# Trust and Terror: Hazards in Text Reveal Negatively Biased Credulity and
Partisan Negativity Bias

Keith Burghardt1, Daniel M.T. Fessler2,3,4, Chyna Tang2, Anne Pisor5, Kristina
Lerman1

###### Abstract

Socio-linguistic indicators of text, such as emotion or sentiment, are often
extracted using neural networks in order to better understand features of
social media. One indicator that is often overlooked, however, is the presence
of hazards within text. Recent psychological research suggests that statements
about hazards are more believable than statements about benefits (a property
known as negatively biased credulity), and that political liberals and
conservatives differ in how often they share hazards. Here, we develop a new
model to detect information concerning hazards, trained on a new collection of
annotated X posts, as well as urban legends annotated in previous work. We
show that not only does this model perform well (outperforming, e.g., zero-
shot human annotator proxies, such as GPT-4) but that the hazard information
it extracts is not strongly correlated with other indicators, namely moral
outrage, sentiment, emotions, and threat words. (That said, consonant with
expectations, hazard information does correlate positively with such emotions
as fear, and negatively with emotions like joy.) We then apply this model to
three datasets: X posts about COVID-19, X posts about the 2023 Hamas-Israel
war, and a new expanded collection of urban legends. From these data, we
uncover words associated with hazards unique to each dataset as well as
differences in this language between groups of users, such as conservatives
and liberals, which informs what these groups perceive as hazards. We further
show that information about hazards peaks in frequency after major hazard
events, and therefore acts as an automated indicator of such events. Finally,
we find that information about hazards is especially prevalent in urban
legends, which is consistent with previous work that finds that reports of
hazards are more likely to be both believed and transmitted.

Figure 1: Detecting hazards. Data used to train a hazard model are either X
posts or a previously annotated set of urban legends (Fessler, Pisor, and
Navarrete 2014). Qualtrics surveys are sent to workers on Cloud Research to
each annotate ten X posts such that there are typically three (and at least 2)
annotators per post. We take the majority decision to annotate the gold
standard label for each text. Data are then fed through a multi-lingual text
embedding and trained on a support vector machine (Hearst et al. 1998)
(alternative and more sophisticated models show poorer performance).

## Introduction

Humans evolved to pay attention to negative information that could signal
potential dangers or hazards in their environment. Evolutionary psychologists
suggest that this “negativity bias” evolved as an adaptive mechanism to
enhance survival by promptly detecting and responding to hazards (Öhman and
Mineka 2001). Researchers have demonstrated that negative information has a
stronger influence on human attention, cognition, and emotion compared to
positive information. Negative events, emotions, and feedback tend to have
more potent and lasting effects than their positive counterparts (Baumeister
et al. 2001; Rozin and Royzman 2001). Additionally, people tend to have
greater credulity towards hazard information compared to benefit information,
even when the information is equivalent in truth value (Fessler, Pisor, and
Navarrete 2014), a phenomenon that has been dubbed “negatively biased
credulity.” Recent work has identified an ideological asymmetry, with social
conservatives more likely to pay attention to and believe information about
hazards than their social liberal counterparts (Fessler 2019). For example,
language highlighting threats has a long history of being used to rally public
support for political measures and causes (Choi et al. 2022).

The same psychological mechanisms also operate in the realm of online
information sharing. Researchers found that negatively-valenced information,
which conveys outrage or sadness, spreads quickly and widely online (Ferrara
and Yang 2015; Brady et al. 2017), and is a factor in the spread of
misinformation (Vosoughi, Roy, and Aral 2018; Ecker et al. 2022), suggesting
that it could be a potent tool for information operations aimed at influencing
public opinion online. However, to date, researchers have not used machine
learning to analyze the presence of information concerning hazards, which can
be potent indicator of negative information (Choi et al. 2022).

We address this gap by presenting a multi-lingual transformer-based model to
recognize information concerning hazards expressed in text messages on X
(formerly Twitter) and within hundreds of urban legends. Analysis of X posts
allows us to understand how major events affect users’ propensity to use
hazard language, and how different users employ such language. Urban legend
analysis, meanwhile, allows us to better test for the hypothesized impact of
negatively-biased credulity: that negative events are more likely to be
believed (Fessler, Pisor, and Navarrete 2014; Donavan, Mowen, and Chakraborty
1999). Urban legends are broadly believed by the sharers (Donavan, Mowen, and
Chakraborty 1999), so a high frequency of information concerning hazards would
support prior findings. We validate the model on human-annotated data, and
show substantially higher performance over baselines.

The model is highly scalable, enabling us to quantify the likelihood that
information concerning a hazard (hereafter, for simplicity, “hazards”) is
present in millions of posts. We use the model to quantify the amount of
hazards present in diverse multi-lingual conversations on X about topics such
as the COVID-19 pandemic and the Hamas-Israel war. We show that, while hazards
are somewhat correlated with negative emotions like anger and fear, our
identification of hazards goes beyond such indices, capturing additional
salient information about harms. Further, binning COVID-19 posts by the
political orientation of the user, we find, in agreement with expectations
(Hibbing, Smith, and Alford 2014; Lilienfeld and Latzman 2014), that social
conservatives discuss hazards at a greater rate than do social liberals, even
when split across various concerns (Rao et al. 2023). Moreover, these data
reveal differences in the language liberals and conservatives associate with
hazards – differences that a dictionary approach to identifying hazards would
not necessarily capture. When posts about the Hamas-Israel war are split by
whether accounts are associated with information operations (coordinated
accounts), we see notable differences in how these accounts describe threats
(emphasizing harms in Gaza and not in Israel) while also using hazard posts to
promote petitions that try to end the war. These results show that identifying
hazards can reveal salient features, potentially unmasking information
operation tactics. Finally, when we apply these data to urban legends, we find
that hazards are very common, which scales up prior work (Fessler, Pisor, and
Navarrete 2014), and provides evidence for (albeit not proof of) the
postulated aggregate effects of negatively-biased credulity in a population.

To summarize, our contributions are as follows:

  * •

We curate a new dataset to classify hazards at scale.

  * •

We develop a new model to extract hazards from both social media posts and
urban legends.

  * •

We apply this model to newly collected online data, and contrast hazards to
alternative indicators of affect.

  * •

We show distinct ways hazards are shared within different datasets, even as
their overall prevalence in urban legends confirms the psychological theory of
negatively-biased credulity.

Code, data, and annotations are available in the following repository:
https://github.com/KeithBurghardt/Hazards.

Overall, these results demonstrate the need for, and utility of, a new tool to
detect information concerning hazards. When applied at scale, this tool can
help replicate and validate prior findings in psychology, and may prove useful
in revealing influence operations.

## Related Work

### Linguistic indicators in text

Detecting indicators from text has a long history. One of the earliest methods
to do this was The General Inquirer (Stone, Dunphy, and Smith 1966), but more
recently text indicators have been analyzed with LIWC (Pennebaker, Francis,
and Booth 2001). These dictionary-based approaches have been superseded by
rule-and-word approaches, such as VADER (Hutto and Gilbert 2014), which have
themselves been superseded by text embedding approaches, including SeerNet
(Duppada, Jain, and Hiray 2018), a moral outrage classifier (Brady et al.
2021), and DeepMoji (Felbo et al. 2017). The advantages of these approaches
are that embeddings can learn how semantically similar content has similar
indicator values (e.g., “I am sad” and “This day was awful” would be viewed as
sad statements even when the two sentences do not have any words in common).
These approaches, however, were based on GRUs (Dey and Salem 2017) or LSTMs
(Hochreiter and Schmidhuber 1997), while more recently transformer-based
models (Vaswani et al. 2017), such as BERT (Devlin et al. 2018) or Sentence-
BERT (Reimers and Gurevych 2019), have become popular for uses such as
detecting hate speech (Davani, Díaz, and Prabhakaran 2022), or emotions
(Acheampong, Nunoo-Mensah, and Chen 2021). Unlike prior approaches,
transformer models account for context. Finally, these methods have then been
improved upon with SpanEmo (Alhuzali and Ananiadou 2021) or Demux (Chochlakis
et al. 2023), that account for correlations between labels, while recent
research has experimented with Large Language Models (LLMs) to similarly
detect emotion (Peng et al. 2024).

In contrast to these approaches, we develop a transformer-based tool to
extract hazards from social media and urban legends, a previously under-
studied indicator. We apply a range of models to Sentence-BERT embeddings and
compare these against LLMs (GPT-3.5 and GPT-4 (Achiam et al. 2023)) to detect
hazards in text.

### Negatively biased credulity

One key use of a hazard detection model is to assess negatively biased
credulity and negativity bias. Negatively-biased credulity is the idea that,
over the course of human history, the costs of erroneously believing false
information about danger will, on average, have been less than the costs of
erroneously rejecting true information about danger, and, as a consequence,
information concerning hazards is viewed as more believable than many other
types of information (Fessler, Pisor, and Navarrete 2014; Fessler, Pisor, and
Holbrook 2017b; Fessler 2019; Samore et al. 2018; Forgas 2019). This builds on
earlier investigations of the role of credulity in recipients’ susceptibility
to manipulation (Little 2017; Kartik, Ottaviani, and Squintani 2007).
Likewise, explorations of negatively-biased credulity partially overlap with
previous work examining how negative content (including content unrelated to
hazards) is more likely to be shared (Martel, Pennycook, and Rand 2020;
Youngblood et al. 2023; Ferrara and Yang 2015). For example, moral-emotional
language spreads most for partisan discussions (Brady et al. 2017), negative
sentiment posts spread faster (Ferrara and Yang 2015), and moral outrage makes
posts more viral (Brady et al. 2021). In contrast to this prior research,
however, not all negative information confers similar survival advantage.
Specifically, recognizing threats swiftly (Öhman and Mineka 2001) is more
critical for survival than recognizing sadness or anger. While negative
emotions may reduce belief in a false claim (Phillips et al. 2024), discussion
of hazards seems to increase it (Fessler 2019). As a metric of content,
hazards are therefore distinct from emotions or other features of language.

Credulity toward hazard information may differ by ideology as well. As we
mention in the Introduction, belief in a dangerous world differs by political
ideology (Clifton and Kerry ; Holbrook et al. 2022; Federico, Hunt, and Ergun
2009), consistent with other differences across ideologies (Rao, Morstatter,
and Lerman 2022). Social conservatives also display greater negativity bias
(Hibbing, Smith, and Alford 2014), and pay more attention to threats
(Lilienfeld and Latzman 2014) than do social liberals. Previous work in
psychology indicates that hazard content is treated with greater credulity by
social conservatives than by social liberals (Samore et al. 2018; Fessler,
Pisor, and Holbrook 2017a).

We expand on previous research about negativity bias and negatively biased
credulity by analyzing hazards at scale in social media posts and urban
legends. The large datasets provide enough statistical power for us to assess
how hazards vary after major events, how they relate to other indicators, and
how diverse groups discuss hazards. This allows us to replicate at scale
research on ideological differences in negativity, and evidence of negatively
biased credulity in urban legends. Moreover, we can develop new hypotheses
about how hazards are harnessed when attempting to influence social media
users.

## Research Methods

All data collected and analyzed were considered exempt by the lead author’s
institutional review board, where all modeling, data collection, and data
analysis was conducted. All annotations were non-human subject research. In
addition, all data were anonymized prior to analysis or annotation to minimize
privacy risks.

### Hazards Benchmark: Data and Model

We curate a ground truth dataset for training models to recognize hazards.
This process involves collecting and annotating posts for the presence of
hazards. This ground truth data is then used to train a language model to
classify hazards posts. The methods used to collect data, annotate data, and
train hazard detection models are shown in Fig. 1.

##### Ground truth data

To create the benchmark X post dataset, we first extract 1,338 X posts
containing at least one word from the Threat Dictionary (Choi et al. 2022),
shown in Fig. 1, left panel. Data is collected via X’s Academic API between
March, 2006 (when X is founded) and late 2022 in order to have a
representative sample of data. We then randomize the order of these posts and
recruit Cloud Research annotators to label these hazards via a Qualtrics
survey. For each post, workers answer, “Does the tweet describe a hazard
(something that could impose harm or other costs on the author of the tweet or
on others)?”. Workers are paid $2 for each assignment, in which they annotate
10 random posts (on average this took 11 minutes to complete, meaning workers
were paid $12 an hour). To account for workers who do not meaningfully
complete the assignment, we add an easy question mid-way through the survey
and remove any workers who did not complete it. We also remove annotations
that are unfinished or take less than 200 seconds to complete (this was an
arbitrary cutoff to better ensure that the annotations were not completed
carelessly). As all data are annotated before 2023, we believe the prevalence
of workers annotating with LLMs (Veselovsky, Ribeiro, and West 2023) was
minimal. All posts annotated by fewer than two crowd workers are also
discarded, resulting in a dataset with 1131 posts for training, validation,
and testing. In addition, we use Python’s demoji library
(https://pypi.org/project/demoji/) to convert all emojis to words in order to
reduce artefacts in the embeddings. These collected data all follow the FAIR
principles: Findable (these data are directly available via the repository
link at the end of the Introduction section, and contain a unique identifier
with all metadata are described), Accessible (the link is accessible to
anyone), Interoperable (metadata use a formal and broadly applicable
language), and Reusable (all data are described in detail). In the repository
link, we also include a datasheet for the dataset Gebru et al. (2021).

We also use data from (Fessler, Pisor, and Navarrete 2014) (from the website
SNOPES) to train a model to detect hazards in urban legends. While these data
contain 278 legends in total, there are only 221 unique legend types. For
example, SNOPES lists three different examples of the story “Bill Gates Wants
to Share His Fortune With You”. We keep the first example listed in “Urban
Legend Data_ 2022 Snopes, Encyclopedia & 2014.xlsx” located within our data
repository.

##### Model training

We use a 90% training-10% testing split for X posts. If validation is needed
(such as for early stopping) we split these data into 81% training, 9%
validation, and 10% testing. We split urban legends data into 86% (190
legends) for training and 14% (31 legends) for testing. When validation is
needed, these data are split 76%, 10%, and 14% for training, validation, and
testing respectively. We randomize these splits 1000 times to come up with a
mean and variance in our performance metrics.

We use the stsb-xlm-r-multilingual sentence-transformers model from
huggingface (https://huggingface.co/sentence-transformers/stsb-xlm-r-
multilingual) to embed all text, shown in Fig. 1, right panel. This model is
used to ensure multi-lingual data common in social media text can be embedded.
We then apply several supervised models to these embeddings including XGBoost
(Chen and Guestrin 2016) (via https://xgboost.readthedocs.io/), neural
networks (Abadi and et al. 2015), random forest (Ho 1995), and support vector
machines (SVMs) (Hearst et al. 1998), the latter two are trained via scikit-
learn (Pedregosa et al. 2011). While XGBoost, random forest, and SVM are
trained with default hyperparameters, the neural network contains 4 fully-
connected layer, with 256 nodes, 85 nodes, 28 nodes, and 1 node, respectively,
with ReLu activation and dropout layers. Alternative models performed
similarly. We also experimented with augmenting the human-annotated posts with
5K GPT-3.5 annotating posts that contained words from the Threat Dictionary
(Choi et al. 2022), and 5K posts collected at random (containing popular
English keywords, namely any of the top 100 lemmas within a large corpus
(https://www.wordfrequency.info/samples.asp) that X does not consider stop
words. This augmentation did not significantly change the performance of the
model (which we show in the Results section). All models are trained via
NVIDIA Tesla K80 GPU with 12GB of VRAM; we predict text via GeForce RTX 2080
GPU with 8GB VRAM. As a comparison, we also use GPT-3.5 and GPT-4 (Achiam et
al. 2023) with chain-of-thought prompting (Wei et al. 2022) via the prompt,
“Does the tweet [story] describe a hazard (something that could impose harm or
other costs on the author of the tweet or on others)? Please answer ‘yes’ or
‘no’ and explain your thought process.” All text with “yes” are labeled
“hazard” and otherwise labeled “not hazard” (therefore rare ambiguous
situations where the LLMs do not know if a hazard exits are labeled “not
hazard”). We validate this technique on 100 posts at random and 100 legends at
random, and we find this technique extracts every post GPT-N considers a
hazard.

### Comparison to alternative text indicators

We compare our hazard detection model to alternative text indicators of
affect: moral outrage (Brady et al. 2021), sentiment (VADER) (Hutto and
Gilbert 2014), emotion detection (Demux) (Chochlakis et al. 2023), and threat
words (Choi et al. 2022). These are state-of-the-art methods to detect each
indicator. These code are under a MIT license (VADER and Demux), and Creative
Commons Attribution-NonCommercial-ShareAlike 2.0 license (moral outrage),
respectively, and adapted as needed to run on a GeForce RTX 2080 GPU with 8GB
VRAM. All model outputs we analyze are continuous (such as confidence values
for emotions) with the exception of the Threat Dictionary (Choi et al. 2022),
in which we either indicate if a threat word is (1) or is not (0), in a post
or legend.

### X Data

##### Hamas-Israel War

Our analysis uses a corpus of 3.6M posts about the 2023 Hamas attack on Israel
and Israel’s subsequent invasion of Gaza from 1.3M accounts. The posts were
collected by querying X with a set of keywords related to the war: e.g.,
“Israel”, “Hamas”, “Gaza”, etc. These data are multilingual, with
approximately 93% of posts in English, 6.5% in Arabic, and a small proportion
in other languages. We further split by a common metric of coordination
(Burghardt et al. 2023; Luceri et al. 2023), in which accounts are coordinated
if they post near duplicate sequences of hashtags (which is strongly
associated with near-duplicate messages). This extracted 9.4K inauthentic
accounts that posted 61K posts. We share the coordinated network in the
repository, but, because X’s terms of service do not allow us to share post
text, and because we have removed all text IDs and user IDs, we do not share
additional data. Because these data are public and anonymized, we did not need
to obtain consent from users to extract these data. We show the frequency of
posts over time for coordinated and non-coordinated accounts in Supplementary
Materials Fig 13.

##### COVID-19

We use a public corpus of posts about the COVID-19 pandemic (Chen, Lerman, and
Ferrara 2020) posted between January 21, 2020 and December 31, 2020. These
posts include one or more terms related to the pandemic, such as
“coronavirus”, “pandemic”, and “outbreak”, among others. We use Carmen (Dredze
et al. 2013), a geo-location tool for X data, to link posts to locations
within the United States. Carmen relies on metadata in posts, such as “place”
and “coordinates” objects that encode location information, as well as
mentions of locations in a user’s X bio, to infer their location. This
approach is applied to filter out users whose home location is not one of the
50 U.S. states. We take a random 1% subsample to reduce the dataset size. This
process yields 12.7M posts from 5.7M users. We show the post frequency in
Supplementary Materials Fig. 12. Data are then split by keywords related to
various concerns created by (Rao et al. 2023): education, healthcare,
lockdowns, masking, origins, therapeutics, and vaccines. To compare hazards
against other text indicators, we took a complete set of posts shared early in
the dataset, from January 22 at 12 a.m. GMT until January 26, 21:00:40 GMT.
This result allows the topics in the post to be more coherent (and not change
in time), allowing for a more consistent comparison of what each indicator
captures (e.g., there might be a lower correlation if each indicator picked up
something different about each topic shared in 2020). We do not believe this
choice affects the results significantly given the similarity of correlations
with other datasets, shown in the Results section.

### Urban Legends

We collect three sets of urban legends: Snopes.com data from 2014 (Fessler,
Pisor, and Navarrete 2014), a new Snopes.com dataset collection, and legends
from the Encyclopedia of Urban Legends (Brunvand 2002). To collect new
Snopes.com data, we had several undergraduate research assistants scrape a
summary and example text of all stories under the “fact check” category of
Snopes.com as of mid-2022. This category includes several types of legends,
such as frauds and scams, history legends, sports legends, and “old wives’
tales”. In total, we find 996 legends. We similarly extract every urban legend
described in the Encyclopedia of Urban Legends (255 in total). Because urban
legends can be long (up to 1,600 words) we use GPT-3.5 from ca. April, 2023 to
summarize each urban legend. This keeps the text below 118 words, well within
the 128 token limit for the sentence embedding model that we employ. We use
these summaries to train our models and then detect hazards. Potential
artefacts with this approach include hallucinations, but we generally found
that these summaries faithfully represented the longer text. Alike to the
annotated data, these collected urban legends all follow the FAIR principles.
In the repository link, we also include a datasheet for the dataset Gebru et
al. (2021).

Figure 2: Performance of competing models. (a) Performance of models on human-
annotated X posts. We show the ROC-AUC of XG-Boost (XGB) (Chen and Guestrin
2016), a neutral network (NN) (Abadi and et al. 2015), random forest (RF) (Ho
1995), and a support vector machine (SVM) (Hearst et al. 1998) trained on
human annotated posts. We also show the ROC-AUC of a neutral network pre-
trained on 10K LLM-annotated posts and then trained on human-annotated posts
(NN-LLM-pre); GPT-3.5 and GPT-4, meanwhile, are zero-shot predictions. (b)
Model performance on urban legends. We show ROC-AUC for the same types of
models, except “[Model]-X-only” represents models only trained on human-
annotated X posts, while all other supervised models are trained exclusively
on urban legends. GPT-3.5 and GPT-4 are zero-shot predictions, as before.

## Results

### Hazards Model and Validation

Performance of the hazards detection model on benchmark data is shown in Fig.
2. Despite the simplicity of the SVM model, its performance exceeds all
others, with an area under the receiver operating characteristic curve (ROC-
AUC) of 0.74 (Fig. 2a). This dataset uses posts that contain words from the
Threat Dictionary (Choi et al. 2022), making the dataset especially
challenging as, despite the posts containing threat words, only a subset show
hazards. Notably, this implies that simple lexical models, like those based on
the Threat Dictionary, are a poorer tool, as the Threat Dictionary would have
an ROC-AUC of exactly 0.5 (all posts would contain at least one threat word).
LLMs, namely GPT-3.5 and GPT-4, also perform markedly worse than supervised
models.

We separately trained this model on urban legends in Fig. 2b. If the model is
only trained on X posts, we see the models struggle to generalize, thus
motivating a separate set of models trained on urban legends. The best model
is the SVM model again, with an ROC-AUC of 0.83. In contrast to supervised
models, we see that GPT-4’s performance is comparable to the urban legend-
trained models, and far exceeds the models trained on X posts. GPT-4 therefore
generalizes remarkably well in the X post and urban legend domains, although
it does not exceed the performance of models trained to each dataset. Other
studies have also noted that LLMs’ versatility comes at the expense of
performance (Chochlakis et al. 2024).

Figure 3: Spearman correlation between hazards moral outrage (Brady et al.
2021), sentiment (Hutto and Gilbert 2014), and emotions (Chochlakis et al.
2023). One indicator, urban legends “love” emotion is not statistically
significantly correlated with hazard (p-value $>0.05$).

### Hazards in Real-world Data

Next, we show in Fig. 3 how hazard confidences compare to other linguistic
indicators of affect in three datasets: X posts on COVID-19, X posts on the
Hamas-Israel war, and urban legends. We then show the words most often seen in
high-hazard posts or legends in each dataset, clarifying how hazards are
typically described. Focusing on X posts, we show the ideological asymmetry in
how hazards are discussed, as well as the asymmetry in how coordinated and
authentic accounts share hazards. While the former tests the greater
negativity bias of conservatives compared to liberals (Hibbing, Smith, and
Alford 2014), the latter offers insight into information operation tactics
meant to drive authentic user behavior. Finally, we show the prevalence of
hazards in urban legends, which offers further evidence, although not proof,
of the postulated aggregate effects of negatively biased credulity (Fessler,
Pisor, and Navarrete 2014).

#### Linguistic Analysis of Hazards

Figure 3 shows post and story-level correlations between hazard confidences
and other text indicators of affect expressed by the post author, including
moral outrage (Brady et al. 2021), sentiment (Hutto and Gilbert 2014), emotion
confidence (Chochlakis et al. 2023), and threat words (Choi et al. 2022). In
all cases the absolute value of Spearman correlations are all at or below 0.4,
suggesting that alternative indicators do not fully capture information about
hazards in text. The positive or negative direction of the correlations,
however, make intuitive sense. For example, moral outrage is weakly positively
correlated with hazards, perhaps because people share their moral outrage at
some hazards, such as those that harm innocent people. Similarly, negative
sentiment, as well as most negative emotions and posts containing Threat
Dictionary words, show positive correlations with hazards. This is consistent
with hazard posts being given a negative framing (cf. example hazard
annotations and predictions in Supplementary Materials Table 1).

Figure 4: Word frequency from highest 10% hazard confidence posts and bottom
10% hazard confidence posts for (a) COVID-19 (Chen, Lerman, and Ferrara 2020),
(b) the Hamas-Israel war, and urban legends. Figure 5: High and low-hazard
words associated with liberals and conservatives in the COVID-19 dataset.
Figure 6: Hazard confidence for different COVID-19 concerns, split by
conservative and liberal accounts. Differences are all significant. Error bars
are too small to see, and are therefore removed for clarity. Figure 7: Hazards
over time among COVID-19 posts (Chen, Lerman, and Ferrara 2020), split by
liberal and conservative, as well as by type of post. Also shown is the
proportion of posts with threat words from the Threat Dictionary (Choi et al.
2022).

To understand what language is associated with hazards, we plot relative word
frequency of high-hazard posts (in the top 10% of hazard confidences) and low-
hazard posts (in the bottom 10%) in Fig. 4. In Fig. 4a we see that the high-
hazard posts about COVID-19 are often associated with dangers (e.g., “sick”,
“killing”, “danger”) compared to low-hazard (“excited” or “stayhome”).
Similarly, in Fig. 4b, the high-hazard words relate to “weapons”, “crimes”,
“abuse”, while low-hazard words include “neutral”. Finally urban legends
discuss “dead” or “car” (urban legends often involve dangerous scenarios
related to cars), versus “2020” or “virus”. In all cases, we see hazard words
are similar to threat words seen in (Choi et al. 2022), yet some words could
not easily be captured with the Threat Dictionary, such as “vaccines” or
“cdc”. Within the Hamas-Israel war dataset (Fig. 4b), we also notice
“civilians” being mentioned, due to posts on the risk of harm to civilians. We
show in Fig. 4c, meanwhile, the prevalence of high- and low-confidence hazard
words (top 20% and bottom 20% hazard confidence due to the smaller dataset)
for the three urban legend datasets combined.

#### Partisan Asymmetry in Hazard Discussions

The reason we see “vaccines” or “cdc” as words associated with hazards is
better illustrated in Fig. 5, where we plot words associated with high-
confidence and low-confidence hazard posts disaggregated by ideology, as
identified using the technique developed in (Rao et al. 2023). We notice that,
consistent with commonsensical observation of public discourse at the time,
conservatives and liberals each discuss vaccines in hazard posts, but, when
reviewing the posts, conservatives often use it in the context of vaccines
being harmful, while liberals discuss it in the context of protecting from a
harmful virus. We also see protests associated with high-confidence hazard
posts among conservative users, possibly because we find conservative users
associate the protests that occurred in 2020 as a hazard (e.g., looting). We
also find that the mean hazard confidences are slightly higher for
conservatives even when data are split by individual concerns, as shown in
Fig. 6. The only exception is for COVID-19 origins. To better understand
ideological differences in hazards discussions, we plot mean hazard confidence
over time among conservative and liberal users in Fig. 7. This plot shows that
hazard prediction confidence is higher for conservatives almost every day in
which posts are collected. Thus, overall we see conservatives discuss hazards
at a greater rate, consistent with conservatives having greater negativity
bias (Hibbing, Smith, and Alford 2014; Samore et al. 2018; Fessler, Pisor, and
Holbrook 2017a).

That being said, we notice that both liberals and conservatives show similar
shifts in hazard confidence over time. In all cases, hazards appear to be
highest at the very beginning of the COVID-19 dataset, shown in Fig. 7, with a
rapid decline starting in March. When we calculate mean daily hazard level
versus CDC weekly case count (https://covid19.who.int/WHO-COVID-19-global-
data.csv) the Spearman correlation is -0.65 (p-value $<0.001$), and the
correlation with death count is -0.49 (p-value $<0.001$), indicating, counter-
intuitively, an increasingly muted response to the COVID-19 hazard, which is
also seen with negative emotions (Guo et al. 2022). The authors of (Guo et al.
2022) suggest that this may be due to a reduction in uncertainty. We also
notice a strong spike in hazards just after the George Floyd killing,
consistent with a greater discussion about police brutality (by liberals) and
property destruction by protesters (by conservatives). Finally, we see a
notable increase in hazard confidence before, and a sharp decline after, the
U.S. presidential election. Due to the sharp partisan divide in the U.S., we
presume that both parties were anticipating some harm associated with the
election depending on which candidate won. Once again, the reduction in hazard
language appears to be associated with reduction in uncertainty about the
outcome. Complementing these results, we show the proportion of posts with
threat words (Choi et al. 2022). We show that results are similar, although
notably, threat posts increase in September, and do not rebound after the
election (both in contrast to hazard posts).

Figure 8: Words associated with high-confidence and low-confidence hazard
posts among (a) coordinated and (b) non-coordinated accounts within the Hamas-
Israel war dataset. Figure 9: Hazards and threats over time for the Hamas-
Israel war dataset. The plots show mean hazard confidences each day as well as
the overall mean proportion of posts with at least one word from the Threat
Dictionary (Choi et al. 2022) for (a) non-coordinated authentic accounts and
(b) inauthentic coordinated accounts. Vertical lines correspond to the Hamas
attack on October 7th, 2023, and the Israel Defence Force (IDF) entering Gaza
on October 27th. Figure 10: Top URLs shared in coordinated accounts for (a)
low-confidence hazard posts and (b) high-confidence hazard posts.

#### Hazards in Information Operations

Next, we explore how hazards are used within coordinated influence campaigns,
specifically how they differ within authentic and inauthentic posts in the
Hamas-Israel war dataset. To identify potentially inauthentic (coordinated)
accounts, we use a hashtag-based heuristic that was shown to be highly
accurate in detecting coordinated influence campaigns on X (Burghardt et al.
2023; Luceri et al. 2023). We find in Fig. 8 that likely authentic accounts
emphasize the association between hazards and children, civilians, and terror
(reflecting hazards surrounding the October 7th Hamas attack on Israel). In
contrast, coordinated accounts (Fig. 8b) do not appear to associate the
terrorist attack with a hazard, but instead associate hazards with bombs and
civilians (reflecting the Israel ground war in Gaza). We plot hazard
confidence in posts over time in Fig. 9. Just after the October 7th attack,
there is a sudden increase in mean hazard confidence among authentic accounts
as they discuss the attack on Israeli civilians. There is no such increase in
hazard discussions among coordinated accounts. There is, however, a spike in
the number of posts by coordinated accounts, shown in Supplementary Materials
Fig. 13, which is why hazard confidence for coordinated account posts becomes
much less noisy after the attack. We also see only a minor (or no) change in
the proportion of posts containing threat words (Choi et al. 2022) after the
October 7th attack. This is reflected in the Spearman correlation between
daily mean hazard confidence and daily proportion of posts containing threat
words, which is only $0.25$ (p-value$=0.047$) for coordinated accounts and
0.40 (p-value$=0.001$) for authentic accounts. Hazards can therefore be a
distinct, and in some ways better, indicator of major hazardous events than
the previous state-of-the-art metrics. In separate analysis, coordinated
account posts seem to show elation (promotion of Hamas, Supplementary
Materials Fig. 14, and positive emotions, Supplementary Materials Figs. 15 &
16), thus the insignificant change hazards just after October 7th is despite
the accounts mentioning the attacks, rather than because the accounts ignore
the event.

We do see, however, a steady increase in hazard discussions among coordinated
accounts as Israel prepared to enter Gaza on October 27th. Reviewing posts,
this appears to be because coordinated accounts start to discuss harm to
Palestinian civilians. This is further corroborated in Fig. 10, where we
compare the URLs shared in high and low-confidence hazard posts from
coordinated accounts. The second most popular domain shared within hazard
posts is secure.avaaz.org. These posts promote a petition to end the invasion
of Gaza, such as “My heart is heavy with the horror in #Palestine & #Israel.
There’s a way to help protect the #Gaza kids facing bombardment & free the
Israeli kids hostage and Palestinian kids in prison – but we need massive
pressure. Join the call #ShieldTheChildren”. These accounts appear to use
hazards as a way to promote petitions meant to pressure world leaders to end
the war.

### Urban Legends

Fig. 11 shows the prevalence of hazards among urban legends in the three
separate urban legend datasets from SNOPES and the Urban Legends Encyclopedia
(Brunvand 2002). In all cases, hazard confidence is surprisingly high compared
to X posts shown in Figs. 9 & Fig. 7. In contrast, only 30% of these urban
legends contain at least 1 word from the Threat Dictionary. Urban legends are
often shared because they were believed by the speaker or audience (Donavan,
Mowen, and Chakraborty 1999). For this reason, the greater tendency to discuss
hazards in an urban legend (e.g., razor blades in an apple during Halloween)
is consistent with hazards increasing the believably of stories (Holbrook et
al. 2022; Fessler, Pisor, and Navarrete 2014; Fessler 2019).

Figure 11: The distribution of Hazard confidences within each urban legends
dataset (Fessler, Pisor, and Navarrete 2014; Brunvand 2002).

## Discussion

In conclusion, we create a new model to detect hazards in social media posts
and urban legends that outpaces simple word-based proxies, such as the Threat
Dictionary (Choi et al. 2022), as well as sophisticated LLM-based approaches.
When this model is applied to X posts, we discover what different groups
consider hazards. This uncovers partisan differences (such as conservatives
associating protests with hazards), as well as differences that reveal
information operation tactics. Specifically, we find that hazard content
shared by coordinated accounts is often related to petitions to stop an
ongoing war. Accounts may specifically mention hazards to take advantage of
authentic users negativity bias in which they will be more likely to focus on
a negative post (Fiske 1980), while the moral outrage of harmed children
mentioned in their posts can drive the virality of the post (Brady et al.
2021). All these act as ways to bring more users to the petition webpage in
order to pressure international leaders. When we measure the prevalence of
hazards over time in both X datasets, we see intuitive peaks in hazards
consistent with harmful events, suggesting that hazards are indeed good
automated indicators of such events. In contrast, posts annotated by the
Threat Dictionary show distinctive patterns over time and correlate poorly
with the hazard model predictions. Moreover, while we see a strong change in
hazards during the Hamas attack, we see no such strong change for Threat
Dictionary-annotated posts, suggesting that the Threat Dictionary may not
fully capture all hazardous events.

Our work replicates at scale prior work in psychology by demonstrating a
higher use of hazards among conservatives compared to liberals, consistent
with greater negativity bias (Hibbing, Smith, and Alford 2014), and more
attention towards threats (Lilienfeld and Latzman 2014), among conservatives.
We also see evidence consonant with the notion that negatively-biased
credulity affects the contents of culture over time, as we find a high
prevalence of hazards within urban legends, which are often believed by the
speaker (Donavan, Mowen, and Chakraborty 1999). This is consistent with
(Fessler, Pisor, and Navarrete 2014), as urban legends that contain hazards
may be more believable than urban legends that do not, hence the former should
persist at higher rates than the latter. However, we cannot rule out
alternative interpretations of these data, such as the possibility that urban
legends are especially negative for an entirely different reason. In the
future, we anticipate that this model can help test other predictions of
negatively biased credulity, including whether conservatives display greater
negatively biased credulity (Fessler 2019), a finding that we cannot directly
test in the present data. While there are concerns about the reproducibility
of science, especially in psychology (Collaboration 2015), our work points to
ways that AI can more generally improve reproducibility by testing social
science hypotheses within larger representative samples across a range of time
periods.

### Limitations

While hazards appear to be a useful feature in text, we should caution that
the model’s performance could still be improved. The subjectivity of text
annotations (Davani, Díaz, and Prabhakaran 2022) and a lack of annotations
both reduce the accuracy of many text indicator models (Chochlakis et al.
2023; Brady et al. 2021), a problem likely present in hazard detection. More
research is needed to annotate text, especially multi-lingual text, and to
improve the model’s generalizability. Notably, we see a significant drop in
ROC-AUC when applying the model from X posts to urban legends. LLMs show
promise, however, as GPT-4’s zero-shot performance was more consistent across
X posts and urban legends. Instruction-tuned, and in-context learning-based
LLMs could therefore allow for a better overall model.

Finally, while our work is consistent with prior psychological research, it
offers only indirect support for both the existence of negatively biased
credulity and the relationship between political orientation and negativity
bias. Whether these results replicate in other datasets remains to be seen. At
the most fundamental level, more work is needed to directly measure whether
hazard texts are more likely to be believed by the author or reader.

### Ethical considerations

This work was approved by an IRB. To improve user privacy, we removed all
personally identifiable information in data, such as post IDs and user IDs,
prior to our analysis. While the hazard model performs well at scale, it is
still possible for the model to make mistakes. Therefore, care must be taken
when interpreting the model output, including applying it to detect whether
individual users are sharing information concerning hazards. The confidence of
the model does not guarantee that individual users or posts are sharing such
content, especially given that sarcasm and in-group language is a challenge
for current AI models. So long as users are aware of these limitations,
however, the harm to society is minimal, much like a sentiment or emotion
detection model, therefore we share this model widely via the link presented
at the top of the paper. We believe that the hazard detection model’s utility
to society far outweighs any harmful societal impact.

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## Supplementary Materials

Figure 12: Post frequency over time for coordinated and non-coordinated accounts. Figure 13: Post frequency over time for (a) authentic non-coordinated, and (b) inauthentic coordinated accounts. Vertical lines correspond to the Hamas attack on October 7th, 2023, and the Israel Defence Force (IDF) entering Gaza on October 27th. Figure 14: Percent of posts mentioning Hamas over time for (a) authentic non-coordinated, and (b) inauthentic coordinated accounts. Vertical lines correspond to the Hamas attack on October 7th, 2023, and the Israel Defence Force (IDF) entering Gaza on October 27th. Figure 15: Mean joy emotion confidence over time for (a) authentic non-coordinated, and (b) inauthentic coordinated accounts using Demux (Chochlakis et al. 2023). Vertical lines correspond to the Hamas attack on October 7th, 2023, and the Israel Defence Force (IDF) entering Gaza on October 27th. Figure 16: Mean optimism emotion confidence over time for (a) authentic non-coordinated, and (b) inauthentic coordinated accounts using Demux (Chochlakis et al. 2023). Vertical lines correspond to the Hamas attack on October 7th, 2023, and the Israel Defence Force (IDF) entering Gaza on October 27th. Dataset |  Text | Ground truth label
---|---|---
Random posts |  Aircraft crashes into residential building in Russian city near Crimea, killing at least three https://xxxx | Hazard
Random posts |  @XXXX @YYYY But he is a liar and was thrown out because he is a liar, we know you love him but really you have to be aware that he is a Liar x | No hazard
2014 Urban Legends |  Two friends had been practicing shooting beer cans off each other’s heads in Johannesburg, South Africa, when one of them accidentally shot the other in the face, causing serious injury. | Hazard
2014 Urban Legends |  A job candidate was described as monosyllabic to the boss. The boss questioned where Monosyllabia was, but the person who made the comment played along and said it was south of Elbonia. The boss then incorrectly guessed that it was near Croatia. | No hazard
Dataset |  Text | Hazard model confidence
COVID-19 |  Ever just have a cry for no particular reason? These COVID times eh? Related: I’m listening to Nick Cave’s The Ship Song. | 0.04
COVID-19 |  Mass shootings held in check during the pandemic (tho we saw a huge rise in domestic abuse). Now indications are, it’s going to get worse if Congress doesn’t act. Irrational ängry white malesḧave become the new face of terrorism. We must protect the country from that! | 0.95
2022 Urban Legends |  A man gives a lift to a young girl dressed in an evening gown who claims her car broke down and asks to be taken home. As he pulls up to the address she provided, he realizes she has vanished from the back seat. Upon ringing the doorbell, he discovers that the girl he gave a lift to has been dead for two years and her father has been experiencing similar encounters. | 0.0
2022 Urban Legends |  19-year-old construction worker from north China died of a heart attack after smoking 100 cigarettes in one sitting in order to win a bet with his friend. The teenager collapsed and died as a crowd of passers-by watched. The attending doctor attributed the cause of death to the excessive intake of cigarette smoke and acute nicotine poisoning. The two friends had dreamed up the bet out of boredom, with the loser agreeing to pay for all the tobacco. The deceased continued smoking, emboldened by a growing crowd of onlookers, until he hit the 100-cigarette mark, while his friend gave up after 40 cigarettes. | 1.0
2023 Hamas-Israel War |  [Text was in Arabic] RT @XXXX, Head of the Prisoners Affairs Commission to ’Al-Ghad’: The West has revealed its true face, and we have no choice but to stand firm against Israel. #AlGhadChannel #Palestine #GazaNow #AlAqsaFlood https://t.co/xxxx | 0.05
2023 Hamas-Israel War |  @XXXX Palestinian govt gave American weapons to leaders of Hamas. Look at the countless pictures. Hamas used those weapons armed with hate and rhetoric propaganda against Israel. Women and children were killed by animals whose punishment should be swift and without prejudice. | 0.94
Table 1: Hazard ground truth labels and model prediction examples.


$\mathcal{E}_{N}(\mathbb{H}),$

$\displaystyle\|\mu\|_{\mathcal{E}_{N}(\mathbb{H})}^{2}$
$\displaystyle=\int_{\mathbb{H}\times\mathbb{H}}\log\frac{1}{|z-w|^{2}}\,\mu(z)\overline{\mu(w)}+\int_{\mathbb{H}\times\mathbb{H}}\log\frac{1}{|z-\bar{w}|^{2}}\,\mu(z)\overline{\mu(w)}$
$\displaystyle=\frac{1}{2}\int_{\mathbb{C}\times\mathbb{C}}\log\frac{1}{|z-w|^{2}}\,\nu(z)\overline{\nu(w)}=\frac{1}{2}\|\nu\|_{\mathcal{E}(\widehat{\mathbb{C}})}^{2},$

where $\nu=\mu$ in the upper half-plane $\mathbb{H}$ and $\nu=\mu^{*}$ in the
lower half-plane $\mathbb{H}^{*}$ ($\mu^{*}(E)=\mu(E^{*}),$
$E^{*}=\\{\bar{z},z\in E\\}$). For example, a test function $f$ in
$\mathbb{H}$ with mean zero and Neumann boundary condition extends smoothly to
$\mathbb{C}$ such that $f(\bar{z})=f(z).$

As a Fock space space, the Gaussian free field $N\equiv N_{D}$ in $D$ can be
constructed from the Gaussian free field $\Psi\equiv\Psi_{S}:$

$N(z)=\frac{1}{\sqrt{2}}(\Psi(z)+\Psi(z^{*})).$

This field $N(z)$ is formal and thus we need to require the neutrality
condition $(\mathrm{NC}_{0})$ on $N.$ For example, $N(z,z_{0}):=N(z)-N(z_{0})$
is a 2-variant well-defined Fock space field. The formal field $N$ has the
formal correlation:

$\mathbf{E}\,N(z)N(z_{0})=2G_{N}(z,z_{0}).$

From the Schottky double construction, the current field
$J_{D}^{D}=\partial\Phi_{D}$ in $D$ with Dirichlet boundary condition and the
current field $J_{S}=\partial\Psi_{S}$ are related as follows. For example,

$J_{\mathbb{H}}^{D}(z)=\frac{1}{\sqrt{2}}\big{(}J_{\widehat{\mathbb{C}}}(z)-\overline{J_{\widehat{\mathbb{C}}}(\bar{z})}\big{)}.$

On the other hand, the current field $J_{\mathbb{H}}^{N}$ with Neumann
boundary condition is related to $J_{\mathbb{C}}$ as

$J_{\mathbb{H}}^{N}(z)=\frac{1}{\sqrt{2}}\big{(}J_{\widehat{\mathbb{C}}}(z)+\overline{J_{\widehat{\mathbb{C}}}(\bar{z})}\big{)}.$

In a similar way, stress tensors are related as

$A_{\mathbb{H}}^{D}(z)=-\frac{1}{2}J_{\mathbb{H}}^{D}\odot
J_{\mathbb{H}}^{D}(z)=\frac{1}{2}\big{(}A_{\widehat{\mathbb{C}}}(z)+\overline{A_{\widehat{\mathbb{C}}}(\bar{z})}+J_{\widehat{\mathbb{C}}}(z)\odot\overline{J_{\widehat{\mathbb{C}}}(\bar{z})}\big{)}$

and

$A_{\mathbb{H}}^{N}(z)=-\frac{1}{2}J_{\mathbb{H}}^{N}\odot
J_{\mathbb{H}}^{N}(z)=\frac{1}{2}\big{(}A_{\widehat{\mathbb{C}}}(z)+\overline{A_{\widehat{\mathbb{C}}}(\bar{z})}-J_{\widehat{\mathbb{C}}}(z)\odot\overline{J_{\widehat{\mathbb{C}}}(\bar{z})}\big{)}.$

It is easy to check that $A^{N}$ is a stress tensor for $N.$ Indeed, as
$\zeta\to z,$ we have the following Ward’s OPE for $N:$

$A^{N}(\zeta)N(z)=-\frac{1}{2}J^{N}(\zeta)\odot
J^{N}(\zeta)N(z)=-\mathbf{E}[J^{N}(\zeta)N(z)]J^{N}(\zeta)\sim\frac{J^{N}(\zeta)}{\zeta-z}\sim\frac{\partial
N(z)}{\zeta-z}$

in the identity chart of $\mathbb{H}.$ Here we use

$\mathbf{E}\,J^{N}(\zeta)N(z)=-\frac{1}{\zeta-z}-\frac{1}{\zeta-\bar{z}}$

in $\mathbb{H}.$

Formal fields $N^{\pm}(z)$ with Neumann boundary condition are defined by

$N^{\pm}(z)=\frac{1}{\sqrt{2}}(\Psi^{\pm}(z)+\Psi^{\mp}(z^{*})).$

For example, we have

(9.1)
$\mathbf{E}\,N(z)N^{+}(z_{0})=\log\frac{1}{z-z_{0}}+\log\frac{1}{\bar{z}-z_{0}},\qquad\mathbf{E}\,N(z)N^{-}(z_{0})=\log\frac{1}{z-\bar{z}_{0}}+\log\frac{1}{\bar{z}-\bar{z}_{0}},$

and

$\mathbf{E}\,N^{+}(z)N^{+}(z_{0})=\log\frac{1}{z-z_{0}},\qquad\mathbf{E}\,N^{+}(z)N^{-}(z_{0})=\log\frac{1}{z-\bar{z}_{0}}$

in $\mathbb{H}.$ As in the Dirichlet case, 2-variant fields
$N^{\pm}(z,z_{0}):=N^{\pm}(z)-N^{\pm}(z_{0})$ are well-defined multivalued
Fock space fields.

###  9.2. Modifications and OPE exponentials

We first define background charge modifications of the chiral bosonic fields
with Neumann boundary condition. Given double background charges
$\boldsymbol{\beta}^{\pm}=\sum_{k}\beta_{k}^{\pm}\cdot q_{k}$ with the
neutrality conditions $(\mathrm{NC}_{b^{\pm}})$ in $D,$ we define the formal
fields $N_{\boldsymbol{\beta}}^{\pm}\equiv
N_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}}^{\pm}$
($\boldsymbol{\beta}=\boldsymbol{\beta}^{+}+\boldsymbol{\beta}_{*}^{-}$) by

$N_{\boldsymbol{\beta}}^{\pm}:=N^{\pm}+n_{\boldsymbol{\beta}}^{\pm},$

where $n_{\boldsymbol{\beta}}^{+}\equiv
n_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}}^{+}$ is a
$\mathrm{PPS}(b,0)$ form ($b=b^{+}+b^{-}$) and
$n_{\boldsymbol{\beta}}^{-}\equiv
n_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}}^{-}$ is a
$\mathrm{PPS}(0,b)$ form such that

(9.2) $\displaystyle n_{\boldsymbol{\beta}}^{+}(z)$
$\displaystyle=\sum\beta_{k}^{+}\log\frac{1}{z-q_{k}}+\sum\beta_{k}^{-}\log\frac{1}{z-\bar{q}_{k}}$
$\displaystyle n_{\boldsymbol{\beta}}^{-}(z)$
$\displaystyle=\sum\beta_{k}^{+}\log\frac{1}{\bar{z}-q_{k}}+\sum\beta_{k}^{-}\log\frac{1}{\bar{z}-\bar{q}_{k}}$

in the $\mathbb{H}$-uniformization. Furthermore,
$n_{\boldsymbol{\beta}}:=n_{\boldsymbol{\beta}}^{+}+n_{\boldsymbol{\beta}}^{-}$
is a $\mathrm{PPS}(b,b)$ form. For two divisors $\boldsymbol{\tau}^{\pm}$
satisfying the neutrality condition $(\mathrm{NC}_{0}),$ we define

$N_{\boldsymbol{\beta}}[\boldsymbol{\tau}]\equiv
N_{\boldsymbol{\beta}}[\boldsymbol{\tau}^{+},\boldsymbol{\tau}^{-}]:=\sum_{j}\big{(}\tau_{j}^{+}N_{\boldsymbol{\beta}}^{+}(z_{j})+\tau_{j}^{-}N_{\boldsymbol{\beta}}^{-}(z_{j})\big{)},$

where $\boldsymbol{\tau}=\boldsymbol{\tau}^{+}+\boldsymbol{\tau}_{*}^{-}.$
Compare this definition to the definition of
$\Phi_{\boldsymbol{\beta}}[\boldsymbol{\tau}]$ in the Dirichlet case:

$\Phi_{\boldsymbol{\beta}}[\boldsymbol{\tau}]:=\sum_{j}\big{(}\tau_{j}^{+}\Phi_{\boldsymbol{\beta}}^{+}(z_{j})-\tau_{j}^{-}\Phi_{\boldsymbol{\beta}}^{-}(z_{j})\big{)},$

where $\boldsymbol{\tau}=\boldsymbol{\tau}^{+}+\boldsymbol{\tau}_{*}^{-}$
satisfies the neutrality condition $(\mathrm{NC}_{0}).$ We now explain why the
signs of $\tau_{j}^{-}$ appear differently. The choices of signs are
consistent with the representation of
$\boldsymbol{\tau}=\boldsymbol{\tau}^{+}+\boldsymbol{\tau}_{*}^{-}$ when the
nodes are on the boundary. For $z\in\partial D,$ we have $\Phi(z)=0$ and thus
the formal field $\Phi[1\cdot z]$ can be represented either by $\Phi^{+}(z)$
or by $-\Phi^{-}(z).$ On the other hand, the formal field $N[1\cdot z]$ is
represented by $N^{+}(z)=N^{-}(z).$ The relation $N^{+}(z)=N^{-}(z)$ for
$z\in\partial D$ is obvious:

$\mathbf{E}\,N^{+}(z)N(z_{0})=\log\frac{1}{(z-z_{0})(z-\bar{z}_{0})}=\log\frac{1}{(\bar{z}-z_{0})(\bar{z}-\bar{z}_{0})}=\mathbf{E}\,N^{-}(z)N(z_{0})$

in the $\mathbb{H}$-uniformization.

A stress tensor $A_{\boldsymbol{\beta}}\equiv A_{\boldsymbol{\beta}}^{N}$ for
$N_{\boldsymbol{\beta}}$ is given by

$A_{\boldsymbol{\beta}}=A+(b\partial-j_{\boldsymbol{\beta}})J,\qquad
j_{\boldsymbol{\beta}}:=\mathbf{E}\,J_{\boldsymbol{\beta}},\qquad
J_{\boldsymbol{\beta}}\equiv J_{\boldsymbol{\beta}}^{N}=\partial
N_{\boldsymbol{\beta}}.$

As $\zeta\to z,$ in $\mathbb{H},$ Ward’s OPE for $N_{\boldsymbol{\beta}}$
holds:

$A_{\boldsymbol{\beta}}(\zeta)N_{\boldsymbol{\beta}}(z,z_{0})=\Big{(}A(\zeta)+\big{(}b\partial_{\zeta}-j_{\boldsymbol{\beta}}(\zeta)\big{)}J(\zeta)\Big{)}N(z,z_{0})\\\
\sim\frac{J_{\boldsymbol{\beta}}(z)}{\zeta-z}+\frac{b}{(\zeta-z)^{2}}.$

In the symmetric case $\boldsymbol{\beta}=\mkern
1.5mu\overline{\mkern-1.5mu\boldsymbol{\beta}_{*}\mkern-1.5mu}\mkern 1.5mu,$
the Virasoro field

$T_{\boldsymbol{\beta}}=-\frac{1}{2}J_{\boldsymbol{\beta}}*J_{\boldsymbol{\beta}}+b\partial
J_{\boldsymbol{\beta}}$

gives rise to the Virasoro pair $(T_{\boldsymbol{\beta}},\mkern
1.5mu\overline{\mkern-1.5muT_{\boldsymbol{\beta}}\mkern-1.5mu}\mkern 1.5mu).$
The central charge $c$ in this theory is given by

$c=1+12b^{2}.$

If the parameter $b$ is related to the SLE parameter $\kappa$ as

(9.3) $b=-a(\kappa/4+1),\qquad a=\pm\sqrt{2/\kappa},$

then

$c=13+12(\kappa/8+2/\kappa)\geq 25.$

Modified multi-vertex fields or OPE exponentials
$\mathcal{O}_{\boldsymbol{\beta}}[\boldsymbol{\tau}]$ are defined by

$\mathcal{O}_{\boldsymbol{\beta}}[\boldsymbol{\tau}]=\frac{C_{(-ib)}[-i(\boldsymbol{\tau}+\boldsymbol{\beta})]}{C_{(-ib)}[-i\boldsymbol{\beta}]}\,\mathrm{e}^{\odot
N[\boldsymbol{\tau}]}.$

We denote by $\mathcal{F}^{N}_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}}$
the OPE family of $N_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}}.$

######  Theorem 9.1.

Given two double background charges
$\boldsymbol{\beta}^{\pm},\boldsymbol{\beta}_{0}^{\pm}$ with the neutrality
conditions $(\mathrm{NC}_{b^{\pm}}),$
$\mathcal{F}^{N}_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}}$ is the image
of $\mathcal{F}^{N}_{\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}_{0}^{-}}$
under the insertion of $\mathrm{e}^{\odot
N[\boldsymbol{\beta}^{+}-\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}^{-}-\boldsymbol{\beta}_{0}^{-}]}.$

###### Proof.

By Wick’s calculus, we have

$\mathbf{E}\,N_{\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}_{0}^{-}}(z,z^{\prime})\,\mathrm{e}^{\odot
N[\boldsymbol{\beta}^{+}-\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}^{-}-\boldsymbol{\beta}_{0}^{-}]}=n_{\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}_{0}^{-}}(z,z^{\prime})+\mathbf{E}\,N(z,z^{\prime})N[\boldsymbol{\beta}^{+}-\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}^{-}-\boldsymbol{\beta}_{0}^{-}].$

In the $\mathbb{H}$-uniformization, it follows from (9.1) and (9.2) that

$\mathbf{E}\,N(z,z^{\prime})N[\boldsymbol{\beta}^{+}-\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}^{-}-\boldsymbol{\beta}_{0}^{-}]=n_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}}(z,z^{\prime})-n_{\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}_{0}^{-}}(z,z^{\prime}).$

Since both sides are scalars, the above identity holds for any chart. Thus we
have

$\mathbf{E}\,N_{\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}_{0}^{-}}(z,z^{\prime})\,\mathrm{e}^{\odot
N[\boldsymbol{\beta}^{+}-\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}^{-}-\boldsymbol{\beta}_{0}^{-}]}=\mathbf{E}\,N_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}}(z,z^{\prime}).$

As in the Dirichlet case, we use Wick’s calculus again to derive

$\mathbf{E}\,N_{\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}_{0}^{-}}(z_{1},z_{1}^{\prime})\odot\cdots\odot
N_{\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}_{0}^{-}}(z_{n},z_{n}^{\prime})\,\mathrm{e}^{\odot
N[\boldsymbol{\beta}^{+}-\boldsymbol{\beta}_{0}^{+},\boldsymbol{\beta}^{-}-\boldsymbol{\beta}_{0}^{-}]}=\mathbf{E}\,N_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}}(z_{1},z_{1}^{\prime})\odot\cdots\odot
N_{\boldsymbol{\beta}^{+},\boldsymbol{\beta}^{-}}(z_{n},z_{n}^{\prime}).$

Differentiating with respect to $z_{j},z_{j}^{\prime}$, we complete the proof.
∎

######  Example.

Given a marked boundary point $q\in\partial D,$ we consider a conformal map
$w:(D,q)\to(\mathbb{H},\infty).$ For $\boldsymbol{\beta}^{+}=2b\cdot q$ and
$\boldsymbol{\beta}^{-}=\boldsymbol{0},$

$N_{\boldsymbol{\beta}}(z,z_{0})=N(z,z_{0})+2b(\log|w^{\prime}(z)|-\log|w^{\prime}(z_{0})|).$

The function $(z,z_{0})\mapsto\log|w^{\prime}(z)|-\log|w^{\prime}(z_{0})|$
does not depend on the choice of $w.$

###  9.3. One-leg operators

We consider a simply-connected domain $D$ with two marked points $p\in\partial
D,$ $q\in\partial D$ in the chordal case, and $q\in D$ in the radial case. For
a symmetric background charge $\boldsymbol{\beta}$ on $S=D^{\mathrm{double}}$
with $\boldsymbol{\beta}(p)=a$ and the neutrality condition
$(\mathrm{NC}_{b}),$ the one-leg operator $\Lambda_{p}$ in the chordal case
(the radial case, respectively) is defined by

$\Lambda_{p}\equiv\Lambda(p)=\mathcal{O}_{\check{\boldsymbol{\beta}\,}}[\boldsymbol{\tau}],\qquad\check{\boldsymbol{\beta}\,}=\boldsymbol{\beta}-\boldsymbol{\tau}$

where $\boldsymbol{\tau}=a\cdot p-a\cdot q$ in the chordal case (and
$\boldsymbol{\tau}=a\cdot p-\frac{1}{2}a\cdot q-\frac{1}{2}a\cdot q^{*}$ in
the radial case, respectively). The insertion of
$\Lambda_{p}/\mathbf{E}\,\Lambda_{p}$ is an operator

$\mathcal{X}\mapsto\widehat{\mathcal{X}}$

on Fock space functionals/fields by the rules (5.20) and the formula in the
chordal case

$\widehat{N}_{\check{\boldsymbol{\beta}\,}}(z,z_{0})=N_{\check{\boldsymbol{\beta}\,}}(z,z_{0})-2a\big{(}\log|w(z)|-\log|w(z_{0})|\big{)}\big{)}$

where $w:(D,p,q)\to(\mathbb{H},0,\infty)$ is a conformal map. In the radial
case, the corresponding formula is given by

$\widehat{N}_{\check{\boldsymbol{\beta}\,}}(z,z_{0})=N_{\check{\boldsymbol{\beta}\,}}(z,z_{0})-2a\big{(}\log|1-w(z)|-\log|1-w(z_{0})|\big{)}\big{)}+a\big{(}\log|w(z)|-\log|w(z_{0})|\big{)}\big{)}$

where $w:(D,p,q)\to(\mathbb{D},1,0)$ is a conformal map.

We denote

$\widehat{\mathbf{E}}\,\mathcal{X}:=\frac{\mathbf{E}\,\Lambda_{p}\mathcal{X}}{\mathbf{E}\,\Lambda_{p}}.$

By Theorem 9.1, we have

$\widehat{\mathbf{E}}\,\mathcal{X}=\mathbf{E}\,\widehat{\mathcal{X}}$

for any string $\mathcal{X}$ of fields in
$\mathcal{F}^{N}_{\check{\boldsymbol{\beta}\,}}.$

We now show the level two degeneracy equations for $\Lambda$ if the parameters
$a$ and $b$ satisfy $2a(a+b)=-1.$ We remark that $a$ and $b$ in (9.3) have
such a relation.

######  Proposition 9.2.

Provided that $2a(a+b)=-1,$ we have

(9.4)
$T_{\check{\boldsymbol{\beta}\,}}*\Lambda=-\frac{1}{2a^{2}}\partial^{2}\Lambda.$

###### Proof.

Let $\tilde{a}=-ia,\tilde{b}=-ib.$ In terms of the action of Virasoro
generators and current generators, the one-leg operators are Virasoro primary
holomorphic fields of conformal dimension
$\lambda=\frac{1}{2}\tilde{a}^{2}-\tilde{a}\tilde{b}$ and current primary with
charge $q=\tilde{a}.$ This implies the level two degeneracy equation (9.4) for
the one-leg operator provided that $2\tilde{a}(\tilde{a}+\tilde{b})=1.$

We here present an alternate but direct proof using Wick’s calculus. For
simplicity, we consider the standard chordal case
$\check{\boldsymbol{\beta}\,}=2b\cdot q$ only. It is left to the reader as an
exercise to prove (9.4) for general background charges. Since the difference
is a differential, it suffices to show (9.4) in $\mathbb{H}.$ In the
$\mathbb{H}$-uniformization we have

$N_{\check{\boldsymbol{\beta}\,}}=N,\quad\partial
N_{\check{\boldsymbol{\beta}\,}}=J_{\check{\boldsymbol{\beta}\,}}=J,\quad
T_{\check{\boldsymbol{\beta}\,}}=T+b\partial J,\quad T=-\frac{1}{2}J\odot
J,\quad\Lambda(z)=\mathrm{e}^{\odot aN^{+}(z,q)}.$

Let us first compute $T*\Lambda:$

$T*\Lambda=T\odot\Lambda+a\partial J\odot\Lambda.$

Indeed, as $\zeta\to z,$ we have

$T(\zeta)\Lambda(z)=-\frac{1}{2}\sum_{n=0}^{\infty}\frac{a^{n}}{n!}J^{\odot
2}(\zeta)(N^{+})^{\odot n}(z,q)=T\odot\Lambda(z)+\mathrm{I}+\mathrm{II}+o(1),$

where terms $\mathrm{I}$ and $\mathrm{II}$ come from 1 and 2 contractions,
respectively. It follows from $\mathbf{E}\,J(\zeta)N^{+}(z,q)=-1/(\zeta-z)$
that

$\mathrm{I}=\frac{a}{\zeta-z}J(\zeta)\odot N^{+}(z,q).$

Thus its contribution to $T*\Lambda$ is $a\partial J\odot\Lambda.$ On the
other hand, we have

$\mathrm{II}=-\frac{1}{2}a^{2}\big{(}\mathbf{E}\,J(\zeta)N^{+}(z,q)\big{)}^{2}\Lambda(z)=-\frac{1}{2}\frac{a^{2}}{(\zeta-z)^{2}}\Lambda(z).$

It has no contribution to $T*\Lambda.$ To compute
$T_{\check{\boldsymbol{\beta}\,}}*\Lambda,$ we need to compute $\partial
J*\Lambda.$ We have

$\partial J(\zeta)\Lambda(z)=\partial J\odot\Lambda(z)+o(1).$

This implies that $\partial J*\Lambda=\partial J\odot\Lambda$ and

$T_{\check{\boldsymbol{\beta}\,}}*\Lambda=T\odot\Lambda+(a+b)\partial
J\odot\Lambda.$

Differentiating $\Lambda(z)=\mathrm{e}^{\odot aN^{+}(z,q)},$ we have

$-\frac{1}{2a^{2}}\partial^{2}\Lambda=-\frac{1}{2a^{2}}\partial(aJ\odot\Lambda)=-\frac{1}{2a^{2}}(a\partial
J\odot\Lambda+a^{2}J\odot J\odot\Lambda)=-\frac{1}{2a}\partial
J\odot\Lambda+T\odot\Lambda.$

Now (9.4) follows provided that $2a(a+b)=-1.$ ∎

###  9.4. Ward’s equations and BPZ-Cardy equations

We define the puncture operators by
$\mathcal{P}_{\boldsymbol{\beta}}:=C_{(-ib)}[-i\boldsymbol{\beta}].$ As in the
Dirichlet case, Ward’s equations hold for the extended OPE family
$\mathcal{F}^{N}_{\boldsymbol{\beta}}$ of $N_{\boldsymbol{\beta}}.$ Since its
proof is similar to the Dirichlet case, we leave it to the reader as an
exercise.

######  Theorem 9.3 (Ward’s equations).

Let $Y,X_{1},\cdots,X_{n}\in\mathcal{F}^{N}_{\boldsymbol{\beta}}$ and let $X$
be the tensor product of $X_{j}$’s. Then

$\mathbf{E}\,T_{\boldsymbol{\beta}}(z)\,X=\mathbf{E}\,\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{+}_{k_{z}}\mathcal{P}_{\boldsymbol{\beta}}X+\mathbf{E}\,\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{-}_{k_{\bar{z}}}\mathcal{P}_{\boldsymbol{\beta}}X,$

where all fields are evaluated in the identity chart of $\mathbb{H}$ and

$2z^{2}\mathbf{E}\,T_{\boldsymbol{\beta}}(z)\,X=\mathbf{E}\,\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{+}_{v_{z}}\mathcal{P}_{\boldsymbol{\beta}}X+\mathbf{E}\,\mathcal{P}_{\boldsymbol{\beta}}^{-1}\mathcal{L}^{-}_{v_{z^{*}}}\mathcal{P}_{\boldsymbol{\beta}}X,$

where all fields are evaluated in the identity chart of $\mathbb{D}.$

For a symmetric background charge $\boldsymbol{\beta}$ with the neutrality
condition $(\mathrm{NC}_{b})$ and a specific charge $a$ at a marked boundary
point $p\in\partial D,$ we define the backward SLE partition function
$Z_{\boldsymbol{\beta}}$ by

$Z_{\boldsymbol{\beta}}:=|\mathcal{P}_{\boldsymbol{\beta}}|=|C_{(-ib)}[-i\boldsymbol{\beta}]|$

and the effective one-leg operators by
$\Lambda_{p}^{\mathrm{eff}}:=\mathcal{P}_{\boldsymbol{\beta}}\,\Lambda_{p}.$
Denote

(9.5)
$R_{\xi}\equiv\widehat{\mathbf{E}}_{\xi}\,\mathcal{X}:=\frac{\mathbf{E}\,\Lambda_{\xi}\mathcal{X}}{\mathbf{E}\,\Lambda_{\xi}}=\frac{\mathbf{E}\,\Lambda_{\xi}^{\mathrm{eff}}\mathcal{X}}{\mathbf{E}\,\Lambda_{\xi}^{\mathrm{eff}}}.$

Let $\xi\in\partial D$ and $\boldsymbol{\beta}_{\xi}=\boldsymbol{\beta}-a\cdot
p+a\cdot\xi.$ Let $X$ be the tensor product $X=X_{1}(z_{1})\cdots
X_{n}(z_{n})$ of fields $X_{j}$ in
$\mathcal{F}_{\check{\boldsymbol{\beta}\,}}$ $(z_{j}\in D).$ Combining
Proposition 9.2 with Theorem 9.3, we obtain the BPZ equations:

(9.6)
$-\frac{1}{2a^{2}}\partial_{\xi}^{2}\mathbf{E}\Lambda_{\xi}^{\mathrm{eff}}X=\mathbf{E}\,\check{\mathcal{L}}_{k_{\xi}}\Lambda_{\xi}^{\mathrm{eff}}X,\qquad
k_{\xi}(z):=\frac{1}{\xi-z}$

in the $\mathbb{H}$-uniformization. In particular, the partition function
$Z_{\xi}\equiv
Z_{\boldsymbol{\beta}_{\xi}}:=\big{|}\mathbf{E}\,\Lambda^{\mathrm{eff}}_{\xi}\big{|}=\big{|}C_{(-ib)}[-i\boldsymbol{\beta}_{\xi}]\big{|}$
satisfies the null vector equation

(9.7)
$-\frac{1}{2a^{2}}\partial_{\xi}^{2}Z_{\xi}=\check{\mathcal{L}}_{k_{\xi}}Z_{\xi}$

in the $\mathbb{H}$-uniformization. Here
$\partial_{\xi}=\partial+\bar{\partial}$ is the operator of differentiation
with respect to the real variable $\xi$ and and
$\check{\mathcal{L}}_{k_{\xi}}$ is taken over the finite notes of $X$ and
$\mathrm{supp}\,\boldsymbol{\beta}_{\xi}\setminus\\{\xi\\}.$ We obtain the
following form of BPZ-Cardy equations in the chordal case. The radial case can
be derived in a similar way.

######  Theorem 9.4 (BPZ-Cardy equations).

Suppose that the parameters $a$ and $b$ satisfy

$2a(a+b)=-1.$

If $X=X_{1}(z_{1})\cdots X_{n}(z_{n})$ with
$X_{j}\in\mathcal{F}^{N}_{\check{\boldsymbol{\beta}\,}},$ then we have

(9.8)
$-\frac{1}{2a^{2}}\big{(}\partial_{\xi}^{2}\widehat{\mathbf{E}}_{\xi}\,X+2(\partial_{\xi}\log
Z_{\xi})\partial_{\xi}\widehat{\mathbf{E}}_{\xi}\,X\big{)}=\check{\mathcal{L}}_{k_{\xi}}\widehat{\mathbf{E}}_{\xi}\,X$

in the identity chart of $\mathbb{H}.$ In the $(\mathbb{D},0)$-uniformization
we have

(9.9) $\kappa\Big{(}\frac{1}{2}\partial_{\theta}^{2}+(\partial_{\theta}\log
Z_{\zeta}\big{)}\partial_{\theta}\Big{)}\widehat{\mathbf{E}}_{\zeta}X=\check{\mathcal{L}}_{v_{\zeta}}\widehat{\mathbf{E}}_{\zeta}X,\qquad\zeta=\mathrm{e}^{i\theta}.$

###  9.5. Connection to backward SLE theory

We now prove Theorem 1.6 and present Sheffield’s observables. By definition,
the backward radial (see (1.11) for the chordal case)
$\mathrm{SLE}[\boldsymbol{\beta}]$ map $f_{t}$ from $\mathbb{D}$ satisfies the
equation

(9.10)
$\partial_{t}f_{t}(z)=-f_{t}(z)\frac{\zeta_{t}+f_{t}(z)}{\zeta_{t}-f_{t}(z)},\qquad\zeta_{t}=\mathrm{e}^{i\theta_{t}}$

driven by the real process $\theta_{t}:$

$\mathrm{d}\theta_{t}=\sqrt{\kappa}\,\mathrm{d}B_{t}+\lambda(t)\,\mathrm{d}t,\quad\lambda(t)=(\lambda\,\|\,f_{t}^{-1}),\quad\lambda=\kappa\,\partial_{\theta}\log
Z_{\boldsymbol{\beta}_{\zeta}},$

where the partition function $Z_{\boldsymbol{\beta}_{\zeta}}$ is given by
$Z_{\boldsymbol{\beta}_{\zeta}}=\big{|}C_{(-ib)}[-i\boldsymbol{\beta}_{\zeta}]\big{|},$
$\boldsymbol{\beta}_{\zeta}=\boldsymbol{\beta}+a\cdot\zeta-a\cdot p.$

###### Proof of Theorem 1.6.

We first consider the chordal case. For $\mathcal{X}=X_{1}(z_{1})\cdots
X_{n}(z_{n}),X_{j}\in\mathcal{F}_{\check{\boldsymbol{\beta}\,}}(D),$ denote

$R_{\xi}(z_{1},\cdots,z_{n})\equiv\widehat{\mathbf{E}}_{\xi}[X_{1}(z_{1})\cdots
X_{n}(z_{n})].$

We want to show that

$M_{t}=m(\xi_{t},t),\quad m(\xi,t)=(R_{\xi}\,\|\,f_{t}^{-1})$

is a local martingale on backward chordal SLE probability space. By Itô’s
formula, we compute $\mathrm{d}M_{t}$ as follows:

$\displaystyle\mathrm{d}M_{t}$
$\displaystyle=\sqrt{\kappa}\,\partial_{\xi}\big{|}_{\xi=\xi_{t}}m(\xi,t)\,\mathrm{d}B_{t}+\kappa\,\partial_{\xi}\big{|}_{\xi=\xi_{t}}m(\xi,t)\,\partial_{\xi}\big{|}_{\xi=\xi_{t}}(\log
Z_{\xi}\,\|\,f_{t}^{-1})$
$\displaystyle+\frac{\kappa}{2}\,\partial_{\xi}^{2}\big{|}_{\xi=\xi_{t}}m(\xi,t)\,\mathrm{d}t+\frac{\mathrm{d}}{\mathrm{d}s}\Big{|}_{s=0}\big{(}R_{\xi_{t}}\,\|\,f_{t+s}^{-1}\big{)}\,\mathrm{d}t.$

Using the same argument in the proof of Theorem 1.5, the last term can be
rewritten as

$\frac{\mathrm{d}}{\mathrm{d}s}\Big{|}_{s=0}\big{(}R_{\xi_{t}}\,\|\,f_{t+s}^{-1}\big{)}\,\mathrm{d}t=2\big{(}\check{\mathcal{L}}_{k_{\xi_{t}}}R_{\xi_{t}}\,\|\,f_{t}^{-1}\big{)}.$

Thus we find the drift term of $\mathrm{d}M_{t}$ as

$\Big{(}\frac{\kappa}{2}\,\partial_{\xi}^{2}\big{|}_{\xi=\xi_{t}}m(\xi,t)+\kappa\,\partial_{\xi}\big{|}_{\xi=\xi_{t}}m(\xi,t)\frac{\partial_{\xi}|_{\xi=\xi_{t}}(Z_{\xi}\,\|\,f_{t}^{-1})}{(Z_{\xi}\,\|\,f_{t}^{-1})}+2\big{(}\check{\mathcal{L}}_{k_{\xi_{t}}}R_{\xi_{t}}\,\|\,f_{t}^{-1}\big{)}\Big{)}\,\mathrm{d}t=0$

employing the BPZ-Cardy equations (9.8).

Next, we consider the radial case. For $\mathcal{X}=X_{1}(z_{1})\cdots
X_{n}(z_{n}),X_{j}\in\mathcal{F}_{\check{\boldsymbol{\beta}\,}}(D),$ denote

$R_{\zeta}(z_{1},\cdots,z_{n})\equiv\widehat{\mathbf{E}}_{\zeta}[X_{1}(z_{1})\cdots
X_{n}(z_{n})].$

As in the chordal case, the process

$M_{t}=m(\zeta_{t},t),\qquad
m(\zeta,t)=\big{(}R_{\zeta}\,\|\,f_{t}^{-1}\big{)}$

is a local martingale by Itô’s formula and the BPZ-Cardy equations (9.9).
Indeed, we find the drift term of $\mathrm{d}M_{t}$ as

$\Big{(}\frac{\kappa}{2}\partial_{\theta}^{2}+\kappa\frac{\partial_{\theta}(Z_{\zeta}\,\|\,f_{t}^{-1})}{(Z_{\zeta}\,\|\,f_{t}^{-1})}\partial_{\theta}\Big{)}\Big{|}_{\zeta=\zeta_{t}}~{}m(\zeta,t)\,\mathrm{d}t\\\
-\
\big{(}\check{\mathcal{L}}_{v_{\zeta_{t}}}R_{\zeta_{t}}\,\|\,f_{t}^{-1}\big{)}\mathrm{d}t=0.$

∎

######  Example (Sheffield’s observables).

In the standard chordal case with $\boldsymbol{\beta}=a\cdot p+(2b-a)\cdot q,$
the 1-point functions of the bosonic fields

$\displaystyle n_{t}(z)$
$\displaystyle=\mathbf{E}[N_{\boldsymbol{\beta}}(z)\,\|\,f_{t}^{-1}]=-2a\log|f_{t}(z)-\xi_{t}|+2b\log|f_{t}^{\prime}(z)|$
$\displaystyle=-2a\log|z|+2\sqrt{2}\int_{0}^{t}\mathrm{Re}\,\frac{1}{f_{s}(z)-\xi_{s}}\,\mathrm{d}B_{s}$

were introduced as backward SLE martingale-observables in [34]. Due to the
following special case of Hadamard’s variation formula

(9.11)
$\mathrm{d}G^{N}(f_{t}(z_{1}),f_{t}(z_{2}))=-4\,\mathrm{Re}\,\frac{1}{f_{t}(z_{1})-\xi_{t}}\,\mathrm{Re}\,\frac{1}{f_{t}(z_{2})-\xi_{t}}\,\mathrm{d}t=-\frac{1}{2}\,\mathrm{d}\langle
n_{\boldsymbol{\beta}}(z_{1}),n_{\boldsymbol{\beta}}(z_{2})\rangle_{t},$

the formal $2$-point functions

$\mathbf{E}[N_{\boldsymbol{\beta}}(z_{1})N_{\boldsymbol{\beta}}(z_{2})]=2G^{N}(z_{1},z_{2})+n_{\boldsymbol{\beta}}(z_{1})n_{\boldsymbol{\beta}}(z_{2})$

are martingale-observables for backward SLE. Sheffield used (9.11) to
construct a coupling of backward SLE and the Gaussian free field with Neumann
boundary condition, see [34].

### Acknowledgements

We would like to thank Tom Alberts, and Sung-Soo Byun for careful reading and
much-appreciated help improving this manuscript.

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