119–126

# Establishing the impact of luminous AGN with multi-wavelength observations
and simulations

C.M. Harrison1 A. Girdhar1,2,3 and S.R. Ward2,3,4 1School of Mathematics,
Statistics and Physics, Newcastle University, U.K.
email<EMAIL_ADDRESS>
2European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching
bei München, Germany
3Ludwig Maximilian Universität, Professor-Huber-Platz 2, 80539 München,
Germany
4Excellence Cluster ORIGINS, Boltzmannstraße 2, 85748 Garching bei München,
Germany

(2023)

###### Abstract

Cosmological simulations fail to reproduce realistic galaxy populations
without energy injection from active galactic nuclei (AGN) into the
interstellar medium (ISM) and circumgalactic medium (CGM); a process called
‘AGN feedback’. Consequently, observational work searches for evidence that
luminous AGN impact their host galaxies. Here, we review some of this work.
Multi-phase AGN outflows are common, some with potential for significant
impact. Additionally, multiple feedback channels can be observed
simultaneously; e.g., radio jets from ‘radio quiet’ quasars can inject
turbulence on ISM scales, and displace CGM-scale molecular gas. However,
caution must be taken comparing outflows to simulations (e.g., kinetic
coupling efficiencies) to infer feedback potential, due to a lack of
comparable predictions. Furthermore, some work claims limited evidence for
feedback because AGN live in gas-rich, star-forming galaxies. However,
simulations do not predict instantaneous, global impact on molecular gas or
star formation. The impact is expected to be cumulative, over multiple
episodes.

###### keywords:

active galactic nuclei, quasars, feedback, galaxy evolution

††volume: 378††journal: Title of your IAU Symposium††editors: G. Bruni, M.
Diaz-Trigo, K. Fukumura, S. Laha, eds.

##  1 Introduction

There is a consensus across galaxy formation models and simulations that
considerable energy injection, into the interstellar medium (ISM) and beyond,
from active galactic nuclei (AGN) is required to regulate star formation and
to reproduce many of the observable properties of massive galaxies (e.g.,
[Schaye15, Schaye et al. 2015]; [Pillepich18, Pillepich et al. 2018]; [Davé et
al. (2019), Davé et al. 2019]). This theoretical work has motivated a
multitude of observational work, searching for ‘direct’ evidence that AGN
impact upon the properties of the ISM, or circumgalactic medium (CGM), and/or
the host galaxy’s star formation (see e.g., [Harrison (2017), Harrison et al.
2017]).

We provide a brief overview of some work studying the most luminous AGN (i.e.,
roughly with $L_{\rm AGN}>$1043 erg s-1), which are associated with rapidly
accreting supermassive black holes and that are identified mostly using
optical, infrared or X-ray wavelengths. Three popular approaches that
observers take to investigate if luminous AGN impact their host galaxies are
discussed: (1) measuring mass outflow rates and kinetic coupling efficiencies
(Section 2); (2) comparing the location of jets and outflows to spatially-
resolved ISM properties and sites of star formation (Section 3); and (3)
investigating the star formation rates and molecular gas properties of AGN
host galaxies (Section 4). Section 5 finishes this article by discussing what
can be learnt about the impact of luminous AGN from these observations, within
the context of galaxy formation simulations.

##  2 Outflow rates and kinetic coupling efficiencies

AGN are known to drive multi-phase ‘outflows’ into the interstellar medium
(ISM) and circumgalactic medium (CGM), as evidenced by a variety of emission-
line and absorption-line studies (see e.g., [Cicone et al. (2019), Cicone et
al. 2018]; [Harrison18, Harrison et al. 2018]; [Veilleux et al. (2020),
Veilleux et al. 2020]). These AGN outflows can result in increased turbulence
and/or can expel material from the host galaxy’s ISM (at least temporarily).

A common approach to assess the potential impact of AGN outflows is to measure
their properties, including the mass outflow rates and kinetic powers. These
calculations typically involve: identifying which gas is outflowing;
converting fluxes to gas masses; measuring the velocities and spatial
distribution of the outflowing gas; and then applying models or basic
assumptions to calculate outflow rates ([Harrison18, Harrison et al. 2018]).

The inferred mass outflow rates in some sources are found to exceed the star
formation rates (see discussion in e.g., [Harrison (2017), Harrison 2017];
[Fiore et al. (2017), Fiore et al. 2017]). This means that gas is being
removed more rapidly than it is forming stars, implying the potential for a
significant impact on the host galaxy. However, it is very challenging to
infer the long-term impact of such outflows, including understanding what
fraction of this material ultimately escapes the host galaxy. Furthermore,
these extreme rates are not ubiquitous. The inferred outflow rates of some
more ‘typical’ systems can be very low to negligible (e.g., [Davies et al.
(2020), Davies et al. 2020]; [Ramos Almeida et al. (2022), Ramos Almeida et
al. 2022]; [Villar Martín et al. (2023), Villar Martin et al. 2023]). There
are also analysis challenges, with very different results possible when
applying different assumptions or using different gas tracers (e.g.,
[Harrison18, Harrison et al. 2018]; [Davies et al. (2020), Davies et al.
2020]).

Another common approach in the literature is to measure the ratio of outflow’s
kinetic power to AGN luminosity to derive the ‘kinetic coupling efficiency’
(see e.g., [Fiore et al. (2017), Fiore et al. 2017]). In some cases, these
measurements have been compared to simulations, which invoke AGN feedback to
establish if the observations are consistent – or not – with feedback models.
However, these comparisons are very challenging. It is important to note that
the observations are typically measuring an outflow rate in one particular gas
phase and often without information on how the properties vary spatially.
Furthermore, the AGN luminosities are ‘instantaneous’ and do not contain
information about earlier times, including when the outflows may have been
initially launched. Therefore, these measured kinetic coupling efficiencies
are not comparable quantities to the AGN coupling efficiencies invoked in sub-
grid models of simulations (see discussion in [Harrison18, Harrison et al.
2018]). Indeed, if simulations track the kinetic powers of outflows as they
propagate through the ISM, they can be significantly lower than their initial
energy, and it remains extremely challenging to simulate the phases of
outflows that are captured by observations (e.g., [Costa, Pakmor, & Springel
(2020), Costa et al. 2020]).

##  3 Spatially-resolved investigation of impact

Figure 1: (a) Molecular gas velocities and radial extents for two types of
AGN-ISM interaction, traced via CO emission: (1) broad emission-line
components for a sample of AGN and star-forming galaxies ([Fluetsch2019,
Fluetsch et al. 2019]): (2) filaments around radio lobes for BCGs
([Tamhane2022, Tamhane et al. 2022]). The larger symbols represent data for
four $z<0.2$ ‘radio quiet’ quasars, two of which exhibit both types of
structure. (b) Example CO emission-line profile for one of these quasars. (c)
Example CO velocity map of the same quasar, with radio contours, and
highlighting CO structures associated with $\approx$10 kpc radio lobes.
Figures adapted from Girdhar et al. (in prep).

Recent numerical simulations have highlighted that multiple feedback ‘modes’
can exist simultaneously, caused by AGN-driven winds, jets and/or radiation
pressure. This can include: injected turbulence to reduce star formation
efficiency; removal of high-density nuclear gas (i.e., star-forming material);
halting accretion from the halo (i.e., regulating future star formation); and
even the localised boosting of star formation due to shocks compressing the
gas to higher densities (e.g., [Costa, Pakmor, & Springel (2020), Costa et al.
2020]; [Mandal21, Mandal et al. 2021]).

Observations now also show that AGN can impact upon the ISM, and wider-scale
environment, in multiple different ways. For example, luminous ‘radio quiet’
quasars, which have considerable radiative output and are often associated
with powerful winds, are also able to interact with the star-forming material
in their host galaxies via low-power radio jets and lobes (e.g., [Girdhar22,
Girdhar et al. 2022]; [Audibert et al. (2023), Audibert et al. 2023]; [Cresci
et al. (2023), Cresci et al. 2023]). An example of this is showcased in Figure
1, where broad, and asymmetric CO emission-line profiles (tracing molecular
outflows in the inner regions of galaxies) and molecular filaments entrained
by expanding radio lobes are observed simultaneously (also see [Morganti et
al. (2023), Morganti et al. 2023]). These processes are typically searched for
asynchronously in observations of luminous AGN (e.g., [Fluetsch, Fluetsch et
al. 2019]) and brightest cluster galaxies (BCGs; e.g., [Russel19, Russel et
al. 2019]; [Tamhane22, Tamhane et al. 2022]), respectively.

The aforementioned observations are, at least qualitatively, similar to
expectations of simulations of individual galaxies, which predict AGN will
ultimately have a negative impact upon a host galaxy (e.g., [Costa, Pakmor, &
Springel (2020), Costa et al. 2020]; [Mandal21, Mandal et al. 2021]). However,
this global, and long-term suppression of star formation, may not easily be
associated with a single AGN episode, or a localised ISM interaction. This
could help explain why ‘direct’ observational evidence of star formation
suppression continues to be scarce, tentative or inconclusive (e.g.,
[Scholtz21, Scholtz et al. 2021]; [Cresci et al. (2023), Cresci et al. 2023]).
On the other hand, observations that show depleted and/or excited molecular
gas within the ISM at the locations of low-power jets or outflows can indicate
that the star-forming material is directly affected by the AGN (e.g., [Rosario
et al. (2019), Rosario et al. 2019]; [Girdhar22, Girdhar et al. 2022];
[Audibert et al. (2023), Audibert et al. 2023]). Nonetheless, the long-term
and global, galaxy-wide impact of such processes remains more speculative.
This motivates a statistical approach, as described below, that measures the
global molecular gas and star formation properties of large samples of AGN
host galaxies.

##  4 Gas content and star formation rates

Figure 2: Ratio of molecular gas to stellar mass versus specific star
formation rate. Galaxies from [Tacconi et al. (2018), Tacconi et al. (2018)]
with and without an identified AGN are represented by density contours (also
green points) and by magenta squares, respectively. Type 2 $z<0.2$ quasars,
with known outflows and/or jets are shown with black circles. Adapted from
[Jarvis et al. (2020), Jarvis et al. (2020)].

One popular approach to investigate the impact of AGN on their host galaxies,
is to take galaxy samples and measure how the star formation properties and/or
the molecular gas properties (i.e., the ‘fuel’ for star formation) are related
(or not) to the presence of an AGN. These studies come in various forms,
including: (1) investigating trends of star formation rates or molecular gas
content with AGN luminosities (e.g., [Stanley et al. (2015), Stanley et al.
2015]; [Ramasawmy et al. (2019), Ramasawmy et al. 2019]; [Shangguan et al.
(2020), Shangguan et al. 2020]); and (2) establishing if AGN host galaxies
have different distributions of star formation and/or molecular gas properties
to non-AGN host galaxies (e.g., [Scholtz et al. (2018), Scholtz et al. 2018];
[Koss et al. (2021), Koss et al. 2021]; [Valentino21, Valentino et al. 2021]).

One specific example of this type of analysis is shown in Figure 2. The
molecular gas fractions (derived from CO observations) are plotted as a
function of specific star formation rate (i.e., star formation rate divided by
stellar mass) for low redshift galaxies and AGN. The star-forming galaxies and
AGN from [Tacconi et al. (2018), Tacconi et al. (2018)] are shown to occupy
the same area of parameter space. Furthermore, quasars with known radio jets
and powerful ionised outflows also have the same molecular gas and star
formation properties than their less active galaxy counterparts ([Jarvis et
al. (2020), Jarvis et al. 2020]).

With only a few exceptions (e.g., [Circosta et al. (2021), Circosta et al.
2021]; [Bischetti et al. (2021), Bischetti et al. 2021], which investigate
molecular gas content at $z\sim 2$), studies of AGN across multiple epochs and
using a variety of samples and approaches tend towards a consensus that
luminous AGN typically live in gas-rich, star-forming galaxies. It can be
tempting to conclude that this is in conflict with predictions that luminous
AGN regulate star formation (as suggested by some observational studies),
however, as we demonstrate below, this empirical result is not conflicting
with predictions of cosmological simulations including AGN feedback.

##  5 Discussion and conclusions

Figure 3: Ratio of molecular gas to stellar mass versus stellar mass, for
$z=0$ galaxies in the IllustrisTNG100 simulation. Colouring represents the
mean AGN bolometric luminosity within each bin. Contour lines represent number
density, and the triangles show the median values in bins of stellar mass for
galaxies classified as AGN (blue) and non-AGN (red), based on their
luminosity. The dashed line is the observed main sequence from [Tacconi et al.
(2018), Tacconi et al. (2018)]. The histograms show the distributions of gas
fractions for sources classified as AGN (blue) or non-AGN (red), based on
their Eddington ratios. Adapted from [ward22, Ward et al. (2022)].

We have given a very brief overview of some of the observational work
investigating the impact of luminous AGN on their host galaxies (in particular
the molecular gas content and the star formation). On the one hand, there is
overwhelming evidence that AGN are able to modify the distribution and
properties of the ISM and CGM of their host galaxies. This is due to the ever-
growing body of observations that AGN radiation, winds and/or jets are able to
influence the molecular gas by injecting turbulence, driving it away, and/or
exciting it (Section 2 and 3). On the other hand, the long-term and global
significance of these processes is less well-established, with the possibility
that their impact is very localised, lasts only on short timescales, and/or is
insignificant. Indeed, the observation that luminous AGN tend to live in gas-
rich, star-forming galaxies has sometimes been presented as evidence against
effective feedback (Section 4).

With this in mind, it is worth revisiting the cosmological simulations, which
invoke AGN feedback, and motivated much of the observational work. Whilst
these simulations typically lack the resolution to include a detailed physical
prescription of AGN feedback, they are useful for comparing to the statistical
studies of AGN and the molecular gas and star formation properties presented
in Section 4. For example, [Ward22, Ward et al. (2022)] investigated
predictions from the cosmological simulations: (1) IllustrisTNG100
([Pillepich18, Pillepich et al. 2018]); (2) EAGLE ([Schaye15, Schaye et al.
2015]) and (3) SIMBA ([Davé et al. (2019), Davé et al. 2019]). These are
broadly similar in scope, with $\sim$100 Mpc3 boxes and simulation outputs
that enable predictions of AGN luminosities, stellar masses, star formation
rates and molecular gas content as a function of cosmic time. [Ward22, Ward et
al. (2022)] used these to reproduce the type of experiments performed by
observers, as discussed in Section 4. One example of this analysis is
presented in Figure 3, where the molecular gas fractions of the $z=0$ galaxies
from IllustrisTNG are presented as a function of stellar mass. The luminous
AGN (selected on either luminosity or Eddington-ratio) are found to be
preferentially located in gas rich galaxies. Broadly speaking, across all
three simulations investigated, and at both $z=2$ and $z=0$, AGN are predicted
to live in gas-rich, star-forming galaxies, in qualitative agreement with most
observational work, although it is worth noting that there are distinct
quantitative differences in the predictions across the three simulations due
to the different feedback models used ([Ward22, Ward et al. 2022]).

We conclude that we may not expect to have a ‘smoking gun’ observational
signature that luminous AGN have a long-term, significant impact on molecular
gas content or star formation. The reasons for this are many, with some
discussion on this presented in e.g., [Scholtz et al. (2018), Scholtz et al.
(2018)]; [Ward22, Ward et al. (2022)]; [Piotrowska et al. (2022), Piotrowska
et al. (2022)]. For example, an AGN may stop being visible before they have
had an appreciable impact on their host galaxies. Furthermore, the significant
impact, which ultimately results in a ‘quenched’ galaxy, may not be attributed
to a single luminous AGN episode, but instead due to the cumulative effect of
many AGN episodes. Indeed, according to [Piotrowska et al. (2022), Piotrowska
et al. (2022)], the strongest predictor of a galaxy as having quenched star
formation (or not), is considered to be the black hole mass, rather than AGN
luminosity, in both cosmological simulations and observations. In this
context, black hole mass can be considered as an indirect tracer of the
integrated power output of the AGN across the history of the galaxy.

In summary, the current body of evidence does not appear to be in conflict
with AGN feedback models. However, work is still needed to test the different
assumed models of AGN feedback. This can be done by comparing the quantitative
predictions of the distributions of properties of galaxy populations from the
different cosmological simulations (see e.g., [Ward22, Ward et al. 2022]).
Furthermore, the current and forthcoming generation of high-resolution
individual galaxy simulations should make it possible to extract meaningful
predictions of spatially-resolved outflow properties in different phases
(e.g., mass outflow rates, kinetic powers), compare these to observations, and
to assess the ultimate impact that they can have on galaxy evolution.

CMH acknowledges a United Kingdom Research and Innovation grant
(MR/V022830/1). SRW acknowledges the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy
(EXC-2094-390783311).

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# Relations in Twisted Quantum $K$-Rings

Irit Huq-Kuruvilla

###### Abstract

We introduce twisted quantum $K$-rings, defined via twisted $K$-theoretic
Gromov-Witten invariants. We develop a toolkit for computing relations by
adapting some results about ordinary quantum K rings to our setting, and
discuss some applications, including Ruan-Zhang’s quantum $K$-theory with
level structure, and complete intersections inside projective space.

In addition, we formulate a ring-theoretic abelian/non-abelian correspondence
conjecture, relating the quantum K-ring of a GIT quotient $X//G$ to a certain
twist of the quantum K-ring of $X//T$, the quotient by the maximal torus. We
prove this conjecture for the case of Grassmanians, and use this to give
another proof of the Whitney relations of Mihalcea-Gu-Sharpe-Zhou.

##  1 Introduction

The quantum $K$-ring (introduced by Givental and Lee) of smooth projective
variety $X$ is a deformation of its topological $K$-ring involving
$K$-theoretic Gromov-Witten invariants of $X$. These invariants are defined as
certain holomorphic Euler characteristics on Kontsevich’s moduli space of
stable maps $\overline{M}_{0,n,d}(X)$.

It is a $K$-theoretic counterpart to the quantum cohomology ring introduced by
Witten and Kontsevich. Many recent developments in physics (including those of
[15], [21]) have resulted in connections to quantum $K$-theory, and
predictions for relations in quantum $K$-rings. However, quantum $K$-rings are
in general much harder to compute than their cohomological counterparts, so
many physical predictions go untested.

It was observed by Givental that one can calculate $K$-theoretic Gromov-Witten
invariants of the quintic by doing intersection theory of
$\overline{M}_{0,n,d}(\mathbb{P}^{4})$, with the virtual fundamental class
modified by the Euler class of an index bundle, corresponding to the
restriction of $\mathcal{O}(5)$ to the source curve. This introduced the
theory of twisted Gromov-Witten invariants, fully developed for cohomology
theory in [3]. In general, twisted invariants can be computed from their
untwisted counterparts. Other examples twisted invariants can be used to
compute Gromov-Witten invariants of cotangent bundles, and to define an
”abelian/non-abelian correspondence”, relating Gromov-Witten invariants of a
GIT quotient $X//G$ to those of the quotient $X//T$, where $T$ is the maximal
torus in $G$.]

The $K$-theoretic counterpart of twisted invariants were introduced in [20],
and generalizations were developed in [7] and [13]. In this paper, we will
consider the most common type of twisting (type I), and develop a theory of
quantum $K$-rings based on these invariants.

In full generality, given $V\in K^{*}(X)$ and a multiplicative $K$-theoretic
characteristic class $C$, twisted $K$-theoretic Gromov-Witten invariants with
twisting $C(V)$ are defined as holomorphic Euler characteristis on
$\overline{M}_{0,n,d}(X)$ with respect to a twisted virtual structure sheaf:

$\mathcal{O}^{tw}:=\mathcal{O}^{vir}\times C(ft_{*}ev_{n+1}^{*}V)$

As it turns out, type-I twisted invariants satisfy very similar properties to
their untwisted counterparts, and as such, many results on quantum $K$-rings
pass unchanged to the twisted case, in particular, we show the following:

######  Theorem.

Type I twisted invariants satisfy analogues of the Kontsevich-Manin axioms,
and they are the structure constants to a ring, the (big or small) _twisted
quantum $K$ ring_ of $X$.

Furthermore we introduce the twisted $J$ function $\mathcal{J}^{tw}(q,t)$, a
certain kind of generating function for twisted invariants, and we prove the
following theorem (which was proven by Iritani-Milanov-Tonita in [14] in the
untwisted case):

######  Theorem.

If $K^{*}(X)$ is generated by line bundles whose 1st Chern classes are
effective curve classes, then a polynomial difference/differential operator in
$q,t$ annihilating $\mathcal{J}^{tw}$ yields a relation in the twisted quantum
$K$-ring of $X$.

Using the above theorem in general is difficult, as the relations are obtained
in terms of operators $A_{i,com}$ which arise as solutions to a Lax-type
equation. We give some conditions on which these operators can be simply
interpreted in terms of multiplication by classes in $K^{*}(X)$, generalizing
some results of [1]. The results we adapt to this setting form a robust
toolkit for computing many examples of quantum $K$-rings.

To demonstrate the effectiveness of these results, we calculate relations in
the (small) quantum $K$-rings for complete intersections in $\mathbb{P}^{N}$,
as well as Ruan-Zhang’s quantum $K$-theory with level structure, and we give a
stable-map analogue of the quasimap quantum $K$-ring for $T^{*}\mathbb{P}^{N}$
introduced in [19].

We also introduce an abelian/non-abelian correspondence conjecture for small
quantum $K$-rings. Our setting will be some reductive group $G$ acting on a
vector space $X$. The classical abelian/non-abelian correspondence proven by
Martin constructs a ring map $sp:H^{*}(X//T)^{W}\to H^{*}(X//G)$, and
identifies the Poincare pairing on the target with the Poincare pairing on the
source modified by the Euler class of the vector bundle
$V^{M}:=\sum_{\alpha\in\text{roots($G$)}}L_{\alpha}$. The analogous statement
in $K$-theory was proven by Harada-Landweber [12].

######  Conjecture.

The map $x\mapsto\lim_{\lambda\to 1}sp(x)$, extended to Novikov variables, is
a surjective ring homomorphism from $QK^{tw}(X//T)$ to $QK^{*}(X//G)$, where
the twisting is determined by $Eu_{\lambda}(V_{M})$, where $Eu_{\lambda}$ is
the $\mathbb{C}^{*}$-equivariant $K$-theoretic Euler class.

We note that this is not a $K$-theoretic version of the correspondence for
quantum cohomology rings conjectured by Ciocan-Fontanine–Kim–Sabbah in [2],
that conjecture relates untwisted rings. There is a likely a $K$-theoretic
counterpart to that conjecture, however the version we work with matches up
well with the kind of formulas that appear in physics for relations in quantum
$K$-rings, as we will see for the Grassmanian. We show this conjecture is a
consequence of other abelian/non-abelian-type conjectures for $J$-functions,
and as a result, using theorems from Givental-Yan in [8] on the $J$-function
of the Grassmanian, we prove the conjecture for Grassmanians. Using this, we
reprove the presentation for $QK^{*}(Gr(k,n))$ from [11]. Our arguments in
principle extend to all partial flag varieties, where a presentation for the
quantum $K$-ring is otherwise not known is not known, however due to the
details involved, we save that case for a future work.

##  2 Twisted $K$-Theoretic Gromov-witten Invariants

$K$-theoretic Gromov-Witten invariants are defined as certain holomorphic
Euler characteristics on the Kontsevich moduli space of stable maps
$X_{g,n,d}(X)$ (this will be shortened to $X_{g,n,d}$ for the remainder of
this text). Given Laurent polynomials $f_{i}(q)$ with coefficients in
$K^{*}(X)$, the associated Gromov-Witten invariant is written in correlator
notation as $\langle f_{1},\dots,f_{n}\rangle_{g,n,d}$ and is defined to be:

$\chi(X_{g,n,d};\mathcal{O}^{vir}\otimes\prod_{i}ev_{i}^{*}f_{i}(L_{i}))$

Here the notation $ev_{i}^{*}f_{i}(L_{i})$ means to pull back the coefficients
of $f_{i}$ by $ev_{i}$, and evaluate the resulting polynomial at $q=L_{i}$.

A _twisting_ is a modification of the virtual structure sheaf
$\mathcal{O}^{vir}$ of the form $C(ft_{*}\alpha)$, where $C$ is some
invertible multiplicative characteristic class in $K$-theory, expressible via
Adams operations as:

$C(E)=e^{\sum_{k\neq 0}\Psi^{k}(s_{k}E)},$

$ft$ is the forgetting map from the universal curve $X_{g,n+1,d}$, and
$\alpha$ can be one of three types of elements:

  * •

Type I: $\alpha=ev_{n+1}^{*}V$, where $V\in K^{*}(X)$ (introduced in [20]).

  * •

Type II: $\alpha=ev_{n+1}^{*}f(L_{n+1})$, where $f$ is a Laurent polynomial
with coefficients in $K^{*}(X)$ satisfying $f(1)=0$. (The other cases are
handled by splitting into type II and type I components) (introduced in [7]).

  * •

Type III: $\alpha=i_{*}f(L_{+},L_{-})$, where $f$ is a symmetric bivariate
Laurent polynomial, $i$ is the inclusion of $\mathcal{Z}$, the codimension-2
locus of nodes in the universal family, and $L_{+},L_{-}$ are the cotangent
lines to each branch at the node (introduced in [13]).

Twistings of different kinds can be taken in succession, and theorems relating
twisted invariants of each type to their untwisted counterparts were proven in
[7] and [13]. This paper will only be concerned with twistings of type I.

###  2.1 Examples of Twistings of Type I

The rest of this text will be concerned with twistings of type I. These
encompass many important geometric situations. All of these twisting will be
some variation of an Euler class, so we include some facts about $K$-theoretic
Euler classes here:

The $K$-theoretic Euler class $Eu$ is defined on line bundles $L$ by
$Eu(L)=1-L^{-1}$. We can write this as $e^{\sum_{k<0}\Psi^{k}(L)/k}$, using
the Taylor series for the logarithm. If we let $\mathbb{C}^{*}$ act by scaling
on $L$ with equivariant parameter $\lambda$, this class becomes invertible. By
multiplicativity and the splitting principle, the expression
$e^{\sum_{k<0}\Psi^{k}(V)/k}$ computes the Euler class for any bundle $V$.

####  2.1.1 Vector Bundles

Let $V$ be a vector bundle over $X$. Since $V$ is not compact, we cannot
define Gromov-Witten invariants of the total space of $V$ directly using the
theory of stable maps.

However, we can take the following approach: Given the standard
$\mathbb{C}^{*}$-action scaling the fibers of $V$, we work equivariantly. In
the hypothetical moduli space $V_{g,n,d}$. the $\mathbb{C}^{*}$ fixed locus is
exactly $X_{g,n,d}$, corresponding to curves that land in $X$. Given such a
map $f:C\to X$, the normal bundle would be $f^{*}V$. Thus the normal sheaf to
$X_{g,n,d}$ is $ft_{*}ev_{n+1}^{*}V$. Thus equivariant holomorphic Euler
characteristics on $V_{g,n,d}$ could be defined on $X_{g,n,d}$ via
heuristically emulating the approach of fixed point localization, which
involves dividing by the equivariant Euler class of the normal sheaf. So the
stable map Gromov-Witten invariants of $V$ are defined as the Gromov-Witten
invariants of $X$, twisted by the class $Eu^{-1}(ev^{*}ft_{*}V)$.

####  2.1.2 Complete Intersections

If $V$ was an ample vector bundle, its sections define a complete intersection
$Y\in X$. If $V$ is furthermore convex, $ft_{*}ev^{*}(V)$ is a bundle over
$X_{0,n,d}$ whose zero locus is $Y_{0,n,d}$. So we can compute $K$-theoretic
Gromov-Witten invariants of $Y$ by inserting the Euler class of
$ft_{*}ev^{*}(V)$ into $X$. To make it invertible, we use the equivariant
Euler class as before.

####  2.1.3 Abelian/non-Abelian Correspondence

As stated in the introduction, our setting will be some reductive group $G$
acting on a vector space $X$. The classical Abelian/non-Abelian correspondence
proven by Martin relates $H^{*}(X//G)$ with $H^{*}(X//T)^{W}$, and uses the
vector bundle $V^{M}:=\sum_{\alpha\in\text{roots($G$)}}L_{\alpha}$. The
analogous statement statement in $K$-theory reads as follows:

######  Theorem 2.1 (Harada-Landweber, [12]).

  * •

$K^{*}(X//G)\cong\frac{K^{*}(X//T)^{W}}{ann(Eu(V^{M}))}$. We refer to this
isomorphism by $sp$

  * •

For $\alpha\in K^{*}(X//T)$,
$\frac{1}{|W|}\chi(X//T;Eu(V^{M})\alpha)=\chi(X//T;sp(\alpha))$.

######  Remark.

This is an equivalent formulation to the one given in Harada-Landweber, but it
doesn’t correspond exactly to their main theorem. Our formulation is closer to
Martin’s, but in $K$-theory rather than cohomology. Their proof applies to
equivariant contexts as well, so we can work in equivariant $K$-theory with
respect to a torus action compatible with $G$

The quantum version of this correspondence relates the $K$-theoretic Gromov-
Witten theory of $X//G$ with Gromov-Witten theory of $X//T$ twisted by
$Eu_{\lambda}(V^{M})$, the $S^{1}$-equivariant Euler class of $V^{M}$. We will
outline the precise statements in a later section of this text.

####  2.1.4 Level Structures

Quantum $K$-theory with level structure was introduced by Ruan and Zhang, and
corresponds to twisting $K$-theoretic Gromov-Witten invariants twisted by the
class $det(ft_{*}ev_{n+1}^{*}V)^{-\ell}$, here $V$ is some vector bundle on
$X$ and $\ell$ is a positive integer known as the _Ruan-Zhang level_. While
the characteristic class given by the determinant is invertible, it cannot be
expressed convergently as $e^{\sum_{k}s_{k}\Psi^{k}(L)}$, so to include it
into the framework of twisted invariants we use the following approach:

First observe that for $V$ a vector bundle with nonvanishing Euler class, we
have:

######  Proposition 2.2.

$\frac{Eu(V)}{Eu(V^{*})}=(-1)^{rk(E)}det(E)^{-1}$

###### Proof.

For $V$ a line bundle, this is $\frac{1-L^{-1}}{1-L}=-L^{-1}$. Using the
splitting principle, write $V=\bigoplus_{i=1}^{rk(V)}L_{i}$, which yields:

$\frac{Eu(V)}{Eu(V^{*})}=\prod_{i}\frac{1-L_{i}^{-1}}{1-L_{i}}=\prod_{i}-L_{i}^{-1}=-1^{rk(V)}det(V)^{-1}$

∎

So if we introduce two variables $Y_{1}$ and $Y_{2}$, the class
$C(V):=(-1)^{rk(\ell V)}\frac{\Lambda_{Y_{1}}(\ell V)}{\Lambda_{Y_{2}}(\ell
V^{*})}$ is an invertible multiplicative characteristic class whose limit as
$Y_{1},Y_{2}\to 1$ becomes $-1^{\ell\times rk(V)}det(V)^{-\ell}$. Replacing
$V$ with $-V$ corrects for the sign issues, but in practice it is often useful
to use the above formulation, and then make ad hoc corrections to account for
the sign.

######  Remark.

In fact, Ruan and Zhang show directly that the properties we use in the next
section also hold for the determinant, with no extra variables necessary. So
in the contexts that appear in this work, the failure of $det$ to be
expressible in standard form can actually be ignored.

##  3 Basic Properties of Type I Theories

Given that many of the above examples represent Gromov-Witten theories of
different spaces, it is not particularly surprising that type I twisted
$K$-theoretic Gromov-Witten invariants satisfy essentially the same set of
properties as their untwisted counterparts. Before stating and proving these
properties we introduce some definitions and notation.

Given a twisting $C(V)$, the classical $K$-ring is just $K^{*}(X)$, with
Poincare pairing modified by $C(V)$. More precisely, given some basis
$\phi_{\alpha}$ for $K^{*}(X)$, the Poincare pairing is defined by

$(\phi_{\alpha},\phi_{\beta})=\chi(X;\phi_{\alpha}\phi_{\beta}C(V))$

. We denote this metric by $g$, with matrix coefficients $g_{\alpha\beta}$,
and inverse matrix $g^{\alpha\beta}$.

Let $F$ be the exponential generating function of the genus-0 $K$-theoretic
Gromov-Witten invariants twisted by $C(V)$, i.e:

$F:=\sum_{n,d}\frac{Q^{d}}{n!}\langle t,\dots,t\rangle_{0,n,d}^{tw}$

Here $t$ stands for $\sum_{\alpha}t_{\alpha}\phi_{\alpha}$. Differentiation of
$F$ with respect to $t_{\alpha}$ corresponds to insertion of $\phi_{\alpha}$
in one of the inputs, we denote $\frac{\partial}{\partial t_{\alpha}}F$ by
$F_{\alpha}$ (and similarly for other functions of $t$).

The twisted quantum $K$-product will be based on the function

$G:=\frac{1}{2}g_{\alpha\beta}t_{\alpha}t_{\beta}+F$

We first define a deformation of the Poincare pairing, the quantum pairing is
defined by:

$((\phi_{\alpha},\phi_{\beta})):=G_{\alpha\beta}$

$H_{g,n,d}(\alpha_{1},\dots,\alpha_{n})$ denotes the $K$-class
$\prod_{i=1}^{n}ev_{i}^{*}\alpha_{i}(L_{i})\otimes\mathcal{O}^{tw}_{g,n,d}$,
i.e. the contents of a twisted $K$-theoretic Gromov-Witten invariant before
the holomorphic Euler characteristic is taken.

For $V\in K^{*}(X)$, if we denote $V_{g,n,d}$ to be $ft_{*}ev_{n+1}^{*}V$,
then the following are of the theory with twisting determined by $V$ and a
multiplicative characteristic class $C$ (these results go back to Coates in
[3]):

######  Theorem 3.1 ([3]).

  * •

$ft^{*}\mathcal{O}^{tw}_{g,n,d}=\mathcal{O}^{tw}_{g,n+1,d}$

  * •

The restriction of $O^{tw}_{g,n,d}$ to the preimage of the gluing map
$X_{g_{1},n_{1}+1,d_{1}}\times_{\Delta}X_{g_{2},n_{2}+1,d_{2}}$ is:

$\frac{\mathcal{O}^{tw}_{g_{1},n_{1}+1,d_{1}}\mathcal{O}^{tw}_{g_{2},n_{2}+1,d_{2}}}{ev_{\Delta}^{*}C(V)}\to
X_{g,n,d}$

  * •

The restriction of $O^{tw}_{g,n,d}$ to the preimage of the gluing map
$X_{g-1,n+1,d}\otimes_{X\times X}\Delta\to X_{g,n,d}$ is
$\frac{O^{tw}_{g-1,n+2,d}}{ev_{\Delta}^{*}(C(V))}$.

Property i) here implies the following:

######  Lemma 3.2.

If $ft$ denotes the map forgetting the $n+1$st marked point, then:

$ft^{*}O_{g,n,d}^{tw}=O_{g,n+1,d}^{tw}$

These facts are sufficient to show that a type-I twisted theory satisfies
analogues of the $K$-theoretic Kontsevich-Manin axioms, which we state below:

######  Theorem 3.3.

Twisted $K$-theoretic Gromov-Witten invariants satisfy the following set of
axioms:

  * •

$S_{n}$ covariance: For $\sigma\in S_{n}$ acting on $X_{g,n,d}$:
$\sigma(H_{g,n,d}(\alpha_{1},\dots,\alpha_{n}))=H_{g,n,d}(\alpha_{\sigma(1)},\dots,\alpha_{\sigma(n)})$.

  * •

Fundamental class: For nondescendant inputs $\alpha_{i}$,
$\langle\alpha_{1},\dots,\alpha_{n},1\rangle_{g,n+1,d}^{tw}=\langle\alpha_{1},\dots,\alpha_{n}\rangle_{g,n,d}^{tw}$

  * •

Mapping to a point:
$\mathcal{O}^{tw,vir}_{g,n,0}\cong\mathcal{O}_{\overline{M}_{g,n}}\boxtimes
C(V)\Lambda_{-1}(T_{X})$

  * •

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

$\Phi:\bigcup_{d_{1}+d_{2}=d}X_{g_{1},n_{1}+1,d_{1}}\times
X_{g_{2},n_{2}+1,d_{2}}\to X_{g,n,d}$

be the contraction map which glues the last marked point of
$\overline{M}_{g_{1},n_{1}+1}$ to the first marked point of
$\overline{M}_{g_{2},n_{2}+1}$ Then

$\begin{split}&\Phi^{*}\sum_{k,d}Q^{d}\frac{1}{k!}\text{ft}^{k}_{*}H^{X}_{g,n+k,d}(\gamma_{1},\ldots,\gamma_{n},t,\ldots,t)\\\
=&\sum_{\alpha,\beta}\left(\sum_{k_{1},d_{1}}Q^{d_{1}}\frac{1}{k_{1}!}\text{ft}^{k_{1}}_{*}H^{X}_{g_{1},n_{1}+k_{1}+1,d_{1}}(\gamma_{1},\ldots\gamma_{n_{1}},t,\ldots,t,\phi_{\alpha})\right)\\\
&G^{\alpha\beta}(t)\left(\sum_{k_{2},d_{2}}Q^{d_{2}}\frac{1}{k_{2}!}\text{ft}^{k_{2}}_{*}H^{X}_{g_{2},n_{2}+k_{2}+1,d_{2}}(\phi_{\beta},\gamma_{n_{1}+1},\ldots\gamma_{n},t,\ldots,t)\right)\end{split}$

where the notation $\text{ft}^{k}$ stands for the forgetful map which forget
the additional $k$ marked points.

  * •

Genus reduction: Let

$\Phi:X_{g-1,n+2,d}\to X_{g,n,d}$

be the contraction map which glues the last two marked points. Then

$\begin{split}&\Phi^{*}\sum_{k,d}Q^{d}\frac{1}{k!}\text{ft}^{k}_{*}H^{X}_{g,n+k,d}(\gamma_{1},\ldots,\gamma_{n},t,\ldots,t)\\\
=&\sum_{\alpha\beta}\left(\sum_{k,d}Q^{d}\frac{1}{k!}\text{ft}^{k}_{*}H^{X}_{g+1,n+k+2,d}(\gamma_{1},\ldots\gamma_{n},t,\ldots,t,\phi_{\alpha},\phi_{\beta})\right)G^{\alpha\beta}(t)\end{split}$

The proofs of the first three axioms are identical to those for the untwisted
theory in [16]. We prove the latter two in the following section.

######  Remark.

Our statements use $H_{g,n,d}$, classes in $K(X_{g,n,d})$, the axioms in [16]
use $I_{g,n,d}$, which are the pushforwards to $\overline{M}_{g,n}$ of
$H_{g,n,d}$, all proofs are equivalent once translated.

###  3.1 Splitting and Genus Reduction Axioms

We first recall the statement and proof for the splitting axiom for untwisted
theories, and then cover the necessary changes for the twisted version.

The preimage of the divisor $D$ is the locus in $X_{g,n+k,d}$ consisting of
two curves of genus $g_{1}$ and $g_{2}$, each containing $n_{1},n_{2}$ marked
points respectively, with $n_{1}+n_{2}=n$. The curves are connected by a chain
of $m$ $\mathbb{P}^{1}$s. The $k$ additional marked points are distributed on
each irreducible component, such that there is at least one marked point in
each curve in the chain. The degrees are chosen to add up to $d$. Call $m+1$
the depth of such a curve. Let $M_{s,k_{\alpha},d_{\alpha}}$ be the space of
curves of depth at least $s$, with a fixed distribution of degrees and marked
points.

The map $\Phi:\bigcup M_{1,k_{\alpha},d_{\alpha}}\to D$ gives $D$ the
structure of a divisor with normal crossings, so the structure sheaf of $D$
can be obtained from the structure sheaf of each $M_{1,\dots}$ via the
inclusion-exclusion principle. However, the intersection of any two depth-1
strata is a unique depth-2 stratum, etc. So if $\mathcal{O}_{m}$ denotes the
virtual structure sheaf of the union of the depth $m$ strata, we have: The
intersection of the loci of depth 1 is the union of the loci of depth 2, etc.
So we can calculate the structure sheaf $\mathcal{O}_{D}$ in terms of these by
the inclusion-exclusion principle. This is a (virtual) normal crossing
divisor, so if we call $D_{m}$ the set of curve configurations of depth $m$,
we have:

$\mathcal{O}_{D}=\sum_{m}(-1)^{m}\sum_{C\in D_{m}}\mathcal{O}_{C}$

Since the map $ft$ is (virtually) traverse to the inclusion of $D$, we can
calculate $\text{ft}^{k}_{*}I(\dots)$ by first restricting to $\bigcup_{C\in
D_{1}}C$, and then pushing forward.

This means for untwisted invariants, we obtain:

$\begin{split}&\Phi^{*}ft^{k}_{*}H_{g,n+k,d}^{X}(\gamma_{1},\dots,\gamma_{n},t\dots,t)=\\\
&\sum_{\alpha,\beta,k_{1}+k_{2}=k}I_{g_{1},n_{1}+k_{1}+1,d_{1}}(\gamma_{1},\dots,\gamma_{n},t\dots,\phi_{\alpha})g^{\alpha\beta}H_{0,k_{2}+2,d_{2}}(\phi_{\beta},t,\dots,\phi_{k})g^{kl}H_{g_{2},n_{2}+k_{2}+1,d_{2}}-\\\
&\sum_{\alpha,\beta,\nu,\mu,d_{i},k_{i}}H_{g_{1},n_{1}+k_{1}+1,d_{1}}(\gamma_{1},\dots,\gamma_{n},t\dots,\phi_{\alpha})g^{\alpha\beta}H_{0,k_{2}+2,d_{2}}(\phi_{\beta},t,\dots,\phi_{\nu})g^{\nu\mu}H_{g_{2},n_{2}+k_{2}+1,d_{2}}(\phi_{\mu}\gamma_{1},\dots,\gamma_{n},t\dots)\\\
+&\dots\end{split}$ (3.1)

A shorthand way of writing this is using the quantum pairing. Since the matrix
inverse of $1+F$ is $1-F+F^{2}+\dots$, since
$G_{\alpha\beta}=g_{\alpha\beta}+F_{\alpha\beta}$ be the matrix of the quantum
pairing, we can sum the above expression over $k$ to yield:

$\langle t,\dots,t\rangle_{0,n,d}^{\Phi}=\langle
t,\dots,\phi^{\alpha}\rangle_{n_{1}+1,g_{1},d_{1}}G^{\alpha\beta}\langle\phi^{\beta},t,\dots,\rangle_{n_{2}+1,g_{2},d_{2}}$
(3.2)

The twisted version works essentially the same way. Using the restriction
theorem for twisting class cited earlier, the same formula is true, replacing
all correlators by their twisted counterparts. The difference is that, by
Theorem 3.1, there is a factor of $\frac{1}{C(V)}$ for each node in the
stratum. We can remove this factor by using the twisted Poincare pairing
rather than the untwisted one, since the extra factor of $C(V)$ is cancelled
by the $\frac{1}{C(V)}$ appearing at each node, yielding the desired result
of:

$\langle t,\dots,t\rangle_{0,n,d}^{tw,\Phi}=\langle
t,\dots,\phi^{\alpha}\rangle_{n_{1}+1,g_{1},d_{1}}^{tw}G^{\alpha\beta,tw}\langle\phi^{\beta},t,\dots,\rangle_{n_{2}+1,g_{2},d_{2}}^{tw}$

The proof of the genus reduction axiom is identical.

###  3.2 String, Dilaton, and WDVV Equations

We also have the following additional properties:

######  Theorem 3.4.

  * •

String Equation: If $\mathcal{H}$ denotes the Hodge bundle, then (for $g>1$:

$ft_{*}\mathcal{O}^{tw}_{g,n,d}(\frac{1}{1-qH}\prod_{i}\frac{1}{1-q_{i}L_{i}})=*\mathcal{O}^{tw}_{g,n,d}(\frac{1}{1-qH}(1+H^{*}+\sum_{i}\frac{q_{i}}{1-q_{i}}\prod_{i}\frac{1}{1-q_{i}L_{i}}$

For $g=0$. the same equation is true with $H^{*}$ set to 0, which leads to the
following, perhaps more familiar, expression in terms of correlators:

$\langle a_{1}(L),\dots,a_{n}(L),1\rangle_{0,n+1,d}^{tw}=\sum_{i}\langle
a_{1}(L),\dots,\frac{a_{i}(L)-a_{i}(1)}{L-1},a_{n}(L)\rangle_{0,n,d}^{tw}$

  * •

Dilaton Equation: With the same setup as for the string equation:

$ft_{*}\mathcal{O}^{tw}_{g,n,d}(\frac{1}{1-qH}\prod_{i}\frac{1}{1-q_{i}L_{i}})L_{n+1}=*\mathcal{O}^{tw}_{g,n,d}(\frac{1}{1-qH}(1+H+\sum_{i}\frac{q_{i}}{1-q_{i}}\prod_{i}\frac{1}{1-q_{i}L_{i}}$

As consequence $ft_{*}(1+L_{n+1})=H+H^{*}+n-2$.

In genus 0, this yields the following relation among correlators:

$\langle a_{1},\dots,a_{n},L_{n+1}-1\rangle_{g,n+1,d}=(n-2)\langle
a_{1},\dots,a_{n}\rangle_{0,n,d}$

Both of these proofs are identical to the ones in [16].

######  Theorem 3.5 (WDVV Identity for $K$-Theory).

$\sum_{d_{1},d_{2},n_{1}.n_{2},\alpha,\beta}\langle
A,B,t,\dots,\phi_{\alpha}\rangle_{0,n_{1}+3,d_{1}}g^{\alpha\beta}\langle
C,D,t,\dots,\phi_{\beta}\rangle_{0,n_{2}+3,d_{2}}-$ $\sum\langle
A,B,t,\dots,\phi_{\alpha}\rangle_{0,n_{1}+3,d_{1}}g^{\alpha\beta}\langle\phi_{\beta},t,\dots,\phi_{\gamma}\rangle_{0,n_{2}+2,d+2}g^{\gamma\delta}\langle
C,D,t,\dots,\phi_{\delta}\rangle_{0,n_{3}+3,d_{3}}+\sum\dots$

is symmetric in $A,B,C,D$.

There is a contraction map
$ct:X_{0,n+4,d}\to\overline{M}_{0,4}\cong\mathbb{P}^{1}$, which forgets the
map to $X$, all marked points but the first 4, and stabilizes the resulting
curve by contracting unstable components. The points
$0,1,\infty\in\mathbb{P}^{1}$ are the boundary points of $\overline{M}_{0,4}$,
parametrized by cross-ratio. They correspond to 2-component curves where the
4th marked point shares a component with the first, second, and third marked
points respectively. Using the splitting axiom, the expression above is
equivalent to restricting to a boundary point of $\overline{M}_{0,4}$, and
calculating the resulting Gromov-Witten invariant. Permuting $A,B,C,D$ chooses
a different boundary point.

##  4 Twisted Quantum Products

For a compact Kahler $X$, the quantum $K$-ring of $X$ is a deformation of the
ordinary $K$-ring, using genus-0 $K$-theoretic Gromov-Witten invariants as
structure constants. The same construction can be applied to (type-I) twisted
invariants.

The quantum product is defined via structure constants, and is determined by
the formula:

$((\phi_{\alpha}*\phi_{\beta},\phi_{\gamma}))=G_{\alpha\beta\gamma}$

Solving for the quantum product gives:

$\phi_{\alpha}*\phi_{\beta}=\sum_{\gamma}G_{\alpha\beta\gamma}\sum_{\delta}G^{\delta\gamma}\phi_{\delta}$

We now verify some basic facts about the quantum product:

######  Theorem 4.1.

The quantum product is associative.

###### Proof.

This is identical to the proof in the untwisted case, but we recount it for
the sake of completeness.

We can write the coefficients $G^{\alpha\beta}$ explicitly in the following
way. Call the matrices with coefficients $G_{\alpha\beta},F_{\alpha\beta}$
$\mathcal{G}$ and $\mathcal{F}$ respectively:

We have: $\mathcal{G}=g+\mathcal{F}$, where $g$ is the matrix of the twisted
Poincare pairing.

Using the fact that $g$ is invertible and $(1+M)^{-1}=1-M+M^{2}-\dots$, as in
the proof of the splitting axiom, we have:

$\mathcal{G}^{-1}=(1-g^{-1}\mathcal{F}+g^{-1}\mathcal{F}g^{-1}\mathcal{F}+\dots)g^{-1}$

So a particular matrix coefficient looks like:

$G^{\alpha\beta}=g^{\alpha\beta}-\sum_{\delta,\gamma}g^{\alpha\gamma}F_{\gamma\delta}g^{\delta\beta}+\dots$

The statement of associativity is equivalent to the symmetry in
$\alpha,\beta,\gamma,\delta$ of:

$\sum_{\mu,\nu}G_{\alpha\beta\mu}G^{\mu\nu}G_{\nu\gamma\delta}$

Using the expression for $G^{\mu\nu}$ calculated previously, and the fact that
third derivatives of $G$ agrees with those of $F$. we can rewrite this as:

$\sum_{\mu\nu}F_{\alpha\beta\mu}g^{\mu\nu}F_{\nu\gamma\delta}-F_{\alpha\beta\mu}\sum_{\mu,\nu,\zeta,\eta}g^{\mu\zeta}F_{\zeta\eta}g^{\eta\nu}F_{\nu\gamma\delta}+\dots$

The symnmetry of this expression is precisely the generating function version
of the WDVV identity for twisted $K$-theoretic invariants.

∎

######  Theorem 4.2.

The quantum product has unit 1

###### Proof.

This is equivalent to the statement

$G_{\alpha,\beta,1}=G_{\alpha,\beta}$

The left hand side is equal to:

$\sum_{n,d}\frac{1}{n!}Q^{d}\langle\phi_{\alpha},\phi_{\beta},1,t,\dots,t\rangle_{0,n+3,d}^{tw}$

The right hand side is:

$g_{\alpha\beta}+\sum_{n,d}\frac{1}{n!}Q^{d}\langle\phi_{\alpha},\phi_{\beta},t,\dots,t\rangle_{0,n+2,d}^{tw}$

Using the string equation, we can equate all but one term on the left hand
side with a term on the right hand side by observing that:

$\frac{1}{n!}Q^{d}\langle\phi_{\alpha},\phi_{\beta},1,t,\dots,t\rangle_{0,n+3,d}^{tw}=\frac{1}{n!}Q^{d}\langle\phi_{\alpha},\phi_{\beta},t,\dots,t\rangle_{0,n+2,d}^{tw}$

This handles all terms except
$\langle\phi_{\alpha},\phi_{\beta},1\rangle_{0,3,0}$, since there is no
forgetful map to apply here. However $X_{0,3,0}$ is just $X$, so this
correlator is equal to $(\phi_{\alpha},\phi_{\beta})=g_{\alpha\beta}$,
identifying it with the constant term (in both $t$ and $Q$) on the right hand
side. ∎

##  5 $q$-difference Modules

Having established some basic properties of twisted Gromov-Witten invariants,
we now illustrate the connection with $q-$difference equations, which will be
the main tool we use to compute ring relations. Many of the results here have
proofs that are completely identical to their untwisted counterparts, so we
only include proofs that are particularly illustrative, or are different in
the twisted case.

###  5.1 Quantum Connection

As in the untwisted case, and for the exact same reasons, quantum $K$-theory
determines a Frobenius structure on $K(X)$

######  Theorem 5.1.

$K(X)\otimes\mathbb{C}$, equipped with the metric $G$ and the quantum product
$\partial_{ijk}G$, has the structure of a Frobenius manifold.

In particular, the Dubrovin connection:

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

is flat

###### Proof.

This is completely a formal consequence of the WDVV and string equations, see
[16] for details. ∎

######  Theorem 5.2.

The twisted quantum differential equation:

$\nabla_{q}S=0$

has fundamental solution matrix $S\in End(K(X))[q,Q]$ given by:

$S_{ij}=g_{ij}+\sum_{n,d}\frac{Q^{d}}{n!}\langle\phi_{i},t,\dots,t,\frac{\phi_{j}}{1-qL}\rangle^{tw}_{0,n,d}$

###### Proof.

This is also a completely formal consequence of WDVV and string equations,
just refer to [16].

∎

######  Theorem 5.3.

Regarding $S$ as an endomorphism valued function in $End(K^{*}(X))(q)[[Q,t]]$,
it is invertible, and $S^{-1}$ is the adjoint of $S$ with respect to the
twisted pairing, and is a fundamental solution to:

$(1-q)\partial_{\alpha}T=T(\phi_{\alpha}*_{Q})$

######  Remark.

By Kawasaki’s Riemann-Roch Theorem, each entry of $S$ is a rational function
of $q$ with poles at roots of unity and vanishes at $\infty$.

######  Definition 5.4.

The twisted big $J$-function, denoted $\mathcal{J}^{tw}(q,t)$ is defined by:

$\mathcal{J}^{tw}=(1-q)S^{-1}\mathbf{1}$

Actually expanding this yields:

$\mathcal{J}^{tw}=(1-q)+t+\sum_{d}\sum_{n}\frac{Q^{d}}{n!}\langle
t,\dots,t,\frac{\phi^{\alpha}}{1-qL}\rangle_{0,n+1,d}\phi_{\alpha}$

The value at $t=0$, denoted $J^{tw}$ is the twisted small $J$-function.

###  5.2 The $J$-function and Symplectic Loop Space

The proper setting to think about generating functions for quantum
$K$-theoretic invariants is within the context of Givental’s symplectic loop
space $\mathcal{K}$. We give a brief overview of the theory below:

The symplectic loop space $\mathcal{K}$ is a $K^{*}(X)[Q]-$module consisting
of ($Q$-adic completions of) rational functions in $q$ with coefficients in
$K^{*}(X)$, with poles allowed only at $0,\infty,$ or roots of unity. It has a
symplectic form given by:

$\Omega(f,g)=-\text{Res}_{0,\infty}(f(q),g(q^{-1}))\frac{dq}{q}$

Here $\text{Res}_{S}$ denotes the sum of the residues at values in the set
$S$, and $(,)$ denotes the $K$-theoretic Poincare pairing.

$\mathcal{K}$ is also equipped with a polarization into two Lagrangian
subspaces:

  * •

$\mathcal{K}^{+}:=\text{Laurent polynomials in $q$}$

  * •

$\mathcal{K}^{-}:=\\{f|f(\infty)=0,f(0)\neq\infty\\}$

As a consequence of the Kawasaki-Riemann-Roch theorems for stacks the
$Q^{d>0}$ terms of the big $J$-function lie in $\mathcal{K}^{-}$, its
projection to $\mathcal{K}^{+}$ is $1-q+t$.

######  Theorem 5.5 (Givental-Tonita).

The range of the big $J$-function is an overruled Lagrangian cone (denoted
$\mathcal{L}$) in $\mathcal{K}$. This means it is a cone that has the
following properties:

  * •

Tangent spaces are invariant with respect to multiplication by $1-q$

  * •

For $T=T_{p}\mathcal{L}$, $(1-q)T\subset\mathcal{L}$

  * •

The tangent spaces for all $p\in(1-q)T$ are equal to $T$.

###  5.3 Relations and Shift Operators

We now make the assumption that we have a basis $p_{i}$ of $H_{2}(X)$
corresponding to the Novikov variables $Q_{i}$, and there exist a set of line
bundles $P_{i}$ with $c_{1}(P_{i})=-p_{i}$.

$\mathcal{L}$ has the structure of a $q$-difference module in the following
way:

######  Theorem 5.6 (Givental-Tonita).

Let $p_{i}$ be some basis for $H^{2}(X)$ consisting of first Chern classes of
line bundles $P_{i}$. $\mathcal{L}$ and each of its tangent spaces is
invariant under the algebra of finite difference operators in $Q_{i}$
generated by $P_{i},Q^{d}$ (for $d$ in the Mori cone of $X$), where $P_{i}$
acts as $P_{i}q^{Q_{i}\partial_{Q_{i}}}$

Givental showed in [7] that essentially the same things holds in the context
of twisted invariants of type I, the graph of $\mathcal{J}^{tw}$ is a
Lagrangian cone in the space $\mathcal{K}^{tw}$ defined the same way as
$\mathcal{K}$ but using the twisted Poincare pairing, and the ruling spaces
have the same difference module structure.

######  Theorem 5.7.

$S^{-1}(K^{*}(X)[q^{\pm},Q,t])$ is a tangent space to the twisted Lagrangian
cone $\mathcal{L}^{tw}$

This is a twisted version of a corresponding theorem of Givental-Tonita and
the proof is the same, it is a direct consequence of the string and dilaton
equations.

By the invariance property, $S^{-1}(K^{*}(X)[q^{\pm},Q,t])$ is thus invariant
under action of $P_{i}q^{\partial Q_{i}}$.

This means the operator $A_{i}:=S^{-1}P_{i}q^{\partial Q_{i}}S$ is a
polynomial in $q$, and lies in $End(K(X))[q,Q,t]$.

We define the $q$-shift operator $\mathcal{A}_{i}$ as
$A_{i}q^{\partial_{Q_{i}}}$, it satisfies the following properties (again the
proofs are the same as the proofs of the untwisted versions):

  * •

$S\circ P_{i}q^{\partial_{Q_{i}}}=\mathcal{A}_{i}\circ S$

  * •

$P_{i}q^{\partial_{Q_{i}}}\circ T=T\circ\mathcal{A}_{i}$

  * •

$[\nabla_{q,\alpha},\nabla_{q,\beta}]=[\mathcal{A}_{i},\mathcal{A}_{j}]=[\nabla_{q,\alpha},A_{i}]=0$

  * •

Given $s_{1},s_{2}\in\mathcal{K}^{+}$:

  * •

$\partial_{\alpha}G(\overline{s_{1}},s_{2})=G(\overline{\frac{\nabla_{q,\alpha}}{1-q}s_{1}},s_{2})+G(\overline{s_{1}},frac{\nabla_{q,\alpha}}{1-q}s_{2})$

  * •

$q^{Q_{i}\partial_{Q_{i}}}G(\overline{s_{1}},s_{2})=G(\overline{\mathcal{A_{i}}^{-1}s_{1}},\mathcal{A}_{i}s_{2})$

Proofs: Entirely formal consequences of string and divisor equations and the
quantum differential equation for $S$. As a result, we have the following
theorem:

######  Theorem 5.8.

  * •

$(1-q)\partial_{\alpha}A_{i}=A_{i}(q^{Q_{i}\partial_{Q_{i}}}\phi_{\alpha}*_{Q})-(\phi_{\alpha}*_{Q})A_{i}$

  * •

$G(\phi_{\alpha},A_{i}\phi_{\beta})=q^{Q_{i}\partial_{Q_{i}}}G(\overline{A_{i}}\phi_{\alpha},\phi_{\beta})$

  * •

Let $A_{i,com}$ denote $A_{i}|_{q=1}$, then $A_{i,com}$ commutes with quantum
multiplication.

  * •

$A_{i}=P_{i}*_{Q}+\sum_{d|d_{i}>0}\sum_{d=0}^{d_{i}-1}Q^{d}A_{i,d,k}(t)(1-q)^{k}Q^{d}$
Where $A_{i,d,k}(t)$ does not depend on $t_{0}$. All expansions here are in
$End(K(X))\otimes Q[q^{\pm}][[Q,t]]$.

It is sometimes more convenient to work not with $S$ and $\mathcal{J}^{tw}$
but with their logarithmic-modified counterparts $\tilde{S}$ and
$\tilde{\mathcal{J}}^{tw}$, obtained by replacing $S$ with
$S(\prod_{i}P_{i}^{\frac{\ln(Q_{i})}{\ln(q)}})$. (This results in left
multiplying $J$ by $\prod_{i}P_{i}^{-\frac{\ln(Q_{i})}{\ln(q)}}$). This
expression is defined by rewriting $P_{i}$ as $(1-(1-P_{i}))$ and using the
binomial expansion. The result converges as a power series in $Q_{i}$ since
$(1-P_{i})$ is nilpotent.

The purpose of this is to remove the factors of $P_{i}$ from theorems about
$\mathcal{A}_{i}$, due to the observation:

$q^{Q_{i}\partial_{Q_{i}}}(\prod_{i}P_{i}^{\frac{\ln(Q_{i})}{\ln(q)}})=P_{i}(\prod_{i}P_{i}^{\frac{\ln(Q_{i})}{\ln(q)}})$

Applying this to the properties written earlier yields:

######  Theorem 5.9.

  * •

$\tilde{S}\circ q^{Q_{i}\partial_{Q_{i}}}=\mathcal{A}_{i}\circ\tilde{S}$

  * •

$q^{Q_{i}\partial_{Q_{i}}}\tilde{S}^{-1}=\tilde{S}^{-1}\circ\mathcal{A}_{i}$

Combining 5.2 and 5.3 yield a twisted version of Iritani-Milanov-Tonita’s
theorem on relations in big quantum $K$-theory:

######  Theorem 5.10.

Given some polynomial difference/differential operator
$D(q,Q,q^{Q_{i}\partial_{Q_{i}}},\partial_{t_{\alpha}})$, we have:

$D\tilde{\mathcal{J}}=0\text{ implies that the following relation holds in the
big quantum $K$-ring:}$

$D(1,Q,A_{i,com},\phi_{\alpha}*_{Q})=0\in QK_{big}^{*}(X)$

###### Proof.

We apply $D$ to $\tilde{J}^{tw}=(1-q)\tilde{S}^{-1}\mathbf{1}$:

If $D=(1-q)\partial_{\alpha}$, then
$D\tilde{J}^{tw}=(1-q)\tilde{S}^{-1})(\phi_{\alpha})$.

If $D=q^{Q_{i}\partial_{Q_{i}}}$, then
$D\tilde{J}^{tw}=(1-q)\tilde{S}^{-1}(\mathcal{A}_{i}(\mathbf{1}))$.

Since the two kinds of operators commute, for a general $D$ if
$D\mathcal{J}^{tw}=0$, then
$\tilde{S}^{-1}D(q,Q,\mathcal{A}_{i},\phi_{\alpha}*_{Q})\mathbf{1}=0$

Multiplying both sides by $\tilde{S}^{-1}$ and setting $q=1$ gives the desired
result.

∎

The same theorem holds for the small $J$-function.

######  Theorem 5.11.

Given some polynomial difference operator $D(q,Q,q^{Q_{i}\partial_{Q_{i}}})$,
we have:

$DJ^{tw}=0\text{ implies that the following relation holds in the small
quantum $K$-ring:}$

$D(1,Q,A_{i,com}|_{t=0})=0\in QK^{*}(X)$

If we replace $J^{tw}$ with $\mathcal{J}^{tw}|_{t=t_{0}}$, the same theorem
holds for twisted quantum $K$-theory with bulk shift $t_{0}$.

###### Proof.

We note that this is not strictly a corollary of the previous theorem but
rather a consequence of its proof. We can set $t=0$ (or some other value) in
each line of the proof and it remains true. (We remark that $\tilde{S}_{t=0}$
is invertible since we can specialize $\tilde{S}\tilde{S}^{-1}=I$ to $t=0$,
and the RHS does not depend on $t$).

∎

###  5.4 Calculating the Shift Operators

In practice, applying the theorem above only yields useful relations when the
$\mathcal{A}_{i}s$ (or their limits $A_{i}$) can be calculated directly. As a
consequence of the theorems of Iritani, Milanov, and Tonita (which all
generalize to this setting), there is a recursive procedure for calculating
them, given the small $J$-function, if the $K$-ring is generated by divisors.

However, the following approach, used in the case of $G/P$ by Anderson-Chen-
Tseng-Iritani[1] is often more practical:

######  Theorem 5.12 (Quantum Triviality Theorem).

Given some polynomial $f(P_{i})$, if $\frac{1}{1-q}J^{tw}$ has the property
that all terms of positive $Q$-degree in $f(q^{Q_{i}\partial_{Q_{i}}})J$
vanish at $q=\infty$, then $f(A_{i,com})|_{t=0}$ acts as quantum
multiplication by $f(P_{i})$ in the small twisted quantum $K$-ring of $X$.

###### Proof.

We make the simplification that $f=P_{i}$ here to save notation, but the proof
is essentially identical in the case of nontrivial $f$.

$P_{i}q^{Q_{i}\partial_{Q_{i}}}T(1_{X})=T(\mathcal{A}_{i}q^{Q_{i}\partial_{Q_{i}}}(1_{X}))$

The left hand side is
$P_{i}q^{Q_{i}\partial_{Q_{i}}}\frac{1}{1-q}\mathcal{J}^{tw}_{X}$. We first
restrict to $t=0$, then expand at $q=\infty$, since $T=I+\text{terms vanishing
at $q=\infty$}$, the expansion becomes:

$P_{i}q^{Q_{i}\partial_{Q_{i}}}\frac{1}{1-q}J^{tw}_{X}=c_{0}+\sum_{d}c_{d}Q^{d}+\dots$

For some terms $c_{d}$ given by the expansion of $A_{i}$ at $q=1,t=0$, with
$c_{0}$ being the operator of (small) quantum multiplication by $P_{i}$.

Clearly, if the left-hand side has no $Q^{>0}$ terms that do not vanish at
infinity, then neither does the right hand side, hence $c_{d}=0$ for all
$d\neq 0$.

∎

What this means is that provided we know the small $J$-function for some
theory, provided the $q-$degrees satisfy the above conditions, we can convert
difference operators annihilating the $J$-function to relations without having
to solve the quantum Lax equation.

In practice, finding these operators is often not difficult. However, if we
wish to compute a presentation for a (twisted) quantum $K$-ring, we need some
way of telling when we have found enough relations.

######  Proposition 5.13.

If we know a set $I$ of relations holds in $QK^{tw}(X)$, and $I|_{Q=0}$ form a
complete set of relations for $K^{*}(X)$, then $I$ form a complete set of
relations for $QK^{tw}(X)$.

###### Proof.

Since $WK^{tw}(X)$ is complete with respect to $Q$, this follows from a
general result in commutative algebra, Proposition A.3 in [11].

∎

We note that we should only expect to get a full set of relations when $P_{i}$
generate $K^{*}(X)$, as $q-$difference operators can only give us relations in
terms of functions of $P_{i}$.

In addition, finding ring relations using these methods requires knowing the
small $J$-function, the next section gives some details on cases where it can
be calculated.

###  5.5 $S_{n}$-invariant Theory

In a series of papers, Givental introduced a generalization of $K$-theoretic
Gromov-Witten invariants that involve the $S_{n}$-action on the marked points.
Generating functions for such invariants generally behave better with respect
to twisting and fixed-point localization.

Theorems about cohomological Gromov-Witten invariants often find natural
generalizations in this setting, rather than in the ”ordinary” quantum
$K$-theory we have considered so far.

The main way this affects the current topic of this paper is that, for many
spaces, it is difficult to find values for the (twisted or untwisted) big
$J$-function. However, it is much easier if we instead use the symmetrized big
$J$ function, denoted $\mathcal{J}^{sym}$, defined in the following way:

$\mathcal{J}^{sym}=(1-q)+t+\sum_{d}\sum_{n}\frac{Q^{d}}{n!}\langle
t,\dots,t,\frac{\phi^{\alpha}}{1-qL}\rangle_{0,n+1,d}^{sym}\phi_{\alpha}$

The correlator $\langle\rangle^{sym}$ means we take pushforwards from the
spaces $X_{0,n+1,d}/S_{n}$. These invariants are a special case of the
permutation-equivariant $K$-theoretic Gromov-Witten invariants introduced in
[5]. The important thing for us is that at $t=0$, the $S_{n}$-action is
irrelevant and we recover the usual small $J$-function. Twistings are defined
for this theory in exactly the same way.

We do not consider any ring structure involving the $S_{n}$-invariant
correlators in this paper.

For the examples of twistings we have covered, we can construct values of
$\mathcal{J}^{sym}$:

######  Theorem 5.14.

Let $V$ be a vector bundle that splits into a sum of $r$ line bundles $V_{i}$.
Let $J:=\sum_{d}Q_{d}J_{d}$. If $(1-q)J$ is a value of $\mathcal{J}^{sym}$,
then:

  * •

$(1-q)\sum_{d}Q^{d}J_{d}\prod_{i=1}^{r}\frac{\prod_{\ell\leq(c_{1}(V_{i}),d)}(1-V_{i}^{-1}q^{\ell})}{prod_{\ell\leq
0}(1-\lambda V_{i}^{-1}q^{\ell})}$ is a value of the symmetrized big
$J$-function of the theory twisted by $Eu_{\lambda}(V)$.

  * •

$(1-q)\sum_{d}Q^{d}J_{d}\prod_{i=1}^{r}\frac{\prod_{\ell<(c_{1}(V_{i}),d)}(1-\lambda
V_{i}q^{\ell})}{prod_{\ell<0}(1-\lambda V_{i}^{-1}q^{\ell})}$ is a value of
the symmetrized big $J$-function of the theory twisted by
$\frac{1}{Eu_{\lambda}(V^{*})}$.

  * •

$(1-q)\sum_{d}Q^{d}J_{d}\prod_{i}(V_{i}^{-(d_{i},c_{1}(V_{i}))}q^{\binom{(d_{i},c_{1}(V_{i})+1)}{2}})^{\ell}$
is a value of the symmetrized big $J$-function of the theory with level
structure given by $V,\ell$.

If these values turn out to be the small $J$-function, which can be tested by
varifying that their projection to $K_{+}$ is equal to $1-q$, then we can
calculate relations in the associated twisted quantum $K$-rings via finding
appropriate $q-$difference operators.

######  Remark.

There is a ring structure involving symmetrized invariants (or more general
$S_{n}-$equivariant ones), but we do not discuss it in this work.

##  6 Reconstruction for Twisted Theories

Another fact about genus-$0$ twisted quantum $K$-theoretic invariants is that,
in nice situations, they are completely determined by the invariants for
$n=1$. In particular, if $K^{*}(X)$ is generated by line bundles. This can be
seen at the level of Lagrangian cones from twisted versions of results in [14]
and [6] (the proofs are identical to the ones in the untwisted case). However,
for use in the following section, we will use the result in terms of
correlators due to Lee-Pandharipande [17].

######  Lemma 6.1 (Lee-Pandharipande).

The following relations hold in $Pic(X_{0,n,d})$, for $\phi$ a line bundle on
$X$:

$ev_{i}^{*}(\phi)=ev_{j}^{*}(\phi)+\langle d,\phi\rangle
L_{j}-\sum_{d_{1}+d_{2}=d}\langle d_{1},\phi\rangle D_{i,d_{1}|j,d_{2}}$ (6.1)

$L_{1}+L_{2}=D_{i|j}$ (6.2)

Noting that a relationship between divisors and line bundles in the Picard
group is equivalent to the same relation between their structure sheaves and
Euler classes in $K$-theory, the lemma has the following $K$-theoretic
interpretation:

######  Lemma 6.2.

The following relations hold in $K^{*}(X_{0,n,d})$, for $\phi$ a line bundle
on $X$:

$(1-ev_{i}^{*}(\phi)^{-1})\mathcal{O}^{vir}_{0,n,d}=(1-ev_{j}^{*}(\phi)^{-1}L_{j}^{-\langle
d,\phi\rangle})\mathcal{O}^{vir}_{0,n,d}-\sum_{d_{1}+d_{2}=d}\langle
d_{1},\phi\rangle\mathcal{O}^{vir}_{D_{i,d_{1}|j,d_{2}}}$ (6.3)

$1-L_{i}^{-1}L_{j}^{-1}=\mathcal{O}^{vir}_{D_{i|j}}$ (6.4)

By applying these relations, coupled with the splitting axiom of the virtual
structure sheaf, any $n$-pointed degree $d$ invariant whose inputs are written
in terms of line bundles on $X$ can be expressed in terms of invariants of
lower degree and/or fewer marked points. For details on how this
reconstruction looks concretely, see [17]

######  Lemma 6.3.

Given an arbitrary twisting of type 1, denote the modified virtual structure
sheaf by $\mathcal{O}^{vir,tw}$, then the following relations hold in
$K^{*}(X_{0,n,d}$, for $\phi$ a line bundle on $X$:

$(1-ev_{i}^{*}(\phi))\mathcal{O}^{vir,tw}_{0,n,d}=(1-ev_{j}^{*}(\phi)^{-1}L_{j}^{-\langle
d,\phi\rangle})\mathcal{O}^{vir,tw}_{0,n,d}-\sum_{d_{1}+d_{2}=d}ev_{node}^{*}C(V)\langle
d_{1},\phi\rangle\mathcal{O}^{vir,tw}_{D_{i,d_{1}|j,d_{2}}}$ (6.5)

$(1-L_{i}^{-1}L_{j}^{-1})\mathcal{O}^{vir,tw}_{0,n,d}=ev_{node}^{*}C(V)\mathcal{O}^{vir,tw}_{D_{i|j}}$
(6.6)

Both of these lemmas follow from their untwisted counterparts paired with the
restriction formula for twisting classes.

This allows us to prove the following theorem:

######  Theorem 6.4.

By applying the lemma sequentially, all twisted Gromov-Witten invariants can
be reconstructed from the one-pointed descendant invariants.

Proof: Choose a generating set $\phi_{i}$ of line bundles. We simoultaneously
induct on $d$, the $\phi_{i}$-degree of an input, and the absolute value of
the $L_{i}$-degree.

For our base cases, when the $\phi_{i}$ and $L_{i}$ degrees are 0, the input
is 1, and we can lower the number of marked points by virtue of the string
equation.

If the $i$th input is a monomial, it is $P(\phi_{i})L_{i}^{a}$ for $\phi_{i}$
some line bundle, we can use the first relation to write the correlator in
terms of one with a lower the $\phi_{i}$-degree of $P$, and ones with lower
degree and number of marked points.

If we have two different inputs with descendants with negative degrees, we can
use the second relation to remove one $L_{i}^{-1},L_{j}^{-1}$ from each of
them, and calculate the invariant via the splitting axiom.

For an input with positive $L_{i}$ degree, we can use a rearranged version of
the first relation to lower the degree.

######  Remark.

This procedure is often not practical to actually carry out in practice.

##  7 Quantum Abelian-Non/Abelian Correspondence

Our setting is the same as in the section discussing the classical
abelian/non-Abelian correspondence. Let $X,Y$ be $X//G$ and $X//T$
respectively. The correspondence relates invariants of $X$ to twisted
invariants of $Y$, with twisting given by $Eu_{\lambda}(V)$, where $V$ is
Martin’s bundle.

Let the small $J$-function of the twisted theory of $Y$ be denoted
$J_{Y}^{tw}$, let the corresponding Lagrangian cone be denoted
$\mathcal{L}_{Y}^{tw}$. We introduce the following morphisms:

  * •

Let $sp$ denote the usual specialization map $K^{*}(Y)^{W}\to K^{*}(X)$

  * •

Let $\nu_{Q}$ denote the specialization map on Novikov variables.

  * •

Let $sp_{Q}$ denote ${}^{\circ}nu_{Q}$, which maps $K^{*}(Y)[[Q]]$ to
$K*(X)[[Q]]$.

We will give 4 different conjectures that go under the name abelian/non-
abelian correspondence:

######  Conjecture 7.1.

$sp_{Q}$ maps a formal germ $N\subset\mathcal{L}^{tw}_{Y}$ to
$\mathcal{L}_{X}$.

######  Conjecture 7.2.

$sp_{Q}$ maps a formal germ $N\subset\mathcal{L}^{tw,sym}_{Y}$ to
$\mathcal{L}^{sym}_{X}$.

######  Conjecture 7.3.

$\lim_{\lambda\to 1}sp_{Q}(J_{Y}^{tw}=J_{X}$

######  Conjecture 7.4.

$\langle a_{1},\dots,a_{n}\rangle_{g,n,d}^{X}=lim_{\lambda\to
1}\sum_{d^{\prime}\mapsto
d}\langle\tilde{a}_{1},\tilde{a_{2}},\dots,\rangle^{tw,Y}_{g,n,d^{\prime}}$

######  Conjecture 7.5.

The map $\phi:x\mapsto lim_{y\mapsto 1}sp_{Q}(x)$ is a surjective ring
homomorphism from $QK^{tw}(Y)^{W}$ to $QK^{*}(X)$.

We will give some relationships between the conjectures below:

######  Theorem 7.6.

If $(0,J_{Y}^{tw})\in N$, then Conjecture 7.1 or Conjecture 7.2 implies
Conjecture 7.3

###### Proof.

Obvious ∎

######  Theorem 7.7.

Conjecture 7.4 implies Conjecture 7.5.

###### Proof.

First, we show that $sp$ (and $sp_{Q}$) respects the quantum pairing in the
following sense.

$\lim_{\lambda\to 1}\frac{1}{|W|}\nu_{Q}(((a,b))^{Y,tw})=((sp(a),sp(b)))^{X}$

The $Q^{d}$ coefficient of the left hand side is equal to:

$\lim_{\lambda\to 1}\frac{1}{|W|}\sum_{d^{\prime}\mapsto d}\langle
a,b\rangle_{0,2,d^{\prime}}^{tw}Q^{d}$

This equal to the RHS by Conjecture 7.4.

Next, we show that $\phi$ is a ring homomorphism. Let $a,b$ be Weyl-invariant
classes in $K^{*}(Y)$. For $\psi_{i}$ a basis in $K^{*}(Y)$, we have the
following:

$a\circ_{Y}b=\sum_{d\in H^{2}(X//T),i}\langle
a,b,\psi_{i}\rangle_{0,3,d}^{tw}\psi_{i}^{*}Q^{d}$

Observe that $W$ induces an action on $\coprod_{d^{\prime}\mapsto
d}Y_{g,n,d^{\prime}}$. Since $W$ acts on $Y$ by automorphisms, it induces
isomorphisms from $Y_{g,n,d}$ and $Y_{g,n,d^{w}}$, which preserve the twisting
class, we conclude that

$\langle a^{w},b^{w},c^{w}\rangle^{tw}_{0,3,d^{w}}=\langle
a,b,c\rangle^{tw}_{0,3,d},$

this means $a\circ b$ is $W$-invariant, so applying $sp_{Q}$ is possible.

For $t\in K^{*}(X)$, with a lift $\tilde{t}\in K^{*}(Y)$, we have:

$((\phi(a\circ_{Y}b),t))=\lim_{\lambda\to
1}((sp_{Q}(a\circ_{Y}b),t))=\lim_{\lambda\to
1}\frac{1}{|W|}\nu_{Q}(((a\circ_{Y}b,\tilde{t}))^{tw})=\lim_{\lambda\to
1}\sum_{d\mapsto d,d\in H_{2}(X)}Q^{d}\langle
a,b,\tilde{t}\rangle_{0,3,d^{\prime}}^{tw}=((\phi(a)\circ_{X}\phi(b),t))$

Thus $\phi(a\circ_{Y}b)=\phi(a)\circ_{X}\phi(b)$. surjectivity of $\phi$ is
obvious. Since $sp$ is a map of $Q$-modules (if we interpret $Q$ as the
Novikov variables of $X//T$), the kernel of $sp$ is the ideal generated by the
kernel of $sp$ on $K^{*}(X)$, which is generated by $Eu(V^{M})$ and kernel of
$sp$ on the Novikov variables. ∎

######  Theorem 7.8.

Conjecture 7.3 implies Conjecture 7.4 for $n=1$

###### Proof.

The correspondence for descendent invariants is equivalent to the following
statement for all $t\in K^{*}(X)$:

$\langle\frac{t}{1-q}\rangle_{0,1,d}=\frac{1}{|W|}\sum_{d^{\prime}\mapsto
d}\langle\frac{\tilde{t}}{1-q}\rangle_{0,1,d^{\prime}}^{tw}$

The left hand side is $(J_{X},t)$, the right-hand side is $lim_{y\to
1}sp_{Q}(J_{Y}^{tw},\tilde{t})$, these are equal by the hypothesis, and the
classical abelian/non-abelian correspondence.

∎

######  Theorem 7.9.

If $K^{*}(X)$ and $K^{*}(Y)$ are generated by line bundles, then Conjecture
Conjecture 7.3 implies Conjecture Conjecture 7.4 for all values of $n$.

###### Proof.

This is a consequence of the twisted reconstruction theorem. We induct on
$n,d$ and the degree of any line bundles appearing in the inputs, as well as
the degree of descendants.

Beginning with $\sum_{d^{\prime}\mapsto
d}\langle\tilde{a_{1}},\dots\rangle_{0,n,d}$ we can use the reconstruction
relations to reduce the degrees of the inputs, $n$, and $d$, and apply the
induction hypothesis.

The nontrivial thing to check is applying the 3rd relation and using the
splitting axiom. What we need to prove is that (assuming the induction
hypothesis):

$\langle a_{1},\dots,\rangle_{0,n,d}^{\Phi}=\sum_{d^{\prime}\mapsto
d}\langle\tilde{a}_{1},\dots,\rangle_{0,n,d^{\prime}}^{tw,\Phi}$

Where $\Phi$ is the contraction map appearing in the proof of the splitting
axiom.

$\sum_{d^{\prime}\mapsto
d,n_{1}+n_{2}=n,d_{1}+d_{2}=d^{\prime}}\sum_{i,j}\langle\tilde{a}_{1},\dots,\phi_{i}\rangle_{0,n_{1}+1,d_{1}}g^{ij}\langle\phi_{j},\tilde{a}_{n_{1}+1},\dots\rangle_{0,n_{2}+1,d_{2}}$

Similarly, the left hand side is:

$\sum_{i,j,d_{1}+d_{2}=d,n_{1}+n_{2}=n}\langle
a_{1},\dots,a_{n_{1}},\phi_{i}\rangle_{0,n_{1}+1,d_{1}}g^{ij}\langle
a_{1},\dots,a_{n_{1}},\phi_{j}\rangle_{0,n_{2}+1,d_{2}}.$

First we note that if $d_{1}+d_{2}=d^{\prime}$, then $sp(d_{1})+sp(d_{2})=d$,
hence for any $w_{1},w_{2}\in W$, $sp(d_{1}^{w_{1}})+sp(d_{2}^{w_{2}})=d$. So
we can rewrite the right hand side as:

$\sum_{n_{1}+n_{2}=n,d_{1}+d_{2}=d}\sum_{i,j}(\sum_{d_{1}^{\prime}\mapsto
d_{1}}\langle\tilde{a}_{1},\dots,\phi_{i}\rangle_{0,n_{1}+1,d_{1}^{\prime}})g^{ij}(\sum_{d_{2}^{\prime}\mapsto
d_{2}}\langle\phi_{j},\tilde{a}_{n_{1}+1},\dots\rangle_{0,n_{2}+1,d_{2}^{\prime}})$

We can write the LHS as $T_{1}(\sum g_{ij}\phi_{i}\otimes\phi_{j}))$, where

$T_{1}=\sum_{d_{1}+d_{2}=d,n_{1}+n_{2}=n}\langle
a_{1},\dots,\cdot\rangle_{0,n_{1}+1,d_{1}}\otimes\langle\cdot,a_{n_{1}+1},\dots\rangle_{0,n_{2}+1,d_{2}}.$

Similarly we can write the left hand side as
$T_{2}(\sum_{i,j}g^{ij,tw}\psi_{i}\otimes\psi_{j})$, where $T_{2}$ is the
tensor

$T_{2}:=\frac{1}{|W|}\sum_{n_{1}+n_{2}=n,d_{1}+d_{2}=d}(\sum_{d_{1}^{\prime}\mapsto
d_{1}}\langle
a_{1},\dots,\cdot\rangle_{0,n_{1}+1,d_{1}^{\prime}})\otimes(\sum_{d_{2}^{\prime}\mapsto
d_{2}}\langle\cdot,a_{n_{1}+1},\dots\rangle_{0,n_{2}+1,d_{2}^{\prime}})$

The induction hypothesis allows us to relate the tensors $T_{1}$ and $T_{2}$,
as linear functions on $K^{*}(X)\otimes K^{*}(X)$ and $(K^{*}(Y)\otimes
K^{*}(Y))^{W}$ respectively. Applying it yields:

$sp^{*}T_{1}=\frac{1}{|W|}\lim_{\lambda\to 1}T_{2}$

The class $\sum_{i,j}g^{ij,tw}\psi_{i}\otimes\psi_{j}$ is the Poincare dual to
the diagonal in $X\times X$. The class
$\sum_{i,j}g^{ij,tw}\psi_{i}\otimes\psi_{j}$ is the twisted Poincare dual to
the diagonal in $Y\times Y$.

Since $sp$ identifies the structure sheaves of the diagonals of $X$ and $Y$,
we have: $\lim_{\lambda
to1}PD^{tw}(\Delta_{Y})=\frac{1}{|W|}sp^{*}PD(\Delta_{X})$. The factor
$\frac{1}{|W|}$ comes from the difference between the Poincare pairings. Thus
the two quantities are equal, as desired. ∎

######  Remark.

The conditions fails for $X$ a Grassmanian or flag variety, but we remedy this
in the following way: We work equivariantly with respect to the usual torus
action, then after localizating at the ideal generated by $1-\Lambda_{i}$,
$K^{*}(X)$ is generated by line bundles, and the argument above goes through.

###  7.1 Twistings

A $G$ representation $E$ induces a vector bundles $E_{X}$ and $E_{Y}$ on $X$
and $Y$ respectively. Their classes in $K$-theory are related by the classical
abelian/non-abelian correspondence.

One can make the exact same conjectures relating invariants of $X$ twisted by
$C(E_{X})$ to twisted invariants of $Y$, further twisted by $C(E_{Y})$, and
the same equivalences hold.

######  Theorem 7.10.

Theorems 7.8 and 7.9 also apply in the setting of twisted invariants.

There is also a direct relationship between the conjecture for Lagrangian
cones and twisted Lagragangian cones.

######  Theorem 7.11.

For any choice of $C,E$, Conjecture 7.2 and Conjecture 7.1 for untwisted
invariants imply Conjecture 7.2, Conjecture 7.1 for twisted invariants.

This is a consequence of Givental’s Quantum Adams-Riemann-Roch theorem, which
relates the Lagrangian cones of the twisted theory to the untwisted theory via
a symplectomorphism $\Delta$. It applies in both the ordinary and
$S_{n}$-invariant settings. Let $\Delta_{X},\Delta_{Y}$ be the maps coming
from the twistings on $X,Y$, respectively. We observe that
$sp_{Q}(\Delta_{Y})=\Delta_{X}$, thus proving the theorem. (The same result
for cohomology is theorem in [4]).

##  8 Summary: Toolkit for Computing Quantum $K$-rings

In view of the results of the previous sections, we have the following toolkit
for calculating relations in small quantum $K$-rings (twisted or not).

  1. 1.

Interpret the ring as a twisted quantum $K$-ring of a simpler space if
possible. (i.e. move from $X//G$ to $X//T$, or consider the Gromov-Witten
invariants of a complete intersection as twisted Gromov-Witten invariants of
the ambient space).

  2. 2.

Determine the small $J$-function of the theory. (In many cases, this is doable
by using Givental’s quantum Riemann-Roch theorem to get some value of
$\mathcal{J}^{sym}$, if it is the value at $t=0$ we are done.)

  3. 3.

Find a bound for the $q-$ degrees of each term in $J$

  4. 4.

Find difference equations annihilating $J$

  5. 5.

Convert these into relations, using the degree bound and quantum triviality
theorem (if possible) to simplify functions of $A_{i,com}|_{t=0}$.

##  9 Applications

###  9.1 Level Structures of Projective Space

We first consider $X=\mathbb{P}^{N}$ with the level structure twisting
determined by the bundle $P:=O(-1)$ and the integer $\ell$. The small
$J$-function of this theory, for $-1<\ell\leq N+1$, was calculated by
Givental-Yan in [8] and is equal to:

$J^{\ell}:=(1-q)\sum_{d}Q^{d}\frac{P^{\ell}q^{\binom{d}{2}}}{\prod_{i}\prod_{j=1}^{d}(1-Pq^{j}/\Lambda_{i})}$

Now, we can find

######  Proposition 9.1.

$\tilde{J}^{\ell}$ satisfies the following difference equation:

$\prod_{i}(1-\Lambda_{i}^{-1}q^{Q\partial_{Q}})\tilde{J}^{\ell}=Qq^{\ell
Q\partial_{Q}}J^{\ell}_{1,N+1}$

###### Proof.

$\begin{split}&\prod_{i}(1-\Lambda_{i}^{-1}q^{Q\partial_{Q}})P^{\frac{\ln(Q)}{\ln(q)}}J^{\ell}_{1,N+1}=(1-q)\prod_{i}(1-q^{d}P/\Lambda_{i})P^{\frac{\ln(Q)}{\ln(q)}}\sum_{d\geq
0}\frac{Q^{d}P^{\ell
d}q^{\ell\binom{d}{2}}}{\prod_{i}\prod_{m=1}^{d}(1-q^{m}P/\Lambda_{i})}\\\
=&(1-q)P^{\frac{\ln(Q)}{\ln(q)}}\sum_{d\geq 1}\frac{Q^{d}P^{\ell
d}q^{\ell\binom{d}{2}}}{\prod_{i}\prod_{m=1}^{d-1}(1-q^{m}P/\Lambda_{i})}=(1-q)P^{\frac{\ln(Q)}{\ln(q)}}\sum_{d\geq
1}q^{\ell(d-1)}P^{\ell
d}\frac{Q^{d}P^{\ell(d-1)}q^{\ell\binom{d-1}{2}}}{\prod_{i}\prod_{m=1}^{d-1}(1-q^{m}P/\Lambda_{i})}\\\
=&(1-q)Qq^{\ell Q\partial_{Q}}P^{\frac{\ln(Q)}{\ln(q)}}\sum_{d\geq
1}\frac{Q^{d-1}P^{\ell(d-1)}q^{\ell\binom{d-1}{2}}}{\prod_{i}\prod_{m=1}^{d-1}(1-q^{m}P/\Lambda_{i})}=Qq^{\ell
Q\partial_{Q}}\tilde{J}_{1,N+1}\end{split}$ (9.1)

∎

Writing $J^{\ell}$ as $(1-q)\sum Q^{d}J_{d}$, we have that
$deg(J_{d})=\ell\binom{d}{2}-(N+1)\binom{d+1}{2}$, this is always strictly
less than $-d$, so by the quantum triviality theorem $A_{com}|_{t=0}=P$. So we
can conclude:

######  Corollary 9.2.

The quantum $K$-ring with level structure of $\mathbb{P}^{N}$ with respect to
the bundle $P$ and the level $\ell$ s determined by the relation:

$\prod_{i}(1-P/\Lambda_{i})=QP^{\ell}$ (9.2)

######  Remark.

The physical derivation of the quantum $K$-ring of projective space, done in
e.g. [11], relies on a specific choice of parameters called the Chern-Simons
levels. Ordinary quantum $K$-theory comes from choosing the parameter
$\tilde{k}_{1}$ (in the notation of [11]) to be $-\frac{N+1}{2}$. The
prediction corresponding to $\tilde{k}_{1}=\ell-\frac{N+1}{2}$ are exactly the
relations we have computed here for level $\ell$.

###  9.2 Complete Intersections in Projective Space

If $P=O(1)$, the $J$-function of $\mathbb{P}^{N}$ is:

$\sum_{d\geq 0}\frac{Q^{d}}{\prod_{m=1}^{d}(1-Pq^{m})^{N+1}}$

Given some complete intersection $Y$ of hypersurfaces of degree
$\ell_{1},\dots,\ell_{r}$, we can apply Givental’s quantum Lefschetz to get
that that the following function $I_{Y}$ is a value of the $J$-function for
invariants twisted by the Euler class of $\bigoplus O(\ell_{i})$:

$I_{Y}:\sum_{d\geq
0}\frac{Q^{d}}{\prod_{m=1}^{d}(1-Pq^{m})^{N+1}}\prod_{i}\prod_{0\leq
b\leq\ell_{i}d}(1-\lambda P^{\ell_{i}}q^{b})$

$I_{Y}$ has no poles at $q=0$, and $I_{Y}=(1-q)\sum_{d}Q^{d}I_{d}$, with
$deg(I_{d})=\sum_{i}\binom{\ell_{i}d}{2}-(N+1)\binom{d+1}{2}$.

######  Proposition 9.3.

If $\sum_{i}\ell_{i}^{2}\leq N+1$, then $I_{Y}$ is the small $J$-function of
this twisted theory, i.e. it is the pushforward to $X$ of $J_{Y}$, furthermore
all $Q^{\neq 0}$ terms of $Pq^{Q\partial_{Q}}\frac{I_{Y}}{1-q}$ vanish at
$q=\infty$.

###### Proof.

First we note that $I_{Y}$ has no poles at $q=0$, and if we write
$I_{Y}=(1-q)\sum_{d}Q^{d}I_{d}$, then the remaining two conditions are both
checked by bounds on
$deg(I_{d})=sum_{i}\binom{\ell_{i}d}{2}-(N+1)\binom{d+1}{2}$. We want to check
that $deg(I_{d})<-1$ so $I_{Y}$ is the small $J$-function, and that
$deg(I_{d})<-d$ so the required terms vanish at infinity. We have:

$deg(I_{d})=\frac{d}{2}\big{(}d(\sum_{i}\ell_{i}^{2}-(N+1))+\sum_{i}\ell_{i}-(N+1)\big{)}$

Our desired result is equivalent to the inequality

$d(\sum_{i}\ell_{i}^{2}-(N+1))+\sum_{i}\ell_{i}-(N+1)<-2$ (9.3)

By hypothesis, $\sum_{i}\ell_{i}^{2}\leq N+1$, it is sufficient to check it
for $d=1$, i.e. to show:

$\sum_{i}\big{(}\ell_{i}^{2}-(N+1)\big{)}+\big{(}\sum_{i}\ell_{i}-(N+1))<-2$
(9.4)

Equivalently:

$\sum_{i}2\ell_{i}^{2}+\sum_{i}(\ell_{i}-\ell_{i}^{2})<2N$ (9.5)

However we observe that unless at most one $\ell_{i}=2$, and all the others
are equal to 1, we have:

$\sum_{i}(\ell_{i}-\ell_{i}^{2})<-2$ (9.6)

Thus when (9.6) is true:

$\sum_{i}2\ell_{i}^{2}+\sum_{i}(\ell_{i}-\ell_{i}^{2})\leq
2N+2-\sum_{i}(\ell_{i}-\ell_{i}^{2})<2N$ (9.7)

In those exceptional cases, we can check by hand that (9.5) is satisfied,
noting that $r<N$, so it is impossible to have both
$\sum_{i}\ell_{i}^{2}-(N+1)$ and $\sum_{i}\ell_{i}-(N+1)$ lie in the set
$\\{0,-1\\}$.

∎

######  Theorem 9.4.

If $\sum_{i}\ell_{i}^{2}\leq N+1$, then the restriction to $Y$ of the
following expression holds in $QK^{*}(Y)$:

$(1-P)^{N+1-r}=\prod_{i}\frac{1-P^{\ell_{i}}}{1-P}Q$

###### Proof.

We have the equation:

$(1-Pq^{Q\partial_{Q}})^{N+1}I_{Y}=\prod_{i}(1-\lambda
P^{\ell}_{i}q^{\ell_{i}Q\partial_{Q}})QI_{Y}$ (9.8)

After normalizing, this corresponds to the equation for $\tilde{I}_{Y}$ (we
omit this step in subsequent examples):

$(1-q^{Q\partial_{Q}})^{N+1}\tilde{I}_{Y}=\prod_{i}(1-\lambda
q^{\ell_{i}Q\partial_{Q}})Q\tilde{I}_{Y}$ (9.9)

Which gives us the relation in the twisted quantum $K$-theory of
$\mathbb{P}^{N}$, since $I_{Y}$ is the small $J$-function of this theory and
$A_{com}|_{t=0}=P$ by the quantum triviality theorem:

$(1-P)^{N+1}=\prod_{i}(1-\lambda P^{\ell_{i}})Q,$

equivalently:

$(1-P)^{N+1-r}=\prod_{i}\frac{1-\lambda P^{\ell_{i}}}{1-P}Q$

Taking non-equivariant limits and pulling back to $QK^{*}(Y)$ gives the
relation:

$(1-P)^{N+1-r}=\prod_{i}\frac{1-P^{\ell_{i}}}{1-P}Q$

∎

######  Remark.

In the case that $r=1$, i.e. when $Y$ is a hypersurface, these results are
mathematical proofs of the physical predictions made in section 5 of [10].

###  9.3 Quasimaps to $T^{*}\mathbb{P}^{N}$

An alternative formulation of quantum $K$-theory has been developed using the
quasimap compactifications instead of the stable map ones. In particular,
there have been many results relating relations in these quasimap rings to
integrable systems. The quasimap $K$-ring of $T^{*}\mathbb{P}^{N}$ is
determined by the relation ([19]):

$P^{N+1}\hbar^{(-N-1)/2}\prod_{i}(\Lambda_{i}^{-1})Q=\prod_{i=1}^{N+1}\frac{1-P/\Lambda_{i}}{1-\hbar
P/\Lambda_{i}}$

An important question to ask is whether there is a relationship between the
quasimap quantum ring and some version of the stable map one. We will give a
partial answer this for the case of $T^{*}\mathbb{P}^{N-1}$.

The quasimap $I$-function that describes the invariants
$T^{*}\mathbb{P}^{N-1}$ has the following form:

$\hat{I}^{T}=\sum_{0\leq
d}Q^{d}\prod_{j=1}^{N+1}\frac{\prod_{m=0}^{d-1}1-q^{m}yX/\Lambda_{j}}{\prod_{m=0}^{d}1-q^{m}X/\Lambda_{j}}$

It was shown in [18] that $(1-q)\hat{I}^{T}$ is the small $J$ function for a
particular set of twisted invariants of $\mathbb{P}^{N-1}$. The twisting is
the inverse Euler class of the tangent bundle, followed by the determinant of
the cotangent bundle. (Equivalently, it’s the stable map theory corresponding
to $T^{*}\mathbb{P}^{N}$ with level structure determined by
$T^{*}\mathbb{P}^{N}$ and $\ell=1$).

Using the twisted version of the Iritani-Milanov-Tonita theorem, we can
calculate the quantum $K$-ring with respect to this twisting, and compare it
to the quasimap relations.

Let $I^{T}=\prod_{j}1-y\frac{X}{\Lambda_{j}}q^{Q\partial_{Q}}\hat{I}^{T}$. The
action of this operator replaces the $d-1$ in the product with a $d$. If we
denote the operator taking $\hat{I}^{T}$ to $I^{T}$ by $D$, we have:

$\prod_{j-1}^{N+1}\frac{1-\frac{X}{\Lambda_{j}}q^{Q\partial_{Q}}}{1-y\frac{X}{\Lambda_{j}}q^{Q\partial_{Q}}}D\tilde{I}^{T}=QD\tilde{I}^{T}$
(9.10)

Since the $q-$degree of the $Q^{d}$th term is at most $-d$, we have that the
operator $q^{Q\partial Q}$ gets converted into multiplication by $P_{i}$ via
the quantum triviality theorem. This yields the relation (after dividing both
sides by $D|_{q=1}$):

$\prod_{j=1}^{N+1}(1-X/\Lambda_{j})=Q\prod_{j=1}^{N+1}(1-yX/\Lambda_{j})$
(9.11)

We can rewrite this as:

$\prod_{j=1}^{N+1}\frac{(1-X/\Lambda_{j})}{1-yX/\Lambda_{j}}=Q$

We can multiply both sides by $\prod_{j}Xy/\Lambda_{j}$ to get:

$\prod_{j=1}^{N+1}\frac{(1-X/\Lambda_{j})}{\Lambda_{j}/yX-1}=Q\prod_{j}\frac{X}{\Lambda_{j}}y^{N}$

Replacing $y$ with $\hbar^{-1}$, this is similar to what is obtained in the
quasimap case, up to a factor of $\hbar^{(N+1)/2}$. We leave the explanation
behind this discrepancy, and some more detailed comparisons between the
quasimap and stable map rings to a future work.

###  9.4 Grassmanians

We will use the ring-theoretic abelian/non-abelian correspondence to recover
some relations for the case $X=Gr(k,n)$, the Grassmanian of $k$-planes in
$\mathbb{C}^{n}$.

Recall the Grassmanian can be written as the GIT quotient
$Hom(\mathbb{C}^{k},\mathbb{C}^{n})//GL(k)$. The corresponding abelian
quotient $Y$ is a product of $k$ copies of $\mathbb{P}^{n-1}$s.

The classical abelian/non-abelian correspondence takes the following form in
this case.
$K^{*}(Y)\cong\prod_{i=1}^{n}\frac{\mathbb{C}[P]}{\prod_{i}(1-P_{i}/\Lambda_{i})}$,
where $P_{i}$ is the tautological bundle of the $i$th projective space.

The ring map $K^{*}(Y)^{W}\to K^{*}(Gr(k,n))$ is given by sending symmetric
functions of the $P_{i}$ to the same symmetric functions of the Chern roots of
the tautological bundle $V$. We will thus abusively also write
$V=\sum_{i}P_{i}$.

Givental-Yan proved Conjecture 7.2 for Grassmanians, in particular they
established the following:

######  Theorem 9.5.

[8] The small $J$-function of the Grassmanian is equal to:

$(1-q)\sum_{d_{1},\dots,d_{n}\leq 0}\frac{Q^{\sum
d_{i}}}{\prod_{i=1}^{n}\prod_{j=1}^{n}\prod_{m=1}^{d_{i}}(1-q^{m}\frac{P_{i}}{\Lambda_{j}})}\prod_{i\neq
j}\frac{\prod_{m=-\infty}^{d_{i}-d_{j}}(1-q^{m}\frac{P_{i}}{P_{j}})}{\prod_{m=-\infty}^{0}(1-q^{m}\frac{P_{i}}{P_{j}})}$

It is the image under $sp_{Q}$ of $J^{tw}_{Y}$, which is a value of the big
$S_{n}$-equivariant $J$-function of the twisted theory of $Y:$:

$J^{tw}_{Y}(q):=(1-q)\sum_{d_{1},\dots,d_{n}\leq
0}\frac{\prod_{i}Q_{i}^{d_{i}}}{\prod_{i=1}^{n}\prod_{j=1}^{n}\prod_{m=1}^{d_{i}}(1-q^{m}\frac{P_{i}}{\Lambda_{j}})}\prod_{i\neq
j}\frac{\prod_{m=-\infty}^{d_{i}-d_{j}}(1-q^{m}\Lambda_{0}\frac{P_{i}}{P_{j}})}{\prod_{m=-\infty}^{0}(1-q^{m}\Lambda_{0}\frac{P_{i}}{P_{j}})}$

In fact, $J^{tw}_{Y}$ is actually the small $J$-function of this theory. To
prove this, it is sufficient to show that its projection to $K_{+}$ is equal
to $1-q$, so it corresponds to the input $t=0$. This means the terms for
$d\neq 0$ must be reduced rational functions, i.e. with no poles at
$q=0,\infty$ and with negative $q$-degree outside the $Q^{0}$ term.

It is easy to see this since if we write $J^{tw}_{Y}$ as
$(1-q)\sum_{d}Q^{d}J_{d}$, $J_{d}$ can only have $q^{-1}$ terms in the
denominator, so there are no poles at 0. In terms of degree, we have:

$deg(J_{d})=\sum_{i,j}\binom{d_{i}-d_{j}+1}{2}-n\sum_{i}\binom{d_{i}+1}{2}$

We can rewrite this as

$\sum_{i}\big{(}\sum_{j}\binom{d_{i}-d_{j}+1}{2}-n\binom{d_{i}+1}{2}\big{)}$

Each term $\sum_{j}\binom{d_{i}-d_{j}+1}{2}-n\binom{d_{i}+1}{2}$ must be
strictly negative unless $d_{i}=0$ since $N>k$ and
$\binom{d_{i}-d_{j}}{2}<\binom{d_{i}+1}{2}$. In fact we can say each term
contributes degree at most $k-n$, hence $deg(J_{d})<-1$.

Thus, we can apply Conjecture 7.5 in this case.

In fact, for the purpose of using the quantum triviality theorem, we will
prove two stronger bounds on $deg(J_{d})$

######  Lemma 9.6.

  * •

$deg(J_{d})<(k-n)max(d_{i})$

  * •

$deg(J_{d})<\sum_{i}d_{i}$

###### Proof.

From before we have:

$deg(J_{d})=\sum_{i}\big{(}\sum_{j}\binom{d_{i}-d_{j}+1}{2}-n\binom{d_{i}+1}{2}\big{)}$

Each summand is less than $(k-n)max(d_{i})$, which establishes both
inequalities.

∎

This gives us the following corollaries as a result of the quantum triviality
theorem:

######  Corollary 9.7.

  * •

$A_{i,com}|_{t=0}=P_{i}$

  * •

Letting $e_{\ell}(P)$ denote the $\ell$th symmetric polynomial applied to
$P_{1},\dots,P_{k}$, the expression $e_{\ell}(P)$ is equal to its classical
product, i.e. $\wedge^{\ell}(\bigoplus_{i}P_{i})$.

  * •

The same is true for $h_{\ell}(P)$ with $\ell\leq n-k$, where $h_{\ell}$
denotes the complete homogenous symmetric polynomial.

We can now find the relations in $QK^{tw}(Y)$.

######  Theorem 9.8.

The following relations are true in $QK^{tw}(Y)$:

$F_{\lambda}(Q_{i},P_{i})=0$

Where

$F_{\lambda}(x,t):=t^{k}e_{n}(\Lambda)\prod_{j}\frac{(P_{j}-\lambda
t)}{(t-\lambda P_{j})}-\prod_{a}(\Lambda_{a}-t)\prod_{j}P_{j}$

###### Proof.

$J^{tw}_{Y}$ satisfies the following $q-$difference equations:

$\prod_{j\neq i}(1-\lambda
qP_{j}P_{i}^{-1}q^{Q_{j}\partial_{Q_{j}}-Q_{i}\partial_{Q_{i}}})\prod_{a}(1-P_{i}q^{Q_{i}\partial_{Q_{i}}}/\Lambda_{a})\mathcal{J}=Q_{i}\prod_{j\neq
i}(1-\lambda
qP_{i}P_{j}^{-1}q^{Q_{i}\partial_{Q_{i}}-Q_{j}\partial_{Q_{j}}})\mathcal{J}$

These translate to the relations (using the fact that $A_{i,com}=P_{i}$ by the
quantum triviality theorem):

$\prod_{a}(1-P_{i}/\Lambda_{a})=Q_{i}\prod_{j\neq i}\frac{1-\lambda
P_{i}/P_{j}}{1-\lambda P_{j}/P_{i}}$

Since

$\frac{1-\lambda P_{i}/P_{j}}{1-\lambda P_{j}/P_{i}}=\frac{P_{i}(P_{j}-\lambda
P_{i})}{P_{j}(P_{i}-\lambda P_{j})},$

the above relation is equivalent to:

$\prod_{j}P_{j}(P_{i}-\lambda
P_{j})\prod_{a}(1-P_{i}/\Lambda_{a})=P_{i}^{k}Q_{i}\prod_{j}(P_{j}-\lambda
P_{i})$

Multiplying by $\prod_{i}\Lambda_{i}$ gives the desired result.

∎

We have established that elements of $QK^{tw}(Y)^{W}|_{Q_{i}=Q}$ have well-
defined limits as $\lambda$ approaches 1. Call the ring thus obtained $S$,
which maps to $QK^{*}(X)$ via $sp$.

If we denote $F$ as $\lim_{\lambda\to 1}F_{\lambda}(Q,t)$, then symmetric
combinations of $F(P_{i})=0$ become relations in $S$. As before, we will
denote the set of $P_{i}$ collectively by $P$, with the notation that
$f(P)=f(P_{1},\dots,P_{k})$ for $f$ any symmetric function. Similarly, we will
denote the equivariant parameters collectively by $\Lambda$.

Calculating the limit explicitly yields:

$F=\sum_{\ell=0}^{n-k}(-1)^{\ell}t[e_{k}(P)e_{\ell}(\Lambda)+Qe_{n}(\Lambda)e_{\ell-n+k}(0)]$

######  Remark.

The equations $F(P_{i})=0$ are exactly the Bethe Ansatz equations obtained
from physical models of $QK^{*}(X)$ in [11].

We will now use Vieta’s formulas to explicitly compute relations in $S$. The
algebraic manipulations done here are identical to those in [9] for the case
of Grassmanians. However, in our situation, we can give each relation a
concrete mathematical meaning since we are working in $QK^{tw}(Y)$. Denote the
entire set of roots of $F$ by $w$, and denote the subset of these roots not in
$P$ by $\bar{P}$, as consequence:

$e_{\ell}(w)=\sum_{i=0}^{n-k}e_{\ell-i}(P)e_{i}(\bar{P})$ (9.12)

Denote the $t^{\ell}$ coefficient of $F$ by $F_{\ell}$, we have that:

$F_{n-\ell}=(-1)^{\ell}(e_{k}(P)e_{\ell}(\Lambda)$

Thus, applying Vieta’s formula to $F$ gives:

$e_{\ell}(w)=(-1)^{\ell}\frac{e_{k}(P)e_{\ell}(T)}{e_{k}(P)}$ (9.13)

Combining the (9.12) and (9.13) gives:

$e_{\ell}(w)=e_{\ell}(T)+Qe_{\ell}(\bar{P})e_{\ell-n+k}(0)$ (9.14)

Using (9.14) to replace $e_{\ell}(w)$ gives us:

$e_{\ell}(T)+Qe_{\ell}(\bar{P})e_{\ell-n+k}(0)=\sum_{i=0}^{n-k}e_{\ell-i}(P)e_{i}(\bar{P})$
(9.15)

We would like to use (9.15) to solve for $e_{\ell}(\bar{P})$, to do this,
first note that $\eqref{d}$ is the $y^{\ell}$ term in the equation:

$(\sum_{i=1}^{k}y^{i}e_{i}(P))(\sum_{j=1}^{k}y^{j}e_{j}(\bar{P}))=\sum_{s=0}^{n}y^{s}e_{s}(\Lambda)+Qe^{n-k}e_{n-k}(\Lambda)$
(9.16)

Since the inverse of the series $\sum y^{i}e_{i}(S)$ is
$\sum(-1)^{i}y^{i}h_{i}(S)$, where $h_{i}$ are the total homogenous symmetric
polynomials, we can rewrite (9.16) as:

$(\sum_{j=1}^{k}y^{j}e_{j}(\bar{P}))=((\sum_{i=1}^{\infty}(-y^{i})h_{i}(P)))(\sum_{s=0}^{n}y^{s}e_{s}(\Lambda)+Qe^{n-k}e_{n-k}(\Lambda))$
(9.17)

Extracting the degree $\ell$ part yields:

$e_{\ell}(\bar{P})=\sum_{i=0}^{n-k}(-1)^{i}e_{\ell-i}(\Lambda)h_{i}(P)+Qe_{n-k}(\bar{X})e_{\ell-n+k}(0)$
(9.18)

Now, having established these relations, we can specialize them into
$QK^{*}(X)$, noting that $e_{\ell}(P_{i})$ and $h_{\ell}(P_{i})$ can be
interpreted as their classical products. Since specialization sends $P_{i}$ to
the $i$th Chern root of the tautological bundle $S$, we have the following:

$sp(e_{\ell}(\bar{P}))=\begin{cases}\wedge^{\ell}(\mathbb{C}^{n}/S)&\ell<n-k\\\
\frac{1}{1-Q}\wedge^{\ell}(\mathbb{C}^{n}/S)&\ell=n-k\end{cases}$ (9.19)

$sp(e_{\ell}(P))=\wedge^{\ell}S$ (9.20)

Thus specializing the relation (9.15) yields:

$\sum_{r=0}^{n-k}\wedge^{\ell-r}{\cal
S}\,\times_{Q}\,\wedge^{r}({\mathbb{C}}^{n}/{\cal
S})\>=\>\wedge^{\ell}{\mathbb{C}}^{n}\>-\>\frac{q}{1-q}\det({\mathbb{C}}^{n}/{\cal
S})\,\times_{Q}\,\left(\wedge^{\ell-n+k}{\cal S}\>-\>{\cal
O}\delta_{\ell,n-k}\right).$ (9.21)

Adding factors of $y$, this yields the Whitney relations:

$\Lambda_{y}({\cal S})\times_{Q}\Lambda_{y}({\mathbb{C}}^{n}/{\cal
S})\>=\>\Lambda_{y}({\mathbb{C}}^{n})\>-\>y^{n-k}\frac{q}{1-q}\det({\mathbb{C}}^{n}/{\cal
S})\times_{Q}\left(\Lambda_{y}({\cal S})-1\right).$ (9.22)

These are deformations of the usual Whitney relations, which generate
$K^{*}(X)$, hence this is a presentation for $QK^{*}(X)$.

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