# Quantum $K$-Theory I: Foundations

Y.-P. Lee  Princeton University
Department of Mathematics
Fine Hall, Washington Road
Princeton, NJ 08544-1000<EMAIL_ADDRESS>Department of Mathematics
University of Utah
Salt Lake City, Utah 84112-0090<EMAIL_ADDRESS>
###### Abstract.

This work is devoted to the study of the foundations of _quantum $K$-theory_,
a $K$-theoretic version of quantum cohomology theory. In particular, it gives
a deformation of the ordinary $K$-ring $K(X)$ of a smooth projective variety
$X$, analogous to the relation between quantum cohomology and ordinary
cohomology. This new quantum product also gives a new class of Frobenius
manifolds.

Research partially supported by NSF grant DMS-0072547.

##  1\. Introduction

This work is devoted to the study of _quantum $K$-theory_, a $K$-theoretic
version of quantum cohomology theory. In particular, it gives a deformation of
the ordinary $K$-ring $K(X)$ of a smooth variety $X$, analogous to the
relation between quantum cohomology and ordinary cohomology.

In order to understand quantum $K$-theory, it is helpful to give a brief
review of quantum cohomology theory. Quantum cohomology studies the
intersection theory (of tautological classes) on
${\overline{M}}_{g,n}(X,\beta)$, the moduli spaces of stable maps from curves
$C$ to a smooth projective variety $X$. The intersection numbers, or
Gromov–Witten invariants, look like

$\int_{[{\overline{M}}_{g,n}(X,\beta)]^{\operatorname{vir}}}{\operatorname{st}}^{*}(\alpha)\prod_{i=1}^{n}{\operatorname{ev}}_{i}^{*}(\gamma_{i})\psi_{i}^{d_{i}}$

where $\gamma_{i}\in H^{*}(X),\alpha\in H^{*}({\overline{M}}_{g,n})$ and
$\psi_{i}$ are the cotangent classes. These notations will be defined in the
next section. For now these numbers are pairings between natural cohomology
classes and the (virtual) fundamental class of
${\overline{M}}_{g,n}(X,\beta)$. These invariants at $genus(C)=0$ determines a
deformation of the product structure of the ordinary cohomology ring. The
associativity of the quantum cohomology ring structure is equivalent to the
_WDVV equation_ , which is basically a degeneration argument by considering a
flat family of solutions parameterized by $\mathbb{P}^{1}$ and equating two
different degeneration points of the solutions.

We propose to apply the same philosophy to $K$-theory. The functorial
interpretation of the integration over the (virtual) fundamental classes
$[{\overline{M}}_{g,n}(X,\beta)]^{\operatorname{vir}}$ is the push-forward of
cohomologies from ${\overline{M}}_{g,n}(X,\beta)$ to the point
$\operatorname{{{Spec}}}\mathbb{C}$ (via the orientation defined by virtual
fundamental class). In fact the push-forward operation can be performed in the
relative setting for any proper morphisms. With this in mind, we define
_quantum $K$-invariants_ as the $K$-theoretic push-forward to
$\operatorname{{{Spec}}}\mathbb{C}$ of some natural vector bundles on
${\overline{M}}_{g,n}(X,\beta)$ (via the orientation defined by the virtual
structure sheaf). More precisely,

$\chi\Bigl{(}{\overline{M}}_{g,n}(X,\beta),\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}_{g,n}(X,\beta)}{\operatorname{st}}^{*}(\alpha)\prod_{i=1}^{n}{\operatorname{ev}}_{i}^{*}(\gamma_{i})\mathcal{L}_{i}^{d_{i}}\Bigr{)}$

where $\gamma_{i}\in K(X),\alpha\in K({\overline{M}}_{g,n})$ and
$\mathcal{L}_{i}$ are the cotangent line bundles. The $K$-theoretic push-
forward $\chi$ will be defined in Section 2.1 and Section 4.1 for algebraic
and topological $K$-theory respectively. The quantum $K$-product is defined by
quantum $K$-invariants and a _deformed_ metric (equation (17)). The
associativity of quantum $K$-product will be established by a sheaf-theoretic
version of WDVV-type argument. A considerable difference in $K$-theory and
intersection theory arises here.

Our main motivations for studying this theory come from two sources. The first
is to use this theory to study the geometry of the moduli space of maps. For
example, in joint works with R. Pandharipande and with I. Ciocan-Fontanine, a
computation of quantum $K$-invariants of $X=\mathbb{P}^{1}$ are used to show
that the Gromov–Witten loci
$\cap_{i}{\operatorname{ev}}^{-1}_{i}(pt_{i})\subset{\overline{M}}_{0,n}(\mathbb{P}^{1},d)$
are not rational when $n$ is large. This thus answers a question raised in
[24] Section 2.2.

The other main motivation comes from the relation between Gromov–Witten theory
and integrable systems. There are many celebrated examples of this sort in
quantum cohomology theory. Two of them are most relevant. The first one is
Witten–Kontsevich’s theory of two dimensional gravity [18] [28]. This theory
builds a link between (full) Gromov–Witten theory of a point to KdV hierarchy.
The second example is Givental–Kim’s theory: It states that genus zero
Gromov–Witten theory on flag manifolds is governed by quantum Toda lattices
[13] [16]. Our goal is therefore to extend this relation to quantum
$K$-theory. The analogous relation between (full) quantum $K$-theory of a
point and _discrete_ KdV hierarchy is yet to be spelled out. See [21] [22] for
some computations in this direction. The relation between genus zero quantum
$K$-theory on flag manifolds and _finite difference_ Toda lattices turns out
to be a very interesting one. The construction of finite difference Toda
lattices involved quantum groups. The interested reader is referred to [14]
for the details.

Besides these, we believe that the quantum $K$-invariants for symplectic
manifolds are symplectic invariants. This assertion, if verified, could have
some implications in symplectic geometry. Also, this theory could be
understood as one manifestation of J. Morava’s quantum generalized cohomology
theory [27].

######  Acknowledgements.

I am grateful to A. Givental and R. Pandharipande for much advice in the
course of this work. The referees’ reports have greatly improved the
exposition of this work. In particular, the proof of Proposition 11 is
drastically simplified. Thanks are also due to T. Graber, G. Kennedy, J.
Morava, Y. Ruan and V. Srinivas for useful discussions. Part of the work is
written during my stay in NCTS, whose hospitality is appreciated.

##  2\. Virtual structure sheaf

The main purpose of this section is to define an element
${\mathcal{O}}^{\operatorname{vir}}$ in the Grothendieck group of coherent
sheaves on the moduli space of stable maps to a smooth projective variety $X$.

###  2.1. Preliminaries on algebraic $K$-theory

The basic reference for algebraic $K$-theory discussed here is [1].

Let $K^{\circ}(V)$ denote the Grothendieck group of locally free sheaves on
$V$ and $K_{\circ}(V)$ denote the Grothendieck group of coherent sheaves on
$V$. There is a cap-product

$K^{\circ}(V)\otimes K_{\circ}(V)\to K_{\circ}(V)$

defined by tensor product

$[E]\otimes[F]\mapsto[E\otimes_{{\mathcal{O}}_{V}}F],$

making $K_{\circ}(V)$ a $K^{\circ}(V)$-module.

For any proper morphism $f:V^{\prime}\to V$, there is a _push-forward_
homomorphism

$f_{*}:K_{\circ}(V^{\prime})\to K_{\circ}(V)$

defined by

$f_{*}([F]):=\sum_{i}(-1)^{i}[R^{i}f_{*}F].$

######  Convention.

If $f$ is an embedding, we will sometimes denote $f_{*}([F])$ simply by $[F]$.

A morphism $f:V^{\prime}\to V$ is a _perfect morphism_ if there is an $N$ such
that

$Tor^{V}_{i}({\mathcal{O}}_{V^{\prime}},F)=0$

for all $i>N$ and for all coherent sheaves $F$ on $V$. For a perfect morphism
$f:V^{\prime}\to V$, one can define the _pull-back_

$f^{*}:K_{\circ}(V)\to K_{\circ}(V^{\prime})$

by

$f^{*}([F])=\sum_{i=0}^{d}(-1)^{i}[Tor^{V}_{i}({\mathcal{O}}_{V^{\prime}},F)].$

It is easy to see that regular embeddings and flat morphisms are perfect.

Let

(1) $\begin{CD}V^{\prime}@>{u}>{}>V\\\ @V{a}V{}V@V{b}V{}V\\\
B^{\prime}@>{v}>{}>B\end{CD}$

be a fibre square with $v$ a regular embedding. The _refined Gysin map_

$v^{!}:K_{\circ}(V)\to K_{\circ}(V^{\prime})$

can be defined as follows. Since $v$ is a regular embedding,
$v_{*}({\mathcal{O}}_{B^{\prime}})$ has a finite resolution of vector bundles
on $B$. That is, ${\mathcal{O}}_{B^{\prime}}$ determines an element
$K^{\circ}_{B^{\prime}}(B)$, the Grothendieck group of vector bundles on $B$
whose homologies are supported on $B^{\prime}$. This then determines an
element, denoted by $[{\mathcal{O}}_{B^{\prime}}^{V}]$, in
$K^{\circ}_{V^{\prime}}(V)$ by pull-back. Define $K_{\circ}^{V^{\prime}}(V)$
to be the Grothendieck group of the coherent sheaves on $V$ with homologies
supported on $V^{\prime}$ (i.e. exact off $V^{\prime}$). There is a canonical
isomorphism
$\varphi:K_{\circ}^{V^{\prime}}(V)\overset{\cong}{\to}K_{\circ}(V^{\prime})$.
One can introduce a generalized cap product

$K^{\circ}_{V^{\prime}}(V)\otimes
K_{\circ}(V)\overset{\otimes}{\to}K_{\circ}^{V^{\prime}}(V)\overset{\varphi}{\to}K_{\circ}(V^{\prime})$

by composing the isomorphism $\varphi$ and the tensor product. The refined
Gysin map is:

$v^{!}([F])=\varphi([{\mathcal{O}}_{B^{\prime}}^{V}\otimes_{{\mathcal{O}}_{V}}F]).$

######  Remark 1.

One can also apply the deformation to the normal cone argument to define the
Gysin pull-back. That is, one can replace the fibre square (1) with

$\begin{CD}V^{\prime}@>{u_{C}}>{}>C_{V^{\prime}}V\\\ @V{a}V{}V@V{b}V{}V\\\
B^{\prime}@>{v_{N}}>{}>N_{B^{\prime}}B\end{CD}$

where $C_{V^{\prime}}V,N_{B^{\prime}}B$ are the normal cone and normal bundle
of the embeddings $u,v$. The specialization to the normal cone argument works
for $K$-theory as well, i.e. one has a map

$\sigma:K_{\circ}(V)\to K_{\circ}(C_{V^{\prime}}V)$

such that the Gysin map can be defined by combining $\sigma_{V}$ with the
Gysin map $v_{N}^{!}$ of the above fibre square. That is

$v^{!}=v_{N}^{!}\circ\sigma.$

######  Remark 2.

If $b$ is also a regular embedding, then
$v^{!}[{\mathcal{O}}_{V}]=b^{!}[{\mathcal{O}}_{B^{\prime}}]$, which will also
be denoted
$[{\mathcal{O}}_{B^{\prime}}]\otimes^{\text{der}}_{{\mathcal{O}}_{V}}[{\mathcal{O}}_{V}^{\prime}]$
to emphasize the symmetry.

If $v:B^{\prime}\to B$ can be factored through a regular embedding
$i:B^{\prime}\to Y$ followed by a smooth morphism $p:Y\to B$, i.e. $v$ is a
_local complete intersection morphism_. One can form the following fibre
diagram

$\begin{CD}V^{\prime}@>{j}>{}>Y^{\prime}@>{q}>{}>V\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ B^{\prime}<EMAIL_ADDRESS>
Then $q$ is smooth and we define $v^{!}([F]):=i^{!}(q^{*}[F])$. This
definition is actually independent of the factorization.

######  Convention.

To simplify the notation, we will often denote the class $[F]$ simply by $F$
and use the operations (like $\otimes$) as if $[F]$ is a complex of sheaves.

###  2.2. Moduli spaces of stable maps

This section serves only the purpose of fixing the notations. The reader
should consult [10] [6] for details.

######  Definition.

A _stable map_ $(\mathcal{C};x,f)$ from a pre-stable curve
$(C;x_{1},\ldots,x_{n})$ of genus $g$ curves with $n$ marked points to $X$,
which represents a class $\beta\in H_{2}(X)$, is a morphism $f:C\to X$
satisfying:

1\. The homological push-forward of $C$ satisfies ${f}_{*}([C])=\beta$.

2\. _Stability_ : The following inequality holds on the normalization
$E^{\prime}$ of each irreducible component $E$:

$\text{$f(E)$ is a point}\Rightarrow 2g(E^{\prime})+\\#\\{\text{special points
on $E^{\prime}$}\\}\geq 3$

It is obvious from the definition that the notion of stable maps is a natural
generalization of the notion of stable curves. The essential point being that
they only allow only _finite_ automorphisms. This makes the following theorem
possible:

######  Theorem 1.

([18]) The stack ${\overline{M}}_{g,n}(X,\beta)$ of stable maps to $X$, a
smooth projective scheme of finite type over $\mathbb{C}$, is a proper
Deligne-Mumford stack.

Between the various spaces in Gromov–Witten theory there are many interesting
morphisms. The first such class of morphisms is called the _forgetful_
morphisms

$\operatorname{ft}_{k}:{\overline{M}}_{g,n+1}(X,\beta)\to{\overline{M}}_{g,n}(X,\beta)$

by forgetting $k$-th marked points and stabilizing if necessary.

The second class is called the _stabilization_ morphisms

${\operatorname{st}}:{\overline{M}}_{g,n}(X,\beta)\to{\overline{M}}_{g,n},$

where ${\overline{M}}_{g,n}$ is the moduli stack of stable curves. It is
defined by forgetting the map and retaining the pre-stable curve, then
stabilizing the curve. Of course one can just forget the map without
stabilization. This leads to the _prestabilization_ morphism [6]

$\operatorname{pres}:{\overline{M}}_{g,n}(X,\beta)\to\mathfrak{M}_{g,n},$

with $\mathfrak{M}_{g,n}$ being the moduli stack of prestable curves.

The third class is called the _evaluation_ morphisms

${\operatorname{ev}}_{i}:{\overline{M}}_{g,n}(X,\beta)\to X$

by evaluating the stable map at the $i$-th marked point. That is,
${\operatorname{ev}}_{i}(f)=f(x_{i})$.

Another object which we will use quite often is the _universal cotangent line
bundle_ $\mathcal{L}_{i}$ on ${\overline{M}}_{g,n}(X,\beta)$. It is defined as
$\mathcal{L}_{i}=x_{i}^{*}(\omega_{\mathcal{C}/M})$, where
$\omega_{\mathcal{C}/M}$ is the relative dualizing sheaf of the universal
curve $\mathcal{C}\to{\overline{M}}_{g,n}(X,\beta)$ and $x_{i}$ are the marked
points.

There are combinatorial generalizations of ${\overline{M}}_{g,n}(X,\beta)$ by
incorporating the modular graphs $\tau$ with degree $\beta$ [6]. The idea is
to use the graphs to keep track of the combinatorics of degeneration of
curves. This bookkeeping simplifies the notations in the proof of functorial
properties of the virtual structure sheaf. We briefly recall the definitions
here.

A _graph_ $\tau$ is a quadruple $(F_{\tau},V_{\tau},j_{\tau},\partial_{\tau})$
where $F_{\tau}$ (_flags_) and $V_{\tau}$ (_vertices_) are finite sets,
$j:F_{\tau}\to F_{\tau}$ is an involution and $\partial:F_{\tau}\to V_{\tau}$
is a map.

$S_{\tau}:=\\{f\in F_{\tau}|jf=f\\}$

is the set of _tails_ (or _marked points_) and

$E_{\tau}:=\\{\\{f_{1},f_{2}\\}\subset F_{\tau}|f_{2}=jf_{1},f_{2}\neq
f_{1}\\}$

is the set of _edges_. A _modular graph_ is a pair $(\tau,g)$, where $\tau$ is
a graph and $g:V_{\tau}\to\mathbb{Z}_{\geq 0}$ is a map. We call $g(v)$ the
_genus_ of the vertex $v$. A modular graph is _stable_ if

$2g(v)+\\#F_{\tau}(v)\geq 3$

for all vertices $v$, where $F_{\tau}(v):=j_{\tau}^{-1}(v)$ and $\\#F(v)$ is
the valence of $v$.

One defines the moduli stack of prestable curves associated to a modular graph
$(\tau,g)$ as

$\mathfrak{M}_{\tau,g}:=\prod_{v\in
V_{\tau}}\mathfrak{M}_{g(v),\\#F_{\tau}(v)}$

where $\mathfrak{M}_{g,n}$ is the usual moduli stack of prestable curves. The
moduli stack of stable curves ${\overline{M}}_{\tau,g}$ associated to a
_stable_ modular graph $(\tau,g)$ is defined similarly

${\overline{M}}_{\tau,g}:=\prod_{v\in
V_{\tau}}{\overline{M}}_{g(v),\\#F_{\tau}(v)}.$

Geometrically, the moduli stack of (pre)stable curves associated to a
(connected) modular graph $(\tau,g)$ which has at least one edge consists of
(connected) singular curves with singularity prescribed by $\tau$. In other
words, the stack of universal curves over the stack of the (pre)stable curves
has singular generic fibres. For details, see [6].

Let $(\tau,g)$ be a modular graph and $\beta:V_{\tau}\to H_{2}(X)$ be the
_degree_. 111We have abused the notations by using $\beta$ (resp. $g$)
sometimes as map $\beta:V_{\tau}\to H_{2}(X)$ ($g:V_{\tau}\to\mathbb{Z}_{\geq
0}$) and sometimes as the total degree $\sum_{v}\beta(v)$ (total genus). It
should be clear in the context which is which. $(\tau,g,\beta)$ is a _stable
modular graph with degree_ if

$\beta(v)=0\Rightarrow 2g(v)+\\#F(v)\geq 3.$

The moduli stack of stable maps ${\overline{M}}(X,\tau,g,\beta)$ 222For
brevity, $\tau$ will sometimes stand for $(\tau,g)$ or $(\tau,g,\beta)$.
associated to $(\tau,g,\beta)$ is defined by the following three conditions:

  1. (1)

If $\tau$ is a graph with only one vertex $v$, and set of flags $F_{v}$ are
all tails, then

${\overline{M}}(X,\tau,g,\beta):={\overline{M}}_{g(v),\\#F_{v}}(X,\beta).$

  2. (2)

${\overline{M}}(X,\tau_{1}\times\tau_{2}):={\overline{M}}(X,\tau_{1})\times{\overline{M}}(X,\tau_{2}),$

where $\tau_{1}\times\tau_{2}$ is the disjoint union of two graphs and $\tau$
stands for $(\tau,g,\beta)$, by abusing the notation.

  3. (3)

If $\sigma$ is obtained from $\tau$ by cutting an edge, then
${\overline{M}}(X,\tau)$ is defined by the following fibre square

(2) $\begin{CD}{\overline{M}}(X,\tau)@>{}>{}>{\overline{M}}(X,\sigma)\\\
@V{}V{}V@V{}V{{\operatorname{ev}}_{i}\times{\operatorname{ev}}_{j}}V\\\
X@>{\Delta}>{}>X\times X,\end{CD}$

where $\Delta$ is the diagonal morphism, and $i,j$ are the corresponding
marked points where the edge is cut.

It is easy to see that the morphisms and operations on
${\overline{M}}_{g,n}(X,\beta)$ generalize to ${\overline{M}}(X,\tau)$. See
[6] for the details.

###  2.3. The construction of virtual structure sheaf

We shall now introduce the notion of _virtual structure sheaf_ , whose
construction is parallel to that of virtual fundamental class as defined in
[25] [5].

Let $V\to B$ be a morphism from a DM-stack to a smooth Artin stack with
constant dimension. A _relative perfect obstruction theory_ is a homomorphism
$\phi$ in the derived category (of quasi-coherent sheaves bounded from above)
of a two term complex of vector bundles $E^{\bullet}=[E^{-1}\to E^{0}]$ to the
relative cotangent complex $L^{\bullet}_{V/B}$ of $V\to B$ such that:

1\. $H^{0}(\phi)$ is an isomorphism.

2\. $H^{-1}(\phi)$ is surjective.

If $V$ admits a global embedding $i:V\to Y$ over $B$ into a smooth Deligne-
Mumford stack, then the two term cutoff of $L^{\bullet}_{V/B}$ at $-1$ is
given by the first two terms in second exact sequence of relative Kähler
differential: $[I/I^{2}\to i^{*}(\Omega_{Y/B})]$, where $I$ is the relative
ideal sheaf of $V$ in $Y$ over $B$.

Let us now recall the definition of _relative intrinsic normal cone_. Choose a
closed $B$-embedding of $V$, $i:V\hookrightarrow Y$, to a smooth DM stack $Y$
over $B$. Since the normal cone $C$ is a $T_{Y}$-cone over $B$ ([5]), one
could associate a cone stack $\mathfrak{C}_{V|B}:=[C/i^{*}T_{Y}]$, which is
independent of the choice of embedding. $E^{\bullet}\to L^{\bullet}_{V}/B$
being a perfect obstruction theory implies ([5] Theorem 4.5) that
$\mathfrak{C}\to[E_{1}/E_{0}]$ is a closed embedding, which induces a cone
$C_{1}\subset E_{1}$. Here $E_{0}\to E_{1}$ is the dual complex of $E^{-1}\to
E^{0}$. Note that even if $V$ does not allow a global embedding, one can still
define the intrinsic normal cone by (étale) local embedding. It is proved in
[5] that the local construction glues together to form a global cone stack.

######  Definition.

Let $E^{\bullet}\to L^{\bullet}_{V/B}$ be a perfect obstruction theory. The
virtual structure sheaf ${\mathcal{O}}^{\operatorname{vir}}_{[V,E]}$ for a
(relative) perfect obstruction theory is an element in $K_{\circ}(V)$ defined
as the pull-back of ${\mathcal{O}}_{C_{1}}\in K_{\circ}(E_{1})$ by the zero
section $0:V\to E_{1}$, i.e.,

${\mathcal{O}}^{\operatorname{vir}}_{[V,E]}=\sum_{i}(-1)^{i}[Tor_{i}^{E_{1}}({\mathcal{O}}_{V}\otimes{\mathcal{O}}_{C_{1}})].$

######  Remark 3.

One could also consider this as an element in the derived category of $V$
defined to be

${\mathcal{O}}^{\operatorname{vir}}_{V}:={\mathcal{O}}_{C_{1}}\otimes^{\text{der}}_{{\mathcal{O}}_{E_{1}}}{\mathcal{O}}_{V},$

where $\otimes^{\text{der}}$ means the derived tensor. Either way, virtual
structure sheaf is in general not a bona fide sheaf.

Apply the above setting to Gromov–Witten theory. Let $\mathfrak{M}_{\tau}$ be
the corresponding moduli stack associated with modular graph $\tau$. Recall
that there is a prestabilization morphism
$\operatorname{pres}:{\overline{M}}(X,\tau)\to\mathfrak{M}_{\tau}$, which will
be our structure morphism $V\to B$. Consider the universal stable map

(3) $\begin{CD}\mathcal{C}@>{f}>{}>X\\\ @V{\pi}V{}V\\\
{\overline{M}}(X,\tau).\end{CD}$

One then get a complex $R^{\bullet}\pi_{*}f^{*}T_{X}$, which can be realized
as a two term complex $[E_{0}\to E_{1}]$ ($E_{i}^{\vee}=E^{-i}$) of vector
bundles on ${\overline{M}}_{g,n}(X,\tau)$ ([4]). In fact, there is a
homomorphism

$\phi:(R^{\bullet}\pi_{*}f^{*}T_{X})^{\vee}\to
L^{\bullet}_{{\overline{M}}(X,\tau)/\mathfrak{M}_{\tau}}$

which is a relative perfect obstruction theory. One therefore defines the
virtual structure sheaf
${\mathcal{O}}_{{\overline{M}}(X,\tau)}^{\operatorname{vir}}$. Note that in
the present case ${\overline{M}}(X,\tau)$ admits a global embedding (Appendix
A of [15]). One might identify
$L^{\bullet}_{{\overline{M}}(X,\tau)/\mathfrak{M}_{\tau}}$ with its two-term
cutoff.

###  2.4. Basic properties of virtual structure sheaf

Some basic properties of virtual structure sheaf is established here as a
$K$-theoretic version of [5]. Because $B$ will always be $\mathfrak{M}_{\tau}$
in all our applications, the readers should feel free to make this assumption,
although all the results will hold for $B$ any smooth equidimensional Artin
stack.

The first step to check the consistency of this definition is the following
two propositions:

######  Proposition 1.

(Consistency) The virtual structure sheaf is independent of the choice of the
global resolutions $E^{0}\to E^{1}$. That is, if $E^{0}\to E^{1}$ is quasi-
isomorphic to $F^{0}\to F^{1}$, then
$\mathcal{O}^{\operatorname{vir}}_{[V,E]}=\mathcal{O}^{\operatorname{vir}}_{[V,F]}$.

###### Proof.

The proof of this proposition is exactly the same as the proof of [4]
Proposition 5.3. ∎

######  Proposition 2.

(Expected dimension) If $V$ has the _expected dimension_ $\dim
B+\operatorname{rank}(E^{\bullet})$, then $V$ is a local complete intersection
and the virtual structure sheaf is equal to ordinary structure sheaf. Note
that the dimension is called “expected dimension” because it is the dimension
in the (“generic”) case of no obstruction.

###### Proof.

The question can be reduced to the absolute case by the argument given in [15]
Appendix B. Thus we will set $B=pt$. Since the question is local (in étale
topology), one may assume that

$X=\operatorname{{{Spec}}}R,\quad
R=\mathbb{C}[x_{1},x_{2},\ldots,x_{r}]/(g_{1},\ldots,g_{s}).$

In fact, it is enough to consider the local ring

$R_{p}=\left(\dfrac{\mathbb{C}[x_{1},x_{2},\ldots,x_{r}]}{(g_{1},\ldots,g_{s})}\right)_{(x_{1},\ldots,x_{r})}$

with maximal ideal $\mathfrak{m}$. Denote $I=(g_{1},\ldots,g_{s})$, then the
two-term cut-off of cotangent complex is

$I/I^{2}\to\Omega_{\mathbb{C}[x_{1},\ldots,x_{r}]}\otimes_{\mathbb{C}[x_{1},\ldots,x_{r}]}R.$

This induces a complex of $R_{p}$ modules

(*) $I/\mathfrak{m}I\overset{\varphi}{\to}\mathfrak{m}/\mathfrak{m}^{2}.$

One may assume that $g_{i}$ contains no linear terms in $x_{i}$, or otherwise
the presentation of $R_{p}$ could be changed to eliminate terms linear in
$x_{i}$. That is, $I\subset\mathfrak{m}^{2}$ and the above $\varphi$ is a zero
map. This implies that the homology of the above complex is

$H^{1}(*)=I/\mathfrak{m}I,\quad H^{0}(*)=\mathfrak{m}/\mathfrak{m}^{2}.$

Because $E^{\bullet}\to L^{\bullet}$ is a perfect obstruction theory,

$H^{0}(*)=H^{0}(E^{\bullet\vee}\otimes\mathbb{C}),\quad
H^{1}(*)\hookrightarrow H^{1}(E^{\bullet\vee}\otimes\mathbb{C}).$

Therefore

$\operatorname{rank}(E^{\bullet})=h^{0}(E^{\bullet}\otimes\mathbb{C})-h^{1}(E^{\bullet}\otimes\mathbb{C})\leq
h^{0}(*)-h^{1}(*)=r-s$

where the last equality is obvious. This shows, combining with the assumption,
that $\dim\operatorname{{{Spec}}}R_{p}\leq r-s$. However,
$\dim\operatorname{{{Spec}}}R_{p}$ is obviously greater or equal to $r-s$.
Thus $\dim\operatorname{{{Spec}}}R_{p}=r-s$ and $\\{g_{i}\\}$ form a regular
sequence. Therefore $V$ is a local complete intersection.

On the other hand, $\dim\operatorname{{{Spec}}}R_{p}=r-s$ implies that
$h^{1}(E^{\bullet\vee}\otimes\mathbb{C})=s$ and $E^{\bullet}\to L^{\bullet}$
is an isomorphism. Hence $C=E_{1}$, and virtual structure sheaf is the
ordinary structure sheaf on $V$. ∎

The following two propositions form the technical heart of the virtual
structure sheaves.

Consider a fibre square

(4) $\begin{CD}V^{\prime}@>{u}>{}>V\\\ @V{a}V{}V@V{b}V{}V\\\
B^{\prime}@>{v}>{}>B\end{CD}$

where $V^{\prime},V$ are separated DM stacks and $B^{\prime},B$ are smooth
(Artin) stacks of constant dimensions.

######  Proposition 3.

(Pull-back) Suppose that $E^{\bullet}\to L^{\bullet}_{V/B}$ is a perfect
obstruction theory for $V\to B$, and $v$ is flat or is a regular embedding.
Then $u^{*}E^{\bullet}\to L^{\bullet}_{V^{\prime}/B^{\prime}}$ is a perfect
obstruction theory for $V^{\prime}\to B^{\prime}$ and

$\mathcal{O}^{\operatorname{vir}}_{[V^{\prime},u^{*}E^{\bullet}]}=v^{!}\mathcal{O}^{\operatorname{vir}}_{[V,E^{\bullet}]}.$

Let $E$ be a perfect relative obstruction theory for $U$ over $B$ and
$E^{\prime}$ be a perfect relative obstruction theory for $U^{\prime}$ over
$B$, where $U,U^{\prime}$ are DM stacks and $B$ is a smooth (Artin) stack of
constant dimension. 333The superscript $\bullet$ in $E^{\bullet}$ will be
omitted for simplicity. Consider the following fibre square of DM $B$-stacks

(5) $\begin{CD}U^{\prime}@>{\mu}>{}>U\\\ @V{\alpha}V{}V@V{\beta}V{}V\\\
<EMAIL_ADDRESS>
Suppose that the morphism $\nu$ is a local complete intersection morphism of
$B$-stacks that have finite unramified diagonal over $B$. Recall ([5] Sect. 7)
that $E$ and $E^{\prime}$ are said to be _compatible over $\nu$_ if there
exists a homomorphism of distinguished triangles

$\begin{CD}\mu^{*}E@>{}>{}>E^{\prime}@>{}>{}>\alpha^{*}L_{V^{\prime}/V}@>{}>{}>\mu^{*}E[1]\\\
@V{}V{}V@V{}V{}V@V{}V{}V@V{}V{}V\\\
\mu^{*}L_{U/B}@>{}>{}>L_{U^{\prime}/B}<EMAIL_ADDRESS>
The compatibility of obstruction theory means that we have the following short
exact sequence of vector bundle stacks

(6)
$\alpha^{*}\mathfrak{N}_{V^{\prime}/V}\to[E^{\prime}_{1}/E^{\prime}_{0}]\to[\mu^{*}E_{1}/\mu^{*}E_{0}],$

where $\mathfrak{N}_{V^{\prime}/V}:=h^{1}/h^{0}(T^{\bullet}_{V^{\prime}/V})$.
(In general a vector bundle stack can be locally represented as a quotient of
a vector bundle by another vector bundle.) In particular when $\nu$ is a local
complete intersection morphism, one can find a smooth stack $Y$ and
$V^{\prime}\to V$ factors through $Y$:

$V^{\prime}\overset{i}{\to}Y\overset{p}{\to}V$

such that $i$ is regular embedding and $p$ is smooth. Then

$\mathfrak{N}_{V^{\prime}/V}:=[N_{V^{\prime}/Y}/i^{*}T_{Y/V}].$

######  Proposition 4.

(Functoriality) If $E^{\prime}$ and $E$ are compatible over $\nu$, then

$\nu^{!}\mathcal{O}^{\operatorname{vir}}_{[U,E]}=\mathcal{O}^{\operatorname{vir}}_{[U^{\prime},E^{\prime}]}$

when $\nu$ is smooth or $V^{\prime}$ and $V$ are smooth.

The rest of this subsection is devoted to the proof of the above two
propositions. It can be skipped without losing the logic flow.

Consider the following fibre diagram

(7)
$\begin{CD}C_{X_{1}}Y\times_{Y}C_{X_{2}}Y@>{}>{}>C_{X_{2}}Y\times_{Y}X_{1}@>{}>{}>C_{X_{2}}Y\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ C_{X_{1}}Y\times_{Y}X_{2}@>{}>{}>Z@>{}>{}>X_{2}\\\
@V{}V{}V@V{}V{}V@V{i}V{}V\\\ C_{X_{1}}Y<EMAIL_ADDRESS>
where $i,j$ are local embeddings.

######  Lemma 1.

$[{\mathcal{O}}_{C_{1}}]=[{\mathcal{O}}_{C_{2}}]$ in
$K_{\circ}(C_{X_{1}}Y\times_{Y}C_{X_{2}}Y)$, where $C_{1}:=C_{C_{X_{1}}Y\times
X_{2}}(C_{X_{1}}Y)$ and $C_{2}$ is similarly defined by switching indices $1$
and $2$. $C_{i}$ are naturally embedded in $C_{X_{1}}Y\times_{Y}C_{X_{2}}Y$.

###### Proof.

The proof of Proposition 4 in [20] also proves this lemma: It is constructed
there two principal Cartier divisors $D$ and $E$ in
$M^{\circ}_{X_{1}}Y\times_{Y}M^{\circ}_{X_{2}}Y$ (with notations like [7] Ch.
5). The proof there shows that
$i_{E}^{!}[{\mathcal{O}}_{D}]=[{\mathcal{O}}_{C_{1}}]$ and
$i_{D}^{!}[{\mathcal{O}}_{E}]=[{\mathcal{O}}_{C_{2}}]$. By Remark 2,
$[{\mathcal{O}}_{C_{1}}]=[{\mathcal{O}}_{C_{2}}]$. ∎

Now apply the above to the diagram (5). (Similar arguments applies to diagram
(4).) Assuming that $\nu$ is a regular embedding and $\beta$ is an embedding,
one has

$\begin{CD}N\times_{V}D@>{}>{}>D^{\prime}@>{}>{}>D\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ N\times_{V}U@>{}>{}>U^{\prime}@>{\mu}>{}>U\\\
@V{}V{}V@V{\alpha}V{}V@V{\beta}V{}V\\\
N@>{\rho}>{}>V^{\prime}@>{\nu}>{}>V,\end{CD}$

where $D=C_{U}V$, $D^{\prime}=C_{U^{\prime}}V^{\prime}$ and
$N=N_{V^{\prime}}V$. The Lemma 1 states that

$[{\mathcal{O}}_{C_{D^{\prime}}D}]=[C_{N\times_{V}U}N]=[\rho^{*}{\mathcal{O}}_{D^{\prime}}],$

where the $\rho^{*}$ is the pull-back on cones induced from $\rho$. Let
$0:D^{\prime}\to N\times_{V}D$ be the zero section. By the deformation to
normal cone argument

$\nu^{!}[{\mathcal{O}}_{D}]=0^{*}[{\mathcal{O}}_{C_{D^{\prime}}D}]\in
K_{\circ}(D^{\prime}).$

Also

$0^{*}[\rho^{*}{\mathcal{O}}_{C_{U^{\prime}}V^{\prime}}]=[{\mathcal{O}}_{C_{U^{\prime}}V^{\prime}}]\in
K_{\circ}(D^{\prime}).$

This implies that

######  Lemma 2.

$\nu^{!}[{\mathcal{O}}_{D}]=[{\mathcal{O}}_{C_{U^{\prime}}V^{\prime}}].$

For the readers who are familiar with the arguments in [5], the above lemmas
easily implies the propositions. In the following we reproduce their arguments
in $K$-theory.

Let us start with the proof of Functoriality. For simplicity, let $B=pt$. We
will follow closely the proof of Proposition 5.10 in [5]. By Proposition 1 we
may choose the global resolution $[E^{\prime}_{0}\to E^{\prime}_{1}]$ of
$E^{\prime}$ such that the right square of the following diagram

$\begin{CD}0@>{}>{}>F_{0}@>{}>{}>E^{\prime}_{0}@>{\phi_{0}}>{}>\mu^{*}E_{0}@>{}>{}>0\\\
&&@V{}V{}V@V{}V{}V@V{}V{}V\\\
0@>{}>{}>F_{1}@>{}>{}>E^{\prime}_{1}@>{\phi_{1}}>{}>\mu^{*}E_{1}@>{}>{}>0\end{CD}$

commutes and $\phi_{i}$ are surjective. $F_{i}:=\ker(\phi_{i})$ are vector
bundles. This induces short exact sequence of vector bundle stacks

$[F_{1}/F_{0}]\to[E^{\prime}_{1}/E^{\prime}_{0}]\to[\mu^{*}E_{1}/\mu^{*}E_{0}].$

The compatibility of obstruction theory means that we have the following short
exact sequence of vector bundle stacks (6)

$\alpha^{*}\mathfrak{N}_{V^{\prime}/V}\to[E^{\prime}_{1}/E^{\prime}_{0}]\to[\mu^{*}E_{1}/\mu^{*}E_{0}].$

This implies that $[F_{1}/F_{0}]\cong\alpha^{*}\mathfrak{N}_{V^{\prime}/V}$.
Let $C_{1}=\mathfrak{C}_{U}\times_{\mathfrak{E}}E_{1}$ and
$C^{\prime}_{1}=\mathfrak{C}_{U}^{\prime}\times_{\mathfrak{E}^{\prime}}E^{\prime}_{1}$.
Then
$\mathcal{O}^{\operatorname{vir}}_{[U,E]}=0_{E_{1}}^{*}{\mathcal{O}}_{C_{1}}$
and
$\mathcal{O}^{\operatorname{vir}}_{[U^{\prime},E^{\prime}]}=0_{E^{\prime}_{1}}^{*}{\mathcal{O}}_{C^{\prime}_{1}}$.

When $\nu$ is smooth or a regular embedding, Proposition 3.14 of [5] implies
that we have a fibre square

$\begin{CD}C^{\prime}_{1}@>{}>{}>\mu^{*}C_{1}\\\ @V{}V{}V@V{}V{}V\\\
<EMAIL_ADDRESS>
If $\nu$ is smooth, $\nu^{!}=\mu^{*}$ and we have

$\nu^{!}\mathcal{O}^{\operatorname{vir}}_{[V,E]}=\nu^{!}0^{*}_{E_{1}}[{\mathcal{O}}_{C_{1}}]=0^{!}_{\mu^{*}E_{1}}[\mu^{*}{\mathcal{O}}_{C_{1}}]=0^{!}_{E^{\prime}_{1}}[{\mathcal{O}}_{C^{\prime}_{1}}]=\mathcal{O}^{\operatorname{vir}}_{[V^{\prime},E^{\prime}]}.$

If $\nu$ is a regular embedding of smooth stacks, then

$\nu^{!}\mathcal{O}^{\operatorname{vir}}_{[V,E]}=\nu^{!}0^{*}_{E_{1}}[{\mathcal{O}}_{C_{1}}]=0^{!}_{E^{\prime}_{1}}\nu^{!}[{\mathcal{O}}_{C_{1}}]=0^{!}_{E^{\prime}_{1}}[{\mathcal{O}}_{C^{\prime}_{1}}]=\mathcal{O}^{\operatorname{vir}}_{[V^{\prime},E^{\prime}]}$

by Lemma 2. In general, when only smoothness of $V^{\prime}$ and $V$ are
assumed, one factors $v$ as

$\begin{CD}U^{\prime}@>{}>{}>U^{\prime}\times U@>{}>{}>U\\\
@V{}V{}V@V{}V{}V@V{}V{}V\\\ V^{\prime}@>{}>{}>V^{\prime}\times
<EMAIL_ADDRESS>
Apply the smooth case to the right square and the regular embedding case to
the left square.

One can generalize the functoriality property to the case $\beta$ an arbitrary
morphism by applying the arguments in [5] Lemma 5.9. That is, to replace the
morphism $j:U\to V$ by $U\to V\times Y$ such that $U\to Y$ (and therefore
$U\to V\times Y$) is an embedding. The fibre square (5) will look like

$\begin{CD}U^{\prime}@>{\mu}>{}>U\\\ @V{}V{}V@V{}V{}V\\\ V^{\prime}\times
Y@>{\nu}>{}>V\times Y.\end{CD}$

This ends the proof of Proposition4.

The proof of Proposition 3 is similar. Let $E^{\bullet}=E^{-1}\to E^{0}$. In
the case $v$ is flat, by the standard properties of (intrinsic) normal cones
$\mathfrak{C}_{V^{\prime}/B^{\prime}}=\mathfrak{C}_{V/B}\times_{B}B^{\prime}$
([5] Proposition 7.1) and $C^{\prime}_{1}\subset u^{*}E_{1}$ is the pull-back
of $C_{1}$. Therefore
${\mathcal{O}}_{C^{\prime}_{1}}=u^{*}{\mathcal{O}}_{C_{1}}=v^{!}{\mathcal{O}}_{C_{1}}$.
Since $0^{*}$ commutes with $v^{!}$,

$\mathcal{O}^{\operatorname{vir}}_{[V^{\prime},u^{*}E]}=0^{*}{\mathcal{O}}_{C^{\prime}_{1}}=0^{*}v^{!}{\mathcal{O}}_{C_{1}}=v^{!}0^{*}{\mathcal{O}}_{C_{1}}=v^{!}\mathcal{O}^{\operatorname{vir}}_{[V,E]}.$

If $v$ is a regular embedding, apply Lemma 2 to get
$[{\mathcal{O}}_{C^{\prime}_{1}}]=v^{!}[{\mathcal{O}}_{C_{1}}]$ in $K$-group
and proceed as above.

##  3\. Axioms of virtual structure sheaves in quantum $K$-theory

The axioms listed here are $K$-theoretic version of the Behrend–Manin axioms
[6].

###  3.1. Mapping to a point

Suppose that $\beta=0$. In this case
${\overline{M}}(X,\tau)={\overline{M}}_{\tau}\times X$ and

$R^{1}\pi_{*}f^{*}T_{X}=R^{1}\pi_{*}{\mathcal{O}}_{\mathcal{C}}\boxtimes
T_{X},$

where the notations are defined in (3). Then

######  Proposition 5.

$\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau)}=\lambda_{-1}(R^{1}\pi_{*}{\mathcal{O}}_{\mathcal{C}}\boxtimes
T_{X})^{\vee},$

where $\lambda_{-1}(F):=\sum_{i}(-1)^{i}\wedge^{i}F$ is the alternating sum of
the exterior power of $F$.

###### Proof.

The (dual of the) obstruction theory $E_{\bullet}$ is quasi-isomorphic to
$R^{0}\pi_{*}{\mathcal{O}}_{C}\boxtimes T_{X}\to
R^{1}\pi_{*}{\mathcal{O}}_{C}\boxtimes T_{X}$, which is a complex of locally
free sheaves with zero differential. Since ${\overline{M}}_{\tau}\times X$ is
smooth, $C_{1}=\operatorname{Image}(E_{0}\to
E_{1})={\overline{M}}_{\tau}\times X$. Now use the self-intersection formula
[8] Chapter VI. ∎

###  3.2. Products

######  Proposition 6.

$\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau_{1})\times{\overline{M}}(X,\tau_{2})}=\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau_{1})}\boxtimes\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau_{2})},$

here
$\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau_{1})\times{\overline{M}}(X,\tau_{2})}$
makes sense as

${\overline{M}}(X,\tau_{1})\times{\overline{M}}(X,\tau_{2})={\overline{M}}(X,\tau_{1}\times\tau_{2}).$

See Section 2.2.

###### Proof.

It is obvious that the perfect obstruction theory of the product is the
product of perfect obstruction theories on each component. ∎

###  3.3. Cutting edges

Let $\sigma$ be a modular graph obtained from $\tau$ by cutting an edge as in
the fibre diagram (2). In this case
$\mathfrak{M}_{\sigma}=\mathfrak{M}_{\tau}=:\mathfrak{M}$. Consider the fibre
square

$\begin{CD}{\overline{M}}(X,\tau)@>{}>{}>{\overline{M}}(X,\sigma)\\\
@V{}V{}V@V{}V{{\operatorname{ev}}_{i}\times{\operatorname{ev}}_{j}}V\\\
\mathfrak{M}\times X@>{\Delta}>{}>\mathfrak{M}\times X\times X,\end{CD}$

where $i,j$ are the markings created by the cut.

The cutting edges axiom says that

######  Proposition 7.

$\Delta^{!}\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\sigma)}=\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau)}.$

###### Proof.

It is proved in [4] that the perfect obstruction theories of
${{\overline{M}}(X,\sigma)}$ and ${{\overline{M}}(X,\tau)}$ are compatible
with respect to $\Delta$. It remains to apply Proposition 4. ∎

###  3.4. Forgetting tails

Let $\sigma$ be obtained from $\tau$ by forgetting a tail (marked point) and
let

$\pi:{\overline{M}}(X,\tau)\to{\overline{M}}(X,\sigma)$

be the forgetful map. Then

######  Proposition 8.

$\pi^{*}\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\sigma)}=\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau)}.$

###### Proof.

The proof in [4] p. 610 shows that

$\begin{CD}\mathfrak{C}_{{\overline{M}}(X,\tau)/\mathfrak{M}_{\tau}}@>{\cong}>{}>\pi^{*}\mathfrak{C}_{{\overline{M}}(X,\sigma)/\mathfrak{M}_{\sigma}}\\\
@V{}V{\cap}V@V{}V{\cap}V\\\
\mathfrak{E}(X,\tau)@>{\cong}>{}>\pi^{*}\mathfrak{E}(X,\sigma)\end{CD}$

form a commutative diagram. Therefore,

$\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau)}=\mathcal{O}^{\operatorname{vir}}_{[{\overline{M}}(X,\tau),\pi^{*}E(X,\sigma)]}.$

Now use the pull-back property Proposition 3

$\pi^{*}\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\sigma)}=\mathcal{O}^{\operatorname{vir}}_{[{\overline{M}}(X,\tau),\pi^{*}E(X,\sigma)]}.$

This completes the proof. ∎

######  Remark 4.

The above propositions in this section are the first four of five axioms on
virtual fundamental classes in quantum cohomology theory listed in [6] . The
last one, _isogenies_ axiom, is not literally true in $K$-theory. The reason
is that even though one can identify fundamental classes of two cycles by a
birational morphism, it is not true that the structure sheaves of these two
cycles are identified. We therefore have to modify the isogenies axiom in
quantum $K$-theory. Four types of isogenies are stated in $K$-theory in three
axioms. The contractions axiom has to be modified, or “quantized”, in
$K$-theory. (The meaning of “quantization” will be clear in the next section.)

###  3.5. Fundamental classes

Let $\sigma$ be obtained from $\tau$ by forgetting a tail. The degree of
$\sigma$ is induced from $\tau$ in the obvious way. Consider the commutative
diagram

$\begin{CD}{\overline{M}}(X,\tau)@>{}>{}>{\overline{M}}(X,\sigma)\\\
@V{}V{}V@V{}V{}V\\\
{\overline{M}}_{\tau}@>{\Phi}>{}>{\overline{M}}_{\sigma}\end{CD}$

which induces a morphism

$\Psi:{\overline{M}}(X,\tau)\to{\overline{M}}_{\tau}\times_{{\overline{M}}_{\sigma}}{\overline{M}}(X,\sigma).$

######  Proposition 9.

$\Psi_{*}(\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau)})=\Phi^{!}\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\sigma)}$

###### Proof.

The proof is the same as the proof in [4] pp. 611-3, with the (bivariant)
cycle classes changed to (bivariant) $K$-classes. We summarize the spaces
involved in the following diagram:

$\textstyle{{\overline{M}}(X,\tau)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{{\overline{M}}_{\tau}\times_{{\overline{M}}_{\sigma}}{\overline{M}}(X,\sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\overline{M}}(X,\sigma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\textstyle{\mathcal{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{l}$$\scriptstyle{\pi^{\prime}}$$\textstyle{{\overline{M}}_{\tau}\times_{{\overline{M}}{\sigma}}\mathfrak{M}_{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{m}$$\textstyle{\mathfrak{M}_{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{\sigma}}$$\textstyle{\mathfrak{M}_{\tau}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{\tau}}$$\textstyle{{\overline{M}}_{\tau}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Phi}$$\textstyle{{\overline{M}}_{\sigma}}$

where the squares are cartesian and
$\pi^{\prime}:\mathcal{C}\to\mathfrak{M}_{\sigma}$ is the universal curve.
Since $\pi^{\prime}$, $m$ and $\Phi$ are representable, proper and flat, they
have natural orientations $[\pi^{\prime}],[m],[\Phi]$ (which are bivariant
$K$-classes [9]). Since $s_{\sigma}$ is flat, $[m]=s_{\sigma}^{*}[\Phi]$.
Furthermore, since $l$ is just blow-down of some rational curves,
$l_{*}{\mathcal{O}}_{\mathcal{C}}={\mathcal{O}}_{{\overline{M}}_{\tau}\times_{{\overline{M}}{\sigma}}\mathfrak{M}_{\sigma}}$.
444Alternatively, one can argue this in the following way. It is known that if
$f:U\to V$ is a birational surjective morphism between smooth schemes, then
$R^{i}f_{*}{\mathcal{O}}_{U}=0$ for $i\geq 1$ and
$R^{0}f_{*}{\mathcal{O}}_{U}={\mathcal{O}}_{V}$. This property is preserved
under flat base change, therefore it holds for stacks as well. That implies
$l_{*}{\mathcal{O}}={\mathcal{O}}$. Therefore
$l_{*}[\pi^{\prime}]=s_{\sigma}^{*}[\Phi]$. Then

$\displaystyle\Phi^{!}[\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\sigma)}]=$
$\displaystyle
a^{*}s_{\sigma}^{*}[\Phi]\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\sigma)}$
$\displaystyle=$ $\displaystyle
a^{*}l_{*}[\pi^{\prime}]\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\sigma)}$
$\displaystyle=$
$\displaystyle\Psi_{*}{\pi^{\prime}}^{!}\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\sigma)}$
$\displaystyle=$
$\displaystyle\Psi_{*}\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau)}$

where the last equality follows from Proposition 4. ∎

###  3.6. Isomorphisms

Suppose that $\sigma$ is isomorphic to $\tau$, i.e. $\sigma$ is a relabeling
of $\tau$. There is an induced isomorphism
$\Psi:{\overline{M}}(X,\tau)\to{\overline{M}}(X,\sigma)$.

######  Proposition 10.

$\Psi^{*}\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\sigma)}=\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau)},\quad\Psi_{*}\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau)}=\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\sigma)}.$

The proof is obvious.

###  3.7. Contractions

Let $\phi:\tau\to\sigma$ be a map of stable modular graphs by contracting one
edge or one loop, such that $genus(\sigma)=genus(\tau)$. Denote $e$ the edge
or loop in question (in $\tau$). Let $\tilde{\sigma}$ be a fixed modular graph
obtained from $\sigma$ by adding $k$ tails and $(\tilde{\sigma},\beta)$ is a
stable modular graph with degree.

$(\tau_{1}^{i},\beta_{1}^{i_{j}})$ are the stable modular graphs with degrees
of the following form. $\tau_{1}^{i}$ are obtained from $\tau$ by adding $k$
tails in ways compatible with $\tilde{\sigma}\to\sigma$. Thus there is a
natural map from $\tau_{1}^{i}$ to $\tilde{\sigma}$ by contracting $e$, and
the following diagram is commutative:

$\begin{CD}\tau_{1}^{i}@>{}>{}>\tilde{\sigma}\\\ @V{}V{}V@V{}V{}V\\\ \tau
@>{}>{}>\sigma\end{CD}$

where the two horizontal arrows contracting $e$ and vertical arrows are
forgetful maps (by forgetting $k$ new tails). $\beta_{1}^{i_{j}}$ are the
degrees on $\tau_{1}^{i}$, compatible with $\beta$.

Similarly, $(\tau_{m}^{i},\beta_{m}^{i_{j}})$ are the stable modular graphs
with degrees such that $\tau_{m}^{i}$ is obtained from $\tau$ by first
replacing $e$ with a chain $c_{m}$ of $m$ edges and $m-1$ vertices such that
the genera of these $m-1$ vertices are all zero. One then adds $k$ tails to
this new graph in ways compatible with $\tilde{\sigma}\to\sigma$ and
contraction map $\tau_{m}^{i}\to\tilde{\sigma}$, which contracts $c_{m}$. We
again arrive at the following commutative diagram:

$\begin{CD}\tau_{m}^{i}@>{p_{m}}>{}>\tilde{\sigma}\\\ @V{a_{m}}V{}V@V{}V{}V\\\
\tau @>{}>{}>\sigma\end{CD}$

where $a_{m}$ is the composition of forgetting $k$ tails and contraction. The
degrees $\beta_{m}^{i_{j}}$ are the degrees assigned on $\tau_{m}^{i}$ in ways
compatible with $\beta$ under $p_{m}$.

Thus there is a natural morphism
$\phi_{m}:\coprod{\overline{M}}(X,\tau_{m}^{i},\beta_{m}^{i_{j}})\to{\overline{M}}(X,\tilde{\sigma},\beta)$
and the diagram

(8)
$\begin{CD}\coprod{\overline{M}}(X,\tau_{m}^{i},\beta_{m}^{i_{j}})@>{\phi_{m}}>{}>{\overline{M}}(X,\tilde{\sigma},\beta)\\\
@V{}V{\alpha_{m}}V@V{}V{l}V\\\
{\overline{M}}_{\tau}@>{\Phi}>{}>{\overline{M}}_{\sigma}\end{CD}$

is commutative.

For example, if
${\overline{M}}_{\tau}={\overline{M}}_{0,4}\times{\overline{M}}_{0,5}$ and
${\overline{M}}_{\sigma}={\overline{M}}_{0,7}$. Then
${\overline{M}}(X,\tilde{\sigma},\beta)={\overline{M}}_{0,7+k}(X,\beta)$ and
${\overline{M}}(X,\tau_{2},\beta_{2})$ are of the form
${\overline{M}}_{0,4+k_{a}}(X,\beta^{a})\times_{X}{\overline{M}}_{0,2+k_{b}}(X,\beta^{b})\times_{X}{\overline{M}}_{0,5+k_{c}}(X,\beta^{c})$
such that $k_{a}+k_{b}+k_{c}=k$ and $\beta^{a}+\beta^{b}+\beta^{c}=\beta_{2}$.
Here the notation
${\overline{M}}_{g_{1},n_{1}+1}(X,\beta_{1})\times_{X}{\overline{M}}_{g_{2},n_{2}+1}(X,\beta_{2})$
is the fibre product

$\begin{CD}{\overline{M}}_{g_{1},n_{1}+1}(X,\beta_{1})\times_{X}{\overline{M}}_{g_{2},n_{2}+1}(X,\beta_{2})@>{}>{}>{\overline{M}}_{g_{1},n_{1}+1}(X,\beta_{1})\times{\overline{M}}_{g_{2},n_{2}+1}(X,\beta_{2})\\\
@V{}V{}V@V{}V{{\operatorname{ev}}_{n_{1}+1}\times{\operatorname{ev}}_{1}}V\\\
X@>{\Delta}>{}>X\times X.\end{CD}$

The commutative diagram (8) induces a morphism

(9)
$\Psi_{m}:\coprod{\overline{M}}(X,\tau_{m}^{i},\beta_{m}^{i_{j}})\to{\overline{M}}_{\tau}\times_{{\overline{M}}_{\sigma}}{\overline{M}}(X,\tilde{\sigma},\beta).$

######  Proposition 11.

(10)
$\sum_{m}(-1)^{m+1}{\Psi_{m}}_{*}\sum_{(i,i_{j})}\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau_{m}^{i},\beta^{i_{j}}_{m})}=\Phi^{!}(\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tilde{\sigma},\beta)})$

in
$K_{\circ}({\overline{M}}_{\tau}\times_{{\overline{M}}_{\sigma}}{\overline{M}}(X,\tilde{\sigma},\beta))$.
The operation $\Phi^{!}$ on the RHS of (10) is the refined Gysin map from
$K_{\circ}({\overline{M}}(X,\tilde{\sigma},\beta))$ to
$K_{\circ}({\overline{M}}_{\tau}\times_{{\overline{M}}_{\sigma}}{\overline{M}}(X,\tilde{\sigma},\beta))$.

###### Proof.

Consider the following commutative diagram

$\begin{CD}\coprod\mathfrak{M}_{\tau^{i}_{m}}@>{}>{}>\mathfrak{M}_{\tilde{\sigma}}\\\
@V{}V{}V@V{}V{j}V\\\
<EMAIL_ADDRESS>
It induces a morphism

$\psi_{m}:\coprod\mathfrak{M}_{\tau^{i}_{m}}\to{\overline{M}}_{\tau}\times_{{\overline{M}}_{\sigma}}\mathfrak{M}_{\tilde{\sigma}}.$

We will show an analogue of (10) in this setting. Namely,

(11)
$\sum_{m,i}(-1)^{m+1}{\psi_{m}}_{*}{\mathcal{O}}_{\mathfrak{M}_{\tau_{m}^{i}}}=\Phi^{!}{\mathcal{O}}_{\mathfrak{M}_{\tilde{\sigma}}}.$

Note that the vertical maps only contracts rational components with two marked
points (or less). That is the reason why only chains of rational curves are
relevant.

It is easy to see that
${\overline{M}}_{\tau}\times_{{\overline{M}}_{\sigma}}\mathfrak{M}_{\tilde{\sigma}}\to\mathfrak{M}_{\tilde{\sigma}}$
is a divisor $D_{\tau}$ of normal crossing in $\mathfrak{M}_{\tilde{\sigma}}$,
and the “smoothing map”

$\mu_{1}:\coprod\mathfrak{M}_{\tau_{1}^{i}}\to D_{\tau}$

is a finite, unramified and surjective birational morphism. In other words,
the smooth Artin stack $\coprod\mathfrak{M}_{\tau_{1}^{i}}$ separates the
normal crossing point. Although $D_{\tau}$ is not equal to
$\mu_{*}{\mathcal{O}}_{\coprod\mathfrak{M}_{\tau_{1}^{i}}}$, the difference is
supported on the normal crossing subset. The following lemma shows how one
could obtain the structure sheaf ${\mathcal{O}}_{D_{\tau}}$ by the inclusion-
exclusion principle.

######  Lemma 3.

Let $D=\cup_{i=1}^{k}D_{i}$ be a divisor with normal crossing, such that
$D_{i}$ are smooth, disjoint away from the origin. Furthermore locally at
origin $D$ is defined by $x_{1}\ldots x_{k}=0$. Then

(12)
$0\to{{\mathcal{O}}}_{D}\to\sum_{i}{\mathcal{O}}_{D_{i}}\to\sum_{i<j}{\mathcal{O}}_{D_{i}\cap
D_{j}}\to\ldots\to{\mathcal{O}}_{D_{1}\cap\ldots\cap D_{k}}\to 0$

is an exact sequence.

###### Proof.

Equation (12) is equivalent to the study the exactness of the following
sequence locally at the origin

$0\to{{\mathcal{O}}}/(x_{1}x_{2}\ldots
x_{k})\to\sum_{i}{{\mathcal{O}}}/(x_{i})\to\\\
\sum_{i<j}{\mathcal{O}}/(x_{i},x_{j})\to\ldots\to{\mathcal{O}}/(x_{1},x_{2},\ldots,x_{k})\to
0.$

This equation holds because, e.g. $k=2$, the following sequence

$0\to{{\mathcal{O}}}/{(x_{1}x_{2})}\to{{\mathcal{O}}}/{(x_{1})}\oplus{{\mathcal{O}}}/{(x_{2})}\to{{\mathcal{O}}}/{(x_{1},x_{2})}\to
0$

is exact. In the general case this follows from the inclusion-exclusion
principle. For the exactness of (12) at points away from the origin, where $l$
divisors intersect ($l<k$), it follows from the induction as the divisor
locally defined by the equation $y_{i_{1}}\ldots y_{i_{l}}$. ∎

It remains to note that the normal crossing substack is stratified by the
image of

$\mu_{m}:\coprod\mathfrak{M}_{\tau_{m}^{i}}\to D_{\tau}$

with codimension $m$ in $\mathfrak{M}_{\tilde{\sigma}}$. This concludes the
proof of (11).

Now consider the following diagram

$\begin{CD}{\overline{M}}_{\tau_{m}^{i}}(X,\beta_{m}^{i_{j}})@>{}>{}>{\overline{M}}_{\tilde{\sigma}}(X,\beta)\\\
@V{}V{}V@V{}V{}V\\\
\mathfrak{M}_{\tau_{m}^{i}}@>{w}>{}>\mathfrak{M}_{\tilde{\sigma}}\end{CD}$

which defines a compatible perfect obstruction theory over $w$. Therefore by
Proposition 4

(13)
$w^{!}\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}_{\tilde{\sigma}}(X,\beta)}=\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}_{\tau_{m}^{i}}(X,\beta_{m}^{i_{j}})}.$

Proposition follows from the combination of equations (11) and (13). ∎

##  4\. Quantum $K$-invariants

###  4.1. Preliminaries on topological $K$-theory

The $K$-theory used in this article has four variants:

  * •

$K_{\circ}(V)$: the Grothendieck group of coherent sheaves on $V$.

  * •

$K^{\circ}(V)$: the Grothendieck group of algebraic vector bundles on $V$.

  * •

$K^{*}(V):=K^{0}(V)\oplus K^{1}(V)$: the topological $K$-cohomology theory on
$V$ (of complex vector bundles).

  * •

$K_{*}(V):=K_{0}(V)\oplus K_{1}(V)$: the topological $K$-homology theory on
$V$.

For the benefit of algebraic geometers let us recall some definitions and
useful properties of topological $K$-theory. All the spaces involved are
topological spaces or algebraic schemes. See Remark 5 below for applications
to stacks.

Let $V\hookrightarrow\mathbb{C}^{N}$ be a topological closed embedding. Define

$K_{i}(V):=K^{i}(\mathbb{C}^{N},\mathbb{C}^{N}-V)\overset{\text{def}}{=}K^{i}_{V}(\mathbb{C}^{N}).$

(Remember that there is a Bott periodicity.) Note that this definition is
independent of the closed embedding.

There is a _cap product_ between homology and cohomology theories. In
algebraic $K$-theory, it is just the tensor product. In topological
$K$-theory, it is defined as follows. Let $V\subset\mathbb{C}^{N}$ be the
closed embedding as above and $U$ be a neighborhood of $V$ in $\mathbb{C}^{N}$
such that every element in $K^{*}(V)$ extends to an element in $K^{*}(U)$.
$K_{i}(V)=K^{i}(\mathbb{C}^{N},\mathbb{C}^{N}-V)=K^{i}(U,U-V)$ by excision.
Then the cap product is defined as

$K^{i}(V)\otimes K_{j}(V)\to K^{i}(U)\otimes K^{j}(U,U-V)\\\ \to
K^{i+j}(U,U-V)=K_{j+i}(V)\overset{\cong}{\to}K_{j-i}(V),$

where the second arrow is the cup product and the last arrow is the Bott
periodicity.

One can define the _push-forward_

$f_{*}:K_{i}(V^{\prime})\to K_{i}(V)$

for $f:V^{\prime}\to V$ a continuous proper map. Since $f$ is proper, there is
$\phi^{\prime}:V^{\prime}\to D^{m}$ ($D^{m}$ is a polydisk in
$\mathbb{C}^{m}$) such that

$(f,\phi^{\prime}):V^{\prime}\to V\times D^{m}$

is a closed embedding. Choose a closed embedding $\phi:V\to\mathbb{C}^{N}$ as
above. The push-forward is the composition

$K_{i}(V^{\prime})\cong K^{i}(\mathbb{C}^{N+m},\mathbb{C}^{N+m}-(\phi
f,\phi^{\prime})(X))\\\
\overset{\text{res}}{\to}K^{i}(\mathbb{C}^{N+m},\mathbb{C}^{N}\times(\mathbb{C}^{m}-D^{m})\cup(\mathbb{C}^{N}-\phi(V))\times\mathbb{C}^{m})\\\
\overset{\cong}{\leftarrow}K^{i}(\mathbb{C}^{N},\mathbb{C}^{N}-\phi(V))\cong
K_{i}(V)$

where the $\leftarrow$ is the Thom isomorphism, i.e. multiplying the canonical
element of $K^{0}(\mathbb{C}^{m},\mathbb{C}^{m}-D^{m})$.

There is an obvious homomorphism $K^{\circ}(V)\to K^{0}(V)$ as every algebraic
vector bundle is a topological one. One can also construct a homomorphism

(14) $Top:K_{\circ}(V)\to K_{0}(V)$

as follows. Choose a closed embedding of $V$ to a smooth scheme
$i:V\hookrightarrow Y$. (In our applications, all schemes are quasi-
projective.) For a bounded complex of coherent sheaves $\alpha^{\bullet}$ on
$V$, $i_{*}(\alpha^{\bullet})$ is quasi-isomorphic to a bounded complex
$E^{\bullet}$ of locally free sheaves on $Y$, which is exact out of $V$.
Regard $E^{\bullet}$ as a a topological element and embed $Y$ in some
$\mathbb{C}^{n}$. By Thom isomorphism $E^{\bullet}$ is identified as a bounded
complex of topological vector bundles on $\mathbb{C}^{n}$, exact off $V$,
which is then an element in $K_{0}(V)$.

A relation between the push-forward maps in two $K$-homology theories is given
by the following theorem of Baum–Fulton–MacPherson:

######  Theorem 2.

([2]) For a proper morphism $g:X\to Y$, the following diagram:

$\begin{CD}K_{\circ}(X)@>{}>{}>K_{0}(X)\\\ @V{g_{*}}V{}V@V{g_{*}}V{}V\\\
K_{\circ}(Y)@>{}>{}>K_{0}(Y)\end{CD}$

commutes.

There is again a bivariant topological $K$-theory which unifies the above
setting. The homomorphism (14) is also valid in the bivariant setting [9].

######  Remark 5.

Although the $K$-theory of a moduli stack is quite different from the
$K$-theory of its coarse moduli space, one can define the quantum
$K$-invariants via coarse moduli space in the following way. Let
$p:\mathcal{M}\to M$ be the canonical map from the moduli stack to its coarse
moduli space. First construct the virtual structure sheaf
$\mathcal{O}^{\operatorname{vir}}_{\mathcal{M}}$ on the moduli stack, then
push-forward $p_{*}(\mathcal{O}^{\operatorname{vir}})$ to its coarse moduli
space, which is a topological space. One gets

$\chi(\mathcal{M},{\operatorname{ev}}_{\mathcal{M}}^{*}(\gamma)\,E\,\mathcal{O}^{\operatorname{vir}}_{\mathcal{M}})=\chi(M,{\operatorname{ev}}_{M}^{*}(\gamma)\,p_{*}(E\,\mathcal{O}^{\operatorname{vir}}_{\mathcal{M}}))$

by projection formula as ${\operatorname{ev}}:\mathcal{M}\to X$ factors
through $M$. $E$ is any $K$-element on $\mathcal{M}$, like $\mathcal{L}$ or
$\mathcal{H}$ (Hodge bundle). This means that one gets the same quantum
$K$-invariants on coarse moduli space if one takes the suitable virtual
structure sheaf.

Therefore, one can define quantum $K$-invariants for topological $K$-theory on
_topological spaces_ rather than orbispaces, by using the topological
$K$-homology for topological space associated to coarse moduli space and
$Top(p_{*}(\mathcal{O}^{\operatorname{vir}}))$ as the virtual structure sheaf.
This eliminates the difficulty of constructing topological $K$-homology on
orbispaces, which will be addressed in another paper.

###  4.2. Definition of Quantum $K$-invariants

In this subsection we will propose a construction of $K$-theoretic invariants
555The word ‘invariant’ should be taken as self-contained since we have not
proved that it is a, say, symplectic invariant as in the case of cohomology
theory. This will be discussed in a separate paper. of the Gromov–Witten type,
which we call quantum $K$-invariants.

Let $K^{\bullet}(V)$ be the (algebraic or topological) $K$-cohomology of $V$
and $K_{\bullet}(V)$ be the (algebraic or topological) $K$-homology of $V$.
When $V$ is smooth, the notation $K(V)$ is used to denote both
$K^{\bullet}(V)$ and $K_{\bullet}(V)$ as they are isomorphic.

The quantum $K$-invariants is defined to be

$\langle\gamma_{1},\gamma_{2},\ldots,\gamma_{n};F\rangle_{g,n,\beta}:=\chi\left({\overline{M}}_{g,n}(X,\beta),\mathcal{O}^{\operatorname{vir}}\otimes{\operatorname{ev}}^{*}(\gamma_{1}\otimes\ldots\gamma_{n})\otimes{\operatorname{st}}^{*}(F)\right),$

where $\gamma_{i}\in K(X),F\in K({\overline{M}}_{g,n})$ and $\chi$ is the
push-forward to a point ($\operatorname{{{Spec}}}\mathbb{C}$). Note that the
triple $(g,n,\beta)$ is chosen so that ${\overline{M}}_{g,n}(X,\beta)$ is
defined. Note that due to Baum–Fulton–MacPherson Theorem 2 topological
invariants will be equal to algebraic invariants (whenever applicable). We
will not make distinctions.

One could also include the gravitational descendents and define:

(15)
$\begin{split}&\langle\tau_{k_{1}}(\gamma_{1}),\tau_{k_{2}}(\gamma_{2}),\ldots,\tau_{k_{n}}(\gamma_{n});{\operatorname{st}}^{*}(F)\rangle_{g,n,\beta}\\\
:=&\chi\left({\overline{M}}_{g,n}(X,\beta),{\mathcal{O}}^{\operatorname{vir}}\otimes\bigl{(}\otimes_{i=1}^{n}\mathcal{L}_{i}^{\otimes
k_{i}}{\operatorname{ev}}_{i}^{*}(\gamma_{i})\bigr{)}\otimes{\operatorname{st}}^{*}(F)\right).\end{split}$

_From this point we assume the $K$-theory is the topological $K$-theory._ The
modification to algebraic $K$-theory is in most places straightforward. A key
point will be discussed in Remark 10.

Let $X$ be a smooth projective variety, and $E\subset H_{2}(X,\mathbb{Z})$
denote the semigroup of effective curve classes. Let $\mathbb{C}[E]$ be the
semigroup ring determined by $E$. Since $0\in E$, $\mathbb{C}[E]$ has a unit
element. For $\beta\in E$, the corresponding element of $\mathbb{C}[E]$ will
be denoted by $Q^{\beta}$.

Let $\mathfrak{m}$ be the maximal ideal in $\mathbb{C}[E]$ generated by the
nonzero elements of $E$. The _Novikov ring_ $N(X)$ is defined to be the
completion of $\mathbb{C}[E]$ in the $\mathfrak{m}$-adic topology.
Alternatively, $N(X)$ may be defined by formal series in $Q^{\beta}$:

$N(X)=\\{\sum_{\beta\in E}c_{\beta}Q^{\beta}|c_{\beta}\in\mathbb{C}\\}.$

Let $e_{0}={\mathcal{O}},e_{1},e_{2},\ldots$ be a basis of $K(X)_{\mathbb{Q}}$
and let $t_{i}$ be its dual coordinates. Define $t:=\sum_{i}t_{i}e_{i}$. The
$K$-theoretic Poincaré pairing

$(\cdot,\cdot):K(X)_{\mathbb{Q}}\otimes K(X)_{\mathbb{Q}}\to\mathbb{Q}$

is defined to be $(e_{i},e_{j})=\chi(e_{i}\otimes e_{j})$. This is a perfect
pairing whenever $X$ is smooth. This defines a metric $g_{ij}:=(e_{i},e_{j})$
on $K(X)$.

The _quantum $K$-potential_ of genus $0$ is a generating series of genus zero
quantum $K$-invariants:

(16) $G(t,Q):=\frac{1}{2}(t,t)+\sum_{n=0}^{\infty}\sum_{\beta\in
E}\frac{Q^{\beta}}{n!}\langle t,\ldots,t\rangle_{0,n,\beta}.$

Here $(0,n,\beta)$ is a triple such that ${\overline{M}}_{0,n}(X,\beta)$
exists. It is obvious that $G(t,Q)\in
N(X)\otimes_{\mathbb{C}}\mathbb{C}[[t]]$.

We will see later that the metric in quantum $K$-theory ought to be
“quantized” due to the modified contraction axiom (§ 3.7). Define the _quantum
$K$-metric_ to be

(17) $((e_{i},e_{j}))=G_{ij}:=\partial_{t_{i}}\partial_{t_{j}}G(t).$

$G_{ij}$ is the quantization of $g_{ij}$ because $G_{ij}|_{q=0}=g_{ij}$ by
definition. Define $G^{ij}$ to be the inverse matrix of $G_{ij}$.

One could also define the quantum $K$-classes for $X$ to be a family of linear
maps (cf. [19])

$I^{X}_{g,n,\beta}:K(X)^{\otimes n}\to K({\overline{M}}_{g,n})$

defined by

(18) $I^{X}_{g,n,\beta}(u_{1}\otimes u_{2}\otimes\ldots\otimes
u_{n}):={\operatorname{st}}_{*}({\mathcal{O}}^{\operatorname{vir}}_{{\overline{M}}_{g,n}(X,d)}\otimes{\operatorname{ev}}_{1}^{*}(u_{1})\otimes\ldots\otimes{\operatorname{ev}}_{n}^{*}(u_{n})).$

The equivalence of these two definitions follows from the fact that the
Poincaré duality in $K({\overline{M}}_{g,n},\mathbb{Q})$ is a perfect pairing.

######  Remark 6.

1\. It is obvious how to include the modular graphs into our data and define
quantum $K$-invariants associated to a modular graph.

2\. Finally we mention that there is an _equivariant_ version of these
invariants. If there is an algebraic group (or compact Lie group) $G$ acting
on $X$, then the moduli space ${\overline{M}}_{g,n}(X,\beta)$ (and its virtual
structure sheaf) has a $G$ action too. We can talk about the $G$-equivariant
quantum $K$-invariants, completely parallel to equivariant quantum cohomology
theory.

###  4.3. On Kontsevich–Manin axioms in $K$-theory

The quantum $K$-invariants (without descendents) as defined above obviously
can not satisfy all axioms of cohomological Gromov–Witten invariants [19]. The
_effectivity_ and _motivic_ axioms follow from the construction. The _grading_
and _divisor_ axioms are missing, due to the lack of “dimension counting”. We
will discuss briefly the remaining five axioms below.

$S_{n}$-covariance: The quantum $K$-class is covariant under $S_{n}$ action on
the marked points.

Fundamental class: Let $e_{0}$ be the ($K$-class of) structure sheaf of $X$
and $\pi_{n}:{\overline{M}}_{g,n}\to{\overline{M}}_{g,n-1}$. Then

(19) $I^{X}_{g,n,\beta}(u_{1}\otimes u_{2}\otimes\ldots\otimes u_{n-1}\otimes
e_{0})=\pi_{n}^{*}(I^{X}_{g,n-1,\beta}(u_{1}\otimes u_{2}\otimes\ldots\otimes
u_{n-1})).$

Mapping to a point: Suppose that $\beta=0$. Consider the following diagram

$\begin{CD}{\overline{M}}_{g,n}\times X@>{p_{2}}>{}>X\\\ @V{}V{p_{1}}V\\\
{\overline{M}}_{g,n}\end{CD}$

where $p_{i}$ are the projection morphisms. Then

(20) $I^{X}_{g,n,0}(u_{1}\otimes\ldots\otimes
u_{n})=(p_{1})_{*}p_{2}^{*}(u_{1}\otimes\ldots\otimes
u_{n}\otimes\lambda_{-1}(R^{1}\pi_{*}({\mathcal{O}}_{C}\otimes
T_{V}))^{\vee})$

Splitting: Fix $g_{1},g_{2}$ and $n_{1},n_{2}$ such that
$g=g_{1}+g_{2},n=n_{1}+n_{2}$. Let

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

be the contraction map which glues the last marked point of
${\overline{M}}_{g_{1},n_{1}+1}$ to the first marked point of
${\overline{M}}_{g_{2},n_{2}+1}$ (contracting a non-looping edge). Then

(21)
$\begin{split}&\Phi^{*}\sum_{k,\beta}q^{\beta}\frac{1}{k!}\operatorname{ft}^{k}_{*}I^{X}_{g,n+k,\beta}(\gamma_{1}\otimes\ldots\otimes\gamma_{n}\otimes
t\otimes\ldots\otimes t)\\\
=&\sum_{ij}\left(\sum_{k_{1},\beta^{\prime}}q^{\beta^{\prime}}\frac{1}{k_{1}!}\operatorname{ft}^{k_{1}}_{*}I^{X}_{g_{1},n_{1}+k_{1}+1,\beta^{\prime}}(\gamma_{1}\otimes\ldots\gamma_{n_{1}}\otimes
t\otimes\ldots\otimes t\otimes e_{i})\right)\\\
&G^{ij}(t)\left(\sum_{k_{2},\beta^{\prime\prime}}\frac{1}{k_{2}!}\operatorname{ft}^{k_{2}}_{*}I^{X}_{g_{2},n_{2}+k_{2}+1,\beta^{\prime\prime}}(e_{j}\otimes\gamma_{n_{1}+1}\otimes\ldots\gamma_{n}\otimes
t\otimes\ldots\otimes t)\right),\end{split}$

where the notation
$\operatorname{ft}^{k}:{\overline{M}}_{g,n+k}\to{\overline{M}}_{g,n}$ stands
for the forgetful map which forget the additional $k$ marked points.

Genus reduction: Let

$\Phi:{\overline{M}}_{g-1,n+2}\to{\overline{M}}_{g,n}$

be the contraction map which glues the last two marked points (contracting a
loop). Then

$\begin{split}&\Phi^{*}\sum_{k,\beta}q^{\beta}\frac{1}{k!}\operatorname{ft}^{k}_{*}I^{X}_{g,n+k,\beta}(\gamma_{1}\otimes\ldots\otimes\gamma_{n}\otimes
t\otimes\ldots\otimes t)\\\
=&\sum_{ij}\left(\sum_{k,\beta}q^{\beta}\frac{1}{k!}\operatorname{ft}^{k}_{*}I^{X}_{g+1,n+k+2,\beta}(\gamma_{1}\otimes\ldots\gamma_{n}\otimes
t\otimes\ldots\otimes t\otimes e_{i}\otimes e_{j})\right)G^{ij}(t)\end{split}$

The proofs of these five axioms follows from the corresponding axioms of
virtual structure sheaves. For example, Contractions, Cutting Edges and
Products imply the splitting and genus reduction axioms. The only novelty is
the appearance of $G^{ij}$ in Splitting and genus reduction axioms. This has
its origin in Contractions Axiom (Section 3.7). Let us write
$G_{ij}(t,q)=g_{ij}+F_{ij}(t,q)$. The inverse matrix $G^{ij}(t,q)$ is
therefore

$g^{ij}+\sum_{m\geq 2}(-1)^{m+1}g^{ia_{1}}F_{a_{1}b_{1}}g^{b_{1}a_{2}}\ldots
F_{a_{m-1}b_{m-1}}g^{b_{m-1}j}$

due to the matrix geometric series

$\frac{1}{1+f}=1-f+f^{2}-f^{3}+\ldots.$

Note that the contributions from the contracted chains (denoted $c_{m}$) in
Proposition 11 are exactly $F_{a_{1}b_{1}}g^{b_{1}a_{2}}\ldots
F_{a_{m-1}b_{m-1}}$. The fact that the combinatorics involved can be
simplified by introducing $G^{ij}$ is first observed in [12].

###  4.4. String equation

Let $\pi:{\overline{M}}_{g,n+1}(X,\beta)\to{\overline{M}}_{g,n}(X,\beta)$ be
the forgetful map by forgetting the last marked point. 666It is proved in [6]
that
$\operatorname{ft}_{n+1}:{\overline{M}}_{g,n+1}(X,\beta)\to{\overline{M}}_{g,n}(X,\beta)$
is the universal curve $\pi:\mathcal{C}\to{\overline{M}}_{g,n}(X,\beta)$. It
is therefore convenient not to distinguish these two. String equation asks the
relations between
$\langle\tau_{k_{1}}(\gamma_{1}),\ldots,\tau_{k_{n}}(\gamma_{n}),\tau_{0}(1)\rangle_{g,n+1,\beta}$
and
$\langle\tau_{k_{1}}(\gamma_{1}),\ldots,\tau_{k_{n}}(\gamma_{n})\rangle_{g,n,\beta}$.
In the higher genus case, the Hodge bundle is also included. It is easy to see
that the relation is reduced to the following equations (by projection
formula):

For $g=0$:

(22)
$\pi_{*}\Bigl{(}{\mathcal{O}}^{\operatorname{vir}}\bigl{(}\prod_{i=1}^{n}\frac{1}{1-q_{i}\mathcal{L}_{i}}\bigr{)}\Bigr{)}=\Bigl{(}1+\sum_{i=1}^{n}\frac{q_{i}}{1-q_{i}}\Bigr{)}\biggl{(}{\mathcal{O}}^{\operatorname{vir}}\Bigl{(}\prod_{i=1}^{n}\frac{1}{1-q_{i}\mathcal{L}_{i}}\Bigr{)}\biggr{)},$

where $q_{i}$ are formal variables. The LHS and RHS are equal as formal series
in $q_{i}$’s.

For $g\geq 1$:

(23)
$\begin{split}&\pi_{*}\left({\mathcal{O}}^{\operatorname{vir}}\frac{1}{1-q\mathcal{H}^{-1}}\prod_{i=1}^{n-1}\frac{1}{1-q_{i}\mathcal{L}_{i}}\right)\\\
=&{\mathcal{O}}^{\operatorname{vir}}\frac{1}{1-q\mathcal{H}^{-1}}\Bigg{[}\left(1-\mathcal{H}^{-1}+\sum_{i=1}^{n-1}\frac{q_{i}}{1-q_{i}}\right)\left(\prod_{i=1}^{n-1}\frac{1}{1-q_{i}\mathcal{L}_{i}}\right)\Bigg{]},\end{split}$

where $\mathcal{H}$ is the Hodge bundle, i.e.
$\mathcal{H}:=R^{0}\pi_{*}\omega_{\mathcal{C}/{\overline{M}}}$.

###### Proof.

First notice that the virtual structure sheaves in the above equations can be
simultaneously erased from both sides by the Fundamental class and Forgetting
tails axioms.

First the case $g=0$. Let $D_{i}$ be the divisors on
${\overline{M}}_{0,n+1}(X,\beta)$ such that the generic curves have two
components: one contains the $i$-th and $n+1$-th marked points, and the other
contains the rest. It is well-known that

(24) ${\mathcal{O}}(\mathcal{L}_{i})=\pi^{*}l_{i}\otimes{\mathcal{O}}(D_{i})$

(see e.g. [28]), where for the notational convenience we have used $l_{i}$ and
$\mathcal{L}_{i}$ for $i$-th universal cotangent line bundles on
${\overline{M}}_{0,n}(X,\beta)$ and ${\overline{M}}_{0,n+1}(X,\beta)$
respectively. It is easy to see that ([21] Lemma 2)

(25) $R^{1}\pi_{*}({\mathcal{O}}(\sum_{i}d_{i}D_{i}))=0$

for $d_{i}\geq 0$. By projection formula

$R^{0}\pi_{*}(\otimes_{i=1}^{n}\mathcal{L}^{\otimes
d_{i}}_{i})=(\otimes_{i=1}^{n}l^{d_{i}}_{i})R^{0}\pi_{*}{\mathcal{O}}(\sum_{i}d_{i}D_{i}).$

It remains to compute $H^{0}:=R^{0}\pi_{*}{\mathcal{O}}(\sum_{i}d_{i}D_{i})$,
which is a vector bundle because $R^{1}=0$.

To compute $H^{0}$, we use some local arguments. The fibre $C_{x}$ of $\pi$ is
a rational curve with $n$ marked points $x_{1},\ldots,x_{n}$. An element of
$H^{0}(C_{x},{\mathcal{O}}(\sum_{i}d_{i}D_{i}))$ is a rational function with
poles of order no more than $d_{i}$ at $x_{i}$. Therefore
$H^{0}(C_{x},{\mathcal{O}}(\sum_{i}d_{i}D_{i}))$ is filtered by the degrees of
poles at $x_{i}$:

$\mathcal{F}_{0}\subset\mathcal{F}_{1}\subset\ldots\subset\mathcal{F}_{d_{i}}.$

It is almost by definition that the graded piece
$\mathcal{F}_{k+1}/\mathcal{F}_{k}$ are isomorphic to $T^{\otimes k}_{x_{i}}$.
Summarizing,

(26) $H^{0}\overset{\text{$K$-theory}}{=}{\mathcal{O}}\oplus
l^{-1}_{1}\oplus\ldots l^{-d_{1}}\oplus l^{-1}_{2}\oplus\ldots\oplus
l^{-1}_{n}\oplus\ldots l^{-d_{n}}_{n},$

where “$K$-theory” above means the equation is valid only in $K$-theory as we
have used the graded pieces.

It is now a matter of elementary computation (see Proposition 1 in [21]) to
deduce the genus zero string equation from (26).

In the case $g\geq 1$ the Hodge bundles $\mathcal{H}$ will naturally occur.
Notice that $\mathcal{H}$ is actually equal to ${\pi}^{*}(\mathcal{H})$, and
this implies that the factor $1/(1-q\mathcal{H})$ commutes with $\pi_{*}$ due
to the projection formula. By Grothendieck–Riemann–Roch formula, (23) is a
rational function of $q$ and $q_{i}$’s. Therefore

(27)
$\pi_{*}(\frac{1}{1-q\mathcal{L}})=\pi_{*}(\frac{-q^{-1}\mathcal{L}^{-1}}{1-q^{-1}\mathcal{L}^{-1}}).$

Now for $d_{i}\geq 1$, we want to show

$\displaystyle\pi_{*}\left(\otimes_{i=1}^{n}\mathcal{L}_{i}^{-d_{i}}\right)$
$\displaystyle=$
$\displaystyle\otimes_{i=1}^{n}\mathcal{L}_{i}^{-d_{i}}\left(\underline{1}-\mathcal{H}^{-1}-\sum_{i,\
d_{i}\neq
0}\left(\underline{1}+\mathcal{L}_{i}+\ldots+\mathcal{L}_{i}^{d_{i}-1}\right)\right).$

The above equality can be proved using the same arguments as in the genus zero
case. By (25) and projection formula

$\pi_{*}(\otimes_{i=1}^{n}\mathcal{L}_{i}^{-d_{i}})=(\otimes_{i=1}^{n}(l_{i})^{-d_{i}})\,\pi_{*}{\mathcal{O}}(\sum_{i}-d_{i}D_{i}).$

Because $d_{i}\geq 1$ for all $i$,
$R^{0}\pi_{*}({\mathcal{O}}(\sum_{i}-d_{i}D_{i}))=0$.

$H^{1}:=R^{1}\pi_{*}({\mathcal{O}}(\sum_{i}-d_{i}D_{i}))$

is a vector bundle.

Now by Serre duality $(H^{1})^{*}=R^{0}\pi_{*}\omega(\sum_{i}d_{i}D_{i})$,
where $\omega$ is the dualizing sheaf. Fibrewisely,
$H^{0}(C,\omega(\sum_{i}d_{i}x_{i}))$ is the holomorphic differential with
poles of order at most $d_{i}$ at $x_{i}$. Thus we have a filtration
$\mathcal{F}_{1}\subset\mathcal{F}_{2}\subset\ldots\subset\mathcal{F}_{d_{i}}$
of degrees of poles at each marked point $x_{i}$ as above and the graded
bundles $\mathcal{F}_{k+1}/\mathcal{F}_{k}$ is isomorphic to
$l_{i}^{\otimes-k}$ and $\mathcal{F}_{0}=\mathcal{H}$. Therefore

$\begin{split}&\pi_{*}(\otimes_{i=1}^{n}\mathcal{L}_{i}^{-d_{i}})\\\
=&\otimes_{i=1}^{n}(l_{i})^{-d_{i}}\left(-R^{1}\pi_{*}{\mathcal{O}}(\sum_{i}-d_{i}D_{i})\right)\\\
=&\otimes_{i=1}^{n}(l_{i})^{-d_{i}}\left(-(R^{0}\pi_{*}{\mathcal{O}}(\sum_{i}-d_{i}D_{i}))^{*}\right)\\\
=&\otimes_{i=1}^{n}l_{i}^{-d_{i}}\left(\underline{1}-\mathcal{H}^{-1}-\sum_{i,\
d_{i}\neq
0}\left(\underline{1}+l_{i}+\ldots+l_{i}^{d_{i}-1}\right)\right).\end{split}$

Notice again that the last equality holds only in $K$-theory (using graded
objects). This means

$\displaystyle\pi_{*}\left(\prod_{i=1}^{n}\frac{q_{i}^{-1}\mathcal{L}_{i}^{-1}}{1-q_{i}^{-1}\mathcal{L}_{i}^{-1}}\right)$
$\displaystyle=$
$\displaystyle\left(1-\mathcal{H}^{-1}+\sum_{i=1}^{n}\frac{q_{i}^{-1}}{1-q_{i}^{-1}}\right)\left(\prod_{i=1}^{n-1}\frac{q_{i}^{-1}l_{i}^{-1}}{1-q_{i}^{-1}l_{i}^{-1}}\right),$

which is equivalent to (23). ∎

###  4.5. Dilaton equation

Let $\pi:{\overline{M}}_{g,n+1}(X,\beta)\to{\overline{M}}_{g,n}(X,\beta)$ as
above. The dilaton equation asks the relations between
$\langle\tau_{k_{1}}(\gamma_{1}),\ldots,\tau_{k_{n}}(\gamma_{n}),\tau_{1}(e_{0})\rangle_{g,n+1,\beta}$
and
$\langle\tau_{k_{1}}(\gamma_{1}),\ldots,\tau_{k_{n}}(\gamma_{n})\rangle_{g,n,\beta}$.

(28)
$\begin{split}&\pi_{*}\left({\mathcal{O}}^{\operatorname{vir}}\frac{1}{1-q\mathcal{H}^{-1}}\biggl{(}\prod_{i=1}^{n-1}\frac{1}{1-q_{i}\mathcal{L}_{i}}\biggr{)}\mathcal{L}_{n+1}\right)\\\
=&{\mathcal{O}}^{\operatorname{vir}}\frac{1}{1-q\mathcal{H}^{-1}}\\\
&\Bigg{[}\left(\mathcal{H}^{-1}+\sum_{i=1}^{n-1}\frac{1}{1-q_{i}}\prod_{i=1}^{n}\mathcal{L}_{i}^{-1}\right)\left(\prod_{i=1}^{n-1}\frac{1}{1-q_{i}\mathcal{L}_{i}}\right)\Bigg{]}\end{split}$

###### Proof.

The proof is similar to the above proof of string equation. In the proof we
again use $l_{i}$ for $\mathcal{L}_{i}$ on ${\overline{M}}_{g,n}(X,\beta)$. We
will use Forgetting tail axiom, (24)(25) and projection formula

$\displaystyle\pi_{*}(\otimes_{i=1}^{n}\mathcal{L}_{i}^{d_{i}}\otimes\mathcal{L}_{n+1})=$
$\displaystyle\otimes_{i=1}^{n}l_{i}^{d_{i}}\otimes\pi_{*}\left(\mathcal{L}_{n+1}(\sum_{i}d_{i}D_{i})\right)$
Because of the relation $\mathcal{L}_{n+1}=\omega(x_{1}+\ldots+x_{n})$,
$\displaystyle=$ $\displaystyle\otimes_{i=1}^{n}l_{i}^{d_{i}}\otimes
R^{0}\pi_{*}\omega\Bigl{(}\sum_{i}(d_{i}+1)D_{i}\Bigr{)}$ $\displaystyle=$
$\displaystyle\otimes_{i=1}^{n}l_{i}^{d_{i}}\otimes\Bigl{(}\mathcal{H}+\sum_{i=1}^{n}\sum_{k=1}^{d_{i}+1}l_{i}^{-k}\Bigr{)}.$

It is easy to see that this is equivalent to (28). ∎

######  Remark 7.

The above computation also yields

$\pi_{*}(\mathcal{L}_{n+1}-1)=\mathcal{H}+\mathcal{H}^{-1}+(n-2)1.$

This resembles the cohomological dilaton equation in the sense that the rank
of the RHS is $2g-2+n$.

######  Remark 8.

The techniques used in proving String and Dilaton equations can also be used
to find the push-forward of negative powers of $\mathcal{L}$.

##  5\. Quantum $K$-ring and Frobenius manifold

In this section we generalize Givental’s treatment [12] to non-convex
algebraic manifolds. Many statements already appeared in [12] (albeit with the
convexity condition there) and are included here for completeness.

###  5.1. The ring structure in quantum $K$-theory

The definition of the ring structure of quantum $K$-theory is analogous to
that of quantum cohomology theory.

Let $\\{e_{i}\\}$ be a basis of $K(X)$ and $t_{i}$ its coordinates.
$t:=\sum_{i}t_{i}e_{i}$ as in Section 4.2.

The quantum $K$-product $*$ is defined to be:

(29)
$((e_{i}*e_{j},e_{k})):=\partial_{t_{i}}\partial_{t_{j}}\partial_{t_{k}}G(t),$

where $G(t)$ is the genus zero potential defined in (16) and $((\cdot,\cdot))$
is the quantized metric defined in equation (17).

The main result of this section is

######  Theorem 3.

$(K(X,\mathbb{Q}[[Q]]),*)$ is a commutative and associative algebra, which
deforms the usual ring structure of $K(X)$.

###### Proof.

The deformation properties follow immediately from the definition. Setting
$q=0$ would make $*$ the ordinary tensor product. The (super-)commutativity
also follows from the definition.

The associativity follows from the Splitting Axiom (21). It is easy to see
that the $K$-theoretic WDVV equation

$\sum_{\mu\nu}G_{ij\mu}G^{\mu\nu}G_{\nu
kl}=\sum_{\mu\nu}G_{ik\mu}G^{\mu\nu}G_{\nu
jl}=\sum_{\mu\nu}G_{il\mu}G^{\mu\nu}G_{\nu jk}$

is equivalent to the associativity of the quantum $K$-ring. The WDVV equation
follows from Splitting Axiom: Let $n=4,n_{1}=2,n_{2}=2$ in (21). Different
ways of splitting amount to the same invariants. ∎

In the case $t=0$, we have defined a deformation by quantum three-point
function, which is similar to the _pair-of-pants_ structure. However, in this
case the metric ought to be quantized as well (and is replaced by two-point
function at $t=0$).

######  Remark 9.

For convex $X$, a proof can be found in [12].

######  Remark 10.

The above description is good for topological $K$-theory. For algebraic
$K$-theory, a finite basis does not necessarily exist and the $K$-theoretic
Poincaré pairing has a huge kernel. However, one could proceed by using a
presentation which is basis-free and does not rely on Poincaré pairing.
Namely, one could define the quantum $K$-product (at $t=0$) as

$e_{i}*e_{j}=\sum_{\beta}Q^{\beta}({\operatorname{ev}}_{3})_{*}\left({\operatorname{ev}}_{1}^{*}(e_{i}){\operatorname{ev}}_{2}^{*}(e_{j})\sum_{m}(-1)^{m+1}\sum_{(i,i_{j})}{\Psi_{m}}_{*}\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau^{i}_{m},\beta^{i_{j}}_{m})}\right)$

where ${\operatorname{ev}}_{i}:{\overline{M}}_{0,3}(X,\beta)\to X$ are the
evaluation morphisms and

$\sum_{m}(-1)^{m+1}\sum_{(i,i_{j})}{\Psi_{m}}_{*}\mathcal{O}^{\operatorname{vir}}_{{\overline{M}}(X,\tau^{i}_{m},\beta^{i_{j}}_{m})}$

is the alternating sum of virtual structure sheaves appeared in (10).

###  5.2. Quantum $K$-theory and Frobenius (super-)manifolds

In this section, we show that the construction of quantum $K$-ring produces a
(new) class of Frobenius manifolds, in the sense of [26]. This in particular
gives a positive answer to the question raised by Bayer and Manin ([3] 1.1.1
Question). Note that in this case the identity element $e_{0}$ is _not_ flat.

The data of $K$-theoretic Frobenius structure includes:

(1) A metric $((e_{i},e_{j}))=G_{ij}:=\partial_{i}\partial_{j}G(t)$ on tangent
spaces to $K(X)_{\mathbb{C}}$.

(2) The quantum multiplication with structural constants

$((e_{i}*e_{j},e_{k}))=\partial_{i}\partial_{j}\partial_{k}G(t).$

on the tangent bundle.

(3) A connection on the tangent bundle defined by the operators of quantum
multiplication:

$\nabla_{q}:=d-\frac{1}{1-q}\sum_{i}(e_{i}*)dt_{i}.$

######  Proposition 12.

1\. The metric and the quantum multiplication $*$ define on the tangent bundle
a formal commutative associative Frobenius algebra with unit $1$.

2\. The connection $\nabla_{q}$ are flat for $q\neq 1$.

3\. The operator $\nabla_{-1}$ is the Levi-Civita connection of the metric
$((\cdot,\cdot))$.

4\. The metric $((\cdot,\cdot))$ is flat.

###### Proof.

(see [12]) The first two are formal consequences of WDVV equation. The Levi-
Civita connection of the metric $G_{ij}(t)$ is

$\Gamma_{ij}^{k}=\frac{1}{2}(G_{il;j}+G_{jl;i}-G_{ij;l})G^{lk}=\frac{1}{2}G_{ijl}G^{lk}=\frac{1}{2}(e_{j})_{i}^{k},$

where $G^{ij}$ is the inverse matrix of $G_{ij}$. Thus _3._ holds. _4._ is an
obvious consequence of _3_. ∎

######  Corollary 1.

$QK^{*}(X)_{\mathbb{C}}$ is a formal Frobenius manifold over the Novikov ring
$N(X)$.

###  5.3. Quantum differential equation in $K$-theory

Parallel to the discussion in quantum cohomology, we may introduce quantum
differential equation [11]:

$\nabla_{q}S=0,$

which is a system of linear partial differential equations.

######  Theorem 4.

The matrix $(S_{ij})$

$S_{ij}(t,Q):=g_{ij}+\sum_{n,\beta}\frac{Q^{\beta}}{n!}{\Big{\langle}}e_{i},t,\ldots,t,\frac{e_{j}}{1-q\mathcal{L}}{\Big{\rangle}}_{0,n+2,\beta}$

is the fundamental solution to the $K$-theoretic quantum differential
equation. Namely, the column vectors form a complete set of solutions.

###### Proof.

The proof relies on two ingredients: the string equation and WDVV equation.
However, the WDVV equation here takes a slightly generalized form. Put
$e_{i},e_{j},e_{k},\frac{e_{l}}{1-q\mathcal{L}}$ on the distinguished four
marked points. The proof of WDVV equation goes through and we get

$G_{i,j,\alpha}G^{\alpha\beta}\partial_{k}S_{\beta
l}=G_{i,k,\alpha}G^{\alpha\beta}\partial_{j}S_{\beta l},$

or equivalently,

(30) $(e_{j}*)\partial_{k}S=(e_{k}*)\partial_{j}S.$

Put $j=0$. Since $(e_{0}*)=\operatorname{Id}$, (30) implies that

$\partial_{k}S=(e_{k}*)\partial_{0}S.$

Now by string equation

$\partial_{0}S=\frac{1}{1-q}S.$

∎

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