# Membrane-protein interactions hold the key to understanding amyloid
formation

John E. Straub1 and D. Thirumalai2 1Department of Chemistry, Boston
University, Boston, MA, 02215
2Department of Chemistry and Biochemistry, and Biophysics Program, Institute
for Physical Sciences and Technology, University of Maryland, College Park, MD
20742

Spurred by the appreciation that protein aggregation, leading to amyloid
fibril formation, is linked to a growing list of diseases there has been
intense effort to understand the factors that trigger association between
peptides. This vastly complex field has attracted a variety of approaches
ranging from mutational studies at the genetic level to characterization of
events at the molecular level of specific proteins. It is unclear if
biophysical approaches used to study protein folding in quantitative terms
will have a major impact on leading to a cure or management of any amyloid
disease, including Alzheimer’s Disease (AD). Nevertheless, it is clear that
rigorous biophysical methods are needed to gain a deeper understanding of the
routes to amyloid formation. After nearly fifteen years of in vitro
experimentation and simulations, certain general principles that govern
oligomer (suspected to be the cause of toxicity) and amyloid formation have
emerged. A few of the most important findings are: (1) All proteins,
regardless of whether their aggregation is related to any specific disease,
can form amyloid fibrils with characteristic cross $\beta$ structure Chiti and
Dobson (2006). This finding implies that under suitable conditions favoring
aggregation the folded monomeric state is unlikely to be the most stable.
Although the possibility that folded proteins are metastable was suggested
long ago Honeycutt and Thirumalai (1990), only recently it has been argued
that under nominal protein concentrations found in cells the native state is
metastable with respect to the fibril state Baldwin et al. (2011). (2) The
propensity of a polypeptide chain to aggregate is encoded in the the free
energies of the spectrum of states of the monomer Tarus et al. (2006). There
is a direct correlation between the probability of accessing such aggregation-
prone states and the rate of amyloid formation Li et al. (2010). (3) Although
there are exceptions, solid state NMR experiments Tycko and Wickner (2013) on
a number of systems, show that in the amyloid fibril state the polypeptide
chains are arranged as parallel $\beta$-sheets, which given the conclusion
summarized in (1) implies that the ground state of interacting peptides has
universal structural features. (4) Despite the possible universal structure of
the fibrils, precise sequence plays a major role in the rate of protein
aggregation as well as in the stability of the final product Goldschmidt et
al. (2010). Dramatic variations in aggregation rates of mutants of A$\beta$
peptides Yamamoto et al. (2004), implicated in AD, validates the importance of
sequence.

The general principles that have been discovered, using well-defined systems
summarized above are of great significance, and establish the power of
biophysical methods. However, the studies leading to these conclusions do not
even approximately model the process of in vivo aggregation. For example, it
is well-known that in order to understand amyloid formation in vivo it is
important to consider interactions of aggregating proteins with membranes,
which in the biophysics community has received little attention. In the
interesting Perspective appearing in this issue Bucciantini et. al.
Bucciantini et al. (2014) review the specific roles that membranes play in
promoting oligomer formation in amyloidogenic proteins with special emphasis
on A$\beta$ peptides. The authors have managed to navigate a substantial
portion of the still growing literature to summarize the key membrane
characteristics, which are involved in promoting or suppressing oligomer
formation. Based on general considerations it is clear that membrane-mediated
aggregation should depend on physical properties, such as net charge per unit
area and hydrophobicity and hydrophilicity. Indeed, it has been found that
membranes composed of anionic phospholipids (such as phosphatidylserine and
phosphatidyglycerol) catalyze the formation of amyloids Necula et al. (2003).
The mechanical properties of the fluid membranes also affect interaction
between proteins, as demonstrated using continuum theories. Thus, cholesterol
(Ch) or crowding agents that effect the overall membrane fluidity also affects
aggregation Kim et al. (2005), as summarized succinctly by Bucciantini et al.
(2014). It is likely that theoretical ideas in the soft matter field could be
profitably used to predict generic aspects of membrane-mediated interpeptide
interaction.

The specific details of membrane composition, which cannot be easily taken
into account using using only the general physicochemical properties, requires
a deeper understanding of biological membranes. For example, biological
membranes form lipid rafts, which are best pictured as heterogeneous
structures. In rafts in neuronal cells, GM1 (see Fig. 2 in Bucciantini et al.
(2014)) is the most abundant ganglioside, and is implicated in AD, Parkinson,
and Huntington diseases. Interaction of GM1 with A$\beta$ might be involved in
promoting A$\beta$ aggregation. Further complicating matters, Ch, whose role
in affecting protein aggregation is not fully understood, appears to encourage
the formation of islands that are rich in GM1, reminiscent of microphase
separation. Based on a study of several model systems Tashima et al. (2004),
it has been asserted that in a lipid mixture composition resembling that found
in the cerebral cortex, A$\beta$ peptides were rapidly released resulting in
amyloid fibril formation. However, in membranes devoid of Ch and GM1,
aggregation did not occur, suggesting that both Ch and GM1 control A$\beta$
aggregation in neuronal cells. In an insightful study, it has been shown that
the Ch effect depends roughly on $\lambda$ = [Ch]/[Pl], where [Ch]([PL]) is
the concentration of Ch (phospholipids). At low $\lambda$ values A$\beta$
aggregation is promoted whereas at higher values there is increased insertion
of A$\beta$ into the membranes, which suppresses aggregation. Therefore, it is
important to examine the phase diagram of the ternary (minimum) of Ch, lipids,
and A$\beta$ before any general conclusion can be drawn. It is worth
emphasizing that lipid composition, which greatly complicates the picture, is
an important variable. Because of this model membrane systems are extremely
useful in obtaining insights into the aggregation process.

A critical factor that is receiving increasing attention is the determination
of the structures and dynamics of the type I transmembrane Amyloid Precursor
Protein (APP) found in neural and non-neural cells. The cleavage of APP
resulting in A$\beta$-peptide of varying lengths is achieved by secretases
Wolfe and Guenette (2007). Cleavage occurs in several steps. $\beta$-secretase
cleaves APP at the $\beta$-site, and the extracellular domain of APP
disassociates from the remaining protein (APP-C99). Subsequently,
$\gamma$-secretase cleaves processively with normal termination at the
$\gamma$-site, which is located on the transmembrane domain of APP-C99, and
the product A$\beta$-protein is released to the extracellular region (Fig. 1).
Many of the familial AD-associated mutations occur close to the secretase
cleavage sites (Fig. 1) and, intriguingly, their is evidence that mutations
intermediate to the $\beta$\- and $\gamma$-sites may also influence the
A$\beta$ product distribution. A major focus of current research on AD is to
understand the elementary steps in the cleavage process, with the hope that
the design of drugs that modulate the production of deleterious A$\beta$
products requires a molecular description of the structures of APP-C99 in the
presence of membranes. Simulations as well as experiments have firmly
established that the flexibility of the GG kink Miyashita et al. (2009);
Barrett et al. (2012) near the $\gamma$-site plays an important role in the
production of distinct isoforms of A$\beta$ upon cleavage of APP-C99 by
$\gamma$-secretase. It is clear that a global understanding of AD (and related
diseases) can only be achieved through knowledge of the molecular structures
of APP-C99, the details of the cleavage processes by secretases, and
characterization of subsequent events leading to aggregation. All of these
events involved in AD depend on lipid composition, Ch, and other factors, thus
making the biophysics of AD an extraordinarily rich field, which is sure to
attract physics-based approaches for years to come.

## References

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  * Tarus et al. (2006) B. Tarus, J. E. Straub, and D. Thirumalai, J. Am. Chem. Soc. 128, 16159 (2006).
  * Li et al. (2010) M. S. Li, N. T. Co, G. Reddy, C.-K. Hu, J. E. Straub, and D. Thirumalai, Phys. Rev. Lett. 105, 218101 (2010).
  * Tycko and Wickner (2013) R. Tycko and R. B. Wickner, Acc. Chem. Res. 46, 1487 (2013).
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Figure 1: Schematic picture of the Amyloid Precursor Protein in a membrane. On
the left the sequence of a portion of APP is shown along with the sites for
engaging the secretases. Cleavage by the flexible GG kink in the inserted
portion of APP near the $\gamma$-site produces A$\beta$ peptides with 39 - 43
amino acid residues. On the right a picture of the neuron and the way that APP
interacts with the membrane of the neuron cell is shown along with the
different secretases.


# System Analysis for Responsible Design of Modern AI/ML Systems††thanks:
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of Defense for Research and Engineering under Air Force Contract No.
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detailed above. Use of this work other than as specifically authorized by the
U.S. Government may violate any copyrights that exist in this work.

Virginia H. Goodwin, Rajmonda S. Caceres
MIT Lincoln Laboratory
<EMAIL_ADDRESS>

###### Abstract

The irresponsible use of ML algorithms in practical settings has received a
lot of deserved attention in the recent years. We posit that the traditional
system analysis perspective is needed when designing and implementing ML
algorithms and systems. Such perspective can provide a formal way for
evaluating and enabling responsible ML practices. In this paper, we review
components of the System Analysis methodology and highlight how they connect
and enable responsible practices of ML design.

##  1 Introduction

In recent years the Machine Learning (ML) community has begun to address
serious issues with lack of responsible use of ML algorithms Lazovich et al.
(2022); Molnar ; Ratner & et al. (2019). Researchers are starting to address
the utility and possible overuse or misuse of standard ML challenge data sets,
such as PASCAL VOC or ImageNet in the computer vision (CV) domain, and
GLUE/SuperGLUE in Natural Language Processing (NLP) Raji et al. (2021). Too
frequently, these common and restricted benchmark data sets are taken as the
end-all-be-all of algorithm performance measurement, and yet our experience
deploying ML algorithms in fielded systems has shown time and time again that
this method leaves a lot to be desired in the context of real-world
performance on novel data. We observe that even the algorithms that achieve
best in class performance against the benchmark data do not necessarily result
in sufficient performance in novel environments. As Green says, Green (2019)
“… although most people talk about machine learning’s ability to predict the
future, what it really does is predict the past.” Formalism and mitigation
approaches have been offered for specific problems and algorithms (e.g.,
classification) Brown et al. (2021); Katz et al. (2017); Kleinberg et al.
(2016); Lundberg & Lee (2017); Suresh & Guttag (2019); Suresh et al. (2021).
However, these are currently only applicable in very limited circumstances.

We believe that taking a traditional Systems Analysis approach to ML
assessment offers a coherent pathway out of this problem, which will result in
better outcomes for both algorithm development and performance in real-world
systems. Current ML system development too often overlooks a critical question
addressed by Systems Analysis, i.e., what are the real-world costs of false
alarms and missed detections? This includes costs to the operating entity,
such as the cost of missing an important detection and the cost of responding
to false alarms, but as we have already seen too many times, ethical
developers and users of ML systems must also consider the costs to the
subjects of the ML system. Already we have seen systems deployed for
operational use with no accounting for how false alarms in particular will
impact people’s lives. When the ML system is making decisions about who
deserves an interview Dastin (2018), where police should spend more time
patrolling Green (2019), identifying and even predicting criminal behavior Hao
(2019), and making sentencing recommendations Heaven (2020), the cost to the
individuals impacted by false alarms is profound. “Move fast and break things”
is a terrible and immoral operating philosophy when the “things” being broken
are people’s lives.

In this paper, we first define Systems Analysis; we then go on to draw
parallels between the main points of Systems Analysis and how they can apply
specifically to ML systems.

##  2 What is System Analysis?

Historically, the practice of Systems Analysis is closely linked with that of
Systems Engineering. Systems Engineering began at Bell Labs in the early 1900s
and became a significant field in its own right during the industrialization
driven by WWII Buede (2016). RAND Corporation is also credited with the
initiation of Systems Analysis as a separate field in the fifties Buede
(2016), although systems analysis was being done as part of systems
engineering earlier.

Systems analysis is often a squishy topic, as it is hard to generalize across
a wide array of types of systems, including hardware, software, human-in-the-
loop (HITL) processes, geospatial and temporal processes. For the purposes of
this discussion we break Systems Analysis down into four main steps:

1\. Scoping the Problem Space. This first step includes understanding both the
technical challenges and also the full environment and constraints of who is
going to be using the final system. This should include understanding the
concept of operations (CONOPS), which is, how the operator will use the
system, what rules circumscribe their choices and actions, what response
resources they have, and the cost of mistakes. This is ideally where metrics
of performance and success should be defined. The most common metrics for the
system are the false alarm and missed detection rates that are considered
tolerable by the operator. Scoping and properly defining the problem space is
typically the bulk of systems analysis work.

2\. Scoping the Solution Space. This second step involves understanding or
specifying out what possible solutions exist, frequently termed “the art of
the possible” Delaney (2015). This may include looking at different hardware
and sensor options, as well as understanding what constraints are placed on
your solutions by the operator’s available responses. This may also include
looking at HITL trade-offs, such as whether a user will be available to
provide input or if the system must perform independently.

3\. Creating Candidate Solutions. Once we understand the range of possible
solutions to a problem, the next step is to build one or more solutions. There
are inevitably challenges that must be overcome in making something actually
work that aren’t addressed in the higher-level scoping of the solutions space.
Furthermore, once a prototype exists, the human element (CONOPS, response,
etc.) inevitably changes when the theorized CONOPS don’t work as well as
expected, or new ways to use the system are discovered.

4\. Assessing Performance. Here we circle back to the metrics of performance
specified in Step 1, and assess how well the candidate solution performs
relative to those metrics. These should include not just technical
performance, but how well the system improves operator performance, what are
the cost trade-offs of mistakes (both false alarms and missed detections) and
also what unintentional effects does the system have.

As ML algorithm development is currently practiced, many algorithms are
published, and subsequently reused in a multitude of different contexts, where
the basis of algorithm training, and/or the metrics of performance, are all
built on a published benchmark data set. In these instances, developers are a
priori limiting themselves in both Items 1 and 4 of the above list. They are
not adequately scoping the problem space, because that space may not be
properly captured in the pre-collected and cleaned benchmark data set, and if
they simply rely on the traditional benchmark performance metrics, they are
not looking at whole-system performance. We believe that if the above rubric
is used from the beginning of algorithm development, this will result in
better outcomes for system performance in final systems.

##  3 Parallels to AI/ML Systems

In this section we take a closer look at each step in Systems Analysis, and
make connections to the machine learning process.

###  3.1 Scoping the Problem Space

As noted above, the vast majority of Systems Analysis is performed in Step 1.
We can unpack this into the following pieces: 1) Input data and associated
technical challenges, 2) Output results and associated costs, 3) Constraints
on the system.

Input Data and Associated Technical Challenges. Whereas most ML algorithms
rely on published data sets to train and against which to demonstrate
performance, deployed systems are ingesting data “in the wild”, where there is
typically lots of noise as well as larger data distribution issues, such as
uneven class distribution, or distributions may shift over time, as the
environment changes. A deep understanding to the technical underpinnings of
the data to be analyzed is critical to identify possible solutions, and
possible sources of error. In addition to the details of the collected data,
there are typically technical challenges related to processing power, input
data resolution or quality, and myriad data processing steps that need to be
addressed, such as the raw data from a sensor may be very far from what is
expected as input to the ML system. Each of these steps may introduce some
bias or error in the process that needs to be understood.

Output Results and Associated Costs. Here we need to understand the system-
level output that is required, whether it is an alert to an operator, a
classification result, an image, etc., and most importantly, how those results
are acted on, and what are the costs associated with errors to the system-
level outputs. The cost of false alarms and missed detections is possibly the
most critical piece of the full Systems Analysis. In most systems with an HITL
component, operators like to think about False Alarm Rates and Missed
Detection Rates, i.e. how many false alarms and missed detections can the
operator tolerate per hour, per day, etc. Critically, in addition to the costs
to the operator of responding to false alarms and not responding to missed
detections, there is the cost to the subject of the false alarm or missed
detection. For example, if the response to a false alarm is a law enforcement
intervention, there is a critical cost to the person who is being erroneously
targeted by the response team. For emerging ML-based systems, these costs are
only beginning to be understood and considered.

Constraints on the System. Closely related to the costs of false alarms and
missed detections, there are typically constraints on the system, both
technical and operational. The ML community is familiar with technical
constraints such as processing power, data quality, data transmission rates,
etc. On top of that, for an operational system, we must also consider
operational constraints such as human resources, laws, rules, and social
norms, and other “external” system constraints. In practice, many technically
functional systems don’t get used because they break the operational
constraints of the whole system.

It is critical to clearly define the full scope of the problem space before we
embark on proposing a solution. Details that will emerge in scoping the
solution space, such as choosing a specific algorithm or learning objective,
creating and assessing performance metrics, and data preprocessing and
handling, will all have an impact on the performance of the final system.

###  3.2 Scoping the Solution Space

Scoping the solution space is essentially doing a survey of existing and
potential future technologies, and understanding how various components might
be put together to build out a system that will address the user needs. In the
context of ML, given the planned input data per Section 3.1, there may be
various different ML algorithms that would all result in the desired output,
or potentially multiple algorithms may be used in ensemble to get to the final
desired result. Frequently, the type or quality of available input data, and
other system-level constraints, will restrict the space of potential
solutions. The challenge here in the context of the whole system is
understanding how error or bias may be introduced first from the input data,
then potentially propagated and magnified through the processing of the ML
algorithms.

Additionally, as part of scoping out potential solutions, there may be various
different options for exactly how the output results are generated or
presented. For example, classification system may output different classes
with associated confidence scores, and then there are multiple options at the
system level to assist an operator in making a decision. The system can either
present the top N results if there is an operator available to choose among
the results, or the system can assume less human resources and output the top
single classification as the definitive decision. These types of performance
trade-offs are a large part of the the cost-benefit balancing necessary for
developing useful systems.

###  3.3 Create One or More Candidate Solutions

Based on the outcome of Steps 1 and 2, one or more potential solutions may be
proposed. For example, we may look at a more computationally-intensive
solution that gives better performance, vs. a lighter-weight solution that
gives slightly worse performance but at lower computational cost, and we may
seek to mitigate performance shortfalls at another level of the system, such
as by presenting the operator with multiple choices vs. one choice.

The challenge in this stage is to rigorously understand and assess the
performance of each system component, while never losing sight of the goals of
the full system.

###  3.4 Assessing System Performance

Finally, here we circle back to Step 1, where we discussed the overall system
performance metrics and costs. It is critical to assess the full system
performance that an operator will experience, and then trace the performance
back to individual component performance. This whole process is usually
iterative, where we get a first base-line performance result and then go into
the system, perhaps even re-architecting to include more or different
algorithms to achieve the desired overall system performance. Here we can also
make trade-offs of requiring more HITL time (if available) to offset algorithm
performance.

###  3.5 Systems Analysis Without a System

While all of the preceding has rested on the presumption that a system exists,
that operators of the system exist, and that developers can understand all
relevant aspects of the above, it is also true that ML systems may be
developed in the absence of an existing system from which to glean these
critical details. In the absence of knowing a priori how an algorithm will be
deployed, the alternative solution is documentation. It is imperative that the
developers rigorously document the training data and trained model that they
create, including both the intended use cases and inappropriate or out of
scope applications. As noted in Dunnmon et al. (2021), the use of rigorous
documentation such as Data Sheets Gebru et al. (2020) and Model Cards Mitchell
et al. (2019) is not only best practice for documentation, but also benefits
the final system by eliciting critical questions and issues early in the
design process. In addition, it is critical to create a measurement framework
Holstein et al. (2019) by identifying points within the system at which
results can be assessed at different levels (data and sensing, algorithmic
design, human-AI interaction, and system/application level), and create
‘hooks’ in the software to enable continuous or periodic reassessment of
performance over time.

##  4 Conclusion

While the additional work required to do systems analysis is non-negligible,
we believe that the benefits of doing this level of analysis at each stage of
an AI/ML system development are quite significant to the final performance of
the system. Many people in the ML community are deeply concerned about the
misuse of ML systems in real-world settings with real costs to individuals and
society. Engaging in systems analysis from the very start of the effort can
call out and mitigate against these harms, in particular by making explicit
correct and incorrect use of available data, as well as external constraints
and costs associated with such a system. There is also benefit in clearly
documenting decision-making responsibility and CONOPS between the ML algorithm
and human users of the system to clarify accountability and ownership. In many
cases we recognize that ML algorithm development is divorced from a future
real-world system, and therefore full system metrics cannot be evaluated. In
these cases, we believe that rigorous documentation of the data and model
training can bridge the gap, such that future developers are able to make
informed decisions regarding how appropriate it is to apply the data or model
to their specific problem.

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# Difference equation for the Gromov-Witten potential of the resolved conifold

Murad Alim  Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, 20146,
Hamburg

###### Abstract.

A difference equation is proved for the Gromov-Witten potential of the
resolved conifold. Using the Gopakumar-Vafa resummation of the Gromov-Witten
invariants of any Calabi-Yau threefold, it is further shown that similar
difference equations are satisfied by the part of the resummed potential
containing the contribution of the genus zero GV invariants.

##  1\. Introduction

Gromov-Witten (GW) theory of a non-singular algebraic variety $X$ is concerned
with the study of integrals over the moduli spaces of maps from Riemann
surfaces into $X$. When $X$ is a point, GW theory has been related to $2d$
topological gravity and the KdV integrable hierarchy in the works of Witten
[Wit91] and Kontsevich [Kon92]. For $X=\mathbb{P}^{1}$ the relation of GW
theory to the Toda integrable hierarchy was studied in [Pan00] using a
conjectured difference equation for the GW potential which implied a
corresponding difference equation for Hurwitz numbers. The latter was proved
in [Oko00]. GW theory for Calabi-Yau (CY) threefolds has greatly benefited
from mirror symmetry, which was discovered in String Theory and which puts
forward connections between symplectic and algebraic geometry. Within String
Theory, GW theory is studied using topological strings, which offer many
connections to Chern-Simons theory as well as matrix models. A general
integrable hierarchy structure of topological string theory and in turn GW
theory is expected [ADK+06] and is subject of continued efforts. Recently the
subject has been further stimulated from the combination of ideas from wall-
crossing in Donaldson Thomas (DT) theory and the exact WKB analysis, most
notably in works of Bridgeland [Bri19, Bri20]. In particular, in [Bri20], a
tau function for the resolved conifold is put forward which solves a Riemann-
Hilbert problem associated to DT wall-crossing while also containing the
information of all genus GW invariants in an asymptotic expansion.

This note is concerned with the GW potential of the resolved conifold. In
particular, a difference equation similar to the one conjecturally satisfied
by the GW theory of $\mathbb{P}^{1}$ [Pan00] is proved. The ideas needed for
the proof are adapted from the work of Iwaki, Koike and Takei [IKT18] where a
difference equation for the free energy of the Weber curve is proved. The
latter is a generating function of the Euler characteristics of the moduli
spaces of curves studied in [HZ86]. Similar difference equations appear in
many interesting problems of mathematical physics such as for example in the
study of matrix models and non-perturbative physics, see e.g. [Mar14, Eq.
4.134 and Sec. 5.1] and in the study of partition functions of supersymmetric
$\mathcal{N}=2$ theories where they characterize the perturbative piece of the
partition function, see e.g. [NO06, Appendix A] as well as in the form of
q-deformed Painlevé equations satisfied by suitably constructed topological
string partition functions, see [BGT19] and references therein.

### Acknowledgements

The author would like to thank Florian Beck, Arpan Saha and Jörg Teschner for
comments on the draft as well as the LMU in Munich for hospitality. This work
is supported through the DFG Emmy Noether grant AL 1407/2-1.

##  2\. Gromov-Witten potentials

Let $X$ be a Calabi-Yau threefold. The GW potential of $X$ is the following
formal power series:

$F(\lambda,t)=\sum_{g\geq 0}\lambda^{2g-2}F^{g}(t)=\sum_{g\geq
0}\lambda^{2g-2}\sum_{\beta\in
H_{2}(X,\mathbb{Z})}N^{g}_{\beta}\,q^{\beta}\,,$ (2.1)

where $q^{\beta}:=\exp(2\pi it^{\beta})$ is a formal variable living in a
suitable completion of the effective cone in the group ring of
$H_{2}(X,\mathbb{Z})$, $\lambda$ is a formal parameter corresponding to the
topological string coupling and $N^{g}_{\beta}$ are the GW invariants.

The GW potential can be written as:

$F=F_{\beta=0}+\tilde{F}\,,$ (2.2)

where $F_{\beta=0}$ denotes the contribution from constant maps and
$\tilde{F}$ the contribution from non-constant maps. The constant map
contribution at genus 0 and 1 are $t$ dependent and the higher genus constant
map contributions take the universal form [FP00]:

$F_{\beta=0}^{g}=\frac{\chi(X)(-1)^{g-1}\,B_{2g}\,B_{2g-2}}{4g(2g-2)\,(2g-2)!}\,,\quad
g\geq 2\,,$ (2.3)

where $\chi(X)$ is the Euler character of $X$. This note is concerned with the
GW potential of the CY threefold given by the total space of the rank two
bundle over the projective line:

$\mathcal{O}(-1)\oplus\mathcal{O}(-1)\rightarrow\mathbb{P}^{1}\,,$ (2.4)

which corresponds to the resolution of the conifold singularity in
$\mathbb{C}^{4}$ and is known as the resolved conifold. The GW potential for
this geometry was determined in physics [GV98, GV99], and in mathematics
[FP00] with the following outcome for the non-constant maps:111See also
[MnM99] for the determination of $F^{g}$ from a string theory duality and the
explicit appearance of the polylogarithm expressions.

$\tilde{F}^{0}=\textrm{Li}_{3}(q)\,,\quad\tilde{F}^{g}=\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\textrm{Li}_{3-2g}(q)\,,\quad
g\geq 1\,,$ (2.5)

where $q:=\exp(2\pi i\,t)$ and the polylogarithm ist defined by:

$\textrm{Li}_{s}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{n^{s}}\,,\quad
s\in\mathbb{C}\,.$ (2.6)

##  3\. A difference equation for the Gromov-Witten potential

######  Theorem 1.

The contribution of the non-constant maps $\tilde{F}(\lambda,t)$ to the GW
potential of the resolved conifold satisfies the following difference
equation:

$\tilde{F}(\lambda,t+\check{\lambda})-2\tilde{F}(\lambda,t)+\tilde{F}(\lambda,t-\check{\lambda})=\left(\frac{1}{2\pi}\frac{\partial}{\partial
t}\right)^{2}\,\tilde{F}^{0}(t)\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,.$
(3.1)

###### Proof.

Starting from $\textrm{Li}_{1}(q)=-\log(1-q)$ and using the property:

$\theta_{q}\textrm{Li}_{s}(q)=\textrm{Li}_{s-1}(q)\,,\quad\theta_{q}:=q\,\frac{d}{dq}\,,$
(3.2)

we write

$\tilde{F}^{g}=\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\,\theta_{q}^{2g-2}\textrm{Li}_{1}(q)\,,\quad
g\geq 1\,.$ (3.3)

In the following, the proofs of [IKT18, Thm. 4.7 and 4.9] are adapted to the
current setting. Consider the generating function of Bernoulli numbers:

$\frac{w}{e^{w}-1}=\sum_{n=0}^{\infty}B_{n}\frac{w^{n}}{n!}\,.$ (3.4)

Applying $w\frac{d}{dw}$ to both sides and rearranging gives:

$\frac{w^{2}e^{w}}{(e^{w}-1)^{2}}=B_{0}-\sum_{n=2}^{\infty}\frac{B_{n}}{n(n-2)!}w^{n}=1-\sum_{g=1}^{\infty}\frac{B_{2g}}{2g(2g-2)!}w^{2g}\,,$
(3.5)

where the last equality is obtained by noting that all
$B_{2n+1},n\in\mathbb{N}\setminus\\{0\\}$ vanish. This yields the following:

$(e^{w}-2+e^{-w})\left(\frac{1}{w^{2}}-\sum_{g=1}^{\infty}\frac{B_{2g}}{2g(2g-2)!}w^{2g-2}\right)=1\,.$
(3.6)

In the next step, we replace on both sides of this equation the variable $w$
by an operator acting on functions of $t$ namely:

$w\rightarrow\check{\lambda}\frac{\partial}{\partial
t}=i\lambda\,\theta_{q}\,.$

Acting with both sides on $\textrm{Li}_{1}(q)$ we obtain:

$(e^{\check{\lambda}\partial_{t}}-2\cdot\textrm{id}+e^{-\check{\lambda}\partial_{t}})\left(-\lambda^{-2}\theta_{q}^{-2}-\sum_{g=1}^{\infty}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\lambda^{2g-2}\,\theta_{q}^{2g-2}\right)\textrm{Li}_{1}(q)=\textrm{id}\cdot\textrm{Li}_{1}(q)\,,$
(3.7)

by using
$\theta_{q}^{2}\tilde{F}^{0}=\theta_{q}^{2}\textrm{Li}_{3}(q)=\textrm{Li}_{1}(q)$
and interpreting $\theta_{q}^{-1}$ as an anti-derivative, we obtain:

$(e^{\check{\lambda}\partial_{t}}-2\cdot\textrm{id}+e^{-\check{\lambda}\partial_{t}})\tilde{F}(\lambda,t)=-\theta_{q}^{2}\tilde{F}^{0}(q)\,,$
(3.8)

which proves the theorem. ∎

######  Corollary 2.

For every $g\geq 1$ the difference equation gives a recursive differential
equation which determines $\frac{\partial^{2}}{\partial
t^{2}}\tilde{F}^{g}(t)\,\,g\geq 1$ by:

$\sum_{k=0}^{g}\frac{1}{(2g-2k+2)!}\left(\frac{1}{2\pi}\frac{\partial}{\partial
t}\right)^{2g-2k+2}\tilde{F}^{k}(t)=0\,,\quad g\geq 1\,.$ (3.9)

###### Proof.

This follows from expanding the L.H.S. of the theorem in $\lambda$ and then
matching the coefficients of $\lambda^{2g}$ on both sides. ∎

##  4\. Difference equation for Gopakumar-Vafa resummation

The statement of the theorem is in the same spirit as the conjectured
difference equation [Pan00] for GW theory of $\mathbb{P}^{1}$ except that here
only the genus zero part of the GW potential appears on the R.H.S. We expect
this result to shed further light on the relation of GW theory of the resolved
conifold and the tau function studied in [Bri20]. Furthermore, it may be
useful for proving a conjecture of Brini [Bri12] relating the GW theory of the
resolved conifold to the Ablowitz-Ladik integrable hierarchy.222Note added in
the revision: The proof of Brini’s conjecture as well as the relation to the
special functions appearing in Bridgeland’s tau function has now appeared in
Ref. [AS21]. Moreover, the result may offer new insights into the universal
structure of GW theory on CY threefolds. An indication of this is given in the
following corollary, which requires the Gopakumar-Vafa resummation of the GW
potential [GV98]. The latter conjectures that there exist
$n^{g}_{\beta}\in\mathbb{Z}$ (called GV invariants), such that for any CY
threefold $X$, the GW potential can be written as:

$\tilde{F}(\lambda,t)=\sum_{\beta>0}\sum_{g\geq 0}n^{g}_{\beta}\,\sum_{k\geq
1}\frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{2g-2}q^{k\beta}\,.$
(4.1)

We focus on the contribution of the genus zero GV invariants to the resummed
GW potential, which we denote by:

$\tilde{F}_{GV_{0}}(\lambda,t)=\sum_{\beta>0}n^{0}_{\beta}\,\sum_{k\geq
1}\frac{1}{k}\left(2\sin\left(\frac{k\lambda}{2}\right)\right)^{-2}q^{k\beta}\,.$

######  Corollary 3.

Fix $\alpha\in H_{2}(X,\mathbb{Z}),\alpha>0$, then $\tilde{F}_{GV_{0}}$
satisfies the following difference equation in the corresponding formal
variable $t^{\alpha}$:

$\tilde{F}_{GV_{0}}(\lambda,t^{\circ},t^{\alpha}+\check{\lambda})-2\tilde{F}_{GV_{0}}(\lambda,t)+\tilde{F}_{GV_{0}}(\lambda,t^{\circ},t^{\alpha}-\check{\lambda})=\left(\frac{1}{2\pi}\frac{\partial}{\partial
t^{\alpha}}\right)^{2}\,\tilde{F}^{0}(t)\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,,$
(4.2)

where $t$ refers to the set of all formal variables $t^{\beta}$,
$t^{\circ}=t\setminus\\{t^{\alpha}\\}$ and

$\tilde{F}^{0}(t)=\sum_{\beta>0}n_{\beta}^{0}\textrm{Li}_{3}(q^{\beta})\,,\quad
q^{\beta}=\exp(2\pi it^{\beta})\,.$

###### Proof.

We use the Laurent expansion:

$\left(2\sin\left(\frac{s}{2}\right)\right)^{-2}=\frac{1}{(s)^{2}}-\frac{1}{12}+\sum_{g\geq
2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}s^{2g-2}\,,$ (4.3)

to write:

$\displaystyle\tilde{F}_{GV_{0}}(\lambda,t)$ $\displaystyle=$
$\displaystyle\sum_{\beta>0,\beta\neq\alpha}n_{\beta}^{0}\left(\lambda^{-2}\,\textrm{Li}_{3}(q^{\beta})+\sum_{g=1}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\textrm{Li}_{3-2g}(q^{\beta})\right)\,$
(4.4) $\displaystyle+$ $\displaystyle
n_{\alpha}^{0}\left(\lambda^{-2}\,\textrm{Li}_{3}(q^{\alpha})+\sum_{g=1}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\textrm{Li}_{3-2g}(q^{\alpha})\right)\,.$

The proof of Thm.1 can be repeated for the second part on the R.H.S. The
statement of the corollary follows by noting that the operator giving the
difference equation in $t^{\alpha}$ gives zero when acting on the first,
$t^{\alpha}$ independent, part. ∎

## References

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# The Zariski–Lipman conjecture for toric varieties

Carl Tipler  Univ Brest, UMR CNRS 6205, Laboratoire de Mathématiques de
Bretagne Atlantique, France<EMAIL_ADDRESS>
###### Abstract.

We give a short proof of the Zariski–Lipman conjecture for toric varieties :
any complex toric variety with locally free tangent sheaf is smooth.

##  1\. Introduction

The Zariski–Lipman conjecture states that an algebraic variety $X$ over
$\mathbb{C}$ with locally free tangent sheaf is necessarily smooth. This
conjecture has been proved under various additional assumptions, see for
example [8] in the $\mathbb{N}$-graded case, [9] for complete intersections,
[4] for log canonical varieties and [5, 1, 2] for surfaces. In this note, we
give a direct and simple proof for toric varieties, that is for irreducible
varieties endowed with an algebraic effective action of a complex torus with a
dense open orbit. Recall that the tangent sheaf $\mathcal{T}_{X}$ of an
algebraic variety $X$ is the dual of its sheaf of Kähler differentials, that
is
$\mathcal{T}_{X}=\mathrm{Hom}_{\mathcal{O}_{X}}(\Omega^{1}_{X},\mathcal{O}_{X})$.

######  Theorem 1.1.

Let $X$ be a toric variety over $\mathbb{C}$. Assume that the tangent sheaf of
$X$ is locally free. Then $X$ is smooth.

As we will see in Remark 1.2 and Remark 1.3, Theorem 1.1 can be obtained as a
corollary of the results from [6] or from [11]. Nevertheless, the proof
presented here is fairly simple and relies purely on methods from toric
geometry.

######  Remark 1.2.

If $(X,0)$ is a germ of a rational singularity over $\mathbb{C}$, then $(X,0)$
is Cohen–Macaulay. If moreover $(X,0)$ has a locally free tangent sheaf, its
canonical divisor is Cartier, and thus $(X,0)$ is Gorenstein. As noticed in
[5], rational Gorenstein singularities are canonical, a class of singularities
for which the Zariski–Lipman conjecture has been proved in [6, 4]. Normal
toric singularities being rational [3, Theorem 11.4.2], Theorem 1.1 is a
corollary of the results in [6].

######  Remark 1.3.

Locally, a normal toric variety with no torus factor is a categorical quotient
of a smooth affine toric variety by a reductive group action [3, Theorem
5.1.11]. From [11, Corollary 5.11], this implies that the Nakai conjecture
holds true for normal toric varieties. As the Nakai conjecture implies the
Zariski-Lipman conjecture [14, Proposition 2], Theorem 1.1 follows from [11].
The author would like to thank Thierry Levasseur for pointing to him this
reference.

######  Remark 1.4.

The Zariski–Lipman conjecture is actually stated over any field of
characteristic zero. As noticed in [8], with no loss of generality one may
assume the field to be algebraically closed. In this note, we will work over
the complex numbers, but Theorem 1.1 and its proof hold over any algebraically
closed field of characteristic zero.

### Acknowledgments

The author would like to thank Patrick Graf for explaining Remark 1.2 and
Thierry Levasseur for explaining Remark 1.3, as well as Thibaut Delcroix,
Henri Guenencia and the anonymous referee for useful remarks. The author is
partially supported by the grants MARGE ANR-21-CE40-0011 and BRIDGES
ANR–FAPESP ANR-21-CE40-0017.

##  2\. Klyachko’s description of toric reflexive sheaves

Let $X$ be a toric variety of dimension $n$ over $\mathbb{C}$, with torus
$T_{N}=N\otimes_{\mathbb{Z}}\mathbb{C}^{*}$, for $N$ the rank $n$ lattice of
its one-parameter subgroups. Denote by
$M=\mathrm{Hom}_{\mathbb{Z}}(N,\mathbb{Z})$ its character lattice. Let
$\mathcal{T}_{X}=\mathrm{Hom}_{\mathcal{O}_{X}}(\Omega^{1}_{X},\mathcal{O}_{X})$
be the tangent sheaf of $X$, that is the dual of its sheaf of Kähler
differentials (see e.g. [3, Section 8.0]). According to [12, Theorem 3], if
$\mathcal{T}_{X}$ is locally free, then $X$ is normal, which we will assume
from now on. Then, $X$ is the toric variety associated to a fan $\Sigma$ of
strongly convex rational polyhedral cones in
$N_{\mathbb{R}}=N\otimes_{\mathbb{Z}}\mathbb{R}$ [3, Chapter 3]. In
particular, $X$ is covered by the $T_{N}$-invariant affine varieties
$U_{\sigma}=\mathrm{Spec}(\mathbb{C}[M\cap\sigma^{\vee}])$, for
$\sigma\in\Sigma$.

Recall that a coherent sheaf $\mathcal{F}$ on $X$ is called
$T_{N}$-equivariant if there is an isomorphism
$\varphi:\alpha^{*}\mathcal{F}\to\pi_{2}^{*}\mathcal{F}$ satisfying some
cocycle condition (see e.g. [13, Section 5] or [7]) where $\alpha:T_{N}\times
X\to X$, $\pi_{1}:T_{N}\times X\to T_{N}$ and $\pi_{2}:T_{N}\times X\to X$
stand for respectively the $T_{N}$-action, the projection on $T_{N}$ and the
projection on $X$. The $T_{N}$-action on $X$ induces naturally an equivariant
structure on its sheaf of Kähler differentials given by the composition :

$\alpha^{*}\Omega^{1}_{X}\xrightarrow{d\alpha}\Omega^{1}_{T_{N}\times
X}\xrightarrow{\simeq}\pi_{1}^{*}\Omega^{1}_{T_{N}}\oplus\pi_{2}^{*}\Omega^{1}_{X}\xrightarrow{\mathrm{pr}_{2}}\pi_{2}^{*}\Omega^{1}_{X}$

where $\mathrm{pr}_{2}$ is the projection on the second factor [7, Section 2].
By duality, $\mathcal{T}_{X}$ is $T_{N}$-equivariant, and, being dual to the
coherent sheaf $\Omega^{1}_{X}$, it is reflexive. Klyachko proved that
equivariant reflexive sheaves on toric varieties are described by families of
filtrations [10] (see also [13]). Let us recall briefly this description for
the tangent sheaf. As $\mathcal{T}_{X}$ is reflexive, its sections extend over
codimension $2$ subvarieties. Denote by $\Sigma(l)$ the set of $l$-dimensional
cones in $\Sigma$, and consider

$X_{0}=\bigcup_{\sigma\in\Sigma(0)\cup\Sigma(1)}U_{\sigma}.$

By the orbit-cone correspondence [3, Section 3.2], $X_{0}$ is the complement
of $T_{N}$-orbits of co-dimension greater or equal to $2$, and thus
$\mathcal{T}_{X}=\iota_{*}\mathcal{T}_{X_{0}}$, where $\iota:X_{0}\to X$ is
the inclusion. By equivariance, $\mathcal{T}_{X_{0}}$ is entirely
characterised by the sections $\Gamma(U_{\sigma},\mathcal{T}_{X})$, for
$\sigma\in\Sigma(0)\cup\Sigma(1)$. If $\sigma=\\{0\\}$, $U_{\\{0\\}}=T_{N}$,
and

$\Gamma(U_{\\{0\\}},\mathcal{T}_{X})=N_{\mathbb{C}}\otimes_{\mathbb{C}}\mathbb{C}[M],$

for $N_{\mathbb{C}}:=N\otimes_{\mathbb{Z}}\mathbb{C}$ is the Lie algebra of
$T_{N}$. Then, if $\rho\in\Sigma$ is a ray (i.e. a one-dimensional cone),
$\Gamma(U_{\rho},\mathcal{T}_{X})$ is graded by
$M/(M\cap\rho^{\perp})\simeq\mathbb{Z}$. As $T_{N}$ is a dense open subset of
$U_{\rho}$, the restriction map
$\Gamma(U_{\rho},\mathcal{T}_{X})\to\Gamma(U_{\\{0\\}},\mathcal{T}_{X})$ is
injective and induces a decreasing $\mathbb{Z}$-filtration

$\ldots\subset E^{\rho}(i)\subset E^{\rho}(i-1)\subset\ldots\subset
N_{\mathbb{C}}$

such that one has

$\Gamma(U_{\rho},\mathcal{T}_{X})=\bigoplus_{m\in M}E^{\rho}(-\langle
m,u_{\rho}\rangle)\otimes\chi^{m},$

where we denote by $u_{\rho}$ the primitive generator of $\rho$ and
$\langle\cdot,\cdot\rangle$ the duality pairing. Explicitly, the family of
filtrations $(E^{\rho}(\bullet))_{\rho\in\Sigma(1)}$ for $\mathcal{T}_{X}$ is
given by [10, Example 2.3(5) on page 350]:

$E^{\rho}(i)=\left\\{\begin{array}[]{ccc}N_{\mathbb{C}}&\textrm{ if }&i\leq
0\\\ \mathbb{C}\cdot u_{\rho}&\textrm{ if }&i=1\\\ \\{0\\}&\textrm{ if }&i\geq
2.\end{array}\right.$

Finally, by Klyachko’s compatibility condition [10, Theorem 2.2.1] (see [13,
Section 5] for a detailed treatment on normal toric varieties),
$\mathcal{T}_{X}$ is locally free if and only if the family of filtrations
$(E^{\rho}(\bullet))_{\rho\in\Sigma(1)}$ satisfies that for each
$\sigma\in\Sigma$, there exists a decomposition

$N_{\mathbb{C}}=\bigoplus_{[m]\in M/(M\cap\sigma^{\perp})}E^{\sigma}_{[m]}$

such that for each ray $\rho\subset\sigma$:

$E^{\rho}(i)=\bigoplus_{\langle m,u_{\rho}\rangle\geq i}E^{\sigma}_{[m]}.$

##  3\. Proof of Theorem 1.1

Assume from now on that $\mathcal{T}_{X}$ is locally free. We want to show
that $X$ is smooth. As this is a local condition, we might as well assume $X$
affine, so that $X=U_{\sigma}$ for some strongly convex rational polyhedral
cone $\sigma\subset N_{\mathbb{R}}$. With no loss of generality, we can also
assume that $X$ has no torus factor, so that
$\\{u_{\rho},\,\rho\in\sigma(1)\\}$ spans $N_{\mathbb{R}}$ [3, Proposition
3.3.9]. Then, $X$ is smooth if and only if $\sigma$ is smooth [3, Theorem
1.3.12], which, by definition, is equivalent to the fact that
$\\{u_{\rho},\rho\in\sigma(1)\\}$ is a $\mathbb{Z}$-basis for $N$.

As $\\{u_{\rho},\,\rho\in\sigma(1)\\}$ spans $N_{\mathbb{R}}$,
$\sigma^{\perp}=\\{0\\}$. Then by Klyachko’s compatibility condition, we can
find a decomposition

$N_{\mathbb{C}}=\bigoplus_{m\in M}E^{\sigma}_{m}$

with, for $\rho\in\sigma(1)$, and $i\in\\{2,1,0\\}$,

(1) $E^{\rho}(i)=\bigoplus_{\langle m,u_{\rho}\rangle\geq i}E^{\sigma}_{m}.$

First, from $E^{\rho}(2)=\\{0\\}$, we deduce that $E^{\sigma}_{m}\neq\\{0\\}$
only if for all $\rho$, $\langle m,u_{\rho}\rangle\leq 1$. Secondly, taking
$i=1$ in (1),

$\mathbb{C}\cdot u_{\rho}=\bigoplus_{\langle m,u_{\rho}\rangle\geq
1}E^{\sigma}_{m}.$

Thus, for each $\rho\in\sigma(1)$ we can find $m_{\rho}\in M$ with $\langle
m_{\rho},u_{\rho}\rangle=1$ and such that $\mathbb{C}\cdot
u_{\rho}=E_{m_{\rho}}^{\sigma}$. Note that $\sigma$ contains no line by strong
convexity, so if $\rho\neq\rho^{\prime}$, then $u_{\rho}\notin\mathbb{Z}\cdot
u_{\rho^{\prime}}$ and $\mathbb{C}\cdot u_{\rho}\neq\mathbb{C}\cdot
u_{\rho^{\prime}}$. As $\\{u_{\rho},\rho\in\sigma(1)\\}$ spans
$N_{\mathbb{C}}$ over $\mathbb{C}$,

$N_{\mathbb{C}}=\sum_{\rho\in\sigma(1)}\mathbb{C}\cdot
u_{\rho}=\bigoplus_{\rho\in\sigma(1)}E^{\sigma}_{m_{\rho}}\subset\bigoplus_{m\in
M}E^{\sigma}_{m}=N_{\mathbb{C}}.$

Then,

$N_{\mathbb{C}}=\bigoplus_{\rho\in\sigma(1)}E^{\sigma}_{m_{\rho}}$

and $\sigma(1)$ contains exactly $n$ elements. Let
$\rho,\rho^{\prime}\in\sigma(1)$ be two distinct rays. Necessarily,

$\mathbb{C}\cdot u_{\rho}\cap\mathbb{C}\cdot u_{\rho^{\prime}}=\\{0\\}$

and by (1) with $i=1$, $m_{\rho}$ must satisfy $\langle
m_{\rho},u_{\rho^{\prime}}\rangle\leq 0$.

Last, taking now $i=0$ in (1), we deduce from $E^{\rho}(0)=N_{\mathbb{C}}$
that $\langle m_{\rho},u_{\rho^{\prime}}\rangle=0$. To conclude, for all
$\rho,\rho^{\prime}\in\sigma(1)$,

$\langle
m_{\rho},u_{\rho^{\prime}}\rangle=\left\\{\begin{array}[]{ccc}1&\mathrm{if}&\rho=\rho^{\prime}\\\
0&\mathrm{if}&\rho\neq\rho^{\prime}.\end{array}\right.$

Hence, each element $u\in N$ can be uniquely written

$u=\sum_{\rho\in\sigma(1)}\langle m_{\rho},u\rangle\,u_{\rho}$

with $\langle m_{\rho},u\rangle\in\mathbb{Z}$ for each $\rho\in\sigma(1)$.
Thus, $\\{u_{\rho},\rho\in\sigma(1)\\}$ is a basis of $N$, which ends the
proof of Theorem 1.1.

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