2\. Example: $X=pt$

In this case, $\mathcal{K}^{\infty}$ consists of sequences
${\mathbf{f}}=({\mathbf{f}}_{1},\dots,{\mathbf{f}}_{r},\dots)$ of scalar (i.e.
$\Lambda$-valued) rational functions of $q$ with poles at $q=0,\infty$ or
roots of unity, and $\mathcal{K}_{+}^{\infty}$ consists of sequences
${\mathbf{t}}=({\mathbf{t}}_{1},\dots,{\mathbf{t}}_{r},\dots)$ of scalar
Laurent polynomials in $q$.

Theorem 2. The range $\mathcal{L}_{pt}\subset\mathcal{K}^{\infty}$ of the “big
J-function” consists of sequences
${\mathbf{f}}=({\mathbf{f}}_{1},\dots,{\mathbf{f}}_{r},\dots)\in\mathcal{K}^{\infty}$
of the form

$\displaystyle{\mathbf{f}}_{1}(q)$
$\displaystyle=(1-q)e^{\textstyle\sum_{k>0}\Psi^{k}(\tau_{k})/k(1-q^{k})}\
{\mathbf{t}}_{1}(q,q^{-1})$ $\displaystyle\dots$
$\displaystyle{\mathbf{f}}_{r}(q)$
$\displaystyle=(1-q)e^{\textstyle\sum_{k>0}\Psi^{k}(\tau_{kr})/k(1-q^{k})}\
{\mathbf{t}}_{r}(q,q^{-1})$ $\displaystyle\dots$

where ${\mathbf{t}}_{k}\in\mathcal{K}_{+}$ are scalar Laurent polynomials
$\Lambda_{+}$-close to $1$, and $\tau_{k}\in\Lambda_{+}$ are arbitrary scalar
parameters.

Proof. We derive this by verifying criteria (i), (ii) of Theorem 1.

It is known (see, for instance, [2]) that the $\mathcal{L}^{fake}_{pt}$
consists of Laurent series of the form

$(1-q)e^{\tau/(1-q)}\ t(q-1),$

where $t$ is a power series in $q-1$ which is $\Lambda_{+}$-close to $1$, and
$\tau\in\Lambda_{+}$ is an arbitrary scalar.

First, we expand ${\mathbf{f}}_{r}$ into Laurent series
${\mathbf{f}}_{r}^{(1)}$ in $q-1$:

${\mathbf{f}}_{r}^{(1)}=(1-q)e^{\textstyle\mathcal{T}_{r}/(1-q)}\ t_{r}(q-1),\
\text{where}\ \mathcal{T}_{r}=\sum_{k>0}\Psi^{k}(\tau_{kr})/k^{2},$

and $t_{r}$ are some power series in $q-1$. We use here that
$1/(1-q^{k})=1/k(1-q)+\mathcal{O}(1)$. Thus
${\mathbf{f}}_{r}^{(1)}\in\mathcal{L}_{pt}^{fake}$, and criterion (i) is
fulfilled.

Note that the tangent space to $\mathcal{L}_{pt}^{fake}$ at the point
${\mathbf{f}}_{r}^{(1)}$ has the form
$e^{\mathcal{T}_{r}/(1-q)}\mathcal{K}_{+}^{fake}$.

In the case $X=pt$, the operator $\triangle_{\zeta}$ consists in
multiplication by an invertible power series in $q-1$. Therefore, for a
primitive $m$th root of unity $\zeta$,

$\triangle_{\zeta}\Psi^{m}(T_{{\mathbf{f}}_{rm}^{(1)}}\mathcal{L}_{pt}^{fake})\otimes_{\Psi^{m}(\Lambda)}\Lambda=e^{\textstyle\Psi^{m}[\mathcal{T}_{rm}/(1-q)]}\mathcal{K}^{fake}_{+}.$

We want to check that this subspace contains the Laurent series expansion
${\mathbf{f}}_{r}^{(\zeta)}$ of ${\mathbf{f}}_{r}(q^{1/m}/\zeta)$ near $q=1$.
We have:

${\mathbf{f}}_{r}^{(\zeta)}=(1-q^{1/m}/\zeta)e^{\textstyle\sum_{k>0}\Psi^{k}(\tau_{kr})/k(1-q^{k/m}/\zeta^{k})}\
{\mathbf{t}}_{r}(q^{1/m}/\zeta,q^{-1/m}\zeta).$

The terms in the exponent which have pole at $q=1$ come from the values of $k$
divisible by $m$. For $k=lm$ we have:

$\sum_{l>0}\frac{\Psi^{lm}(\tau_{lmr})}{lm(1-q^{l})}=\Psi^{m}\left(\frac{\mathcal{T}_{rm}}{1-q}\right)+\text{terms
regular at $q=1$}.$

Thus, criterion (ii) is also fulfilled.

Thus, the family of sequences
${\mathbf{f}}=({\mathbf{f}}_{1},\dots,{\mathbf{f}}_{r},\dots)$ described in
Theorem 2 lie in $\mathcal{L}_{pt}$. Note that at $\tau=0$, this family
contains the $\Lambda_{+}$-neighborhood of the dilaton vector in
$\mathcal{K}_{+}^{\infty}$. At ${\mathbf{t}}=\mathbf{1}$, the projection of
${\mathbf{f}}$ to $\mathcal{K}^{\infty}_{+}$ along $\mathcal{K}^{\infty}_{-}$
is $[{\mathbf{f}}]_{+}=(1-q+\tau_{1},\dots,1-q+\tau_{r},\dots)$. Thus, the
tangent space at the dilaton point to the entire domain of $\mathcal{F}_{pt}$
is covered by our family. It follows now from the formal Implicit Function
Theorem, that the family parameterizes the whole of $\mathcal{L}_{pt}$.

##  3\. $\mathcal{D}_{q}$-symmetries

We generalize here results of [5, 4], and Part VIII about symmetries of
genus-0 quantum K-theory induced by finite-difference operators in Novikov’s
variables.

Pick an integer basis $p_{1},\dots,p_{s}$ in $H_{2}(X;\mathbb{Q})$ with
respect to which all degrees of holomorphic curves in $X$ are expressed by
vectors $d=(d_{1},\dots,d_{s})$ with non-negative components $d_{i}$. Let
$P_{i}$ be the line bundle over $X$ whose 1st Chern class is $-p_{i}$. Let
$\mathcal{D}_{q}$ denote the algebra of finite difference operators in
Novikov’s variables. By definition it consists of non-commutative polynomial
expressions formed from multiplication operators $Q_{i}$, translation
operators $q^{Q_{i}\partial_{Q_{i}}}$, and can have Laurent polynomials in $q$
in the role of coefficients. We make $\mathcal{D}_{q}$ act on the space
$\mathcal{K}$ of K-valued rational functions of $q$ so that $Q_{i}$ act
naturally as multiplication by $Q_{i}\in\Lambda_{+}$, while the translation
operators act on $K$-valued functions of $Q$ by
$P_{i}q^{Q_{i}\partial_{Q_{i}}}$, i.e. by

$\mathcal{F}(Q_{1},\dots,Q_{s})\mapsto
P_{i}\mathcal{F}(\dots,Q_{i-1},qQ_{i},Q_{i+1},\dots).$

For convergence purposes we will further assume that our operators
$D(Pq^{Q\partial_{Q}},Q,q)$ have “small free terms”, i.e.
$D(1,0,q)\in\Lambda_{+}[q,q^{-1}]$.

Theorem 3. Let $D=(D_{1},\dots,D_{k},\dots)$ be a sequence of finite
difference operators $D_{k}(Pq^{Q\partial_{Q}},Q,q)$. Then the following
transformation on the space $\mathcal{K}^{\infty}$ of sequences
${\mathbf{f}}=({\mathbf{f}}_{1},\dots,{\mathbf{f}}_{r},\dots)$ of vector-
valued rational function preserves $\mathcal{L}_{X}$:

${\mathbf{f}}_{r}\mapsto
e^{\textstyle\sum_{k>0}\Psi^{k}(D_{kr}(Pq^{kQ\partial_{Q}},Q,q))/k(1-q^{k})}\
{\mathbf{f}}_{r},\ \ r=1,2,\dots.$

Remark. Recall that Adams’ operations $\Psi^{k}$ act naturally on $K^{0}(X)$,
act of functions of $q$ by $\Psi^{k}(q)=q^{k}$, act through the
$\lambda$-algebra structure on the coefficient ring $\Lambda$, and in
particular by $\Psi^{k}(Q^{d})=Q^{kd}$ on Novikov’s variables. Consequently,
multiplication by $P_{i}$ hidden in $\Psi^{k}(D(Pq^{Q\partial_{Q}},Q,q))$
turns after the application of $\Psi^{k}$ into multiplication by $P_{i}^{k}$.
However,

$\Psi^{k}(q^{Q_{i}\partial_{Q_{i}}}Q^{d})=\Psi^{k}(q^{d_{i}}Q^{d})=q^{kd_{i}}Q^{kd}=q^{Q_{i}\partial_{Q_{i}}}\Psi^{k}(Q^{d}),$

i.e. $\Psi^{k}(q^{Q}\partial_{Q})=q^{Q\partial_{Q}}$. Therefore
$\Psi^{k}(Pq^{kQ\partial_{Q}})=(Pq^{Q\partial_{Q}})^{k}$. Thus, the operators
in the exponent act as legitimate finite difference operators in our
representation, i.e. as combinations of multiplications by $Q_{i}$ and twisted
translation operators $P_{i}q^{Q_{i}\partial_{Q_{i}}}$.

Proof. Suppose ${\mathbf{f}}\in\mathcal{L}_{X}$. Then: (i) expansion
${\mathbf{f}}_{r}^{(1)}$ of ${\mathbf{f}}_{r}(q)$ near $q=1$ lie in
$\mathcal{L}_{X}^{fake}$, and (ii) expansion ${\mathbf{f}}_{r}^{(\zeta)}$ of
${\mathbf{f}}_{r}(q^{1/m}/\zeta)$ near $q=1$, where $\zeta\neq 1$ is a
primitive $m$th root of unity, lie in
$\triangle_{\zeta}\Psi^{m}(T_{{\mathbf{f}}_{rm}^{(1)}}\mathcal{L}_{X}^{fake})\otimes_{\Psi^{m}(\Lambda)}\Lambda$.

It is known (see, for instance, [1, 5], or Section 3 in Part IX), that
$\mathcal{L}_{X}^{fake}$ is an overruled Lagrangian cone in the symplectic
loop space $\mathcal{K}^{fake}$, and that it is obtained by a certain loop
group transformation $\triangle:\mathcal{H}\to\mathcal{K}^{fake}$ from the
overruled Lagrangian cone $\mathcal{L}_{X}^{H}$ of quantum cohomology theory
on $X$. Recall that $\mathcal{L}_{X}^{H}$ lies in the appropriate symplectic
loop space $\mathcal{H}=H^{even}(X;\Lambda((z)))$, where $z=\log q$, and
represents the graph of the differential of the genus 0 descendant potential
of cohomological GW-theory of $X$. It follows (see [4]) from the divisor
equations of quantum cohomology theory, that $\mathcal{L}_{X}^{H}$ is
invariant under the action of operators $f\mapsto e^{D/z}f$ where
$D(zQ\partial_{Q}-p,Q,z)$ is any (pseudo) differential operator (with “small
free term”) in Novikov’s variables. Consequently, since the loop group
transformation $\triangle$ does not depend on $Q$, the cone
$\mathcal{L}_{X}^{fake}$ is similarly invariant under the action of operators
$f\mapsto e^{D/(1-q)}f$, where $D((\log q)Q\partial_{Q}-p,Q,q-1)$ is any
differential operator. 222Note that $\log
q=\log(1-(1-q))=-\sum_{k>0}(1-q)^{k}/k$ is a legitimate coefficient in
$\mathcal{K}^{fake}$ (though not in $\mathcal{K}$), and $p_{i}=-\log P_{i}$ is
well-defined under the Chern character identification of
$K^{0}(X)\otimes\mathbb{Q}$ with $H^{even}(X;\mathbb{Q})$. The operator in the
exponent here is required to have at most the 1st order pole at $q=1$.
Besides, the tangent spaces $T_{f}\mathcal{L}_{X}^{fake}$ and the ruling
spaces $(1-q)T_{f}\mathcal{L}_{X}^{fake}\subset\mathcal{L}_{X}^{fake}$ are
$\mathcal{D}$-modules, i.e. stay invariant under the operators $D$, and hence
under the operators $e^{D}$ (with no pole at $q=1$). In particular, since
$Pq^{Q\partial_{Q}}=e^{(\log q)Q\partial_{Q}-p}$, the invariance properties
hold true for finite difference operators.

Now let ${\mathbf{g}}=({\mathbf{g}}_{1},\dots,{\mathbf{g}}_{r},\dots)$ be the
result of the transformation described in the theorem and applied to
${\mathbf{f}}$. We intend to check that ${\mathbf{g}}$ satisfies the criteria
(i), (ii) of Theorem 1. Since $1/(1-q^{k})$ has the 1st order pole at $q=1$,
it follows that ${\mathbf{g}}_{r}^{(1)}\in\mathcal{L}_{X}^{fake}$.
Furthermore, according to the invariance properties described above, the
tangent space to $\mathcal{L}_{X}^{fake}$ at ${\mathbf{f}}_{rm}^{(1)}$ is
transformed to another tangent space (at ${\mathbf{g}}_{rm}^{(1)}$) determined
by the polar part of the operator in the exponent. Therefore

$T_{{\mathbf{g}}_{rm}^{(1)}}\mathcal{L}_{X}^{fake}=e^{\textstyle\sum_{k>0}\Psi^{k}(D_{krm}(Pq^{kQ\partial_{Q}},Q,q)/k^{2}(1-q)}\
T_{{\mathbf{f}}_{rm}^{(1)}}\mathcal{L}_{X}^{fake}.$

On the other hand,

$\displaystyle{\mathbf{g}}_{r}^{(\zeta)}$
$\displaystyle=e^{\sum_{k>0}\Psi^{k}(D_{kr}(P(q^{1/m}/\zeta)^{kQ\partial_{Q}},Q,q^{1/m}/\zeta))/k(1-q^{k/m}/\zeta^{k})}\
{\mathbf{f}}_{r}(q^{1/m}/\zeta)$
$\displaystyle=e^{D(P(q^{1/m}/\zeta)^{Q\partial_{Q}},Q,q-1)}\
e^{\sum_{l>0}\Psi^{lm}(D_{lmr}(Pq^{lQ\partial_{Q}},Q,q))/l^{2}(1-q^{m})}\
{\mathbf{f}}_{r}^{(\zeta)}.$

The transition from the first to the second line uses a version of the
Campbell-Hausdorff formula which rewrites $e^{A+B/(1-q)}$ as
$e^{C}e^{B/(1-q)}$, where by $A$ and $B$ we denote operators without pole at
$q=1$. Since the commutator of $[A,B]$ is divisible by $q-1$, the operator $C$
also comes out without the pole.

Comparing with the formula for
$T_{{\mathbf{g}}_{rm}^{(1)}}\mathcal{L}_{X}^{fake}$, we find that the factor
$e^{B/(1-q)}$ coincides with

$\Psi^{m}\left(e^{\textstyle\sum_{k>0}\Psi^{k}(D_{krm}(Pq^{kQ\partial_{Q}},Q,q))/k^{2}(1-q)}\right).$

Given that
${\mathbf{f}}_{r}^{(\zeta)}\in\triangle_{\zeta}\Psi^{m}(T_{{\mathbf{f}}_{rm}^{(1)}}\mathcal{L}_{X}^{fake})\otimes\Lambda$,
and using the fact that $\triangle_{\zeta}$ does not depend on $Q$, we
conclude, that
$e^{B/(1-q)}{\mathbf{f}}_{r}^{(\zeta)}\in\triangle_{\zeta}\Psi^{m}(T_{{\mathbf{g}}_{rm}^{(1)}}\mathcal{L}_{X}^{fake})\otimes\Lambda$.

As we have discussed, the tangent space
$T_{{\mathbf{g}}_{rm}^{(1)}}\mathcal{L}_{X}^{fake}$ is preserved by
(differential and hence) finite difference operators. However, Adams’
operation $\Psi^{m}$ changes the representation by which such operators act.
Namely,
$\Psi^{m}((P_{i}q^{Q_{i}\partial_{Q_{i}}})^{1/m})=P_{i}(q^{Q_{i}\partial_{Q_{i}}})^{1/m}$.
Also note that $\zeta^{-Q_{i}\partial_{Q_{i}}}$, acting on $Q^{d}$ by
$\zeta^{-d_{i}}$, acts trivially on $\Psi^{m}(Q^{d})=Q^{md}$ when $\zeta$ is
an $m$th root of unity. Therefore the operator
$D(P(q^{1/m}/\zeta)^{Q\partial_{Q}},Q,q-1)$ (nicknamed as $C$ in the Campbell-
Hausdorff formula) acts in $\Psi^{m}(\mathcal{K}^{fake})$ as a legitimate
pseudo-differential operator in this new representation, and preserves the
subspace $\Psi^{m}(T_{{\mathbf{g}}_{rm}^{(1)}}\mathcal{L}_{X}^{fake})$. This
implies that the same is true for the subspace
$\triangle_{\zeta}\Psi^{m}(T_{{\mathbf{g}}_{rm}^{(1)}}\mathcal{L}_{X}^{fake})\otimes\Lambda$
in $\mathcal{K}^{fake}$, and shows that
${\mathbf{g}}_{r}^{(\zeta)}=e^{C}e^{B/(1-q)}{\mathbf{f}}_{r}^{(\zeta)}$ lies
in this subspace.

Corollary (string flows). The transformations

${\mathbf{f}}_{r}\mapsto
e^{\textstyle\sum_{k>0}\Psi^{k}(\tau_{kr})/k(1-q^{k})}\ {\mathbf{f}}_{r},\
r=1,2,\dots,$

where $\tau_{k}\in\Lambda_{+}$, $k=1,2,\dots$, preserve $\mathcal{L}_{X}$.

##  4\. Explicit reconstruction

We generalize “explicit reconstruction” results from [4] and Part VIII. Let us
begin with the following proposition.

Proposition ($\mathcal{D}_{q}$-module structure). If
${\mathbf{f}}=({\mathbf{f}}_{1},\dots,{\mathbf{f}}_{r},\dots)$ lies in
$\mathcal{L}_{X}$, then
$(D_{1}{\mathbf{f}}_{1},\dots,D_{r}{\mathbf{f}}_{r},\dots)$, where
$D_{1},D_{2},\dots$ is any sequence of finite difference operators, also lies
in $\mathcal{L}_{X}$.

Proof. This is a simplified version of the previous arguments. We are given
that ${\mathbf{f}}_{r}^{(\zeta)}$ pass the tests (i), (ii) of Theorem 1, i.e.
${\mathbf{f}}_{r}^{(1)}\in\mathcal{L}_{X}^{fake}$, and for primitive $m$th
roots of unity $\zeta\neq 1$,
${\mathbf{f}}_{r}^{(\zeta)}\in\triangle_{\zeta}\Psi^{m}(T_{{\mathbf{f}}_{rm}^{(1)}}\mathcal{L}_{X}^{fake})\otimes\Lambda$.
Then ${\mathbf{f}}_{r}^{(1)}$ lies in (its own) ruling space
$(1-q)T_{{\mathbf{f}}_{r}^{(1)}}\subset\mathcal{L}_{X}^{fake}$, which is
$D_{r}$-invariant. Therefore, for ${\mathbf{g}}_{r}:=D_{r}{\mathbf{f}}_{r}$,
we find that ${\mathbf{g}}_{r}^{(1)}$ lies in the same ruling space as
${\mathbf{f}}_{r}^{(1)}$, and the tangent spaces $T_{{\mathbf{g}}_{r}^{(1)}}$
and $T_{{\mathbf{f}}_{r}^{(1)}}$ coincide as well. Furthermore,

${\mathbf{g}}_{r}^{(\zeta)}=D_{r}(P(q^{1/m}/\zeta)^{Q\partial_{Q}},Q,q^{1/m}/\zeta)\
f_{r}^{(\zeta)}\in\triangle_{\zeta}\Psi^{m}(T_{{\mathbf{f}}_{rm}^{(1)}})\otimes\Lambda,$

since the operator here commutes with $\triangle_{\zeta}$ and preserves the
subspace $\Psi^{m}(T_{{\mathbf{f}}_{rm}^{(1)}})$ in
$\Psi^{m}(\mathcal{K}^{\operatorname{fake}})$ (as was discussed in the proof
of Theorem 3).

Remark. This Proposition together with the previous Corollary give another
proof of Theorem 1. Namely, it is easy to check that
${\mathbf{t}}={\mathbf{0}}$ is a critical point of $\mathcal{F}_{pt}$, which
implies that the dilaton vector $(1-q)(\mathbf{1},\mathbf{1},\dots)$ lies in
$\mathcal{L}_{pt}$. Then, applying Theorem 4 with
$D_{r}={\mathbf{t}}_{r}(q,q^{-1})$ (multiplication by a Laurent polynomial),
we conclude that $(1-q)\mathcal{K}^{\infty}_{+}$ lies in $\mathcal{L}_{X}$.
Finally, applying the string flow from Corollary, we obtain the whole of
$\mathcal{L}_{pt}$.

Theorem 4. Suppose that the degree-$2$ classes $p_{1},\dots,p_{s}$ generate
the cohomology algebra $H^{even}(X;\mathbb{Q})$, and hence (equivalently) the
line bundles $P_{1},\dots,P_{s}$ multiplicatively generate
$K^{0}(X)\otimes\mathbb{Q}$. Pick Laurent monomials $P^{m_{\alpha}}$ to form a
linear basis in $K^{0}(X)\otimes\mathbb{Q}$. Let
$c_{\alpha,r}\in\Lambda[q,q^{-}1]$, $r=1,2,\dots$, be arbitrary333For
consistency with our earlier understanding of the domain of $\mathcal{F}_{X}$
we should assume that $c_{\alpha,r}$ are $\Lambda_{+}$-close to $1$. scalar
Laurent polynomials in $q$, and $\tau_{\alpha,k}\in\Lambda_{+}$,
$k=1,2,\dots$, be arbitrary “small” parameters. Suppose that
${\mathbf{f}}=({\mathbf{f}}_{1},\dots,{\mathbf{f}}_{r},\dots)$ is a point in
$\mathcal{L}_{X}$, and let ${\mathbf{f}}_{r}=\sum_{d}f_{r,d}Q^{d}$ be the
$Q$-expansion of ${\mathbf{f}}_{r}$ (i.e. the coefficients $f_{r,d}$ do not
depend on $Q$). With this notation, the following family
${\mathbf{g}}=({\mathbf{g}}_{1},\dots,{\mathbf{g}}_{r},\dots)$ of points in
$\mathcal{K}^{\infty}$ lies in $\mathcal{L}_{X}$ and parameterizes the whole
of it:

${\mathbf{g}}_{r}:=\sum_{d}f_{r,d}Q^{d}e^{\sum_{k>0}\sum_{\alpha}\Psi^{k}(\tau_{\alpha,rk})P^{km_{a}}q^{k(m_{\alpha},d)}/k(1-q^{k})}\sum_{\alpha}c_{\alpha,r}(q)P^{m_{\alpha}}q^{(m_{\alpha},d)}.$

Here $(m,d)=\sum_{i}m^{i}d_{i}$ denotes the value of $\sum_{i}m^{i}p_{i}\in
H^{2}(X)$ on $d\in H_{2}(X)$.

Proof. Let us assume first that $\tau_{\alpha,k}$ are free $\lambda$-algebra
generators added to the ground ring $\Lambda$. We apply Theorem 3 and above
Proposition over such an extended ground ring $\widetilde{\Lambda}$ to
conclude that the following family
$\widetilde{{\mathbf{g}}}=(\dots,\widetilde{{\mathbf{g}}}_{r},\dots)$ of
points lies in $\widetilde{\mathcal{L}}_{X}$:

$\widetilde{{\mathbf{g}}}_{r}:=\left(\sum_{\alpha}c_{\alpha,r}(Pq^{Q\partial_{Q}})^{m_{\alpha}}\right)e^{\sum_{k>0}\sum_{\alpha}\Psi^{k}(\tau_{\alpha,rk})(Pq^{Q\partial_{Q}})^{km_{\alpha}}/k(1-q^{k})}\
{\mathbf{f}}_{r}.$

Since $\tau_{\alpha,k}$ do not depend on $Q$, the operators in the exponent
commute and can be computed explicitly on the monomials $Q^{d}$. Namely,
$(q^{Q\partial_{Q}})^{km_{\alpha}}Q^{d}=q^{k(m_{\alpha},d)}Q^{d}$. After that
the operators on the left can be applied explicitly too. The resulting
expression matches the one for ${\mathbf{g}}_{r}$ in the formulation of
Theorem 4. Now we can invoke the change of the ground ring
$\widetilde{\Lambda}\to\Lambda$ defined as the homomorphism of
$\lambda$-algebras induced by the specialization of $\tau_{\alpha,k}$ to their
values in $\Lambda_{+}$. It is easy to see that the descendant potential
$\widetilde{\mathcal{F}}_{X}$ turns into $\mathcal{F}_{X}$ under this
operation, and hence the variety $\widetilde{\mathcal{L}}_{X}$ turns into
$\mathcal{L}_{X}$. Therefore the family ${\mathbf{g}}$ indeed lies in
$\mathcal{L}_{X}$.

We can check now the same way as we did in the proof of Theorem 2 that the
projection of this family to $\mathcal{K}_{+}^{\infty}$ along
$\mathcal{K}_{-}^{\infty}$ covers the $\Lambda_{+}$-neighborhood of the
dilaton vector. For instance, modulo Novikov’s variables, taking all
$\sum_{\alpha}c_{r,\alpha}P^{m_{\alpha}}=1$, we have

$g_{r,0}=f_{r,0}+\sum_{\alpha}\tau_{\alpha,r}P^{m_{\alpha}}+\text{terms with
poles at $q\neq 0,\infty$},$

which gives a family444We assume that $\mathcal{L}_{X}\ni{\mathbf{f}}$ lies in
$\mathcal{K}^{\infty}$ in $\Lambda_{+}$-neighborhood of the dilaton vector,
and in particular $f_{r,0}=(1-q)\mathbf{1}\mod\Lambda_{+}$.
$[g_{r,0}]_{+}=(1-q)\mathbf{1}+t$ with any $q$-independent value of $t$. In
the other extreme, when all $\tau_{\alpha,k}=0$, we have

${\mathbf{g}}_{r,0}=\left(\sum_{\alpha}c_{\alpha,r}(q)|_{Q=0}P^{m_{\alpha}}\right)f_{r,0},$

which yields a family $[g_{r,0}]_{+}$ containing in projection along $(1-q)+t$
all multiples of $(1-q)$. Our claim follows from this by virtue of the formal
Implicit Function Theorem.

##  5\. Derivation of Theorem 1

One way to prove Theorem 1 would be to extend the arguments of [5] where a
recursion relation for the J-function of quantum K-theory is derived from the
virtual Kawasaki-RR formula on moduli spaces of genus 0 stable maps. Here we
are rather interested in obtaining Theorem 1 as a consequence of the higher
genus formula from Part IX for the total descendant potential
$\mathcal{D}_{X}$ of permutation-equivariant quantum K-theory of $X$. However,
we need to warn the reader, that such a derivation may in the end provide few
advantages over the first, more direct method.

The genus 0 descendant potential $\mathcal{F}_{X}$ enters the total descendant
potential $\mathcal{D}_{X}$ in the form

$\mathcal{D}_{X}=e^{\textstyle\sum_{k>0}\hbar^{-k}\Psi^{k}(\mathcal{F}_{X}({\mathbf{t}}_{k},{\mathbf{t}}_{2k},\dots,{\mathbf{t}}_{rk},\dots))/k+\mathcal{O}(1)},$

where $\mathcal{O}(1)$ denotes all terms weighted by non-negative powers of
$\hbar$, and containing contributions of higher genus curves. The adelic
expression given for $\mathcal{D}_{X}$ in Part IX is built from the adelic
tensor product

$\underline{\mathcal{D}}_{X}(\\{t_{r}^{(\zeta)}\\},\hbar,Q):=\bigotimes_{M=1}^{\infty}\mathcal{D}_{X/\mathbb{Z}_{M}}^{tw}\left(\left\\{t_{r(\zeta)}^{(\zeta)}/\sqrt{\hbar^{r(\zeta)}}\right\\}_{\zeta:\,\zeta^{M}=1},1,Q^{M}\right),$

where
$\mathcal{D}_{X/\mathbb{Z}_{M}}^{tw}(\\{t^{(\zeta)}_{r(\zeta)}\\},\hbar,Q)$ is
the total descendant potential of a certain twisted fake quantum K-theory of
the orbifold target space $X/\mathbb{Z}_{M}$. Recall that the inputs
$t_{r(\zeta)}^{(\zeta)}\in\mathcal{K}^{fake}_{+}:=K((q-1))$ of
$\mathcal{D}_{X/\mathbb{Z}_{M}}^{tw}$ are labeled by primitive roots of unity
$\zeta$ of orders $m(\zeta)$ dividing $M$, while the subscript
$r(\zeta):=M/m(\zeta)$. This adelic product, subject to the suitable dilaton
shift, is considered as a quantum state
$\langle\underline{\mathcal{D}}_{X}\rangle$ in the quantization of the adelic
product of symplectic loop spaces:

$(\underline{\mathcal{K}}^{\infty},\underline{\Omega}^{\infty}):=\prod_{r=1}^{\infty}(\underline{\mathcal{K}}^{(r)},\underline{\Omega}^{(r)})$

Likewise, the dilaton-shifted function $\mathcal{D}_{X}$ is considered as a
quantum state $\langle\mathcal{D}_{X}\rangle$ in the quantization of the
symplectic loop space $(\mathcal{K}^{\infty},\Omega^{\infty})$, where
$\mathcal{K}^{\infty}$ consists of sequences
${\mathbf{f}}=({\mathbf{f}}_{1},\dots,{\mathbf{f}}_{r},\dots)$ of $K$-valued
rational functions of $q$.

The expression for $\langle\mathcal{D}_{X}\rangle$ in terms of
$\langle\underline{\mathcal{D}}_{X}\rangle$ is induced by the symplectic
adelic map $\underline{\
}:(\mathcal{K}^{\infty},\Omega^{\infty})\to(\underline{\mathcal{K}}^{\infty},\underline{\Omega}^{\infty})$.
To a sequence ${\mathbf{f}}$ of rational functions rational function of $q$,
it associates a collection $\\{\underline{{\mathbf{f}}}_{r}^{(\zeta)}\\}$ of
$K$-valued Laurent series in $(q-1)$, where
$\underline{{\mathbf{f}}}_{r}^{(\zeta)}$ is the Laurent series expansion near
$q=1$ of $\Psi^{r}({\mathbf{f}}_{r}(q^{1/m(\zeta)}/\zeta))$. Note that
$\underline{{\mathbf{f}}}_{r}^{(\zeta)}$ is related by $\Psi^{r}$ to what we
denoted earlier in this text by ${\mathbf{f}}_{r}^{(\zeta)}$.

The relation between the quantization spaces induced by the symplectic adelic
map involves the change from the standard Lagrangian polarization
$\underline{\mathcal{K}}^{\infty}_{\pm}$ (using which the adelic product
$\otimes_{M}\mathcal{D}_{X/\mathbb{Z}_{M}}^{tw}$ is lifted to a function on
$\underline{K}^{\infty}$) to the uniform polarization, in which the negative
space coincides with adelic image of
$\mathcal{K}^{\infty}_{-}\subset\mathcal{K}^{\infty}$. After the
polarization’s change, $\langle\mathcal{D}_{X}\rangle$ is identified with the
restriction of $\langle\underline{\mathcal{D}}_{X}\rangle$ to the image of
$\mathcal{K}^{\infty}_{+}=\mathcal{K}^{\infty}/\mathcal{K}^{\infty}_{-}$ in
$\underline{\mathcal{K}}^{\infty}_{+}$ under the adelic map.

The whole expression has the structure of Wick’s formula of summation over
graphs, where the factors in
$\otimes_{M}\langle\mathcal{D}_{X/\mathbb{Z}_{M}}^{tw}\rangle$ represent
different types of vertices, and the edge propagators come from the change of
polarization. The latter is equivalent to the application of
$e^{\oplus_{r=1}^{\infty}\hbar^{r}\nabla_{r}/2}$, where $\nabla_{r}$ is a
certain 2nd order differential operator acting only through the inputs
$t_{r}^{(\zeta)}/\sqrt{\hbar^{r}}$ of $\underline{\mathcal{D}}_{X}^{\infty}$
of the same level $r$.

Our current goal is to extract from this expression a formula for the
genus-$0$ part of $\log\mathcal{D}_{X}$, i.e. the terms with negative powers
of $\hbar$:

$\sum_{k>0}\hbar^{-k}\Psi^{k}(\mathcal{F}_{0}({\mathbf{t}}_{k},{\mathbf{t}}_{2k},\dots,{\mathbf{t}}_{rk},\dots))/k.$

Recall that each $\mathcal{D}_{X/\mathbb{Z}_{M}}^{tw}$ has the form of the
genus expansion
$\mathcal{D}({\mathbf{t}})=e^{\sum_{g}\hbar^{g}\mathcal{F}_{g}({\mathbf{t}})}$,
where $\mathcal{F}_{g}$, which counts contributions of genus-$0$ stable maps
to the orbifold $X/\mathbb{Z}_{M}$, is homogeneous of degree $2-2g$ in
${\mathbf{t}}$ after the dilaton shift. Contributions to genus-0 part of
$\log\mathcal{D}_{X}$ come, however, from mappings of those connected
orbicurves to $X/\mathbb{Z}_{M}$ whose ramified $\mathbb{Z}_{M}$-cover is
rational (though not necessarily connected). If $\widehat{\sigma}_{i}$,
$i=1,\dots,n$, are marked points of ramification index $m_{i}=M/r_{i}$ on a
connected orbicurve $\widehat{\Sigma}$, then Hurwitz’ formula for the Euler
characteristic of the covering curve $\Sigma$ gives

$\operatorname{eu}(\Sigma)=M(2-2g-n)+\sum_{i=1}^{n}r_{i}=M\left(2-2g-\sum_{i}\left(1-\frac{1}{m_{i}}\right)\right).$

For $\operatorname{eu}(\Sigma)$ to be positive, we must have $g=0$, all but at
most $3$ of $m_{i}$ equal to $1$, and those $3$ (call them $a,b,c$) satisfy
$1/a+1/b+1/c>1$. Of course, this singles out the $ADE$-orbifold structures on
$\mathbb{C}P^{1}$. However, it is easy to check that in the cyclic group
$\mathbb{Z}_{M}$ there are no elements of orders $a,b,c$ whose product is the
identity, unless it is $A_{m-1}$-case: one of $a,b,c$ equals $1$, and the
other $2$ are equal to $m$. In other words, if $\Sigma\to\mathbb{C}P^{1}$ is a
rational ramified $\mathbb{Z}_{M}$-cover, then base $\mathbb{C}P^{1}$ carries
2 ramification points of the same order $m$, $\Sigma$ consists of $r=M/m$
disjoint copies of $\mathbb{C}P^{1}$, and the generator $h\in\mathbb{Z}_{M}$
acts on $\Sigma$ by cyclically permuting the $r$ copies in such a way that
$h^{r}$ acts on each of them as the multiplication by a primitive $m$th root
of unity $\zeta$ at one of the ramification points (and hence by $\zeta^{-1}$
at the other).

We can now extract from the logarithm of the adelic product the vertex
contribution into the genus-0 part of $\log\mathcal{D}_{X}$. It has the form:

$\sum_{M=1}^{\infty}\left(\mathcal{F}_{X/\mathbb{Z}_{M}}^{tw}\left(\frac{t_{M}^{(1)}}{\sqrt{\hbar^{M}}}\right)+\frac{1}{2}\sum_{\stackrel{{\scriptstyle\zeta^{M}=1}}{{\zeta\neq
1}}}\left\langle
d^{2}\mathcal{F}^{tw}_{X/\mathbb{Z}_{M}}\left(\frac{t_{M}^{(1)}}{\sqrt{\hbar^{M}}}\right)\,\frac{t^{(\zeta)}_{r(\zeta)}}{\sqrt{\hbar^{r(\zeta)}}},\frac{t^{(\zeta^{-1})}_{r(\zeta^{-1})}}{\sqrt{\hbar^{r(\zeta^{-1})}}}\right\rangle\right).$

In this expression we use a number of ad-hoc conventions. By
$\mathcal{F}_{X/\mathbb{Z}_{M}}^{tw}$ we denote here the genus-$0$ descendant
potential of fake twisted quantum K-theory of $X/\mathbb{Z}_{M}$. It is a
function of $\\{t_{r(\zeta)}^{(\zeta)}\\}$, where $\zeta$ is an $M$th root of
unity of primitive order $m(\zeta)$, and $r(\zeta)=M/m(\zeta)$. The input
$t_{M}^{(1)}$ corresponds to the unramified sector ($\zeta=1$). In the left
summand, we assume that the inputs from all other sectors are set to $0$. In
the previous discussion, this term represents contributions of totally
unramified (and hence trivial) $\mathbb{Z}_{M}$-covers
$\Sigma\to\mathbb{C}P^{1}$. The right sum represents covers ramified at $2$
points. Each term comes from the quadratic differential of
$\mathcal{F}_{X/\mathbb{Z}_{M}}^{tw}$, and is a bilinear form in
$t_{r(\zeta)}^{(\zeta)}$ and $t_{r(\zeta^{-1})}^{(\zeta^{-1})}$. The
coefficient matrix of the bilinear form consists, of course, of the 2nd
derivatives of $\mathcal{F}_{X/\mathbb{Z}_{M}}^{tw}$ in the direction of the
appropriate sectors (corresponding to $\zeta$ and $\zeta^{-1}$), and evaluated
at the input $t_{M}^{(1)}/\sqrt{\hbar^{M}}$ (as indicated in the formula),
while all inputs $t_{r(\eta)}^{(\eta)}$ from ramified sectors
($\eta^{M}=1,\eta\neq 1$) are set to zero. The linear operator
$d^{2}\mathcal{F}_{X/\mathbb{Z}_{M}}^{tw}(t_{M}^{(1)}/\sqrt{\hbar^{M}})$ acts
from the space $(\mathcal{K}_{r}^{(\zeta)})_{+}\ni t_{r}^{(\zeta)}$ to
$(\mathcal{K}_{r}^{(\zeta)})_{-}$ which is identified by the symplectic
pairing with the dual of $(\mathcal{K}_{r}^{(\zeta^{-1})})_{+}\ni
t_{r}^{(\zeta^{-1})}$.

Since the (dilaton-shifted) functions $\mathcal{F}_{X/\mathbb{Z}_{M}}^{tw}$
have homogeneity degree $2$, we can rewrite the above sum by powers of
$\hbar$:

$\underline{\mathcal{F}}:=\sum_{r=1}^{\infty}\hbar^{-r}\left(\mathcal{F}_{X/\mathbb{Z}_{r}}^{tw}(t^{(1)}_{r})+\frac{1}{2}\sum_{\zeta\neq
1}\left\langle
d^{2}\mathcal{F}^{tw}_{X/\mathbb{Z}_{m(\zeta)r}}(t_{m(\zeta)r}^{(1)})\,t_{r}^{(\zeta)},t_{r}^{(\zeta^{-1})}\right\rangle\right).$

There are two ways to approach our goal. One is to retain in
$\log(e^{\oplus_{r}\hbar^{r}\nabla_{r}/2}e^{\underline{\mathcal{F}}})$ all
terms of negative order in $\hbar$. This leads to summation over rooted trees.
The other is to introduce the adelic version
$\underline{\mathcal{L}}^{\infty}_{X}\subset(\underline{\mathcal{K}}^{\infty},\underline{\Omega}^{\infty})$
of the variety $\mathcal{L}_{X}$ using $\underline{\mathcal{F}}_{X}$ and the
standard polarization $\underline{\mathcal{K}}_{\pm}^{\infty}$.

Let us start with the first way. Put
$\underline{\mathcal{F}}=\sum_{r}\hbar^{-r}\underline{\mathcal{F}}_{r}$, i.e.
denote by $\underline{\mathcal{F}}_{r}$ the sum of the terms weighted by the
$r$th power of $\hbar^{-1}$, and consider each $\underline{F}_{r}$ as a family
of functions of $t_{r}=\\{t_{r}^{(\zeta)}\\}$ depending on $t_{mr}^{(1)}$ with
$m>1$ as parameters. Let us denote the parameters in
$\underline{\mathcal{F}}_{r}$ by $t_{m,r}^{(1)}$ (thereby making the
parameters with different $r$ independent of each other). The differential
operator $\nabla_{r}$ acts only through the variables $t_{r}$. Therefore

$\log\left(e^{\hbar^{r}\nabla_{r}/2}e^{\hbar^{-r}\underline{\mathcal{F}}_{r}}\right)=\hbar^{-r}\mathcal{F}_{r}+\mathcal{O}(\hbar^{0}),$

where $\mathcal{F}_{r}$ is obtained by Wick’s summation over trees from vertex
contribution $\underline{\mathcal{F}}_{r}$ and edge propagator $\nabla_{r}$.
It still depends on $t_{m,r}^{(1)}$ with $m>1$ as parameters.

Let us now employ our second approach, based on the families of Lagrangian
varieties in adelic symplectic loop spaces.

We have

$\underline{\mathcal{F}}_{r}=\mathcal{F}_{X/\mathbb{Z}_{r}}^{tw}(t^{(1)}_{r})+\frac{1}{2}\sum_{\zeta\neq
1}\left\langle
d^{2}\mathcal{F}^{tw}_{X/\mathbb{Z}_{m(\zeta)r}}(t_{m(\zeta),r}^{(1)})\,t_{r}^{(\zeta)},t_{r}^{(\zeta^{-1})}\right\rangle,$

where $t_{m,r}^{(1)}$ are viewed as parameters. Given their values, the graph
of $d\underline{\mathcal{F}}_{r}$ is a Lagrangian submanifold
$\underline{\mathcal{L}}_{r}$ in the adelic loop space
$\underline{\mathcal{K}}^{(r)}$, which is the Cartesian product555More
precisely, it is the subset in this product, consisting of sequences of
Laurent series in $q-1$ with all but finitely many terms having no pole when
considered modulo any finite term of the filtration in $\Lambda$.

$\underline{\mathcal{K}}^{(r)}:=\prod_{\text{roots of unity $\zeta$}}\
\mathcal{K}_{r}^{(\zeta)},$

where each $\mathcal{K}_{r}^{(\zeta)}$ is a copy of
$\mathcal{K}^{fake}:=K((q-1))$. It is equipped with the Cartesian product
symplectic form

$\underline{\Omega}^{\infty}({\mathbf{f}},{\mathbf{g}})=\sum_{\zeta}\frac{1}{m(\zeta)}\sum_{r=1}^{\infty}\frac{1}{r}\operatorname{Res}_{q=1}(\underline{{\mathbf{f}}}_{r}^{(\zeta)}(q^{-1}),\underline{{\mathbf{g}}}_{r}^{(\zeta^{-1})}(q))^{(r)}\,\frac{dq}{q},$

where the twisted Poincaré pairing $(\cdot,\cdot)^{(r)}$ on
$K:=K^{0}(X)\otimes\Lambda$ is characterized by

$(\Psi^{r}a,\Psi^{r}b)^{(r)}=r\Psi^{r}(a,b).$

Note that the symplectic form pairs the Lagrangian subspace
$(\mathcal{K}_{r}^{(\zeta)})_{\pm}$ with
$(\mathcal{K}_{r}^{(\zeta^{-1})})_{\mp}$ (here $+/-$ refer to “power
series/principal part” respectively), and the standard polarization is
$\underline{\mathcal{K}}^{\infty}_{\pm}:=\prod_{\zeta,r}(\mathcal{K}_{r}^{(\zeta)})_{\pm}$.

Based on the explicit form of $\underline{\mathcal{F}}_{r}$, for each value of
the parameters, we have
$\underline{\mathcal{L}}_{r}=\prod_{\zeta}\mathcal{L}_{r}^{(\zeta)}$, where
$\mathcal{L}_{r}^{(1)}\subset\mathcal{K}_{r}^{(1)}$ is the graph of
$d\mathcal{F}_{X/\mathbb{Z}_{r}}^{tw}(t_{r}^{(1)})$, and for $\zeta\neq 1$,
each
$\mathcal{L}_{r}^{(\zeta)}\subset(\mathcal{K}_{r}^{(\zeta)})_{+}\oplus(\mathcal{K}_{r}^{(\zeta^{-1})})_{-}$
is the graph of the self-adjoint linear map $t_{r}^{(\zeta)}\mapsto
d^{2}\mathcal{F}^{tw}_{X/\mathbb{Z}_{m(\zeta)r}}(t_{m(\zeta),r}^{(1)})\,t_{r}^{(\zeta)}$.

Our reasoning based on summation over trees implies that replacing
$\underline{\mathcal{F}}_{m}$ with $\mathcal{F}_{m}$ can be considered as the
change of the standard polarization into the uniform one without changing the
Lagrangian submanifold
$\underline{\mathcal{L}}_{r}\subset\underline{\mathcal{K}}^{(r)}$ which
represents the graph of the differential.

Our next goal is to see how all $\nabla_{mr}$ intertwine $\mathcal{F}_{r}$
with $\mathcal{F}_{mr}$ (acting through the parameters $t_{m,r}^{(1)}$ on the
former, and through the variables $t_{mr}^{(\zeta)}$ on the latter), and
extract terms of weight $\hbar^{-1}$ from the whole expression. We claim that
the terms of the total weight $\hbar^{-r}$ are obtained by Wick’s summation
over rooted trees, where the root vertex is represented by $\mathcal{F}_{r}$,
all other vertices correspond to $\mathcal{F}_{mr}$ with $m>1$, and the edges,
connecting vertices of lower level $mr$ with higher level $mlr$, come from the
propagators $\nabla_{mlr}$.

Figure 1. A rooted tree

This is a purely combinatorial statement. Orient the edges from lower level to
higher (Figure 1). For a tree just described, each non-root vertex of level
$mr$ comes with the weight $\hbar^{-mr}$ and a unique entering edge weighted
by the inverse factor $\hbar^{mr}$. Therefore the whole rooted tree
contributes with the weight $\hbar^{-r}$ of the root vertex. Conversely,
suppose that a tree has $k+1>1$ “roots” (i.e. vertices with no entering
edges). Then there are $k$ edges in excess of those entering non-root vertices
and canceling their weights as above. Moreover, since the part of the tree
below any fixed level must remain cycle-free, the weights $-n_{i}$,
$n_{0}\leq\dots\leq n_{k}$ of the roots are majorated by the weights
$m_{1}\leq\dots\leq m_{k}$ of the excess edges this way: $n_{i}<m_{i}$. In
effect, due to the divisibility properties of the weights, $2n_{i}\leq m_{i}$.
Therefore $n_{0}+\cdots+n_{k}\leq m_{1}+\cdots+m_{k}$, and hence the total
weight of the tree is non-negative. Finally, adding cycle-generating edges to
a tree (even with one root) makes the weight of the resulting graph non-
negative.

Let us examine now the summation over trees with one root of level $r=1$. It
affects the root contribution $\mathcal{F}_{1}$ by shifting the values of the
parameters $t=(t_{1,1}^{(1)},\dots,t_{m,1}^{(1)},\dots)$. This follows from
Taylor’s formula

$\mathcal{F}_{1}(t+y)=\sum_{\alpha=(\alpha_{1},\dots,\alpha_{m},\dots)}\partial^{\alpha}\mathcal{F}_{1}(t)\frac{y^{\alpha}}{\alpha!},$

where $y_{m}$ signifies the component of the derivation
$\nabla_{m}\mathcal{F}_{m}$ (recall that we view $\nabla_{m}$ as a bi-
derivation) in the direction of the variables $t_{m,1}^{(1)}$ (and dropping
the components in the directions of $t_{m}^{(\eta)}$ with $\eta\neq 1$). Of
course, the variables $t_{m}^{(\zeta)}$ of $\mathcal{F}_{m}$ together with
partial derivatives of $\mathcal{F}_{m}$ in these variables (computed at
certain values of the parameters $t_{l,m}^{(1)}$) form a point on the graph of
the differential of $\mathcal{F}_{m}$ (at these values of the parameters),
which is a Lagrangian submanifold in the symplectic loop space
$\underline{\mathcal{K}}^{(m)}:=\prod_{\zeta}\mathcal{K}_{m}^{(\zeta)}$. Let
us denote this point by $\underline{f}_{m}$, and the Lagrangian submanifold by
$\underline{\mathcal{L}}_{m}$. Since $\nabla_{m}$ generates the transition
between the standard and uniform polarizations on
$\underline{\mathcal{K}}^{(m)}$, we conclude that the shifted value
$t_{m}^{(1)}+y_{m}$ of the parameter $t_{m,1}^{(1)}=t_{m}^{(1)}$ can be
described as the component in the untwisted sector of the projection
$[\underline{f}_{m}]_{+}$ of
$\underline{f}_{m}\in\underline{\mathcal{L}}_{m}\subset\underline{\mathcal{K}}^{(m)}$
to $\underline{\mathcal{K}}^{(m)}_{+}$ with respect to the uniform
polarization (while with respect to the standard one, such projection is
$t_{m}$ by the very definition).

The same is true for the terms weighted by $\hbar^{-r}$, which come from
rooted trees with the root at the level $r$: the effect of all propagators
$\nabla_{mlr}$ on $\mathcal{F}_{r}$ consists in shifting the values of the
parameters $t_{rm}^{(1)}$ into $[\underline{f}_{rm}]_{+}^{(1)}$ where
$\underline{f}_{rm}\in\underline{\mathcal{L}}_{rm}\subset\underline{\mathcal{K}}^{(rm)}$
represents the differential $d\mathcal{F}_{rm}$ at the point
$\\{t_{rm}^{(\eta)}\\}$ and at the appropriate values of the parameters
$t_{rlm}^{(1)}$. The subscript in $[\dots]_{+}^{(1)}$ refers to the
projections along the negative space of the uniform polarization in
$\underline{\mathcal{K}}^{(rm)}$, while the superscript indicates the
component in the untwisted sector $\eta=1$. Note that the “appropriate” values
of the parameters $t_{rlm}^{(1)}$ are likewise determined by the differentials
of $\mathcal{F}_{n}$, where $n$ runs multiples of $rlm$, so that this
description has the form of an infinite recursion relation.

Our next goal is to identify $\mathcal{L}_{r}^{(\zeta)}$ in terms of the
genus-0 descendant potential $\mathcal{F}_{X}^{fake}$ of the fake (non-
twisted) quantum K-theory of $X$. We denote by
$\mathcal{L}_{X}^{fake}\subset\mathcal{K}^{fake}$ the overruled Lagrangian
cone (see [1, 3, 5]) representing the graph of the differential of
$\mathcal{F}_{X}^{fake}$ with respect to the Lagrangian polarization
$\mathcal{K}^{fake}_{\pm}$ on $(\mathcal{K}^{fake},\Omega^{fake})$ (and
dilaton-shifted by $(1-q)\mathbf{1}$). In the symplectic space
$\mathcal{K}_{r}^{(1)}$ (it is equipped with the symplectic form based on the
twisted pairing $(\cdot,\cdot)^{(r)}$, lies the Lagrangian cone
$\Psi^{r}(\mathcal{L}_{X}^{fake})$. In fact, this is $\mathcal{L}_{r}^{(1)}$.

To prove it, let us express in terms of $\mathcal{F}_{X}^{fake}$ the function
$\mathcal{F}_{X/\mathbb{Z}_{r}}^{tw}$ at the unramified input $t^{(1)}_{r}=t$
and with all inputs from ramified sectors set to zero. We claim
that666$\Psi^{r}$ is invertible in fake K-theory.

$\mathcal{F}_{X/\mathbb{Z}_{r}}^{tw}(t,Q^{r})=r^{-1}\Psi^{r}(\mathcal{F}_{X}^{fake}(\Psi^{1/r}t,Q)).$

Indeed, let $\mathcal{M}$ temporarily denote a moduli space, $X_{0,n,d}$, of
genus-0 stable maps to $X$, so that its contribution to
$\mathcal{F}_{X}^{fake}(t)$ is given by

$\chi^{fake}(\mathcal{M};V)=\int_{[\mathcal{M}]}\operatorname{ch}(V)\operatorname{td}(T_{\mathcal{M}}),$

where
$V=\otimes_{i=1}^{n}\left(\sum_{m}\operatorname{ev}_{i}^{*}(t_{m})L_{i}^{m}\right)$,
and $T_{\mathcal{M}}$ is the virtual tangent bundle to $\mathcal{M}$. The same
moduli space parameterizes stable maps to $X/\mathbb{Z}_{r}$, which are
trivial $\mathbb{Z}_{r}$-covers of stable maps to $X$. Its contribution to
$\mathcal{F}_{X/\mathbb{Z}_{r}}$ (non-twisted) equals
$r^{-1}\chi^{fake}(M;V)$, where the factor $r^{-1}$ comes from
$\mathbb{Z}_{r}$-symmetries of the covers. The contribution into
$\mathcal{F}_{X/\mathbb{Z}_{M}}^{tw}$ is different, and is equal to

$r^{-1}\chi^{fake}\left(\mathcal{M};V\otimes\frac{\operatorname{Eu}(T_{\mathcal{M}})}{\operatorname{Eu}(\Psi^{r}(T_{\mathcal{M}}))}\right).$

The twisting fraction
$\operatorname{Eu}(T_{\mathcal{M}})/\operatorname{Eu}(\Psi^{r}(T_{\mathcal{M}}))$
reflects the difference between deformations of a trivial
$\mathbb{Z}_{r}$-cover as stable maps to $X/\mathbb{Z}_{r}$ and as a stable
map to $X$ of a disconnected curve with $r$ components. For each K-theoretic
Chern root $L$ of the virtual tangent bundle, we have (putting $x=c_{1}(L)$):

$\operatorname{td}(L)\left(\operatorname{ch}\frac{Eu(L)}{\operatorname{Eu}(L^{r})}\right)=\frac{x}{1-e^{-x}}\frac{1-e^{-x}}{1-e^{-rx}}=\frac{1}{r}\operatorname{td}(L^{r}).$

Therefore in cohomological terms, the contribution into
$\mathcal{F}_{X/\mathbb{Z}_{r}}^{tw}$ reads:

$r^{-1-\dim_{\mathbb{C}}T_{\mathcal{M}}}\int_{[\mathcal{M}]}\operatorname{ch}(V)\operatorname{td}(\Psi^{r}(T_{\mathcal{M}})).$

The integrand here can be rewritten as
$\Psi^{r}(\operatorname{ch}(\Psi^{1/r}(V))\operatorname{td}(T_{\mathcal{M}}))$.
Note that $\Psi^{r}$ acts on cohomology as multiplication by $r^{degree/2}$.
Since integration over the virtual fundamental cycle $[\mathcal{M}]$ picks
only the terms of degree $2\dim_{\mathbb{C}}T_{\mathcal{M}}$, the result comes
out the same as

$r^{-1}\int_{[\mathcal{M}]}\operatorname{ch}(\Psi^{1/r}(V))\operatorname{td}(T_{\mathcal{M}})=r^{-1}\chi^{fake}(\mathcal{M};\Psi^{1/r}(V)).$

Taking into account that the polarizations in
$\mathcal{K}_{1}^{(1)}=\mathcal{K}^{fake}$ and $\mathcal{K}_{r}^{(1)}$ are
also related by the operation $\Psi^{r}$, we conclude777The coefficient
$r^{-1}$ is absorbed by the fact the symplectic form in
$\underline{\mathcal{K}}^{(r)}$ satisfies
$\underline{\Omega}^{(r)}(\Psi^{r}(f),\Psi^{r}(g))=r^{-1}\Psi^{r}(\Omega^{fake}(f,g))$.
Indeed, due to degree-2 homogeneity of $\mathcal{F}_{X}^{fake}$, for a point
$f\in\mathcal{L}_{X}^{fake}$, we have
$\mathcal{F}_{X}^{\operatorname{fake}}([f]_{+})=\Omega^{fake}(f,[f]_{+})/2$,
and hence
$\underline{\Omega}^{(r)}(\Psi^{r}(f),\Psi^{r}([f]_{+}))/2=r^{-1}\Psi^{r}(\mathcal{F}_{X}^{fake}([f]_{+})$.
that the graph $\mathcal{L}_{r}^{(1)}\subset\mathcal{K}_{r}^{(1)}$ of the
differential of $\mathcal{F}_{X/\mathbb{Z}_{r}}^{tw}$ in the unramified sector
(i.e. after setting $t_{r}^{(\zeta)}=0$ for all $\zeta\neq 1$) coincides with
$\Psi^{r}(\mathcal{L}_{X}^{fake})$.

Next, from Section 3 of Part IX, we have (in dilaton-shifted notation)

$\mathcal{F}_{X/\mathbb{Z}_{M}}\left(\\{{\mathbf{t}}^{(h)}\\}_{h\in\mathbb{Z}_{r}}\right)=\sum_{\chi\in\operatorname{Repr}(\mathbb{Z}_{r})}\mathcal{F}_{X}^{fake}\left(\frac{1}{M}\sum_{\chi}t^{(h)}\chi(h)\right).$

The quadratic differential of $\mathcal{F}_{X/\mathbb{Z}_{M}}$ at an
unramified input is

$\frac{1}{2M}\sum_{h\neq 1}\left\langle d^{2}\mathcal{F}_{X}^{fake}(t^{(1)})\
t^{(h)},t^{(h^{-1})}\right\rangle,$

where we took into account that dilaton-shifted 2nd derivatives of
$\mathcal{F}_{X}^{fake}$ are homogeneous of degree $0$. The Lagrangian
subspace generated by the quadratic form $\frac{1}{2}\left\langle
d^{2}\mathcal{F}_{X}^{fake}(t)u,u\right\rangle$, i.e. the graph of the linear
map $u\mapsto d^{2}\mathcal{F}_{X}^{fake}(t)u$, is the tangent space $T$ to
$\mathcal{L}_{X}^{fake}$ at the point corresponding to
$d\mathcal{F}_{X}^{fake}$ at the input $t$. Now the transformation from (non-
twisted) $\mathcal{F}_{X/\mathbb{Z}_{M}}$ to
$\mathcal{F}_{X/\mathbb{Z}_{M}}^{tw}$ described in Section 4 of Part IX shows
that the graph $\mathcal{L}_{r(\zeta)}^{(\zeta)}$ of the linear map
$u^{(\zeta)}\mapsto d^{2}\mathcal{F}_{X/\mathbb{Z}_{M}}^{tw}(t)u^{(\zeta)}$
(defined by 2nd derivatives of $\mathcal{F}_{X/\mathbb{Z}_{M}}^{tw}$ in
sectors labeled by $\zeta^{\pm 1}$) is obtained from $T$ by the multiplication
operator (where $r=r(\zeta),m=m(\zeta)$)

$\Box_{\zeta,r}=e^{\textstyle\sum_{k>0}\left(\frac{\Psi^{kr}(T^{*}_{X}-1)}{k(1-\zeta^{-k}q^{kr/m})}-\frac{\Psi^{k}(T^{*}_{X}-1)}{k(1-q^{k})}\right)}.$

Note that our previous result
$\mathcal{F}_{X/\mathbb{Z}_{M}}^{tw}(t)=M^{-1}\mathcal{F}_{X}^{fake}(\Psi^{1/M}(t))$
implies that $\Box_{1,M}T=\Psi^{M}(T)$. Therefore

$\mathcal{L}_{r}^{(\zeta)}=\Box_{\zeta,r(\zeta)}\Box_{1,M}^{-1}\Psi^{M}(T)=\Psi^{r}(\triangle_{\zeta})\Psi^{M}(T)=\Psi^{r}(\triangle_{\zeta}\Psi^{m}(T)),$

where $m=m(\zeta)$, $r=r(\zeta)$, $M=mr$, and

$\triangle_{\zeta}:=e^{\textstyle\sum_{k>0}\left(\frac{\Psi^{k}(T^{*}_{X}-1)}{k(1-\zeta^{-k}q^{k/m})}-\frac{\Psi^{km}(T^{*}_{X}-1)}{k(1-q^{km})}\right)}.$

To be more accurate, one needs to tensor the result with $\Lambda$ over
$\Psi^{M}(\Lambda)$ since $\Psi^{M}$ may not be invertible on the coefficient
ring.

Finally, let us combine all our previous observations on the adelic data with
the description of the adelic map which, according to the main theorem of Part
IX, induces $\langle\mathcal{D}_{X}\rangle$ from
$\langle\underline{\mathcal{D}}_{X}\rangle$. The term $\mathcal{F}_{X}$ in
$\log\mathcal{D}_{X}$ is induced this way from terms of weight $\hbar^{-1}$ in
$\log\underline{D}_{X}$ after the change of standard polarization to the
uniform one.

Let
${\mathbf{f}}=({\mathbf{f}}_{1},{\mathbf{f}}_{2},\dots,{\mathbf{f}}_{r},\dots)$
be a point on the graph $\mathcal{L}_{X}\subset\mathcal{K}^{\infty}$ of the
differential of $\mathcal{F}_{X}$. Then for each $r$,
$\underline{{\mathbf{f}}}_{r}=\\{\underline{{\mathbf{f}}}_{r}(q^{1/m(\zeta)}/\zeta)\\}$
lies in $\underline{\mathcal{L}}_{r}=\prod_{\zeta}\mathcal{L}_{r}^{(\zeta)}$,
which is the graph of the differential of $\mathcal{F}_{r}$ (in fact of one of
the functions of the family $\mathcal{F}_{r}$, taken at an appropriate value
of the parameter) with respect to the uniform polarization.

More concretely, decompose each vector-valued rational function
${\mathbf{f}}_{r}$ into the sum of a Laurent polynomial
${\mathbf{t}}_{r}\in\mathcal{K}^{(r)}_{+}$ and simple fractions
${\mathbf{p}}_{r}^{(\eta)}$ with poles (or any order) at a root of unity
$q=\eta$:
${\mathbf{f}}_{r}={\mathbf{t}}_{r}+\sum_{\eta}{\mathbf{p}}_{r}^{(\eta)}$.
Computing the adelic map,
$\underline{{\mathbf{f}}}_{r}=\\{\underline{{\mathbf{f}}}_{r}^{(\zeta)}\\}$,
for a primitive $m$th root of unity $\zeta$, we find888Expansions of rational
functions into Laurent series near $q=1$ are tacitly assumed on the right.

$\displaystyle\underline{{\mathbf{f}}}_{r}^{(\zeta)}$
$\displaystyle=\Psi^{r}({\mathbf{p}}_{r}^{(\zeta)}(q^{1/m}/\zeta))\ \ \
\text{($\mathcal{K}^{(\zeta)}_{-}$-part in the standard polarization)}$
$\displaystyle+\Psi^{r}\left({\mathbf{t}}_{r}(q^{1/m}/\zeta)+\sum_{\eta\neq\zeta}{\mathbf{p}}_{r}^{(\eta)}(q^{1/m}/\zeta)\right)\
\ \ \text{($\mathcal{K}^{(\zeta)}_{+}$-part).}$

In particular, if we put ${\mathbf{t}}_{r}=0$, these expansions would describe
vectors of the negative space in the uniform polarization. For a general value
of ${\mathbf{t}}_{r}$, projections along such vectors yields
$\Psi^{r}({\mathbf{t}}_{r}(q^{1/m}/\zeta))$. This means that
$\underline{{\mathbf{f}}}_{r}$ represents with respect to the uniform
polarization the differential $d\mathcal{F}_{r}$, computed at the input
$\\{t_{r}^{(\zeta)}\\}$ with
$t_{r}^{(\zeta)}=\Psi^{r}({\mathbf{t}}_{r}(q^{1/m}/\zeta))$. Equivalently:
$\underline{{\mathbf{f}}}_{r}$ represents with respect to the standard
polarization the differential of $d\underline{\mathcal{F}}_{r}$, computed at
the point where
$t_{r}^{(\zeta)}=\Psi^{r}({\mathbf{t}}_{r}(q^{1/m}/\zeta)+\sum_{\eta\neq\zeta}{\mathbf{p}}_{r}^{(\eta)}(q^{1/m}/\zeta))$.
Since the graph of $d\underline{\mathcal{F}}_{r}$ is the product
$\underline{\mathcal{L}}_{r}=\prod_{\zeta}\mathcal{L}_{r}^{(\zeta)}$, we
conclude that
$\underline{{\mathbf{f}}}_{r}^{(\zeta)}\in\mathcal{L}_{r}^{(\zeta)}$.

Now recall that $\mathcal{L}_{r}^{(1)}=\Psi^{r}(\mathcal{L}_{X}^{fake})$ to
conclude that
${\mathbf{f}}_{r}^{(1)}=\Psi^{1/r}(\underline{{\mathbf{f}}}_{r}^{(1)})$ lies
in $\mathcal{L}_{X}^{fake}$. This confirms the criterion (i) in Theorem 1.

Furthermore, to verify the criterion (ii), take $\zeta\neq 1$, and recall that
$\mathcal{L}_{r}^{(\zeta)}=\Psi^{r}(\triangle_{\zeta}\Psi^{m}(T))\otimes\Lambda$,
where $T$ is some tangent space to $\mathcal{L}_{X}^{fake}$. We conclude that
${\mathbf{f}}_{r}^{(\zeta)}=\Psi^{1/r}\underline{{\mathbf{f}}}_{r}^{(\zeta)}$
lies in $\triangle_{\zeta}\Psi^{m}(T)\otimes\Lambda$.

How is the space $T$ determined? It is $T_{f}\mathcal{L}_{X}^{fake}$, where
$[\Psi^{M}(f)]_{+}$ ($M=mr$) is the value of the parameter $t_{m,r}^{(1)}$ in
the family of functions $\mathcal{F}_{r}$. How is that value determined? It is
the untwisted component $[\underline{{\mathbf{f}}}_{mr}]_{+}^{(1)}$ of the
point on the graph of $d\mathcal{F}_{mr}$. As we already know,
$\underline{{\mathbf{f}}}_{mr}^{(1)}$ lies in
$\Psi^{mr}(\mathcal{L}_{X}^{fake})$, and hence
${\mathbf{f}}_{mr}^{(1)}=\Psi^{1/mr}(\underline{{\mathbf{f}}}_{mr}^{(1)})$
lies in $\mathcal{L}_{X}^{fake}$. This shows $f={\mathbf{f}}_{mr}^{(1)}$ and
therefore $T=T_{{\mathbf{f}}_{mr}^{(1)}}\mathcal{L}_{X}^{fake}$ as required.

Ultimately we conclude that if
${\mathbf{f}}=({\mathbf{f}}_{1},\dots,{\mathbf{f}}_{r},\dots)$ is a point in
$\mathcal{L}_{X}$, then the criteria (i) and (ii) of Theorem 1 must be
satisfied:

$(i)\ \ {\mathbf{f}}_{r}^{(1)}\in\mathcal{L}_{X}^{fake},\ \ \text{and}\ \
(ii)\ \
{\mathbf{f}}_{r}^{(\zeta)}\in\triangle_{\zeta}\Psi^{m}(T_{{\mathbf{f}}_{rm}^{(1)}}\mathcal{L}_{X}^{fake})\otimes_{\Psi^{m}(\Lambda)}\Lambda$

for every $r=1,2,\dots$, and every primitive $m$th root of unity $\zeta\neq
1$.

From the computational point of view, this is an infinite system of recursion
relations. However, as it is easy to see, modulo any fixed power of the ideal
$\Lambda_{+}$ (or, more explicitly, under a limited degree and/or the number
of marked points on the curves) this yields a finite system of recursion
relations which uniquely determines ${\mathbf{f}}\in\mathcal{L}_{X}$ from the
projection ${\mathbf{t}}$ of ${\mathbf{f}}$ to $\mathcal{K}_{+}^{\infty}$.

Thus, the criteria (i) and (ii) completely characterize points in
$\mathcal{L}_{X}$ which in a $\Lambda_{+}$-neighborhood of the dilaton vector.

## References

  * [1] T. Coates. Riemann–Roch theorems in Gromov–Witten theory. PhD thesis, 2003, available at http://math.harvard.edu/ tomc/thesis.pdf
  * [2] T. Coates, A. Givental. Quantum cobordisms and formal group laws. The unity of mathematics, 155–171, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006.
  * [3] A. Givental. Symplectic geometry of Frobenius structures. Frobenius manifolds, Aspects Math., E36, Vieweg, Wiesbaden, 2004, pp. 91–112.
  * [4] A. Givental. Explicit reconstruction in quantum cohomology and K-theory. Annales de la Faculté des Sciences de Toulouse, Vol. XXV, no. 2–3, 2016 pp. 215–228.
  * [5] A. Givental, V. Tonita. The Hirzebruch-Riemann–Roch theorem in true genus-0 quantum K-theory. Preprint, arXiv:1106.3136
  * [6] Y.-P. Lee. Quantum K-theory I. Foundations. Duke Math. J. 121 (2004), no. 3, 389–424.


Minki Kang${}^{1,2*}\lx@make@thanks{*:EqualContribution}$, Wooseok Han1∗, Sung
Ju Hwang2, Eunho Yang1,2

# ZET-Speech: Zero-shot adaptive Emotion-controllable
Text-to-Speech Synthesis with Diffusion and Style-based Models

###### Abstract

Emotional Text-To-Speech (TTS) is an important task in the development of
systems (e.g., human-like dialogue agents) that require natural and emotional
speech. Existing approaches, however, only aim to produce emotional TTS for
seen speakers during training, without consideration of the generalization to
unseen speakers. In this paper, we propose ZET-Speech, a zero-shot adaptive
emotion-controllable TTS model that allows users to synthesize any speaker's
emotional speech using only a short, neutral speech segment and the target
emotion label. Specifically, to enable a zero-shot adaptive TTS model to
synthesize emotional speech, we propose domain adversarial learning and
guidance methods on the diffusion model. Experimental results demonstrate that
ZET-Speech successfully synthesizes natural and emotional speech with the
desired emotion for both seen and unseen speakers. Samples are at https://ZET-
Speech.github.io/ZET-Speech-Demo/.

Index Terms: Text-to-Speech Synthesis, Emotional TTS

##  1 Introduction

Modern Text-To-Speech (TTS) systems can now synthesize high-quality speech of
a single or multiple speakers [1, 2, 3]. Beyond synthesizing the speech of
seen speakers during training, several works have proposed zero-shot adaptive
TTS models based on style-based generators [4, 5, 6] and diffusion models [7,
8], which can generalize to unseen speakers' voice as well. However, the
existing methods in adaptive TTS face significant limitations when applied to
emotional speech generation, whose objective is to synthesize the speech with
the desired emotion, in zero-shot scenarios. Specifically, existing approaches
for zero-shot adaptive TTS do not consider the emotion-controllable
generation, while the most of existing approaches for emotional TTS [9, 10,
11, 12, 13, 14] do not take zero-shot scenarios into account. These
limitations necessitate the development of new methods that can effectively
address both requirements in TTS systems, as depicted in Figure 1.

In this work, we tackle the aforementioned limitations in building zero-shot
adaptive emotion-controllable TTS systems. We assume that the emotional
speeches of multiple speakers with the emotion label are given in the training
stage, but only the neutral speech of the target speaker is available in the
inference stage. In the context of training, we empirically find that a naive
application of the existing emotion control methods [9, 15] is ineffective
since the emotion feature and speaker identity are highly entangled in the
style vector used by the style-based generator (See Figure 3 for qualitative
analysis). In order to overcome this limitation, we use domain adversarial
training [16] to disentangle the emotional content from the style vector and
make the style-based generator solely pay attention to the specified emotion
condition. We further suggest using the guidance methods [17, 18] on the
diffusion model in the inference stage, to improve the model's ability to
synthesize the emotional speech of the target speaker conditioned on the
emotion label.

Figure 1: Concept. Our objective is to build a zero-shot adaptive TTS model
capable of synthesizing emotional speech for both seen and unseen speakers
based on emotion labels and neutral reference speech of the target speaker.

Combining the above ideas, we propose a Zero-shot adaptive Emotion-
controllable TTS model, ZET-Speech, that can synthesize the emotional speech
of unseen speakers only with their neutral speech. In experiments, we
empirically validate that our method enables the zero-shot adaptive TTS model
to successfully perform the emotional TTS task.

Our contributions are summarized as follows:

  * •

We propose a zero-shot adaptive emotion-controllable TTS model named ZET-
Speech which can produce emotional speech given the neutral reference speech
of the any speaker and the emotion condition.

  * •

Our proposed approach involves domain adversarial training on the style
vector, which enables the TTS model to be controlled by the provided emotional
condition. Moreover, we leverage guidance methods on the diffusion model to
enhance the level of emotional expression in the synthesized speech.

  * •

Our method demonstrates favorable performance in generating emotional speech,
as evidenced by results on the Korean [19, 20] and English [21, 22] speech
dataset.

##  2 Method

As illustrated in Figure 1, the zero-shot adaptive emotion controllable Text-
to-Speech (TTS) synthesis task aims to generate the mel-spectrograms of
emotional speech $\hat{{\bm{Y}}}$ having the target emotion, given the
following inputs: the text (phonemes) ${\bm{x}}$, the reference speech
${\bm{Y}}$, and the emotion label $e$. In this work, we focus on TTS models
with the style-based generator [23, 6] and the diffusion model [17]. As shown
in Figure 2, our ZET-Speech consists of two ideas upon the TTS models: (1)
domain adversarial training on the style-based generator in training (§ 2.2)
and (2) guidance methods on the diffusion model in inference (§ 2.3). We first
briefly recapitulate the basic components in Section 2.1. Then, we introduce
our proposed method in Section 2.2, 2.3

###  2.1 Preliminary: Style-based Generator and Diffusion

A style-based generator [6, 24] consists of two components: the transformer
encoder $f_{\theta}$ and the mel-style encoder $h_{\psi}$. First of all, the
mel-style encoder $h_{\psi}$ takes a mel-spectrogram of the reference speech
${\bm{Y}}$ as input and outputs the style vector
${\bm{s}}=h_{\psi}({\bm{Y}})$. The style vector ${\bm{s}}$ plays an important
role in zero-shot adaptive TTS by embedding the reference speech of any
speaker into the latent space. Then, the transformer model $f_{\theta}$ takes
a phoneme sequence ${\bm{x}}$ and the style vector ${\bm{s}}$ as input and
outputs the reconstructed mel-spectrogram
$\bm{\mu}=f_{\theta}({\bm{x}},{\bm{s}})$111For brevity, we regard the variance
adaptor for the pitch, energy, and duration prediction as a part of the
transformer encoder..

Moreover, some previous works [2, 7] propose to generate high-fidelity speech
with the diffusion models based on $\bm{\mu}$ from the generator.
Specifically, given $\bm{\mu}$, the reverse diffusion process generates a
speech by denoising the noise drawn from the gaussian distribution
${\bm{Y}}_{T}\sim\mathcal{N}(\bm{\mu},{\bm{I}})$. For the reverse diffusion,
we define the differential equation as follows [2, 17]:

$d{\bm{Y}}_{t}=\left(\frac{1}{2}(\bm{\mu}-{\bm{Y}}_{t})-\nabla_{{\bm{Y}}_{t}}\log
p({\bm{Y}}_{t})\right)\beta_{t}dt,$ (1)

where the neural network
$\bm{\epsilon}_{\phi}({\bm{Y}}_{t},t,\bm{\mu},{\bm{s}})$ estimates the score
function of the noisy data distribution $\nabla_{{\bm{Y}}_{t}}\log
p_{t}({\bm{Y}}_{t})$ given timestep $t$, the noisy data ${\bm{Y}}_{t}$ at $t$,
noise prior $\bm{\mu}$, and the style vector ${\bm{s}}$ [7]. For emotion-
controllable TTS, we additionally input emotion vector ${\bm{e}}$ into both
the hierarchical transformer encoder $f_{\theta}({\bm{x}},{\bm{s}},{\bm{e}})$
and the score estimator
$\bm{\epsilon}_{\phi}({\bm{Y}}_{t},t,\bm{\mu},{\bm{s}},{\bm{e}})$.

Figure 2: Method Overview. (Left) During training, we use domain adversarial
training to separate emotion from style vectors. (Right) In inference, we use
guidance methods on the diffusion model to generate speech with more emotional
features.

###  2.2 Domain Adversarial Training

The conventional approach for incorporating emotional information into the TTS
model is to concatenate or add the emotion vector to the style vector, and
then forward the combined vector to the style-based generator [15]. However,
this approach has proven to be ineffective in style-based generators,
primarily due to the fact that the style vector ${\bm{s}}$ contains both the
speaker identity and emotional features of the reference speech. As a result,
the naive method of combining the emotion vector with the style vector fails
to isolate the emotional features of the reference speech and also fails to
incorporate the desired emotion condition, leading to suboptimal emotional TTS
performance. For instance, as depicted in Figure 1, assume the case where the
neutral reference speech of the unseen speaker is given with the desired
emotion label `happy'. If the emotional feature is entangled in the style
vector, the TTS model generates neutral speech conditioned on the neutral
emotion inside the reference speech, ignoring the given emotion condition
`happy'. Hence, it is of utmost importance to disentangle the speaker identity
and emotional features from the style vector, so that the style vector solely
encompasses speaker-related information.

To this end, we use domain adversarial training [16] to make the mel-style
encoder separate the emotional features of the reference speech from the style
vector. Specifically, we introduce an emotion classifier $g_{\lambda}$ which
is trained to predict the emotion label of the speech given the style vector
$p(e|{\bm{s}};\lambda)$, but we insert the gradient reversal layer prior to
the emotion classifier which scales the gradients by the hyperparameter
$-\alpha$. Therefore, given the emotion classification loss function
$\mathcal{L}_{e}=-\log p(\hat{e}|{\bm{s}};\lambda)$ with the emotion label
$\hat{e}$, the gradient for the mel-style encoder parameter $\psi$ with
regards to the emotion classification loss $\mathcal{L}_{e}$ becomes
$-\alpha\frac{\partial\mathcal{L}_{e}}{\partial\psi}$. By employing DAT, the
mel-style encoder embeds the speech into the style vector, which contains much
less emotional information than it would have otherwise.

###  2.3 Guidance Methods on the Diffusion Model

Thanks to domain adversarial training on the style vector, we can synthesize
emotional speech conditioned on the provided emotion label. The remaining
issue is to make the TTS model produce more emotional speech conditioned on
the given emotion. Recently, diffusion models have demonstrated state-of-the-
art performance in conditional generation [25]. The guidance methods –
classifier [26] and classifier-free guidance [18] – which allow the diffusion
model to generate better-conditioned data are at the heart of a powerful
conditional generation performance.

We incorporate guidance methods so that the diffusion model generates more
emotional speech that is controlled by the given condition. For classifier
guidance, we train the emotion-unconditional score estimator
$\bm{\epsilon}_{\phi}({\bm{Y}}_{t},t,\bm{\mu},{\bm{s}})$ and an emotion
classifier on the noisy mel-spectrograms $p(e|{\bm{Y}}_{t})$ afterwards [27].
Then, we add the classifier gradient into the sampling process:

$\displaystyle
d{\bm{Y}}_{t}=\left(\frac{1}{2}(\bm{\mu}-{\bm{Y}}_{t})-\nabla_{{\bm{Y}}_{t}}\log
p({\bm{Y}}_{t}|e)\right)\beta_{t}dt,$ (2)
$\displaystyle\nabla_{{\bm{Y}}_{t}}\log
p({\bm{Y}}_{t}|e)=\gamma*\nabla_{{\bm{Y}}_{t}}\log
p(e|{\bm{Y}}_{t})+\nabla_{{\bm{Y}}_{t}}\log p({\bm{Y}}_{t}),$

where $e$ is the emotion label and $\gamma$ is a hyperparameter for guidance
control.

Table 1: Objective evaluation on Seen speakers (Korean). Experimental results
on speakers where the models are trained on. Note that the classifier guidance
is not applicable along with the Global Style Token.

|  \+ Emotion Label |  \+ Global Style Token (GST)
---|---|---
Model |  ECA ($\uparrow$) |  CER ($\downarrow$) |  SECS ($\uparrow$) |  ECA ($\uparrow$) |  CER ($\downarrow$) |  SECS ($\uparrow$)
Grad-StyleSpeech |  16.35${}_{\pm\text{8.40}}$ |  16.84${}_{\pm\text{11.28}}$ |  0.778 |  12.54${}_{\pm\text{8.20}}$ |  17.60${}_{\pm\text{11.65}}$ |  0.778
\+ Domain Adversarial Training (DAT) |  31.27${}_{\pm\text{15.78}}$ |  11.93${}_{\pm\text{9.48}}$ |  0.757 |  24.44${}_{\pm\text{16.94}}$ |  10.99${}_{\pm\text{9.06}}$ |  0.746
\+ Classifier Guidance (CG) |  28.73${}_{\pm\text{14.04}}$ |  21.51${}_{\pm\text{20.08}}$ |  0.765 |  - |  - |  -
\+ Classifier-free Guidance (CFGγ=1.75) |  19.68${}_{\pm\text{10.12}}$ |  26.38${}_{\pm\text{18.15}}$ |  0.776 |  13.81${}_{\pm\text{9.15}}$ |  23.21${}_{\pm\text{16.33}}$ |  0.776
Ours |  |  |  |  |  |
ZET-Speech (CG) |  51.59${}_{\pm\text{16.45}}$ |  16.22${}_{\pm\text{16.97}}$ |  0.759 |  - |  - |  -
ZET-Speech (CFGγ=1.25) |  41.75${}_{\pm\text{14.64}}$ |  13.26${}_{\pm\text{10.31}}$ |  0.750 |  33.65${}_{\pm\text{15.40}}$ |  11.84${}_{\pm\text{9.66}}$ |  0.770
ZET-Speech (CFGγ=1.75) |  49.84${}_{\pm\text{17.33}}$ |  16.33${}_{\pm\text{11.74}}$ |  0.747 |  38.89${}_{\pm\text{13.81}}$ |  15.95${}_{\pm\text{14.68}}$ |  0.757

Table 2: Objective evaluation on Unseen speakers (Korean). We use unseen
speakers where the models are not trained on.

| \+ Emotion Label | \+ GST
---|---|---
Model |  ECA ($\uparrow$) |  CER ($\downarrow$) |  ECA ($\uparrow$) |  CER ($\downarrow$)
Grad-StyleSpeech |  18.29${}_{\pm\text{8.59}}$ |  10.20${}_{\pm\text{9.44}}$ |  13.72${}_{\pm\text{7.26}}$ | 9.98${}_{\pm\text{9.85}}$
\+ DAT |  26.01${}_{\pm\text{12.99}}$ | 9.32${}_{\pm\text{9.79}}$ |  23.71${}_{\pm\text{13.28}}$ |  10.15${}_{\pm\text{9.35}}$
\+ CG |  23.29${}_{\pm\text{13.65}}$ |  17.83${}_{\pm\text{21.78}}$ | - | -
\+ CFGγ=1.75 |  17.86${}_{\pm\text{9.56}}$ |  15.74${}_{\pm\text{13.67}}$ |  12.43${}_{\pm\text{9.40}}$ |  15.03${}_{\pm\text{14.35}}$
Ours |  |  |  |
ZET-S. (CG) | 39.86${}_{\pm\text{16.57}}$ |  12.15${}_{\pm\text{15.29}}$ | - | -
ZET-S. (CFGγ=1.25) |  31.57${}_{\pm\text{14.18}}$ |  11.26${}_{\pm\text{10.66}}$ |  31.29${}_{\pm\text{15.61}}$ |  10.50${}_{\pm\text{8.75}}$
ZET-S. (CFGγ=1.75) |  34.57${}_{\pm\text{16.56}}$ |  14.01${}_{\pm\text{12.21}}$ | 32.57${}_{\pm\text{16.67}}$ |  13.57${}_{\pm\text{13.78}}$

With regard to classifier-free guidance, we randomly replace the emotion
embedding with a null embedding $\varnothing$ in training. The important point
to consider is that $\bm{\mu}=f_{\theta}({\bm{x}},{\bm{s}},{\bm{e}})$ also
contains emotional information. Therefore, we should also forward
$\bm{\mu}_{\varnothing}$ to estimate the unconditional noise. We forward null
embedding into a transformer encoder to generate
$\bm{\mu}_{\varnothing}=f_{\theta}({\bm{x}},{\bm{s}},\varnothing)$. We then
combine the conditional and unconditional score estimation to perform sampling
with classifier-free guidance. Formally,

$\displaystyle\hat{\bm{\epsilon}}_{\phi}({\bm{Y}}_{t},t,\bm{\mu},{\bm{s}},{\bm{e}})=$
(3)
$\displaystyle\bm{\epsilon}_{\phi}({\bm{Y}}_{t},t,\bm{\mu},{\bm{s}},{\bm{e}})+\gamma(\bm{\epsilon}_{\phi}({\bm{Y}}_{t},t,\bm{\mu},{\bm{s}},{\bm{e}})-\bm{\epsilon}_{\phi}({\bm{Y}}_{t},t,\bm{\mu}_{\varnothing},{\bm{s}},\varnothing)),$

where $\gamma$ is a hyperparameter for guidance control. Then, we modify the
differential equation in Equation 1 with score estimation in Equation 3 as
follows:

$d{\bm{Y}}_{t}=\left(\frac{1}{2}(\bm{\mu}-{\bm{Y}}_{t})-\hat{\bm{\epsilon}}_{\phi}({\bm{Y}}_{t},t,\bm{\mu},{\bm{s}},{\bm{e}})\right)\beta_{t}dt.$
(4)

##  3 Experiment

###  3.1 Experimental Setup

####  3.1.1 Implementation Details

We use Grad-StyleSpeech [7] as the base model due to its promising performance
based on the style-based generator and diffusion model. We start from the TTS
model pre-trained on a neutral-only multi-speaker dataset. Then, we train the
pre-trained TTS model on the emotional speech dataset for 200k steps with the
batch size of 8 on a single TITAN RTX GPU, using an Adam optimizer with a
learning rate of 1e-3 and a linear decay.

####  3.1.2 Dataset

We mainly target Korean speech synthesis since there are high-quality Korean
speech datasets available from the public organization
AIHub222https://aihub.or.kr/. We pre-train the model on the large-scale multi-
speaker Korean [28] or English [22] dataset. For Korean, we train the model on
the multi-emotional dataset [19]. To evaluate the unseen speaker performance,
we use a separate Korean dataset [20] not used in training. For English, we
use ESD [21] dataset for training and LibriTTS Test [22] for evaluation. We
downsample all speech to 16kHz and use HiFi-GAN [29] as a vocoder.

####  3.1.3 Evaluation Setup

In this subsection, we provide detailed evaluation setups.

Baselines. We use Grad-StyleSpeech [7] as the base model for all baselines for
a fair comparison. (1) Grad-StyleSpeech (GSS). Vanilla Grad-StyleSpeech model
where we input the emotion embedding from the hard emotion label or global
style token [9] as input [15]. (2) GSS + Domain Adversarial Training. Only
applying domain adversarial training on the speaker vector. (3) GSS +
Classifier Guidance. Only applying classifier guidance on the diffusion model
similar to EmoDiff [27]. (4) GSS + Classifier-free Guidance. Only applying
classifier-free guidance on the diffusion model. (5) ZET-Speech (CG, ours).
Our ZET-Speech model with classifier guidance ($\gamma=50$). (6) ZET-Speech
(CFG, ours). Our ZET-Speech model with classifier-free guidance
($\gamma=1.25,1.75$).

Evaluation Metric. For objective evaluation, we introduce three metrics. The
first one is Emotion Classifier Accuracy (ECA) which measures how emotional
the synthesized speech is. Specifically, we utilize a pre-built emotion
classification model based on the Audio Spectrogram Transformer (AST) [30] and
predict the emotion of the synthesized speech. If the AST model successfully
predicts the target emotion, we regard this as a successful case of emotional
TTS. We measure Character Error Rate (CER) to evaluate the quality of the
synthesized speech. We leverage Whisper [31] to recognize the text within the
synthesized speech. To check whether the synthesized speech reflects the
target speaker's identity, we measure Speaker Embedding Cosine Similarity
(SECS). For SECS, we measure the similarity between vectors of the synthesized
and gold emotional speech. To embed speech into the vector, we use the speaker
encoder from the resemblyzer library333https://github.com/resemble-
ai/Resemblyzer. If there is no gold emotional speech available, we cannot
measure the SECS (unseen speaker case). For subjective evaluation, we use Mean
Opinion Score (MOS) to measure the emotional naturalness of the synthesized
speech. For MOS, we require the evaluator to assess the degree to which the
given emotion is naturally reflected in the synthesized speech. We also
measure speaker Similarity MOS (SMOS) for seen speakers, by requesting the
evaluator to check the voice similarity between the synthesized speech and the
gold emotional speech. For all experiments, we use neutral speech as a
reference speech and give emotion conditions as a hard label (Emotion Label)
or sampled emotional speech from the training dataset (Global Style Token). We
use 9 seen speakers and 10 unseen speakers with 10 scripts and 7 emotions for
evaluations. We recruit 20 evaluators for subjective evaluations.

Table 3: Objective evaluations on Unseen speakers (English). We use unseen
speakers (LibriTTS Test [22]) for evaluations.

Model |  ECA ($\uparrow$) |  CER ($\downarrow$)
---|---|---
GSS |  40.50${}_{\pm\text{14.82}}$ | 13.55${}_{\pm\text{9.69}}$
ZET-Speech (CFGγ=1.5, ours) | 59.75${}_{\pm\text{15.65}}$ |  21.96${}_{\pm\text{13.86}}$

###  3.2 Experimental Results

In Table 1 and 2, we present the performance of models in terms of objective
evaluation metrics for seen and unseen speakers, respectively. Results show
that our ZET-Speech outperforms baselines in terms of ECA, demonstrating that
ZET-Speech is able to synthesize the emotional speech conditioned on the given
emotion. These findings suggest that our method can be successfully applied to
any speaker even if there is only a limited amount of neutral speech
available. To elaborate, Domain Adversarial Training (DAT) significantly
enhances the model's ability to generate emotional speech. In other words, the
model has lower sensitivity to emotion conditions without DAT. Furthermore,
the ECA scores obtained by comparing ZET-Speech with the GSS \+ DAT baseline
indicate that guidance methods are beneficial in improving the emotional
expressivity of the synthesized speech.

Experimental results show that both guidance methods contribute to improving
ECA scores. Although classifier guidance shows better ECA than classifier-free
guidance, classifier-free guidance has exclusive merits compared to classifier
guidance where it does not require an additional classifier and can be
combined with other methods like global style token [9].

Our empirical findings also suggest that incorporating guidance methods may
have a negative impact on the quality of the synthesized speech, resulting in
a low CER and SECS metric. However, we also observe that there exists a trade-
off between emotional expressivity and speech quality, which can be managed by
adjusting the relevant hyperparameters $\gamma$. We provide an extensive
analysis of this phenomenon in Section 3.3.

In Table 3, we briefly present the object evaluations on the English benchmark
[21, 22]. We also upload speech samples in English on the demo page
(https://ZET-Speech.github.io/ZET-Speech-Demo/).

Table 4 shows subjective evaluation results. Most evaluators found ZET-Speech
more effective in conveying desired emotoin than Grad-StyleSpeech, consistent
with Tables 1 and 2. Interestingly, evaluators found the speech from ours with
classifier-free guidance more natural and emotional than the speech from ours
with classifier guidance, which differs from Tables 1 and 2.

Table 4: Subjective evaluations (Korean). Experimental results on seen and
unseen speakers with human evaluation.

| Seen | Unseen
---|---|---
Model | MOS | SMOS | MOS
GSS |  2.60${}_{\pm\text{.13}}$ |  3.11${}_{\pm\text{.15}}$ |  2.63${}_{\pm\text{.11}}$
ZET-Speech (CG, ours) |  3.02${}_{\pm\text{.13}}$ |  3.12${}_{\pm\text{.14}}$ |  3.06${}_{\pm\text{.13}}$
ZET-Speech (CFG, ours) | 3.44${}_{\pm\text{.14}}$ | 3.22${}_{\pm\text{.13}}$ | 3.31${}_{\pm\text{.15}}$

Figure 3: Style Vector Visualization. To visually analyze the effect of Domain
Adversarial Training (DAT), we plot style vectors from various speakers and
emotions. The same marker denotes the same speaker, and the same color denotes
the same emotion.

###  3.3 Analysis

Figure 4: Spectrum Visualization. We plot the mel-spectrograms, pitch, and
energy contours of the synthesized speech varying the emotion condition, given
the same script and unseen speaker. Figure 5: Guidance Scale Analysis. In
classifier-free guidance, we measure ECA and CER by varying the guidance scale
$\gamma$.

Effect of Domain Adversarial Training. In Figure 3, we plot the style vectors
of the randomly sampled speakers from the training dataset. We sample speech
with four major emotions (Happy, Sad, Angry, and Neutral) from each speaker
and use t-SNE [32] to reduce the dimension. We plot the style vectors from the
model trained with and without Domain Adversarial Training (DAT). The style
vectors from the model trained without DAT are clearly clustered based on
emotion in Figure 3, demonstrating how the emotional features are entangled
with the style vector. On the other hand, the style vectors from the model
trained with DAT are more evenly distributed across the emotion and less
tightly clustered. The results demonstrate that emotional features are
successfully disentangled from the style vector with DAT.

Spectrum Visualization. In Figure 4, we show the mel-spectrograms of the
synthesized speech for different emotions, accompanied by pitch (F0) and
energy contours to highlight the variations among them. The visualizations
clearly show that the speech generated for Happy and Angry emotions typically
has a higher pitch than that of Sad emotion, emphasizing the ability of our
model to produce emotionally expressive speech with distinct variations for
different emotional conditions.

Guidance Scale. Our analysis indicates that the guidance scale has a
significant impact on both the quality and emotional expressiveness of
synthesized speech in guidance methods. To evaluate this trade-off, we measure
the Emotional Classification Accuracy (ECA) and Character Error Rate (CER)
while varying the guidance scale $\gamma$ from $1.25$ to $2.0$ in ZET-Speech
with classifier-free guidance. As illustrated in Figure 5, we observe that
increasing the guidance scale results in higher ECA and CER. This implies that
a higher guidance scale improves emotion expressivity, but at the expense of
reduced speech quality.

##  4 Conclusion

We proposed ZET-Speech, a zero-shot adaptive emotion controllable TTS model
capable of generating emotional speech in any speaker's voice only with
neutral reference speech and emotion labels. Our approach employs domain
adversarial training to disentangle emotion features from the style vector
during training, allowing the synthesized speech to be controlled solely by
the emotion condition. Furthermore, we utilize guidance methods on the
diffusion model to enhance the control of emotion in a synthesized speech.
Experimental results demonstrated that our method significantly outperforms
baselines in objective and subjective measures of emotional expressivity. We
hope that our method enables practitioners to synthesize emotional speech for
desired speakers without the need for emotional recordings.

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