# Electronic paddle-wheels in a solid-state electrolyte

Harender S. Dhattarwal  Rahul Somni  Richard C. Remsing
<EMAIL_ADDRESS>Department of Chemistry and Chemical Biology, Rutgers
University, Piscataway, NJ 08854

###### Abstract

Solid-state superionic conductors (SSICs) are promising alternatives to liquid
electrolytes in batteries and other energy storage technologies. The rational
design of SSICs and ultimately their deployment in battery technologies is
hindered by the lack of a thorough understanding of their ion conduction
mechanisms. In SSICs containing molecular ions, rotational dynamics couple to
translational diffusion to create a ‘paddle-wheel’ effect that facilitates
conduction. The paddle-wheel mechanism explains many important features of
molecular SSICs, but an explanation for ion conduction and anharmonic lattice
dynamics in SSICs composed of monatomic ions is still needed. We predict that
ion conduction in the classic SSIC AgI involves ‘electronic paddle-wheels,’
rotational motion of lone pairs that couple to and facilitate ion diffusion.
The electronic paddle-wheel mechanism creates a universal perspective for
understanding ion conductivity in both monatomic and molecular SSICs that will
create design principles for engineering solid-state electrolytes from the
electronic level up to the macroscale.

Solid-state superionic conductors (SSICs) promise significant improvements
over liquid electrolytes for energy storage technologies 1; 2; 3; 4. Because
of their use as electrolytes in solid-state batteries, SSICs have been studied
extensively over the past 50 years in an effort to understand the molecular-
scale principles underlying conduction 5; 6; 7. Conduction in molecular ionic
solids is well understood. In solids like lithium sulfate, Li+ cations rapidly
diffuse through the rigid lattice made by the molecular SO${}_{4}^{2-}$ anions
with rotational “paddle-wheel” motions of sulfate often facilitating the rapid
cationic diffusion 8; 9; 10; 6; 11; 12. This paddle-wheel motion has since
become a principle for designing molecular SSICs 11; 13. In contrast, similar
principles have not been identified for SSICs composed of monatomic ions, like
the prototypical AgI 6, where fundamental questions concerning the coupling
between the mobile cations and the immobile anionic lattice still remain 14;
15. In this Letter, we provide new insights into the coupled cation-anion
dynamics in SSICs by identifying an electronic analogue of molecular paddle-
wheel motion in AgI.

Our investigation into AgI is motivated by recent ideas concerning “hidden
disorder” in otherwise ordered materials. This hidden disorder is typically
local and electronic in nature, arising from an interplay between electronic
and nuclear degrees of freedom 16. In particular, we exploit the concept of
electronic plastic crystals — phases of crystalline solids in which localized
lone pair electrons are dynamically orientationally disordered 17; 18. In AgI,
localized lone pairs exist on the iodide ions, and we show here that AgI is an
electronic plastic crystal in its superionic phase. The rotational motion of
I- lone pair electrons accompanies the diffusion of Ag+ through the SSIC,
creating electronic paddle-wheels. This electronic paddle-wheel is summarized
by the simulation snapshots in Fig. 1a, where the rotation of I- lone pairs
(light purple) facilitates the motion of an Ag+ cation (orange) from one site
to another.

Before quantifying dynamics in AgI, we first characterize the structural
correlations involving lone pairs to identify cation coordination
environments. We focus on the radial pair distribution function, $g(r)$,
quantifying two-body correlations among cations, anions, and iodide lone
pairs. We represent the positions of lone pairs using maximally localized
Wannier function centers (MLWFCs) 19, which give a reasonable description of
the location of localized electron pairs. The Ag+-lone pair $g(r)$ exhibits
two peaks at short distances, Fig. 1b. The first peak corresponds to close,
direct interactions between lone pairs and Ag+. Integration of this peak
yields a coordination number between two and three, indicating that each Ag+
is coordinated on average by 2 or 3 sets of lone pairs. The second peak
between 3 and 4 Å corresponds to correlations between the central Ag+ and lone
pairs on the iodide that are pointing away from the cation.

Figure 1:  (a) Snapshots from a trajectory of AgI at 750 K illustrating the
electronic paddle-wheel motion that facilitates Ag+ conduction. A single Ag+
(orange sphere) and its initial three closest I- (purple spheres) with their
lone pair MLWFCs (small, light purple spheres) are highlighted. All other Ag+
and I- are shown as small gray and larger dark gray spheres. The lone pairs
rotate concertedly with the motion of Ag+ as it moves from one binding site to
the next, moving from left to right along the panels shown here. Following one
rotation over a 300 fs time interval, the Ag+ stays in a binding site until it
diffuses with another lone pair rotation, with the waiting time and rotation
time totaling approximately 1200 fs. (b) Radial distribution functions,
$g(r)$, between Ag+ cations and iodide lone pairs, represented as maximally
localized Wannier function centers. (c) Joint probability distribution of the
lone pair-Ag+ distance ($r$) and the I--lone pair-Ag+ angle ($\theta$). (d)
Tetrahedral rotor function time correlation functions (TCFs), $C_{2}(t)$,
characterizing I- lone pair rotational dynamics. (e) Ag+ cage TCF, $C_{\rm
R}(t)$, quantifying the diffusion of cations between binding sites.

The coordination structure suggested by $g(r)$ is further supported through
the joint probability distribution of the lone pair-Ag+ distance ($r$) and the
I--lone pair-Ag+ angle ($\theta$), Fig. 1c. The peak near 2 Å in $g(r)$
involves linear I--lone pair-Ag+ arrangements, consistent with direct
coordination of the cation by lone pair electrons. The peak near 3 Å involves
values of $\theta\approx 120^{\circ}$, consistent with the three other
tetrahedal lone pair sites of I- pointing away from the Ag+. This structural
analysis strongly suggests that lone pair electrons are coordinating silver
cations in AgI.

In an electronic plastic crystal, lone pair electrons exhibit dynamic
orientational disorder. To quantify the rotational dynamics of I- lone pairs,
we follow previous work and focus on tetrahedral rotor functions,
$M_{\gamma}$, of order $l=3$, such that $\gamma$ labels the $(2l+1)$ functions
for each $l$ 20; 17; 18. The tetrahedral rotor functions do not distinguish
between the four I- MLWFCs and treat them all identically, as desired for
identical electron pairs. We focus on the $\gamma=2$ rotor function here,
$M_{2}=\frac{3\sqrt{5}}{40}\sum_{i=1}^{4}(5x_{i}^{3}-3x_{i}r_{i}^{2})$, where
$\mathbf{r}_{i}=(x_{i},y_{i},z_{i})$ is a unit vector along I-MLWFC bond $i$,
and $r_{i}=\left|\mathbf{r}_{i}\right|$. The rotational dynamics of I- lone
pairs can be quantified through the time correlation function

$C_{\gamma}(t)=\frac{\left<M_{\gamma}(0)M_{\gamma}(t)\right>}{\left<M_{\gamma}^{2}(0)\right>},$
(1)

where $\left<\cdots\right>$ indicates an ensemble average. In the superionic
phase of AgI at 750 K, the rotational TCF $C_{2}(t)$ decays with a relaxation
time of approximately 0.4-0.5 ps, indicating that MLWFC rotational motion
occurs on a picosecond timescale, Fig. 1d. In comparison, similar MLWFC
rotations in cesium tin halide perovskites occur on faster timescales of
0.04-0.1 ps 17; 18.

Figure 2: Snapshots showing the trajectory of a single Ag+ at (a) 750 K and
(b) 100 K, where the Ag+ is colored by time according to the color bar, and
iodides are shown as purple spheres. The space-time dynamics in the system are
quantified through the distinct van Hove correlation function for cation-lone
pair correlations at (c) 750 K and (f) 100 K, the distinct van Hove
correlation function for cation-cation correlations at (d) 750 K and (g) 100
K, all normalized by the bulk density, $\rho$, and the self van Hove
correlation function for cation correlations at (e) 750 K and (h) 100 K.

When Ag+ diffuses between the various sites in the AgI lattice, the identity
of the iodides making up its coordination cage changes. Therefore, we quantify
Ag+ dynamics through a TCF that probes these changes in coordination cage,
inspired by the cage correlation functions used in the study of supercooled
liquids 21. We define two indicator functions, $h^{\rm
out}_{i}(0,t)=\Theta\left(1-\xi^{\rm out}_{i}(0,t)\right)$ and $h^{\rm
in}_{i}(0,t)=\Theta\left(1-\xi^{\rm in}_{i}(0,t)\right)$, where $\Theta(x)$ is
the Heaviside step function,

$\xi^{\rm
out}_{i}(0,t)=\frac{\mathbf{{\ell}}_{i}(0)\cdot\mathbf{{\ell}}_{i}(t)}{\mathbf{{\ell}}_{i}(0)\cdot\mathbf{{\ell}}_{i}(0)},$
(2) $\xi^{\rm
in}_{i}(0,t)=\frac{\mathbf{{\ell}}_{i}(0)\cdot\mathbf{{\ell}}_{i}(t)}{\mathbf{{\ell}}_{i}(t)\cdot\mathbf{{\ell}}_{i}(t)},$
(3)

and $\mathbf{{\ell}}_{i}(t)$ is the neighbor list of the $i$th Ag+ at time
$t$. Therefore, $h^{\rm out/in}_{i}(0,t)=1$ if the neighbors are the same at
times $t$ and $0$. If the number of neighbors decreases, then $h^{\rm
out}_{i}(0,t)=0$ and $h^{\rm in}_{i}(0,t)=1$. If the number of neighbors
increases, then $h^{\rm in}_{i}(0,t)=0$ and $h^{\rm out}_{i}(0,t)=1$. We then
define a cage TCF as $C_{\rm R}(t)=\left<h^{\rm out}_{i}(0,t)h^{\rm
in}_{i}(0,t)\right>$, which quantifies the diffusion of Ag+ between occupation
sites through changes in its coordination cage, like the motion shown in Fig.
1a. The cage TCF has a relaxation time of approximately 0.6-0.7 ps at 750 K,
Fig. 1e, which is on a similar timescale to lone pair rotations. The similar
timescales for Ag+ diffusion out of a binding site and lone pair rotations
suggests a coupling between electron rotational motion and cation diffusion.

Figure 3: Schematic diagram of the collective diffusion mechanism of Ag+ in
AgI. Thermal vibrational motion weakens some cation-anion interactions, new
cation-anion interactions are formed through lone pair rotation, and these
lone pair rotations are coupled to the diffusion of a coordinated Ag+.

The MLWFC rotational dynamics are temperature dependent, Fig. 1d. At the low
temperature of 100 K, where AgI is no longer conductive (see SI), lone pair
rotational dynamics are significantly slowed and AgI is no longer in an
electronic plastic crystal phase. As a result, the lone pair rotation TCF,
$C_{2}(t)$, and the TCF quantifying Ag+ diffusion, $C_{\rm R}(t)$, both decay
very little on the timescale of the simulations. This slowing of lone pair
rotations and Ag+ diffusion upon cooling suggests that they are intimately
connected; Ag+ cannot diffuse unless I- lone pairs rotate.

To characterize cation diffusion in more detail, we computed the distinct and
self parts of the van Hove correlation function, $G_{\rm d}(r,t)$ and $G_{\rm
s}(r,t)$, respectively, defined according to

$G_{\rm d}(r,t)=\left<\frac{1}{N}\sum_{i}\sum_{j\neq
i}\delta\left(\left|\mathbf{r}_{i}(0)-\mathbf{r}_{j}(t)\right|\right)\right>$
(4)

and

$G_{\rm
s}(r,t)=\left<\frac{1}{N}\sum_{i}\delta\left(\left|\mathbf{r}_{i}(0)-\mathbf{r}_{i}(t)\right|\right)\right>.$
(5)

The distinct van Hove function, $G_{\rm d}(r,t)$, quantifies space-time
correlations of the initial position of one particle ($i$) with other
particles in the system ($j$). The self van Hove function, $G_{\rm s}(r,t)$,
quantifies space-time correlations of a particle with its own initial
position. For $t=0$, $G_{\rm d}(r,0)$ is equal to the $g(r)$ between the two
particle types of interest, and $G_{\rm s}(r,0)$ is a delta function at $r=0$.

The distinct van Hove function for cation-lone pair correlations suggests
electronic paddle-wheels in the SSIC phase of AgI, Fig. 2c,f. At $t=0$,
$G_{\rm d}(r,t)$ is the Ag+-MLWFC pair distribution function. As $t$ increases
at $T=750$ K, the first peak in $G_{\rm d}(r,t)$ broadens on the timescale of
paddle-wheel motion, indicating that Ag+ and the MLWFCs are correlated. This
correlation in cation and lone pair dynamics is reflective of electronic
paddle-wheels. At low temperatures, when AgI is not a SSIC, $G_{\rm d}(r,t)$
hardly changes shape, reflecting a lack of cation diffusion and anion lone
pair rotation.

Cation motion in traditional molecular SSICs often occurs through discrete
hops induced by paddle-wheel motion of molecular anions. Cations rattle in a
binding site and then very rapidly move to another site during a hop. In
contrast, we find that Ag+ dynamics are diffusive in AgI. The Ag+ trajectories
shown in Fig. 2a show no evidence of the rapid jumps indicative of hopping;
the trajectories are smooth and diffusive.

The diffusive nature of cation dynamics in AgI at 750 K is further supported
by $G_{\rm d}(r,t)$ and $G_{\rm s}(r,t)$ shown in Fig. 2d,e, while a lack of
dynamics is suggested by those for AgI at 100 K, Fig. 2g,h. At $t=0$, the
cationic $G_{\rm d}(r,t)$ is equal to the Ag-Ag $g(r)$. As time progresses, a
peak smoothly grows in near $r=0$, the initial location of a Ag+. This peak at
the origin at later times is consistent with a cation leaving its initial
location and another cation taking its place. The smooth growth of this peak
with time suggests a lack of hopping-like motion. A similar lack of hopping
can be observed in $G_{\rm s}(r,t)$, Fig. 2e. Hopping manifests in $G_{\rm
s}(r,t)$ through the appearance of well-defined peaks at typical hopping
distances. Peaks are not observed in the cation $G_{\rm s}(r,t)$ and we
instead find smooth distributions. However, the motion is not purely
diffusive, which would result in the dashed Gaussian distributions. The
deviations from Gaussian diffusion suggest that the dynamics are collective
and dynamically heterogeneous, with some ions moving slower and faster than
Gaussian expectations. Collective cation dynamics can be expected, because a
cation cannot diffuse to another binding site unless another cation leaves
that site.

The mechanism of collective dynamics in AgI is suggested by our simulations,
and it involves a combination of translational and rotational dynamics. We
propose three main contributions that may occur nearly simultaneously, shown
in Fig. 3. Thermally-induced translational fluctuations weaken the strength a
Ag+-I- interaction. The Ag+ is then coordinated by a different I-, and the
newly coordinating I- rotates its lone pairs to form this new interaction. The
rotating I- coordinates other cations, and one of these coordinated cations
diffuses by accompanying the lone pair rotation.

The diffusive dynamics of the Ag+ cations observed here represents a
fundamental difference between electronic paddle-wheels and molecular paddle-
wheels in SSICs. Cation motion in molecular paddle-wheels often occurs through
hopping, but the motion is diffusive in electronic paddle-wheels. Despite this
difference, we expect the analogy between electronic and molecular paddle-
wheels to be useful, such that many of the concepts developed for molecular
paddle-wheels may be transferable to electronic paddle-wheels in monatomic
SSICs.

To summarize, our molecular simulations suggest that the SSIC AgI exhibits
hidden electronic disorder, and this dynamic electronic disorder leads to
electronic paddle-wheels — anion lone pair rotations that couple to cation
diffusion. Electronic paddle-wheels may explain important mysteries about
conduction mechanisms of SSICs composed of monatomic ions. For example,
collective Ag+ diffusion was recently identified using topological analysis
22. Our proposed diffusion mechanism involving lone pair rotations is
consistent with this collective diffusion and provides an electronic origin
for these collective dynamics. We also anticipate that the cation-anion
coupling due to electronic paddle-wheels may be responsible for the anharmonic
coupling observed in spectroscopy 15, similar to anharmonic couplings observed
in molecular plastic crystals 23.

The ionic diffusion mechanism predicted here is analogous to the paddle-wheel
motion observed in molecular solid-state electrolytes. We expect that this
analogy can be leveraged to adapt our understanding of molecular electrolytes
to monatomic electrolytes. As a result, electronic paddle-wheels expand
unifying frameworks for crystalline, amorphous, and liquid molecular
electrolytes to include SSICs composed of monatomonic ions 24. Similarly,
electronic paddle-wheels naturally expand the concept of frustration in SSICs
to include dynamical electronic frustration resulting from lone pair
rotations, in analogy to the dynamical frustration caused by molecular paddle-
wheels 25; 26. This paddle-wheel mechanism has been leveraged to design better
molecular electrolytes 27; 11; 28. Consequently, understanding electronic
paddle-wheels will create rational, electronic-scale design principles that
leverage the coupling between lone pair rotations and cation diffusion to
discover new superionic conductors.

## Acknowledgements

We acknowledge the Office of Advanced Research Computing (OARC) at Rutgers,
The State University of New Jersey for providing access to the Amarel cluster
and associated research computing resources that have contributed to the
results reported here.

## Methods

We performed Born-Oppenheimer molecular dynamics simulations with Kohn-Sham
density functional theory using the CP2K software package employing a hybrid
Gaussian and plane wave (GPW) approach for representing electron density29. By
doing so, we neglect the fast electronic modes that require a detailed
description of quantum dynamics, and instead focus on the slowest electronic
modes that are coupled to the motion of the nuclei and most relevant to ion
conduction. We employed the molecularly optimized double-$\zeta$ with valence
polarization (DZVP) basis set30 with cutoffs of 450 Ry for plane wave energy
and 60 Ry for the reference grid. Goedecker-Teter-Hutter pseudopotentials
compatible with employed basis sets were used to represent core electrons31;
32, while the valence electrons were represented explicitly using the Perdew-
Burke-Ernzerhof (PBE) generalized gradient approximation to the exchange-
correlation functional 33. The $\mathbf{k}$-point sampling was performed only
at the $\Gamma$-point. Long-range dispersion interactions were included using
Grimme’s D3 van der Waals correction34. Our simulations used a $3\times
3\times 3$ supercell of a AgI unit cell with edge length 5.0855 Å 6. Starting
structures were equilibrated for at least 50 ps at 100 K and 750 K in the
canonical ensemble (NVT) using the canonical velocity rescaling (CSVR)
thermostat35. For the calculation of dynamic properties, equations of motion
were propagated for 80 ps in the microcanonical ensemble using velocity Verlet
integrator with a timestep of 0.5 fs. The last 40 ps of these trajectories
were used for the analysis. Maximally localized Wannier functions were
obtained by minimizing their spreads within CP2K19; 36.

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# The one-legged K-theoretic vertex of fourfolds from 3d gauge theory

Nicolò Piazzalunga NHETC, Rutgers University. Research supported by the US
Department of Energy under grant DE-SC0010008.

###### Abstract

We present formulas for the K-theoretic Pandharipande-Thomas vertex of
fourfolds, for the case of one non-trivial leg. They are obtained from
computations in a three-dimensional supersymmetric gauge theory, where we
identify the field content and boundary conditions that correspond to the
vertex with tautological insertions.

##  1 Introduction

Recently, there’s been some interest in gauge theories on Calabi-Yau
fourfolds. These make sense in string theory, as theories on D-branes when
gravity is decoupled, so they’re automatically twisted, and are ultimately
justified as non-commutative gauge theories, by taking the Seiberg-Witten
limit. Generally speaking, the solitons correspond to D-branes wrapping
cycles, and partition functions are Witten indices counting their bound
states.

For toric manifolds, there exists a general machinery due to Nekrasov that
allows to localize such theories and express their partition functions in
terms of simple building blocks, akin to the instanton partition function of
4d $\mathcal{N}=2$ theory. In the present case, this is replaced by a four-
valent K-theoretic vertex. This is where most of the complexity of such
Donaldson-Thomas counts lies.

Another source of complexity, which is new to four complex dimensions, is
that, although the general machinery is more or less straightforward to apply,
and it produces a decomposition into vertices-edges-faces dictated by the
toric polyhedron, one has to choose signs (orientations) for each of these,
which are not easy to fix. In other words, the selfduality equations involve a
real representation of $SU(4)$, and one has to take _half_ of the obstruction
complex. This is mirrored in the fact that one cannot relax the CY condition.

Rather than discussing the general case, here we focus on the one-legged
vertex, where three legs are trivial. This can be reformulated in terms of
K-theoretic quasimaps counts, as maps from the sphere $\mathbb{P}^{1}$ into
the Hilbert scheme of points of $\mathbb{C}^{3}$. While the gauge theory is
naturally DT theory, the quasimap counts are equivalent to Pandharipande-
Thomas theory: this, while being closely related to DT theory, has the
advantage of being more economical, as it factors out contributions from pure
D0-branes.

The corresponding theory for threefolds (not necessarily CY) has been studied
in great generality by the Okounkov school [Oko15], building on prior
milestones [Nak94, NS15]. From there, it is clear that the object of our study
can be looked at from various angles (with some subtle differences): GLSM
partition functions on the disk and D-brane central charges [HR13],
holomorphic blocks [BDP14], and Givental’s I-function [Giv96].

Recently, the mathematical foundations of quasimap theory relevant to DT4 have
been developed [CZ23], building on previous works [CK10, Par21]. Physically,
the K-theoretic PT vertex with one non-trivial leg, with plane partitions
classifying fixed points on $\mathrm{Hilb}^{k}(\mathbb{C}^{3})$, and with
tautological insertions, should be equal [KW13] to the partition function of a
topologically twisted 3d $\mathcal{N}=2$ theory on the cigar. Here, we argue
that one can use the physical $\mathcal{N}=2$ theory on
$D\times\mathbb{S}^{1}$, with $D$ being the disk, to compute certain
observables of the topologically twisted theory, upon making the correct
identifications [JM20]. In the physical theory, it’s clear that tautological
insertions, corresponding to fundamental matter in DT theory, are chiral
fields with Dirichlet boundary conditions. The point is that one needs to
cancel anomalies for Chern-Simons terms, and one way to do it is via anomaly
inflow from the boundary [YS20]. This is crucial to obtain the correct measure
for the PT vertex, which is our main result in eq. 4.3.

Despite their physical origin, our formulas are mathematically rigorous, and
the integration contour is completely determined, so we do not need to make
any ansatz for it.

##  2 The setup

Let us consider stable quasimaps from $C=\mathbb{P}^{1}$ to
$\mathrm{Hilb}^{k}(\mathbb{C}^{3})$, twisted by the line bundles $L_{a}$ over
$C$ of degree $l_{a}$, such that $K_{C}=L_{1}\otimes L_{2}\otimes L_{3}$, and
$l_{1}+l_{2}+l_{3}=-2$. Denote by $N$ the (fixed) vector bundle of rank $n=1$
over $C$. Let $G=U(k)$. A quasimap is the datum of a principal $G$-bundle over
$C$ and a section $f=(B_{1},B_{2},B_{3},I)$ of the associated rank-$k$ complex
vector bundle $K$, where $B_{a}\in H^{0}(C,\operatorname{End}(K)\otimes
L_{a})$ and $I\in H^{0}(C,\operatorname{Hom}(N,K))$. The degree is $\deg
f=c_{1}(K)$.

The corresponding topological field theory is obtained by twisting the
$\mathcal{N}=(2,2)$ theory on $C$. The twist identifies the surviving
R-symmetry with the $U(1)$ Lorentz isometry on $C$. The fields
$X=(B_{1},B_{2},B_{3},I)$ of the 2d cohomological theory satisfy equations

$D_{\bar{z}}X=-G^{-1}\frac{\partial\bar{W}}{\partial\bar{X}}$ (2.1)

which follow from the superpotential $W=\operatorname{tr}B_{1}[B_{2},B_{3}]$.
The fields $B_{a}$ have R-charges $-l_{a}$, so $W$ is quasi-homogeneous of
degree two. Here $z$ is a local coordinate on $C$, and $G$ a Kähler metric. We
can either quotient by $GL(k)$, or quotient by $U(k)$ if we replace the
stability condition by the D-term equation

$*F+\mu(X,\bar{X})=\xi$ (2.2)

where $F$ is the field strength on $C$, and $\mu$ the momentum map associated
to the symplectic form determined by $G$. The K-theoretic counts correspond to
the three-dimensional lift of the 2d twisted theory.

We consider quasimaps with a non-singular boundary condition at
$\infty\in\mathbb{P}^{1}$, and compute the K-theoretic push-forward of
tautological insertions using the evaluation map at $\infty$ and the
orientation defined by the virtual fundamental class, and weighted by $z^{\deg
f}$. We work with the physical three-dimensional $\mathcal{N}=2$ theory on
$D\times\mathbb{S}^{1}$, with gauge group $U(k)$ and the following field
content: 3d vector, 3d chirals with Neumann boundary conditions ($B_{a}$,
$I$), 3d chiral with Dirichlet boundary conditions (corresponding to
fundamental matter, i.e. tautological insertions in the quasimap theory), zero
bare Chern-Simons level, and (crucially) 2d vector to cancel the anomaly. The
$\Omega$-background acts on $\mathbb{R}^{2}\times\mathbb{S}^{1}$ as
$q=e^{\beta\epsilon}$, where $\beta$ is the radius of the circle
$\mathbb{S}^{1}$, and it’s natural to identify this parameter with the
fugacity for the sum of $U(1)$ R-charge and rotations generators, denoted by
$q^{2}$ in ref. [YS20].

##  3 The computation

We work equivariantly with respect to all symmetries. The torus
$(\mathbb{C}^{*})^{3}$ acts on $\mathrm{Hilb}^{k}(\mathbb{C}^{3})$, with fixed
points in one-to-one correspondence with plane partitions of size $k$. We turn
on twisted masses $q_{a}=e^{\beta\epsilon_{a}}$ for $B_{a}$. For the disk, we
do not need to worry about R-charges (while they matter for the sphere case),
as we can reabsorb them in the equivariant parameters. We must impose the CY4
condition $q\prod_{a=1}^{3}q_{a}=1$. The equivariant parameter
$a=e^{\beta\alpha}$ acts on the framing node $N$, while $\mu=e^{\beta m}$ is a
mass for the fundamental matter.

For $|q|<1$, the K-theoretic disk partition is computed by the residue

$Z_{k}=\frac{1}{k!}\oint_{|x_{i}|=1}\phi(-T_{k})\,\prod_{i\neq
j}\theta(x_{i}/x_{j})\prod_{i=1}^{k}\frac{dx_{i}}{2\pi ix_{i}}x_{i}^{-\xi}$
(3.1)

where the functions $\phi$ and $\theta$ are defined in appendix A, and

$T_{k}=\sum_{i}x_{i}(a^{-1}-{\mu}^{-1})+\sum_{i\neq
j}\frac{x_{i}}{x_{j}}(q_{1}^{-1}+q_{2}^{-1}+q_{3}^{-1}-1)$ (3.2)

is determined by the field content. We require that
$|a|,|q_{1}|,|q_{2}|,|q_{3}|>1$, so that the denominator is convergent.

### Contour

For $\xi>0$, the contour picks up poles at one of the hyperplanes

$x_{i}=aq^{-d_{1}},\quad d_{1}\geq 0$ (3.3)

and $k-1$ of the hyperplanes

$\frac{x_{r}}{x_{s}}q_{a}^{-1}=q^{-d_{p}},\quad d_{p}\geq
0,\,a=1,2,3,\,p=2,\ldots,k$ (3.4)

such that $x=x_{cl}$, defined by taking all $d=0$, is the character of a plane
partition in $q_{1}$, $q_{2}$, $q_{3}$. (We call any such $x_{cl}$ a classical
pole.)

For reference, the disk partition function [HR13] reads

$F_{k}=\frac{1}{k!}\oint\prod_{i=1}^{k}\frac{\Gamma(\phi_{i}-\alpha)}{\Gamma(\phi_{i}-m)}\prod_{i\neq
j}\frac{\prod_{a=1}^{3}\Gamma(\phi_{i}-\phi_{j}-\epsilon_{a})}{\Gamma(\phi_{i}-\phi_{j})}\prod_{i=1}^{k}\frac{d\phi_{i}}{2\pi
i}\,e^{-\phi_{i}\xi}$ (3.5)

##  4 The conjecture

Let us define the content of plane partition $\pi$ corresponding to pole
$x_{cl}$

$\mathrm{cont}(\pi)=\prod_{i=1}^{k}x_{cl,i}=\prod_{(i,j,k)\in\pi}aq_{1}^{i-1}q_{2}^{j-1}q_{3}^{k-1}$
(4.1)

and the perturbative prefactor

$Z_{pert}(\pi)=\mathrm{cont}(\pi)^{-\xi}\,\phi(k-T_{k}(x_{cl})-qk)\,\prod_{i\neq
j}\theta(x_{cl,i}/x_{cl,j})$ (4.2)

Upon identifying $z=q^{\xi}$ as the box-counting parameter, we claim that

$Z_{k}=\sum_{|\pi|=k}Z_{pert}(\pi)\,V_{PT}(0,0,0,\pi)$ (4.3)

where $V_{PT}$ is the vertex of PT theory and the sum runs over plane
partitions. We checked this claim against conjectural formulas for the general
four-vertex of Donaldson-Thomas theory in the upcoming work with N.Nekrasov,
for the cases $|\pi|\leq 2$ and the first few instantons.

##  5 Conclusion and outlook

We presented formulas for the K-theoretic PT vertex in the case of one non-
trivial leg, constructed from the supersymmetric gauge theory in three
dimensions. We exploited the physical theory to make sense of fundamental
matter in terms of boundary conditions, and used the physically-motivated
anomaly cancellation to get the correct vertex measure. We recovered the
$\Omega$-background of the twisted theory from the fugacity for the
appropriate combination of R-charge and Lorentz rotation.

Although we focused on the important example of
$\mathrm{Hilb}(\mathbb{C}^{3})$, it is possible to extend this procedure to
other quivers with potential, by either computing the virtual tangent space or
equivalently the corresponding field content.

One topic that deserves further investigations are the difference equations
satisfied by the vertex. The first type comes from the operation of shifting
the Kähler parameter, $\xi\to\xi+1$, which corresponds to inserting the
observable $\prod_{i}x_{i}\sim\det K$. They work in the same fashion as the
abelian case. The second set is more interesting, and it comes from acting on
the equivariant parameters, as $q_{a}\to q_{a}q^{-l_{a}}$, $q\to q^{-1}$, from
the degeneration idea. The $R$-matrix for $\mathrm{Hilb}^{k}(\mathbb{C}^{3})$
is a $p_{k}\times p_{k}$ matrix, where $p_{k}$ is the number of plane
partitions of size $k$, but this is not enlightening, unless we know that it
can be reduced to smaller blocks.

### Acknowledgments

We thank D.-E.Diaconescu, N.Nekrasov and M.Romo for inspiring discussions, as
well as W.-y.Chuang, M.Porta and F.Sala for sharing their knowledge on the
mathematical foundations. We thank A.Grassi for playing with us on the $k=1$
case, where we’ve been stuck for some time.

##  Appendix A Definitions and elementary properties

Let $(a;q)_{n}=\prod_{k=0}^{n-1}(1-aq^{k})$, with $(a;q)_{0}=1$. Let
$\phi(a)=(a;q)_{\infty}$, and extend it on K-theory classes by multiplication.
It satisfies the difference equation $\phi(qx)=\frac{1}{1-x}\phi(x)$.

We extend the definition to all integers by the relation

$(x;q)_{p}:=\frac{\phi(x)}{\phi(xq^{p})}=\begin{dcases*}(x;q)_{p}&if $p\geq
0$\\\ \frac{1}{(xq^{p};q)}_{-p}&if $p<0$\end{dcases*}$ (A.1)

Let us denote $\pi_{d}=\prod_{i=1}^{d}(1-q^{-i})$ for $d\geq 0$ and
$(x)_{p}=(x;q)_{p}$ for $p\in\mathbb{Z}$.

Let us define $\theta(y):=\theta(y;q)=\phi(y)\phi(y^{-1}q)$. This satisfies
the relation

$\theta(yq^{p})=\theta(y)(-1)^{p}y^{-p}q^{-p(p-1)/2}$ (A.2)

for $p\in\mathbb{Z}$, as well as the relation
$\theta(y^{-1})=-y^{-1}\theta(y)$.

The residue evaluation, for $k\in\mathbb{Z}_{\geq 0}$, gives

$\operatorname{Res}_{y=q^{-k}}\frac{dy}{2\pi
iy}\phi(-y)=(-1)\frac{\phi(-q)}{\prod_{i=1}^{k}(1-q^{-i})}=(-1)^{k+1}\frac{\phi(-q)q^{k(k+1)/2}}{(q;q)_{k}}$
(A.3)

## References

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$\\{e_{\lambda^{\prime}}\mid\lambda\in\mathcal{P}_{k}\\}\text{ and
}\\{s_{\lambda}\mid\lambda\in\mathcal{P}_{k}\\}$

where for any partition $\lambda=(\lambda_{1},\ldots,\lambda_{k})$ we have
$e_{\lambda}=e_{\lambda_{1}^{\prime}}\cdots e_{\lambda_{k}^{\prime}}$ and
$s_{\lambda}$ is the Schur polynomial associated to $\lambda$. For any
positive integer $l$, consider $I_{k,l}$ the ideal of $\Lambda_{k}$ defined by

$\mathcal{I}_{k,l}=\langle s_{\lambda}\mid\lambda\text{ has }l+1\text{ columns
of height less than }k\rangle.$

Write $\mathcal{P}_{k}^{l}$ the set of partitions in $\mathcal{P}_{k}$ with at
most $l$ columns of height less than $k$.

######  Lemma 7.4.

(See [18]) We have $I_{k,l}=\langle h_{l+1},\ldots,h_{l+k-1}\rangle$.

We define the ideals

$I_{k,l}^{q}=\langle h_{l+1},\ldots,h_{l+k-1},h_{l+k}=(-1)^{k-1}q\rangle\text{
and }J_{k,l}=\langle h_{l+1},\ldots,h_{l+k-1},e_{k}=1\rangle$

and set

$\mathcal{Q}_{k,l}:=\Lambda_{k}/\mathcal{I}_{k,l},\mathcal{A}_{k,l}^{q}:=\Lambda_{k}/I_{k,l}^{q}\text{
and }\mathcal{S}_{k,l}:=\Lambda_{k}/J_{k,l}.$

Clearly the algebras $\mathcal{Q}_{k,l}$ and $\mathcal{A}_{k,l}^{q}$ are
isomorphic by sending $h_{l+k}\operatorname{mod}\mathcal{I}_{k,l}$ on
$(-1)^{k-1}q$. They were considered in [2] and [18] from where we can extract
the following theorems gathering some important properties of these algebras.

######  Theorem 7.5.

(see [18]) Set $n=l+k$.

  1. (1)

The set
$\mathfrak{B}_{k,l}^{a}=\\{b_{\lambda}:=s_{\lambda}\operatorname{mod}\mathcal{A}_{k,l}^{q}\mid\lambda\subset
l^{k}\\}$ is a $\mathbb{Z}[q]$-basis of $\mathcal{A}_{k,l}^{q}$ so
that$\dim\mathcal{A}_{k,l}^{q}=\binom{n}{k}$.

  2. (2)

When $q=1$, we have in $\mathcal{A}_{k,l}^{1}$ for any $\lambda\subset l^{k}$

$b_{(1)}\cdot b_{\lambda}=\sum_{\nu\subset l^{k}}b_{\nu}$

where $\nu$ is obtained by adding one box to $\lambda$ or deleting the first
row and column when $\lambda_{1}=l$ and $\lambda$ has $k$ rows (i.e. by
removing the unique possible hook of length $l+k-1$ if any). In particular, in
the algebra $\mathcal{A}_{k,l}^{1}$, the matrix of the multiplication by
$b_{(1)}$ in the basis $b_{\lambda}$ is the adjacency matrix of the KR-crystal
$B_{k,n}$.

  3. (3)

The structure constants associated to $\mathfrak{B}_{k,l}^{a}$ in the algebra
$\mathcal{A}_{k,l}^{1}$ are nonnegative integers.

######  Corollary 7.6.

The graph $B_{k,n}^{a}$ with $n=l+k$ is positively multiplicative for the
algebra $\mathcal{A}_{k,l}^{q}$ with the basis $\mathfrak{B}_{k,l}^{a}$.

Now, let us turn to the algebra
$\mathcal{S}_{k,l}:=\Lambda_{k}/J_{k,l}=\mathcal{Q}_{k,l}/\langle
e_{k}=1\rangle$.

######  Theorem 7.7.

(See [1, 2]) Set $n=k$. In the algebra $\mathcal{S}_{k,l}$ the following
statements hold.

  1. (1)

The set
$\mathfrak{B}_{k,l}^{s}=\\{b_{\lambda}\operatorname{mod}J_{k,l}\mid\lambda\subset
l^{k-1}\\}$ is a $\mathbb{Z}$-basis of $\mathcal{S}_{k,l}$ so that
$\dim\mathcal{S}_{k,l}=\binom{l+k-1}{l}$.

  2. (2)

In $\mathcal{S}_{k,l}$, we have for any $\lambda\subset l^{k-1}$

$b_{1}\cdot b_{\lambda}=\sum_{\nu\subset l^{k-1}}b_{\nu}$

where $\nu$ is obtained by adding one box to $\lambda$, next by deleting a
column of height $k$ if this makes appear one. In particular, in the algebra
$\mathcal{S}_{k,l}$, the matrix of the multiplication by $b_{1}$ in the basis
$\mathfrak{B}_{k,l}^{s}$ is the adjacency matrix of the KR-crystal labelled by
the rows of length $l$ on $\\{1,\ldots,k\\}$.

  3. (3)

The structure constants associated to $\mathfrak{B}_{k,n}^{s}$ in the algebra
$\mathcal{S}_{k,l}$ are nonnegative integers.

######  Corollary 7.8.

The graph $B_{l,n}^{s}$ with $n=k$ is positively multiplicative for the
algebra $\mathcal{S}_{k,l}$ with the basis $\mathfrak{B}_{k,n}^{s}$.

###  7.5. An example: the graph $B_{l,2}^{s}$

We consider $KR$-graphs $B_{l,2}^{s}$ with one row and $l$ columns filled with
integers in $\\{1,2\\}$. The first examples of these graphs are :

1

2

1&1

1&2

2&2

1&11

1&12

1&22

2&22

The adjacency matrix of $B_{l,2}^{s}$ is the $(l+1)\times(l+1)$-matrix

$A_{l}=\begin{pmatrix}0&1&0&0&\ldots&0\\\ 1&0&1&0&\ldots&0\\\
0&1&0&1&\ldots&0\\\ 0&0&\ddots&\ddots&\ddots&0\\\ 0&0&\ldots&1&0&1\\\
0&0&\ldots&0&1&0\end{pmatrix}.$

The algebra $\mathcal{S}_{2,l}$ is by construction equal to the quotient

$\mathrm{Sym}[x_{1},x_{2}]/(e_{2}=1,h_{l+1}=0).$

Since $e_{2}=x_{1}x_{2}$, we just have $x_{2}=x_{1}^{-1}$ and by setting
$x=x_{1}$, the elements of $\mathrm{Sym}[x_{1},x_{2}]/(e_{2}=1)$ can be
regarded as the $\mathbb{Q}$-algebra $\mathcal{L}[x,x^{-1}]$ of Laurent
polynomials $P$ in the indeterminate $x$ such that $P(x^{-1})=P(x)$. Now in
$\mathcal{L}[x,x^{-1}]$, we have

$h_{a}(x,x^{-1})=\sum_{1\leq i\leq j\leq l\mid
i+j=a}x^{i}x^{-j}=x^{-a}\sum_{i=0}^{a}x^{2i}=\frac{x^{a+1}-x^{-a-1}}{x-x^{-1}}.$

Therefore $\mathcal{S}_{2,l}$ is isomorphic to the algebra
$\mathcal{L}[x,x^{-1}]/(x^{l+2}-x^{-l-2)})$. A simple computation shows that

$h_{i}(x,x^{-1})\times h_{j}(x,x^{-1})=\sum_{k=0}^{j}h_{i+j-2k}(x,x^{-1})$

Write writing $b_{i},i=0,\ldots,l$ for the image of $h_{a}(x,x^{-1})$ in
$\mathcal{L}[x,x^{-1}]/(x^{l+2}-x^{-(l+2)})$ and define
$\mathfrak{B}=\\{b_{0}=1,\ldots,b_{l}\\}$. By setting
$\overline{x}=x\operatorname{mod}(x^{l+2}-x^{-(l+2)})$ we get
$\overline{x}^{l+2}=\overline{x}^{-l-2}.$ Therefore, for any $0\leq a\leq
l-1$, one has since

$b_{l+1+a}=\frac{\overline{x}^{l+a+2}-\overline{x}^{-l-a-2}}{\overline{x}-\overline{x}^{-1}}=\frac{\overline{x}^{-l-2+a}-\overline{x}^{l+2-a}}{\overline{x}-\overline{x}^{-1}}=-b_{l+1-a}.$

This gives if we assume $i\geq j$

$b_{i}b_{j}=b_{i-j}+b_{i-j+2}+\cdots+b_{i+j}\text{ if }i+j\leq l$

and

$\displaystyle b_{i}b_{j}$
$\displaystyle=\sum_{k=0}^{j}b_{i+j-2k}=b_{i-j}+\cdots+b_{l}+0-b_{l}-\cdots-
b_{i+j-l-1}$ $\displaystyle=b_{i-j}+\cdots+b_{2(l+1)-i-j-1}\text{ if }i+j>l.$

This can be summarize by the rules

$b_{i}b_{j}=\begin{array}[c]{l}b_{\left|i-j\right|}+\cdots+b_{i+j}\text{ if
}i+j\leq l,\\\ b_{\left|i-j\right|}+\cdots+b_{2(l+1)-i-j-1}\text{ if
}i+j>l.\end{array}$

One may recognize the fusion rules for the fusion algebra $\widehat{su}_{2}$
(see [7]) at level $l$. This construction is related to Conformal Field
Theory. The Perron-Frobenius eigenvalue of the adjacency matrix of
$B_{l,2}^{s}$ is $\lambda=2\cos\frac{\pi}{l+2}$ with normalized associated
eigenvector

$\frac{1}{\sin\frac{\pi}{l+2}}(\sin\frac{a\pi}{l+2})_{1\leq a\leq l+1}.$

It is also possible to determine the set $I_{m}$ of maximal indices and its
group structure studied in Section 5 and Theorem 5.11. Indeed, the previous
fusion rules shows that $b_{0}b_{j}=1$ and $b_{l}b_{j}=b_{l-j}$ for any
$j=0,\ldots,l$. Also when $i=1,\ldots,l-1$ the action of $b_{i}$ does not
yields a generalized permutation of the basis $\mathfrak{B}$. Thus
$I_{m}=\\{0,l\\}$ and the group structure is given by

$l\cdot l=0,l\cdot 0=0\cdot l=l,0\cdot 0=0.$

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$\downarrow$ Model | Test on $\rightarrow$ | Card | Coin | Dice |  | Card | Coin | Dice |  | Card | Coin | Dice
BERT |  $<$.001 |  $<$.001 | .011 |  |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001
RoBERTa |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001
DeBERTa |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 | .011 |  |  $<$.001 |  $<$.001 |  $<$.001
BigBird |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001

Table 17: P values corresponding to the ‘Strict’ results in Figure 3, when
comparing to random performance of 12.5%.

Fine-tune on $\rightarrow$ | Card |  |  |  | Coin |  |  |  | Dice |  |
---|---|---|---|---|---|---|---|---|---|---|---
$\downarrow$ Model | Test on $\rightarrow$ | Card | Coin | Dice |  | Card | Coin | Dice |  | Card | Coin | Dice
BERT |  $<$.001 | 1.00 | 1.00 |  |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001
RoBERTa |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001
DeBERTa |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001
BigBird |  $<$.001 | 1.00 | .437 |  | 1.00 |  $<$.001 |  $<$.001 |  | .222 | .222 |  $<$.001

Table 18: P values corresponding to the ‘Positive Gain’ results in Figure 3,
when comparing to random performance of 25%.

Fine-tune on $\rightarrow$ | Card |  |  |  | Coin |  |  |  | Dice |  |
---|---|---|---|---|---|---|---|---|---|---|---
$\downarrow$ Model | Test on $\rightarrow$ | Card | Coin | Dice |  | Card | Coin | Dice |  | Card | Coin | Dice
BERT |  $<$.001 |  $<$.001 | .764 |  |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001
RoBERTa |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001
DeBERTa |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001
BigBird |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001 |  |  $<$.001 |  $<$.001 |  $<$.001

Table 19: P values corresponding to the ‘Non-Negative Gain’ results in Figure
3, when comparing to random performance of 25%.

### Exact values for RQ3

Fine-tune on $\rightarrow$ | Card |  |  |  | Coin |  |  |  | Dice |  |
---|---|---|---|---|---|---|---|---|---|---|---
Test on $\rightarrow$ | Card | Coin | Dice |  | Card | Coin | Dice |  | Card | Coin | Dice
$\downarrow$ Model | Train Set(Acc)/Test Set(Acc)
BERT | 100/100 | 75/76 | 75/77 |  | 75/82 | 100/100 | 50/50 |  | 85/83 | 74/69 | 100/99
RoBERTa | 100/100 | 75/75 | 75/75 |  | 75/72 | 100/97 | 50/42 |  | 75/73 | 75/73 | 100/96
DeBERTa | 100/98 | 96/88 | 99/93 |  | 100/95 | 100/97 | 75/67 |  | 72/58 | 100/83 | 85/85
BigBird | 99/90 | 54/56 | 71/67 |  | 81/59 | 100/100 | 100/97 |  | 65/63 | 51/50 | 97/91

Table 20: Exact quantitative results corresponding to the visualization in
Figure 2