While we were not able to conduct a proper survey, we still wanted to share
some experiences by former members of our community. This is admittedly a
biased sample, as we reached out to specific colleagues who we knew had
experienced some of these frustrations. That being said, we recognize that
these are real experiences of our colleagues and as such are still valid.
Whether or not these anecdotes are representative of a significant number of
researchers must be left to future work.

We provide feedback from two individuals who worked on HEP experiments for
some of their time in graduate school. In an effort to allow them anonymity,
we only mention that their experiences span the past 15 years and that they
both identify as either women or a different gender minority. We received
their permission to relate their story as presented here.

##### Individual #1

>   * •
>
> As an undergraduate researcher, I was fortunate to work in a lab where
> running existing code was discouraged. There was an awareness of the culture
> of HEP to have students work with existing code blocks and software and part
> of my education was to recreate these with my own knowledge and skills. When
> not possible, this allowed for more energy to dig into documentation and
> poorly commented code when necessary. My energy to approach CLUSTER/GRID
> SUBMISSION TOOLS for submitting jobs was high as I hadn’t been working with
> this documentation as my main job.
>
>   * •
>
> When I got to graduate school, I was in a LHC EXPERIMENT lab that did not
> value this. In my first research experience, I attempted to write my own
> code to perform analysis and was chastised for taking longer than it should
> take using existing code. When I attempted to use the existing code, I spent
> all of my time doing research reading documentation, tracing function and
> variable definitions through nested code, and asking questions to other
> graduate students. This did not appeal to my desires for exploration,
> creativity, and fun when performing my research. I wasn’t interested in this
> work I was putting in and the work became draining.
>
>   * •
>
> In my next project, I was running code written by students one and five
> years ahead of me in the program. No documentation was written to pair with
> these files. I was told to ask the students for guidance. I did not want to
> eat up all of their time with endless questions and tutorials, so I spent a
> lot of time debugging on my own. When we did meet, there were diagrams drawn
> and maps for the code but I still had questions that remained unanswered.
> The barriers to exploring the questions I had about my data limited my
> excitement for the work.
>
>   * •
>
> This research group had other cultural issues that made the group
> environment difficult. Aggressive lines of questioning in group meetings
> made students defensive and often ended in arguments. This made approaching
> anyone for assistance mentally taxing. Asking another student, professor, or
> post doc for help was like preparing for battle. I wanted to be excited
> about the work I was doing and instead, it was survival that was driving my
> work.
>
>   * •
>
> Finally, I chose to leave the LHC EXPERIMENT group as I no longer had any
> drive to continue. I was experiencing extreme burnout and doing any amount
> of work drove me to tears. The code block approach to learning was
> compounded by the environment I was in and I was no longer able to continue
> my relationship with LHC EXPERIMENT. At this point, I turned to A NON-
> COLLIDER EXPERIMENT group to continue research. This group was friendly and
> curious. Code was well documented and I wrote all of my own code once more.
> I had a friend in the group who actively made time to work with me when I
> was stuck. My questions were appreciated as a signal of my love of the work.
> The environment at NON-COLLIDER EXPERIMENT is extremely different than THE
> LHC EXPERIMENT GROUP and it was very sad to say good bye to the experiment I
> started my HEP career in.
>
>

##### Individual #2

>   * •
>
> I got into physics because I loved math and learning a bit about cosmology
> and string theory during senior year of high school blew my mind. And when I
> discovered particle physics in college, I was captivated by its theoretical
> elegance, teamwork and collaboration, chances to travel for conferences and
> the experiment, and opportunities for doing physical work fixing and
> upgrading the detector.
>
>   * •
>
> I took only one programming class in college, the intro to C class, which I
> didn’t do great in, and was told by pretty much every advisor I had that I
> didn’t need any more, that I’d pick it up as I go.
>
>   * •
>
> That was terrible advice. Of course I’d pick something up, but the
> fundamental ideas of how to find and solve bugs, how to write clean and
> readable code (to preempt bugs), how to read others’ code, how to think of
> software as a whole, all of that should have been a course. Instead I
> painfully kluged my way through grad school. Stack overflow was also not as
> big as it is now and I didn’t get to actually collaborate with anyone until
> I think the last year of grad school, so I was fully alone and adrift for
> years.
>
>   * •
>
> I hated C++. I couldn’t understand why we didn’t have CS people properly
> writing common libraries that I could just use for my higher-level analysis.
> I even asked about that and was told that CS people wouldn’t understand the
> physics we need.
>
>   * •
>
> I hated having to “write complicated, excruciatingly atomic step-by-step
> instructions for the world’s stupidest toddler who will do exactly
> everything you tell it to do completely literally,” as someone on reddit
> phrased it. I don’t think like that. I don’t. I hate breaking seemingly
> obvious things down step by step in an orderly fashion. I love inspiration
> and creativity and beauty.
>
>   * •
>
> Being handed a topic (I didn’t even get to discuss choosing one!) and left
> fully alone for years, not knowing who to ask for help, being afraid to ask
> for help because I felt that would reveal how dumb I was and how much I
> didn’t deserve to be there, was awful. Being told I wasn’t allowed to travel
> to conferences, even to practice presenting, because I needed to focus on my
> analysis without understanding how exactly, was awful. Being told I wasn’t
> allowed to physically work on the detector, again because I had to “focus on
> my analysis”, was demoralizing.
>
>   * •
>
> Contrasting that experience with being a data scientist, well a few things
> were similar. Not being listened to when I asked for what I want, for
> example. Not having a code review when I know that would preempt my own and
> my colleagues’ bugs. Not getting learning resources when I asked for them.
>
>   * •
>
> But working with Python, and in a team that needed to use my code, is a
> completely different beast. I had weekly or biweekly meetings where I
> described to the client the upgrades in the algorithm, and when I needed
> help I was assigned it. The team discussed appropriate libraries to pull in.
> If someone didn’t know what to do, they’d ask and there’d be no penalty,
> someone would just show them. Mostly. And Python is much more like Matlab,
> i.e. more directly doing math, than C++ was. It’s still not as pleasant as
> doing math directly, but the instructions are slightly less atomic and the
> form of instructing can be a little varied.
>
>

While the details are different, a commonality is that both of them felt that
the software was barrier to doing what they wanted to do, which was physics,
and that they did not always know who to go for help. Again, we stress that
the situation is improving in many experiments and in many groups, but the
improvements are most likely not universal.

###  6.2 Where We Go From Here

We are not experts in this field and so our comments must be taken with a
grain of salt. But we have some minimal recommendations for either individual
experiments or the community as a whole.

  * •

Engage with other researchers. We have identified two scientists who study
large communities who engage in software development, Ford and Ko, and would
strongly encourage the community to reach out to them or similar researchers.

  * •

Consider hiring scientific writers to produce better documentation and
training modules. This would be a very different approach to how things are
normally done where everything is produced “in-house”. But there are people
who learn how to do this and if freed of the demands of performing an
analysis, could probably do this better than many of us.

  * •

Regularly re-evaluate what parts of the analysis workflow should be provided
as “services” and what should be left to analysts. For example, does it
benefit every analyst to have to continually re-run their skims as new
corrections are performed? Or to modify their workflow as new submission tools
are developed? With an eye toward a “greener” compute model, should we work to
minimize how often analysts have to go back upstream and instead provide more
centrally-produced skims, made to-order, so that analysts can focus on the
physics and novel approaches to extracting results? While the shift from C/C++
to python has helped lower the threshold for many researchers, new
developments in accelerators and decorators are adding an increased level of
complexity (though providing significant gain in speed) and analysts need time
to learn these new tools. That means that we might have to offload some of the
work that almost every analyst performs in the early stages of their work

While there is no obvious solution to ensure that all collaboration affiliates
can contribute equally to the software ecosystem and feel supported in those
efforts, we are encouraged that more and more people are realizing this is an
issue to be tackled. There are already many projects and workshops among the
experiments that are trying to address these challenges and we hope this
discussion and this section can be part of those efforts.

##  7 Throughlines and Additional Topics

###  7.1 Expertise and Compensating Experts

We begin this section with what is likely the most important takeaway from
this paper: that physicists should not be doing the job of organizational
psychologists and DEI experts. Our expertise is in high energy particle
physics and astrophysics, and not in the broad fields of sociology, history,
or psychology. Our best (and truthfully, only) solution to addressing both
inequities in HEPA as well as deficiencies within HEPA culture, relies on the
advice and recommendations of paid experts. We also call on funding agencies
to require such coordination as part of funding documentation: inclusion of
climate-related topics into safety components of ‘Operational Readiness
Reviews,” “Conceptual Design Reviews,” or similar documentation.

##### Recommendation F3.1:

Funding should be made available to both engage with and compensate such
experts. This can take the form of independent grants, but more effective
would be the inclusion of climate-related topics into safety components of
collaboration “Operational Readiness Reviews,” “Conceptual Design Reviews,” or
similar documentation submitted to funding agencies.

##### Recommendation C3.1:

Experts should be adequately integrated into HEPA communities, including
collaborations, such that their expertise can be applied effectively. This may
take the form of an official collaboration role like a non-voting member of a
collaboration council.

Putting this call aside for a moment, even within our existing culture,
physicists who do this work are not given proper recognition or respect. Many
institutes and collaborations have ‘Diversity, Equity, and Inclusion’
committees, yet the work done within these bodies tends to not be implemented
and the individuals doing the work are not rewarded in a professional
capacity. Additionally, the current culture of physics categorically places
DEI work as an optional ‘service’ and sometimes as a ‘distraction’ from
research rather than a necessary and important aspect of the job. In fact,
there is evidence that these service activities fall heavily on the shoulders
of underrepresented groups and negatively impact the promotion or advancement
for the engaged individuals.

Finally, we must acknowledge that performing diversity and inclusion
initiatives without expertise can very well harm the populations they are
trying to help. Expertise should be considered a mandatory part of policy
development.

> Too often, people approach transformative justice as a neat idea they’d like
> to experiment with in their community, and end up harming the most
> marginalized members of their community instead of helping them. If you want
> to use transformative justice in your community, we recommend you only do so
> if you have several people who have formal training in transformative
> justice available to work the process, and you only attempt the process with
> members of your community who have a strong incentive to stay inside the
> community and to treat other members of the community well.

###  7.2 Diversity in Leadership and Representation

HEPA communities should be asked how their leadership is designed, how
affiliates are trained and selected for leadership. Who actually ends up in
charge? What are the short- and long-term repercussions of this choice?
101010The authors credit [59], which details ”Responses to 10 common
criticisms of anti-racism action in STEMM” for a substantial portion of this
section’s literature review.

As discussed at length in section 5, inherently hierarchical structures such
as seniority, scientists / engineers (section 5), racial-, and gender-
hierarchies offer to maintain the myth of “meritocracy” in HEPA spaces. These
must be replaced with policies that advance equity over an ill-defined
conception of “merit.” Scientific “meritocracies” do not exist. Even the man
who coined the term “meritocracy” meant it as satire: specifically that a
meritocracy would appear at face value to be equitable but would rather
continue the cycle of disenfranchisement through systemic processes designed
to deprive minoritized populations of education and, therefore, upward
mobility [88, 59]. Instead, success depends strongly on external factors which
include but are not limited to race, gender, socio-economic class, and
disability; the stress of these external factors contribute to greater
cognitive load which in BIPOC has been associated with illness and reduced
productivity[89, 90, 91], which further distances BIPOC researchers in terms
of “merit”.

And within larger collaborations, affiliates can often be rewarded by
“internal promotion” which does not come with monetary benefits but where
individuals are given increased responsibility and greater visibility as an
expert on a certain topic. This, in turn, can be leveraged for professional
advancement. This may include assignment into leadership of working groups,
but also may include the assignment of influential tasks tasks (such as which
graduate student receives the Nature-worthy thesis topic). The manner in which
these decisions are made is in no way uniform and often-times relies heavily
on individual professional networks, thereby not rewarding “good science” but
“who knows who.” It also means that the recognition of junior colleagues is
more directly reliant on the power dynamics with the senior colleagues most
closely connected to their work. This is inequitable and prone to creating
“in” and “out” groups, sometimes referred to as “gatekeeping,” and can be done
intentionally or unintentionally. Leadership drawn from the conception of
“merit” (for example, the number of publications generated or cited, or
admission to a top university) fail to account for such factors and are
inherently biased towards those affiliates whose race, gender, socio-economic
status, or disability did not influence their productivity. These choices
perpetuate inequality in leadership, which is a driving factor among hiring
choices and additional opportunities, and ultimately the next pool of
potential collaboration leaders and field-wide experts.

We call on the community to develop methodologies, in association with experts
in organizational psychology and sociology, to better address hiring and
leadership disparities in HEPA. Collaborations, especially those which host
multiple separate working groups, should be required to perform climate
evaluations to ensure that advancement within the collaboration is equitable
and that interventions are taken where necessary to improve the advancement of
collaboration affiliates.

##### Recommendation C2.1:

Reviews of community climate should include an evaluation of how leadership is
selected within HEPA collaborations (e.g. assignment of high-impact analyses &
theses topics, convenership of working groups, public-facing roles
representing the collaboration such as spokespersons or analysis announcement
seminars)

##### Recommendation C2.2:

Reviews of community climate should include the valuation of sub-community
contributions. This includes the participation of “non-scientists” in
community engagement and authorship, community perceptions of operations and
service work, the development of onboarding and early-career networks, and
implementation of policies toward equity in information sharing and software.

###  7.3 Onboarding / Mentoring Networks (Including Early Career ERG)

The culture of our community cannot be assessed without acknowledging the
existing hierarchical structure and power dynamics in academia that all too
often allow the detrimental behaviors to permeate without recourse. In many
systems, graduate students and postdocs report only to their advisor who
solely controls their funding. Others have instituted mentorship or co-
advisors for graduate students and/or postdocs that offer additional support
throughout the full length of the program. The effectiveness of these programs
can be assessed through questions gauging how included students/postdocs feel,
what their support structures look like, if their needs are being met, and if
they feel like they could report a concern with their advisor to their
department/institution without it negatively impacting their careers.
Including such questions in a climate survey could result in best practices
for institutions to maintain mentorship and retention of graduate students and
postdocs while providing a method to circumvent destructive power dynamics.

This topic is discussed briefly in section 6 with regards to onboarding into
HEPA collaborations, and we’d like to expand on this theme here. Formal
mentorship roles within a collaborative environment would go far — not only
towards addressing discrepancies in career trajectories post-PhD [92] but also
toward a sense of belonging within physics communities. There is extensive
work done on effective mentoring practices in physics (e.g.[93]), that this
work does not have time to review, but could be addressed as part of a
collaboration services office (section 8) or cross-collaboration network.

Early career resource groups have been implemented in HEPA collaborations,
such as in ATLAS111111https://atlas.cern/authors/atlas-early-career-
scientists-board. Further conversations are needed to evaluate the
effectiveness and suggested outcomes of such a resource group, but we do
encourage collaborations to set aside space for early career members to
socialize, network, and collaborate with each other.

###  7.4 Organizational Culture for Diversity, Equity and Inclusion

Institutions and collaborations should work to implement support structures,
response practices, and other mechanisms to address the issues identified in
this report. As identified in section 3, it is increasingly common for
institutions to have one or more committees. Steps need to be taken to ensure
that these are effective in meaningful changes including more careful
organization and planning, structures of accountability, institutional power,
and effective support from community/collaboration leadership. This should
include organization of activities in the institution and high level
involvement of management. Examples include the involvement of executive
sponsors from the directorate in the committees, and leadership by a central
authority such as a Chief Diversity Officer who has power in the organization.
Evidence-based best practices should be sought and implemented by institutions
in order to improve workforce diversity, recruiting, support, inclusion and
equity and ensure that all have the best opportunity to excel once on board.
They include:

  * •

All search committee members required to take implicit bias training and
discuss the lessons learned.

  * •

Hiring committees required to have diverse membership, with e.g. requirements
for female and under-represented group participation, and scientists from
other Divisions are invited to serve in order to give their perspective.

  * •

Job postings contain gender neutral language, have clear qualifications
relevant to the position and do not inadvertently screen out under-represented
minorities.

  * •

All positions are posted on diversity web sites to increase outreach to women
and underrepresented minorities.

Institutions should have active DEI committees, and employees in HEP areas
should seek to serve in Employee Resources Groups (ERGs), such as the Women’s
Science and Engineering Council, Early Career Resource Group, Lambda Alliance,
Asian Pacific Islander, Veterans, African American and others. As part of
increasing awareness and accountability of DEI principles, institutional
leadership and staff should participate in a Equity Reset program with a
equity curriculum addressing race, gender, gender and sexuality minorities and
other aspects. We believe that one of the best ways to have diverse staff is
to invest in the pipeline for talented and diverse students and encourage
their careers. For example, staff in the HEP divisions actively participate in
the SAGE workshops, a summer camp for high school girls that is co-hosted by
SLAC and LBNL.121212https://k12education.lbl.gov/programs/high-school/sage

Implementing the various evidence based practices described in this section,
has increased awareness and importance of DEI efforts in workforce development
and resulted in actions. We employ numerous undergraduates who are in the
pipeline toward STEM careers, and pride ourselves in the further development
and career success of our diverse graduate students and postdocs. We need to
continue these efforts and measure effectiveness of these practices.

The strategic goal: to create at all levels an institution where all employees
feel valued, included, and empowered to reach their full potential.

##  8 Collaboration Services

##### Recommendation F1.4:

Funding and structural aid should be made available to develop “Collaboration
services” offices at host laboratories that can provide HEPA collaborations
and other physics communities with the following: a) advice on topics
including those listed in F1.3, b) training in project management,
ombudsperson roles, and victim-centered approaches to investigations, and c)
logistical tools including facilitation of investigation and mediation.

##### Recommendation C1.2:

The community should prioritize the implementation of best practices networks
across institutions and communities of physics practice. This may be
facilitated through Collaboration Services Offices (F1.4), but may also
include the facilitation of networks between DEI groups at similar
collaborations.

###  8.1 Development of network at host labs

Host laboratories can be assigned responsibility for services for their
collaborations, examples of which include

  * •

Training. Host labs could provide DEI training to the collaborations they
host. Topics include microaggressions, conflict resolution, effective group
management, accessibility (expecially in the post-COVID era). Labs already
provide such training to their employees and employ experts in these fields.
Initiatives such as LBNL Equity Reset[94] have produced documentation for
events that is already being propogated into other communities (namely the
First Friday events have spread to the UC Berkeley Physics Department). It is
important at this point to note that not all collaboration members have access
to training at their home institutions; for example, the DESI collaboration
has invited LBNL diversity experts to speak at its meetings.

  * •

Reporting Inbox. Host labs can intake reports of concerns and violations in an
anonymized way that is firewalled from the collaboration. Fermilab provides
this service for people with an account.

  * •

Personal conflict resolution. Collaborations generally do not have access to
trained experts who can handle personal non-scientific conflicts. The
availability of professional mediators would dramatically decrease the chance
of additional harm compared to internal mediation.

  * •

Hosting Resources. Myriad resources exist on a number of platforms by which
individuals can educate themselves and help form more inclusive communities,
so much so that for many people it may be overwhelming to know where to begin.
For a large number of individuals in our community, rediscovering materials by
word of mouth is likely not to be the most efficient and effective way of
becoming educated. Furthermore, these materials may not be composed in ways
that are conducive to application within particle physics at universities and
national laboratories. Perhaps an individual is planning to hold a conference
at their institution and wants to be inclusive but does not have ample time or
resources to rediscover the most important aspects of inclusivity at a
workshop of their variety. A solution should address questions and desires
such as “How should I educate myself or my group?” and “I’m organizing a
workshop, what am I missing?” By compiling go-to, literature-driven resources
meant specifically for the physics community (“from physicists for
physicists”) that are open source and living/evolving over the course of time,
we can mitigate this problem to catalyze more widespread change and
understanding of the issues.

Other services needed by collaborations may be better served at the community
level, due to considerations of the economy of scale and independence.

  * •

Investigation and adjudication. Trusted impartial and independent bodies are
sometimes needed to resolve incidents within collaborations . Professional
societies and host laboratories should play a role.

  * •

Ombudsperson training Training for collaboration ombudspeople is expensive,
oversubscribed, and tends to be focused on an institutional rather than
collaboration environment (which can differ dramatically, as demonstrated in
sections 3 and 4. A network of community oriented offices can sponsor periodic
collaboration-specific ombudsperson training workshops.

  * •

Climate surveys and external reviews. The American Physical Society offers
site visits for improving the climate for women and other marginalized groups
in physics departments. Additionally, reviews can be prohibitively expensive
for each collaboration, but some reviews are already ongoing at host labs
(section 4.1.3). Formal implementation, funded by host labs, can dramatically
improve our access to data about the status and progress of climate related
initiatives.

##### Recommendation F2.1:

Funding agencies should facilitate Climate Community Studies, instead of
leaving such studies up to individual communities to complete. In line with
F3.2, these studies should be informed by expertise in social and
organizational dynamics.

##### Recommendation F3.2:

In line with F2.1, community studies should receive advice from experts in
sociology and organizational psychology, such that the tools used to evaluate
the climate are adequate, effective, and informative. These studies and
accompanying expertise should be funded at the federal and institutional
levels.

###  8.2 Notes on implementation

Educating and training the community to bring awareness about the DEI issues
is also essential for changing the culture. Again, this is something that is
addressed to varying degrees by individual institutions but is not broadly
accessible throughout the community.

Additionally, institutional training (specifically training on bias) done
without care or the advice of experts has been shown to have little impact on
either implicit or explicit bias [95, 96, 97]; therefore, as with so many
things within the context of this contributed paper, such policies must be
advised by DEI researchers who have expertise in interventions. Furthermore,
training alone should not be represented as institutional change or commitment
to DEI. Unfortunately, as discussed multiple places within this work, modern
concepts and practices performed under the banner of ‘equity’, ‘diversity’,
and ‘inclusion’ often fall far short of directly addressing injustice, and
cannot be entrusted with driving the process for change[10].

Instead, we encourage this collaboration services office to be driven by
and/or staffed with experts in the fields of sociology and
social/organizational psychology, and for policies made by these offices be in
support of DEI beyond what is legally mandated by agency policy.

##  9 Concluding Remarks

We hope that this work has brought up new questions within HEPA communities
and the institutions and agencies that support them — new questions about old
topics. We acknowledge that the themes that have arisen as part of this work
are not novel, and yet we still must speak on them. As such, we will reiterate
our full-throated support of the recommendations provided by Nord et al. in
the “Culture change is necessary, and it requires strategic planning” letter
of intent submitted as part of the Snowmass planning process[1]; these
recommendations are as follows:

##### Recommendation 1.0:

HEPA communities must employ the use of robust strategic planning procedures,
including a full re-envisioning of science workplace norms and culture.

##### Recommendation 2.0:

HEPA communities must implement new modes of community organizing and
decision-making that promote agency and leadership from all stakeholders
within the scientific community.

##### Recommendation 3.0:

HEPA communities must engage in partnership with scholars, professionals, and
other experts in several disciplines, including but not limited to anti-
racism, critical race theory, and social science.

We encourage HEPA communities and the institutions and agencies that support
them to reflect on what is necessary to reinvent the physics workplace, and
then act with policy, with funding, and with person-power toward these ends.
Our community has a chance to, and realistically must go beyond what is
legally mandated, and in fact must go beyond what is “comfortable” in order to
fully realize a field that is equitable, inclusive, and diverse in identities,
ideas, and lived experiences.

Acknowledgements

This work was strongly supported in text and themes by Letters of Interest
submitted to the Snowmass2021 Planning Process. These letters are listed here
(by author-order):

  * •

Assamagan, K.A., et al. (2020a) ‘Climate of the field.’ [98]

  * •

Assamagan, K.A., et al. (2020b) ‘Educational resources for the community.’
[99]

  * •

Assamagan, K.A., et al. (2020c) ‘Recruitment, evaluation, and recognition.’
[100]

  * •

Back, A., et al. (2020) ‘Community support for enforcing collaboration codes
of conduct.’ [101]

  * •

Bard, D. and Bellis, M. (2020) ‘The effect of HEP software culture and
practices on diversity, inclusion, and retention.’ [102]

  * •

Buuck, M., et al. (2020) ‘Creating inclusive collaborations.’ [103]

  * •

Nord, B., et al. (2020) ‘Culture change is necessary, and it requires
strategic planning.’ [1]

  * •

Palladino, K. (2020) ‘A need for alternative collaborative means to address
misconduct.’[104]

  * •

Pronskikh, V. and Meehan, S. (2020) ‘Engineering roles and identities in the
scientific community: towards participatory justice.’ [105]

To that end, we would like to thank the following people for their
contributions: Kétévi A. Assamagan, Ashley Back, Robert Bernstein, Carla
Bonifazi, Johan S. Bonilla, Daniel Bowring, Micha Buuck, Mu-Chun Chen,
Michelle J. Dolinski, Alex Drlica-Wagner, Kirsty Duffy, Bo Jayatilaka, Alvine
Kamaha, L. J. Kaufman, Michael Kirby, Naoko Kurahashi Neilson, Cecilia Levy,
Kendall Mahn, Rachel Mannino, Sam Meehan, Mark D. Messier, Brian Nord,
Kimberly Palladino, Thomas Rainbolt, Bryan Ramson, Kate Scholberg, Sara Simon,
Erica Snider, Jason St. John, Kelly Stifter, Jon Urheim, Justin Vasel, Tammy
Walton, Lindley Winslow, and Jeremy Wolcott.

## References

  * [1] Brian Nord et al.  Culture change is necessary, and it requires strategic planning.  Snowmass2021 LOI, 2020.
https://www.snowmass21.org/docs/files/summaries/CommF/SNOWMASS21-CommF1_CommF6_brian_nord_new-056.pdf.

  * [2] Anne Marie Porter and Rachel Ivie.  Women in Physics and Astronomy, 2019.  Technical report, AIP Statistical Research Center, 2019.
  * [3] Rachel Ivie, Garrett Anderson, and Susan White.  African Americans & Hispanics among Physics & Astronomy Faculty.  Technical report, AIP Statistical Research Center, July 2014.
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potential relation with the work of [104]. When the group is non-simply-laced
there exist other singularities called quasi-minimal [104]. (By $E_{8}[[z]]$
we mean the group of matrices in $E_{8}$ with coefficients in the ring of
formal power series in $z$, usually denoted $\mathbb{C}[[z]]$. For an
introduction to affine Grassmannians accessible to physicists see e.g. [78,
Sec. 2.3].) This is precisely what we found in our
$\text{Hom}(\mathbb{Z}_{k},E_{8})$ Hasse. The connection between affine
Grassmannians of Lie groups and moduli spaces of 3d $\mathcal{N}=4$ quivers
has already been put forward in [78], so we expect the (slices of the) affine
Grassmannian of $E_{8}$ to play a role in Higgs branches of 6d electric
quivers obtained from O8-D8-D6-NS5 brane systems with 8 supercharges. It would
be extremely interesting to explore this further.

An obvious generalization of our work is the construction of hierarchies of RG
flows for D-type orbi-instantons, which involve M5-branes probing an $E_{8}$
wall and the $\mathbb{C}^{2}/\Gamma_{\text{D}}=\mathbb{C}^{2}/\mathbb{D}_{k}$
orbifold, with $\mathbb{D}_{k}$ the binary dihedral group of order $4k$
associated with the $D_{k+2}$ singularity. (The corresponding Type IIA
configurations will contain an O6-plane crossing the O$8^{-}$ [12], as well as
the more exotic ON0 [105].404040The (Type IIA) ON0 wraps the same directions
as an NS5 (see table 1), being (T-dual to) the S-dual of an O5- (the ON-) with
a single NS5 on top. Placed at the left end of an electric quiver without
O6-planes or the O8 as follows ON0$k$$k$$2k$$2k$$2k$ , it engineers an
electric quiver with D-type Dynkin diagram shape,
$\overset{{\mathfrak{su}({2k})}}{2}\,\overset{{\mathfrak{su}({2k})}}{2}\cdots\overset{{\mathfrak{su}({2k})}}{2}\,[\operatorname{SU}(2k)]\
,$$\overset{{\mathfrak{su}({k})}}{2}$$\overset{{\mathfrak{su}({k})}}{2}$ which
is allowed by the general rules in [13, App. D]. Adding O6 and O8, there are
four gauge anomaly free combinations [105, Sec. 6], of which the two relevant
ones for our present purposes are O$8^{-}$-O$6^{\mp}$ with an ON0 stuck at
their intersection in the zeroth segment. (Notice that any two of these
orientifold projections always imply the third, as can be seen by taking three
T-dualities on the system of [86, Sec. 7.4]. We are grateful to A. Hanany and
M. Sperling for pointing this out to us.) The electric quiver engineered by
the former (with all 8 D8’s on top of the O8) can be found in [106, Eq.
(5.1)], whereas the latter in [106, Eq. (5.16)]. Whenever O6 and O8 have the
same charge the quiver is linear (i.e. has A-type Dynkin shape), whereas when
they have opposite charge it has D-type shape [105]. The corresponding 3d
magnetic quivers have been worked out in [107, Sec. 4.2]. (There also exists
an $\widetilde{\text{O}6}^{-}$ which is equivalent to an O$6^{-}$ with a
half-D6 stuck on it; it is allowed only for odd Romans mass [108, 109]. If the
Romans mass is instead even –including the case where it is vanishing– only
the O$6^{-}$ is allowed, whence the above list of possibilities.)) This poses
two immediate difficulties however. First, the work of Kac [58] cannot be
directly generalized to homomorphisms in $\text{Hom}(\mathbb{D}_{k},E_{8})$,
since it is impossible to grade a Lie algebra by a non-Abelian group.414141See
footnote 17. Hence we do not have at our disposal a set of “natural” Kac
labels to define the SCFTs. One may proceed as follows. To find the embeddings
of $\mathbb{D}_{k}$ into the $E_{8}$ group we first embed its cyclic subgroup
$\mathbb{Z}_{2k}\subset\mathbb{D}_{k}$, which is done via Kac labels. For each
label we have an element $a$ of a maximal torus of $E_{8}$, generating a
subgroup of order $k$. We must then find elements $w$ of the Weyl group
$W(E_{8})$ which satisfy $w(a)=a^{-1}$: these are the elements of the form
$-w^{\prime}$ where $w^{\prime}$ belongs to the centralizer of $a$ in
$W(E_{8})$. We can pick one representative $b$ of each conjugacy class $w$,
and ask how many of those have representatives in $E_{8}$ which are of order
2, namely find which of the homomorphisms in
$\text{Hom}(\mathbb{D}_{k},E_{8})$ from [36] (called
$\text{Hom}(\text{Dic}_{k},E_{8})$ there) have kernels consisting of
$b^{2}=a^{n}$ (see [13, Sec. 7.3.3] for their F-theory realization).424242We
are grateful to P. Levy for suggesting this argument to us. Second, the full
set of rules to obtain the 3d ortho-symplectic magnetic quivers of D-type
orbi-instantons coming from O8-D8-O6-D6-NS5 brane systems has not been worked
out yet (to the best of our knowledge), even though this difficulty may more
easily be overcome by extending the rules proposed in [75, 110, 111] to also
include the O8 (see e.g. [107, Sec. 4] for progress in this direction). Once
this is done, we can again apply the quiver subtraction algorithm to construct
all possible RG flows, and check compatibility with the $a$-theorem. For
E-type orbi-instantons, the 3d magnetic quiver technology is currently
unavailable. (We expect there to be a finite number of hierarchies of RG flows
since the binary tetrahedral, octahedral, and icosahedral groups
$\Gamma_{E_{6,7,8}}$ have a finite, non-parametric order. See e.g. [13, Sec.
7.4.1] for the F-theory realization of orbi-instantons in
$\text{Hom}(\Gamma_{E_{8}},E_{8})$.)

Finally, we wish to comment on two less obvious generalizations of the present
work. One direction involves studying [112] the hierarchy of Higgs branch RG
flows between so-called “massive E-string theories” [113, Sec.
5.2],434343These theories have appeared even earlier in [27, 38, 39, 40],
though they were not given a name there. which are generalizations of (2.6)
with nonzero Romans mass, the total number of D8’s being less than 8. These
systems do not allow for an M-theory construction, but do so both in massive
Type IIA (hence the name) and F-theory, where the left flavor symmetry factor
is $[E_{1+(8-n_{0})}]$ for $n_{0}=1,\ldots,8$ ($8-n_{0}$ being the total
number of D8’s, which are close to the O8-):

$[E_{1+(8-n_{0})}]\ {1}\ \overset{\mathfrak{su}(n_{0})}{2}\
\overset{\mathfrak{su}(2n_{0})}{2}\cdots\overset{\mathfrak{su}((N-1)n_{0})}{2}\
[\operatorname{SU}(Nn_{0})]\ .$ (5.1)

As is clear, the difference with (2.6) stems from the absence of a plateau of
$\mathfrak{su}$ algebras. Therefore we take $Nn_{0}$ to mean $k$. Indeed
notice that (5.1) comes from truncating the more general _massless_ quiver in
[37, Eq. (5.71)] with $8-n_{0}+n_{0}=8$ D8’s, namely

$[E_{1+(8-{n}_{0})}]\ 1\ \overset{\mathfrak{su}({n}_{0})}{2}\
\overset{\mathfrak{su}(2{n}_{0})}{2}\
\overset{\mathfrak{su}(3{n}_{0})}{2}\cdots\
\overset{\mathfrak{su}((m-1){n}_{0})}{2}\
\underset{[N_{\text{f}}={n}_{0}]}{\overset{\mathfrak{su}(m{n}_{0})}{2}}\
\overset{\mathfrak{su}(m{n}_{0})}{2}\cdots\overset{\mathfrak{su}(m{n}_{0})}{2}\
[\operatorname{SU}(m{n}_{0})]\ ,$ (5.2)

where obviously $k=mn_{0}$.444444(5.2) is the electric quiver of $[n_{0}^{m}]$
for $n_{0}=1,\ldots,6$, $[4^{\prime m},3^{\prime m}]$ for $n_{0}=7$, and
$[4^{\prime 2m}]$ for $n_{0}=8$. Putting $m=N$ and truncating at the leftmost
$\overset{\mathfrak{su}(m{n}_{0})}{2}$ in position $m$, we obtain (5.1) with
$k=Nn_{0}$. However notice that this $k$ is _not_ the order of the
$\mathbb{C}^{2}/\mathbb{Z}_{k}$ orbifold but rather the number of semi-
infinite D6’s in the rightmost segment, since as we said there is no M-theory
engineering of that theory. Here

$E_{1+(8-n_{0})}=\\{E_{8},E_{7},E_{6},\mathfrak{so}(10),\mathfrak{su}(5),\mathfrak{su}(3)\oplus\mathfrak{su}(2),\mathfrak{su}(2)\oplus\mathfrak{u}(1),\mathfrak{su}(2)\\}_{n_{0}=1}^{8}$
(5.3)

is the list of [70] (excluding the $\tilde{E}_{1}={\mathfrak{u}({1})}$ and
$E_{0}=\emptyset$ cases of [68]). In analogy with (2.2), one can define
$E_{1+(8-{n}_{0})}$ Kac labels for all $n_{0}$ (now associated with
$\mathbb{Z}_{Nn_{0}}$-gradings of $E_{1+(8-n_{0})}$) by once again exploiting
the results of [58]. Although $E_{8}$ has a unique affine form, $E_{8}^{(1)}$
in (2.3), this is not true for some of the algebras in the above list, and one
has to consider both _untwisted and twisted_ affine versions,454545The twisted
affine algebras also make an appearance in [45] in the construction of 4d
$\mathcal{N}=3$ S-folds from A-type orbi-instantons. It would be interesting
to study whether a potential connection exists. namely

$E_{8}^{(1)}\ ,\ E_{7}^{(1)}\ ,\ E_{6}^{(2)}\ ,\ E_{6}^{(1)}\ ,\ D_{5}^{(2)}\
,\ D_{5}^{(1)}\ ,\ A_{4}^{(2)}\ ,\ A_{4}^{(1)}\ ,\ A_{2}^{(2)}\ ,\
A_{2}^{(1)}\ ,\ A_{1}^{(1)}\ ,$ (5.4)

and direct sums (also with ${\mathfrak{u}({1})}$) as per (5.3): the
superscript $k^{\prime}=1$ ($k^{\prime}=2$) selects the untwisted (twisted)
version [59, Sec. 7.9]. (See table 4 for definitions.)

$A_{1}^{(1)}:$ | $\overset{1}{\underset{\alpha_{0}}{\circ}}\Leftrightarrow\overset{1}{\underset{\alpha_{1}}{\circ}}$
---|---
$A_{2}^{(1)}:$ | $\begin{array}[]{c}\raisebox{-5.0pt}{\rotatebox{45.0}{$\rule{10.0pt}{0.4pt}$}}\overset{1}{\underset{\alpha_{0}}{\circ}}\raisebox{2.5pt}{\rotatebox{-45.0}{$\rule{10.0pt}{0.4pt}$}}\\\\[-7.0pt] \overset{1}{\underset{\alpha_{1}}{\circ}}-\overset{1}{\underset{\alpha_{2}}{\circ}}\end{array}$
$A_{2}^{(2)}:$ | $\overset{2}{\underset{\alpha_{0}}{\circ}}\lower 2.0pt\hbox{ \leavevmode\hbox to12.4pt{\vbox to10.4pt{\pgfpicture\makeatletter\hbox{\hskip 12.20001pt\lower-5.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}}{{}}{} {{{}{}}{}}{}{{}{}}{{}}{} {{{}{}}{}}{}{{}{}}{{}}{} {{{}{}}{}}{}{{}{}}{{}}{} {{{}{}}{}}{}{}{{}}{} {{}{{}{}}}{{}} {{{}}{{}}}{{}}{{{}}{{{}{}}}}{{}{}}{{}}{}{}{} {{}{}}{{}} {{{}}{{}}}{{}}{{{}}{{}}}{}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{3.0pt}\pgfsys@lineto{-10.00002pt}{3.0pt}\pgfsys@moveto{0.0pt}{1.0pt}\pgfsys@lineto{-11.00002pt}{1.0pt}\pgfsys@moveto{0.0pt}{-1.0pt}\pgfsys@lineto{-11.00002pt}{-1.0pt}\pgfsys@moveto{0.0pt}{-3.0pt}\pgfsys@lineto{-10.00002pt}{-3.0pt}\pgfsys@moveto{-9.00002pt}{5.0pt}\pgfsys@curveto{-9.58954pt}{2.79987pt}{-10.3894pt}{1.61061pt}{-12.00002pt}{0.0pt}\pgfsys@curveto{-10.3894pt}{-1.61061pt}{-9.58954pt}{-2.79987pt}{-9.00002pt}{-5.0pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hbox to0.0pt{}{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}} }\overset{1}{\underset{\alpha_{1}}{\circ}}$
$A_{4}^{(1)}:$ | $\begin{array}[]{c}\raisebox{-10.5pt}{\rotatebox{30.0}{$\rule{30.0pt}{0.4pt}$}}\overset{1}{\underset{\alpha_{0}}{\circ}}\raisebox{4.0pt}{\rotatebox{-30.0}{$\rule{30.0pt}{0.4pt}$}}\\\\[-7.0pt] \overset{1}{\underset{\alpha_{1}}{\circ}}-\overset{1}{\underset{\alpha_{2}}{\circ}}-\overset{1}{\underset{\alpha_{3}}{\circ}}-\overset{1}{\underset{\alpha_{4}}{\circ}}\end{array}$
$A_{4}^{(2)}:$ | $\overset{2}{\underset{\alpha_{0}}{\circ}}\Leftarrow\overset{2}{\underset{\alpha_{1}}{\circ}}\Leftarrow\overset{1}{\underset{\alpha_{2}}{\circ}}$
$D_{5}^{(1)}:$ | $\overset{1}{\underset{\alpha_{1}}{\circ}}-\overset{2\overset{{\hbox to0.0pt{\hss$\scriptstyle 1$}\displaystyle\circ{\hbox to0.0pt{$\scriptstyle\alpha_{0}$\hss}}}}{\scriptstyle|}}{\underset{\alpha_{2}}{\circ}}-\overset{2\overset{{\hbox to0.0pt{\hss$\scriptstyle 1$}\displaystyle\circ{\hbox to0.0pt{$\scriptstyle\alpha_{5}$\hss}}}}{\scriptstyle|}}{\underset{\alpha_{3}}{\circ}}-\overset{1}{\underset{\alpha_{4}}{\circ}}$
$D_{5}^{(2)}:$ | $\overset{1}{\underset{\alpha_{0}}{\circ}}\Leftarrow\overset{1}{\underset{\alpha_{1}}{\circ}}-\overset{1}{\underset{\alpha_{2}}{\circ}}-\overset{1}{\underset{\alpha_{4}}{\circ}}\Rightarrow\overset{1}{\underset{\alpha_{5}}{\circ}}$
$E_{6}^{(1)}:$ | $\overset{1}{\underset{\alpha_{1}}{\circ}}-\overset{2}{\underset{\alpha_{2}}{\circ}}-\overset{3\overset{\overset{{\hbox to0.0pt{\hss$\scriptstyle 1^{\prime\prime}$}\displaystyle\circ{\hbox to0.0pt{$\scriptstyle\alpha_{0}$\hss}}}}{\scriptstyle|}}{\overset{{\hbox to0.0pt{\hss$\scriptstyle 2^{\prime\prime}$}\displaystyle\circ{\hbox to0.0pt{$\scriptstyle\alpha_{6}$\hss}}}}{\scriptstyle|}}}{\underset{\alpha_{3}}{\circ}}-\overset{2^{\prime}}{\underset{\alpha_{4}}{\circ}}-\overset{1^{\prime}}{\underset{\alpha_{5}}{\circ}}$
$E_{6}^{(2)}:$ | $\overset{1}{\underset{\alpha_{0}}{\circ}}-\overset{2}{\underset{\alpha_{1}}{\circ}}-\overset{3}{\underset{\alpha_{2}}{\circ}}\Leftarrow\overset{2^{\prime}}{\underset{\alpha_{3}}{\circ}}-\overset{1^{\prime}}{\underset{\alpha_{4}}{\circ}}$
$E_{7}^{(1)}:$ | $\overset{1}{\underset{\alpha_{0}}{\circ}}-\overset{2}{\underset{\alpha_{1}}{\circ}}-\overset{3}{\underset{\alpha_{2}}{\circ}}-\overset{4\overset{{\hbox to0.0pt{\hss$\scriptstyle 2^{\prime\prime}$}\displaystyle\circ{\hbox to0.0pt{$\scriptstyle\alpha_{7}$\hss}}}}{\scriptstyle|}}{\underset{\alpha_{3}}{\circ}}-\overset{3^{\prime}}{\underset{\alpha_{4}}{\circ}}-\overset{2^{\prime}}{\underset{\alpha_{5}}{\circ}}-\overset{1^{\prime}}{\underset{\alpha_{6}}{\circ}}$
Table 4: The untwisted ($k^{\prime}=1$) and twisted ($k^{\prime}=2$) affine
Dynkin diagrams of (summands of) the $E_{1+(8-n_{0})}$ algebras listed in
(5.3).

Accordingly, (2.2) gets modified to

$Nn_{0}=k^{\prime}\sum_{j=0}^{\operatorname{rk}\left(E_{1+(8-n_{0})}\right)}a_{j}n_{j+1}\
,$ (5.5)

with $k^{\prime}=2$ being allowed only for even $Nn_{0}$, and $a_{j}$ the
Coxeter labels appearing in table 4 (combining unprimed and primed ones by
abuse of notation). We expect different massive E-string theories (for a given
$E_{1+(8-n_{0})}$) to be classified by Kac labels, and there to be a hierarchy
of RG flows between them.

Second, the original motivation for this work was the study of the stability
of the non-supersymmetric version of the Type IIA AdS7 vacua dual to A-type
orbi-instantons, constructed in full generality in [92, 28, 31]. (The Type IIA
massless vacua come from a reduction to 10d of an orientifolded version of the
11d Freund–Rubin solution, i.e. AdS${}_{7}\times
S^{4}/(\mathbb{Z}_{2}\times\Gamma_{\text{ADE}})$ [27, Sec. 7]. For M5’s
probing just the $E_{8}$ wall, i.e. for $k=1$ in type A, an AdS${}_{7}\times
S^{4}/\mathbb{Z}_{2}$ vacuum was argued to exist already in [114].) Indeed the
O8 is the last missing ingredient in the analysis of [115], which has found
that AdS7 vacua dual to D8-D6-NS5 or D8-O6-D6-NS5 brane systems are unstable
(in their non-supersymmetric version) and non-scale-separated (in their
supersymmetric version). We plan to come back to an extension of that analysis
in presence of the O8- and O8∗ in [116].

## Acknowledgments

We are indebted to G. Bruno de Luca for collaboration in the early stages of
this work, useful comments, and ongoing collaboration on related topics; to
Simone Giacomelli and Noppadol Mekareeya for many useful discussions about
magnetic quivers, and to the former for sharing with us some results from [44]
prior to publication and ongoing collaboration; to Paul Levy and Tom Rudelius
for enlightening email correspondence; and to Alessandro Tomasiello for
bringing to our attention reference [66] and various useful comments. The work
of MF is supported in part by the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No.
754496 - FELLINI. MF would like to thank SISSA, Trieste for hospitality during
the completion of this work. SG is supported in part by INFN and by MIUR-PRIN
contract 2017CC72MK003.

##  Appendix A RG flow Hasse diagrams for $k=2,\ldots,20$

In this appendix we collect the Hasse diagram of homomorphisms in
$\text{Hom}(\mathbb{Z}_{k},E_{8})$ for $k=2,\ldots,20$. The legend for the
various colors and decorations in the RG flows is as follows. (The figures for
$k\geq 7$ have been scaled down to fit in a single page; the interested reader
can zoom in to inspect the details.)

  * •

For $k=2,3,4,5$ (figure 11) the list of colors and decorations is the same as
that in section 3.7. The red nodes and red dashed edges correspond to
nilpotent orbits and KP transitions between them for ${\mathfrak{su}({2})}$,
${\mathfrak{su}({3})}$, ${\mathfrak{su}({4})}$, ${\mathfrak{su}({5})}$
respectively.

In particular in figure 12 we have plotted the proposed hierarchy for $k=4$
from [36, Fig. 1]. Notice the difference with respect to our hierarchy: we
have fixed $N$ whereas the latter reference describes a “mixed” flow between
orbi-instantons with different Kac labels while also changing $N$ (i.e.
performing small instanton transitions):464646We would like to thank T.
Rudelius for email correspondence, and A. Tomasiello for discussion on this
point. over each SCFT in their hierarchy we have superposed the number of
tensor multiplets of the SCFT. The flows are implemented by KP transitions of
the minimal type written besides the edges in figure 12; these transitions are
found by quiver subtraction between the magnetic quivers (associated with the
Kac labels) appearing in table 5. Notice in particular the presence of
$\mathfrak{e}_{i}$ KP transitions: these can only appear in the game whenever
we perform a small $E_{8}$ instanton transition reducing $N$.

  * •

For $k=7$ (figure 13), the colors and decorations correspond once again to
those in section 3.7. The red nodes and red dashed edges correspond to
nilpotent orbits and KP transitions between them for ${\mathfrak{su}({7})}$,
which starts from $[4^{\prime},3^{\prime}]$ as noted in eq:k7start

  * •

For $k=8$ and $k=9$ (figure 13 and 14 respectively), the nodes, edges, edge
labels and the green nodes have the usual meaning as in section 3.7. The red
nodes and red dashed edges correspond to nilpotent orbits and KP transitions
between them for ${\mathfrak{su}({8})}$ and ${\mathfrak{su}({9})}$
respectively. These flows start from $[4^{\prime 2}]$ and $[3^{\prime 3}]$
respectively as noted in equations (4.4) and (4.5). The yellow nodes indicate
a pair of theories which have the same electric quiver but differ by the 6d
$\theta$ angle.

  * •

For $k=10$ and $k=11$ (figure 15 and figure 16 respectively), the nodes,
edges, edge labels and the green nodes have the usual meaning as in section
3.7. Moreover:

    * –

The blue nodes correspond to a pair of theories that also differ by their 6d
$\theta$ angles, but have ${\mathfrak{usp}({2})}$ as gauge algebra on the $-1$
curve.

    * –

the red nodes and red dashed edges correspond to nilpotent orbits and KP
transitions between them for ${\mathfrak{su}({8})}$. For $k=10$, the starting
point is $[4^{\prime 2},2^{\prime}]$, or equivalently $[3^{\prime 2},2^{\prime
2}]$ , as noted in eq:generick-2ai. For $k=11$, the flow starts from
$[3^{\prime 3},2^{\prime}]$ as noted in eq:generick-2b.

    * –

The orange nodes and edges correspond to “parallel flows” between theories
differing by the 6d $\theta$ angle, and have an empty gauge algebra on the
$-1$ curve of their electric quivers.

  * •

For $k=12$ (figure 17), the nodes, edges, edge labels and the green nodes have
the usual meaning as in section 3.7. Moreover:

    * –

the red nodes and red dashed edges correspond to nilpotent orbits and KP
transitions between them for ${\mathfrak{su}({9})}$. This starts from
$[3^{\prime 4}]$ as noted in eq:su9starting.

    * –

The orange nodes and edges correspond to parallel flows between theories
differing by the 6d $\theta$ angle, and have an empty algebra group on the
$-1$ curve of their electric quivers.

    * –

The blue nodes correspond to parallel flows between theories that differ by
their 6d $\theta$ angles, and have ${\mathfrak{usp}({2})}$ as gauge algebra on
the $-1$ curve.

    * –

The purple nodes correspond to a pair of theories that also differ by their 6d
$\theta$ angles, and have a ${\mathfrak{usp}({4})}$ gauge algebra on the $-1$
curve.

  * •

For $k=13$ (figure 18), the decorations are very similar to those to $k=12$,
except that the red nodes and red dashed edges correspond to nilpotent orbits
and KP transitions between them for ${\mathfrak{su}({8})}$. The starting point
of the flow is $[3^{\prime 3},2^{\prime 2}]$ as noted in eq:generick-2b.

  * •

For $k=14$ (figure 19), the nodes, edges, edge labels and the green nodes have
the usual meaning as in section 3.7. Moreover:

    * –

there are now four different ways of embedding the nilpotent orbits Hasse
diagram of ${\mathfrak{su}({8})}$ in this homomorphisms Hasse as discussed in
equations (4.13)-(4.15):

      * *

the red nodes and red dashed edges correspond to ${\mathfrak{su}({8})}$ Hasse
diagrams starting from the electric quiver in (4.14);

      * *

the blue nodes and blue dashed edges correspond to ${\mathfrak{su}({8})}$
Hasse diagrams starting from the electric quiver in (4.15);

      * *

the magenta nodes and magenta dashed edges correspond to
${\mathfrak{su}({8})}$ Hasse diagrams starting from the electric quiver in
(4.13). The two starting points for these flows differ by the 6d $\theta$
angle and are highlighted by black nodes. The node $[4^{\prime 2},2^{\prime
3}]$ is black, magenta as well as green. For simplicity of presentation, we
represent it simply as a green dot in fig:k14.

    * –

There are four families of theories differing by the 6d $\theta$ angle:

      * *

the orange and light blue nodes with the corresponding edges are parallel
flows between theories differing by their 6d $\theta$ angles. These flows have
an empty gauge algebra on the $-1$ curve and are discussed in (4.27);

      * *

the olive green nodes and edges correspond to parallel flows between theories
that have a ${\mathfrak{usp}({2})}$ gauge algebra on the $-1$ curve, and are
discussed in (4.28);

      * *

the brown nodes and edges correspond to parallel flows between theories that
have a ${\mathfrak{usp}({4})}$ gauge algebra on the $-1$ curve, and are
discussed in (4.29);

      * *

the black nodes correspond to the pair of theories that have a
${\mathfrak{usp}({6})}$ gauge group on the $-1$ curve. As remarked above, The
node $[4^{\prime 2},2^{\prime 3}]$ is black, magenta as well as green. For
simplicity of presentation, we represent it simply as a green dot.

  * •

For $k=15$ (figure 20), the nodes, edges, edge labels and the green nodes have
the usual meaning as in section 3.7.

    * –

In addition, the red nodes and dashed edges correspond to the
${\mathfrak{su}({9})}$ Hasse diagram, which starts from $[3^{\prime 5}]$ as
noted in eq:su9starting.

    * –

The four families of theories differing by the 6d $\theta$ angle are
highlighted using the same colors as in the $k=14$ diagram.

  * •

For $k=16$ (figure 21), the nodes, edges, edge labels and the green nodes have
the usual meaning as in section 3.7. Moreover:

    * –

there are three different starting points for the nilpotent orbits Hasse of
${\mathfrak{su}({8})}$ as is evident from equations (4.7) and (4.8). The ones
that follow from eq:generick-2ai are $[4^{\prime 2},2^{\prime 4}]$ and
$[4^{\prime 4},2^{\prime}]$, which additionally have an equivalent node that
differ by the 6d $\theta$ angle namely $[3^{\prime 2},2^{\prime 5}]$ and
$[3^{\prime 4},2^{\prime 2}]$ respectively. eq:generick-2aii gives $[4^{\prime
3},2^{\prime 2}]$. The nilpotent orbits Hasse diagram of
${\mathfrak{su}({8})}$ originating from each of these nodes is highlighted
with magenta (for $[4^{\prime 2},2^{\prime 4}]$ and $[3^{\prime 2},2^{\prime
5}]$), red (for $[4^{\prime 3},2^{\prime 2}]$), blue (for $[4^{\prime 4}]$)
and orange (for $[3^{\prime 4},2^{\prime 2}]$) dashed edges respectively.

    * –

The olive green, brown, black and yellow nodes correspond to theories and
flows that differ by their 6d $\theta$ angles. The electric quivers
corresponding to these theories have
${\mathfrak{usp}({2})},{\mathfrak{usp}({4})},{\mathfrak{usp}({6})}$ and
${\mathfrak{usp}({8})}$ gauge groups on the $-1$ curve respectively.

  * •

For $k=17$ (figure 22), the nodes, edges, edge labels and the green nodes have
the usual meaning as in section 3.7. The rest of the decorations are the same
as those for $k=15$. The nilpotent orbits Hasse diagram of
${\mathfrak{su}({8})}$ starts from $[3^{\prime 3},2^{\prime 4}]$ as expected
from eq:generick-2b.

  * •

For $k=18$ (figure 23), the nodes, edges, edge labels and the green nodes have
the usual meaning as in section 3.7. In addition:

    * –

the red nodes and dashed edges correspond to the ${\mathfrak{su}({9})}$ Hasse
diagram, which starts from $[3^{\prime 6}]$ as noted in eq:su9starting.

    * –

The flows highlighted with cyan and orange correspond to parallel flows among
theories that differ by the 6d $\theta$ angle, and have an empty $-1$ curve in
their electric quiver.

    * –

Nodes and flows highlighted in olive green, brown, black, yellow and blue
correspond to theories that differ by their 6d $\theta$ angle and have the
gauge group
${\mathfrak{usp}({2})},{\mathfrak{usp}({4})},{\mathfrak{usp}({6})},{\mathfrak{usp}({8})}$
and ${\mathfrak{usp}({10})}$ on the $-1$ curve of their electric quiver.

  * •

For $k=19$ (figure 24), the nodes, edges, edge labels and the green nodes have
the usual meaning as in section 3.7. In addition:

    * –

there are two different ways of embedding the nilpotent orbits Hasse diagram
of ${\mathfrak{su}({8})}$ in this homomorphisms Hasse. This follows from
eq:generick-2b – which gives $[3^{\prime 3},2^{\prime 5}]$, and eq:generick-3a
– which gives $[3^{\prime 5},2^{\prime 2}]$. These are highlighted with blue
and red nodes and dashed edges respectively.

    * –

The flows highlighted with cyan and orange correspond to parallel flows among
theories that differ by the 6d $\theta$ angle, and have an empty $-1$ curve in
their electric quiver.

    * –

Nodes and flows highlighted in olive green, brown, black, yellow and blue
correspond to theories that differ by their 6d $\theta$ angle and have the
gauge group
${\mathfrak{usp}({2})},{\mathfrak{usp}({4})},{\mathfrak{usp}({6})},{\mathfrak{usp}({8})}$
and ${\mathfrak{usp}({10})}$ on the $-1$ curve of their electric quiver.

  * •

For $k=20$ (figure 25), the nodes, edges, edge labels and the green nodes have
the usual meaning as in section 3.7. In addition:

    * –

there are four different ways of embedding the nilpotent orbits Hasse diagram
of ${\mathfrak{su}({8})}$ in this homomorphisms Hasse as discussed in
sec:generick_nilpotent_hasse. These flows start from $[4^{\prime 2},2^{\prime
6}]$ (and the equivalent $[3^{\prime 2},2^{\prime 7}]$ that differs by the 6d
$\theta$ angle), $[4^{\prime 3},2^{\prime 4}]$, $[4^{\prime 4},2^{\prime 2}]$
(and the equivalent $[3^{\prime 4},2^{\prime 4}]$ that differs by the 6d
$\theta$ angle) and $[4^{\prime 5}]$. These follow from equations (4.7) and
(4.8). The flows are highlighted with dashed edges and nodes colored in red,
blue, magenta and grey.

    * –

The flows highlighted with cyan and orange correspond to parallel flows among
theories that differ by the 6d $\theta$ angle, and have an empty $-1$ curve in
their electric quiver.

    * –

Nodes and flows highlighted in olive green, brown, black, yellow, blue and
gray correspond to theories that differ by their 6d $\theta$ angle and have
the gauge group
${\mathfrak{usp}({2})},{\mathfrak{usp}({4})},{\mathfrak{usp}({6})},{\mathfrak{usp}({8})},{\mathfrak{usp}({10})}$
and ${\mathfrak{usp}({12})}$ on the $-1$ curve of their electric quiver.

(a) $k=2$

(b) $k=3$

(c) $k=4$

(d) $k=5$

Figure 11: Hierarchy of RG flows between A-type orbi-instantons for $k=2,3,4,5$ at fixed $N$. Green nodes indicate theories that can flow into the top nodes at $N-1$, and therefore generate the full RG flow hierarchy at $N-1$. Red nodes and red dashed lines between them highlight the Hasse diagram of nilpotent orbits of the ${\mathfrak{su}({2})},{\mathfrak{su}({3})},{\mathfrak{su}({4})},{\mathfrak{su}({5})}$ flavor algebras that are realized on a stack of $2,3,4$ and $5$ D8-branes respectively. $\overset{(N)}{[1^{4}]\leftrightarrow\mathfrak{f}=E_{8}}$$\overset{(N-1)}{[2,1^{2}]\leftrightarrow\mathfrak{f}=E_{7}\oplus{\mathfrak{u}({1})}}$$\overset{(N-2)}{[2^{\prime},1^{2}]\leftrightarrow\mathfrak{f}={\mathfrak{so}({14})}}$$\overset{(N-2)}{[3,1]\leftrightarrow\mathfrak{f}=E_{6}\oplus{\mathfrak{su}({2})}\oplus{\mathfrak{u}({1})}}$$\overset{(N-2)}{[2^{2}]\leftrightarrow\mathfrak{f}=E_{7}\oplus{\mathfrak{su}({2})}}$$\overset{(N-3)}{[3^{\prime},1]\leftrightarrow\mathfrak{f}={\mathfrak{su}({8})}\oplus{\mathfrak{u}({1})}}$$\overset{(N-3)}{[2,2^{\prime}]\leftrightarrow\mathfrak{f}={\mathfrak{so}({12})}\oplus{\mathfrak{su}({2})}\oplus{\mathfrak{u}({1})}}$$\overset{(N-3)}{[4]\leftrightarrow\mathfrak{f}={\mathfrak{so}({10})}\oplus{\mathfrak{su}({4})}}$$\overset{(N-4)}{[4^{\prime}]\leftrightarrow\mathfrak{f}={\mathfrak{su}({8})}\oplus{\mathfrak{su}({2})}}$$\overset{(N-4)}{[2^{\prime 2}]\leftrightarrow\mathfrak{f}={\mathfrak{so}({16})}}$$\mathfrak{e}_{8}$$\mathfrak{e}_{7}$$\mathfrak{d}_{7}$$\mathfrak{a}_{1}$$\mathfrak{e}_{6}$$\mathfrak{e}_{7}$$\mathfrak{a}_{7}$$\mathfrak{d}_{6}$$\mathfrak{d}_{5}$$\mathfrak{a}_{1}$ Figure 12: The hierarchy of RG flows for A-type $k=4$ orbi-instantons from [36, Fig. 1] recast in the formalism of the present paper. The two staggered nodes are intended to convey that the higher IR SCFT (i.e. Kac label) has larger 6d $a$ anomaly. Kac label |  Number of tensor multiplets | 3d magnetic quiver
---|---|---
$[1^{4}]$ | $N$ | ${\scriptstyle 1-2-3-4-N-2N-3N-4N-5N-\overset{\overset{\scriptstyle 3N}{|}}{6N}-4N-2N}$
$[2,1^{2}]$ | $N-1$ | ${\scriptstyle 1-2-3-4-N-2(N-1)-3(N-1)-4(N-1)-5(N-1)-\overset{\overset{\scriptstyle 3(N-1)}{|}}{6(N-1)}-4(N-1)-2(N-1)}$
$[2^{\prime},1^{2}]$ | $N-1$ | ${\scriptstyle 1-2-3-4-N-(2N-2)-(3N-4)-(4N-6)-(5N-8)-\overset{\overset{\scriptstyle 3N-5}{|}}{(6N-10)}-(4N-7)-(2N-4)}$
$[3,1]$ | $N-2$ | ${\scriptstyle 1-2-3-4-N-(2N-3)-(3N-6)-(4N-8)-(5N-10)-\overset{\overset{\scriptstyle 3N-6}{|}}{(6N-12)}-(4N-8)-(2N-4)}$
$[2^{2}]$ | $N-2$ | ${\scriptstyle 1-2-3-4-N-2(N-2)-3(N-2)-4(N-2)-5(N-2)-\overset{\overset{\scriptstyle 3(N-2)}{|}}{6(N-2)}-4(N-2)-2(N-2)}$
$[3^{\prime},1]$ | $N-3$ | ${\scriptstyle 1-2-3-4-N-(2N-3)-(3N-6)-(4N-9)-(5N-12)-\overset{\overset{\scriptstyle 3N-8}{|}}{(6N-15)}-(4N-10)-(2N-5)}$
$[2,2^{\prime}]$ | $N-3$ | ${\scriptstyle 1-2-3-4-N-(2N-4)-(3N-7)-(4N-10)-(5N-13)-\overset{\overset{\scriptstyle 3N-8}{|}}{(6N-16)}-(4N-11)-(2N-6)}$
$[4]$ | $N-3$ | ${\scriptstyle 1-2-3-4-N-2(N-2)-(3N-8)-4(N-3)-5(N-3)-\overset{\overset{\scriptstyle 3(N-3)}{|}}{6(N-2)}-4(N-2)-2(N-2)}$
$[4^{\prime}]$ | $N-4$ | ${\scriptstyle 1-2-3-4-N-(2N-4)-(3N-8)-(4N-12)-(5N-16)-\overset{\overset{\scriptstyle 3N-10}{|}}{(6N-20)}-(4N-14)-(2N-7)}$
$[2^{\prime 2}]$ | $N-4$ | ${\scriptstyle 1-2-3-4-N-(2N-4)-(3N-8)-(4N-12)-(5N-16)-\overset{\overset{\scriptstyle 3N-10}{|}}{(6N-20)}-(4N-14)-(2N-8)}$
Table 5: Magnetic quivers corresponding to the $k=4$ Kac labels of figure 12.

(a) $k=7$

(b) $k=8$

Figure 13: Hierarchy of RG flows between A-type orbi-instantons for $k=7$ and
$k=8$ at fixed $N$. Green nodes indicate theories that can flow into all the
top nodes at $N-1$, and therefore generate the full hierarchy at $N-1$. Red
nodes and red dashed lines between them highlight the Hasse diagram of
nilpotent orbits of the ${\mathfrak{su}({7})}$ or ${\mathfrak{su}({8})}$
flavor symmetry algebra that is realized on a stack of $7$ or $8$ D8-branes
respectively. Figure 14: Hierarchy of RG flows between A-type orbi-instantons
for $k=9$ and fixed $N$. Figure 15: Hierarchy of RG flows between A-type orbi-
instantons for $k=10$ and fixed $N$. Figure 16: Hierarchy of RG flows between
A-type orbi-instantons for $k=11$ and fixed $N$. Figure 17: Hierarchy of RG
flows between A-type orbi-instantons for $k=12$ and fixed $N$. Figure 18:
Hierarchy of RG flows between A-type orbi-instantons for $k=13$ and fixed $N$.
Figure 19: Hierarchy of RG flows between A-type orbi-instantons for $k=14$ and
fixed $N$. Figure 20: Hierarchy of RG flows between A-type orbi-instantons for
$k=15$ and fixed $N$. Figure 21: Hierarchy of RG flows between A-type orbi-
instantons for $k=16$ and fixed $N$. Figure 22: Hierarchy of RG flows between
A-type orbi-instantons for $k=17$ and fixed $N$.

Figure 23: Hierarchy of RG flows between A-type orbi-instantons for $k=18$ and
fixed $N$.

Figure 24: Hierarchy of RG flows between A-type orbi-instantons for $k=19$ and
fixed $N$.

Figure 25: Hierarchy of RG flows between A-type orbi-instantons for $k=20$ and
fixed $N$.

## References

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