# Answering Count Queries with Explanatory Evidence

Shrestha Ghosh  Max Planck Institute for InformaticsSaarland
UniversitySaarbrueckenGermany<EMAIL_ADDRESS>,  Simon Razniewski  Max
Planck Institute for InformaticsSaarbrueckenGermany<EMAIL_ADDRESS>and  Gerhard Weikum  Max Planck Institute for InformaticsSaarbrueckenGermany
<EMAIL_ADDRESS>
(2022)

###### Abstract.

A challenging case in web search and question answering are count queries,
such as “number of songs by John Lennon”. Prior methods merely answer these
with a single, and sometimes puzzling number or return a ranked list of text
snippets with different numbers. This paper proposes a methodology for
answering count queries with inference, contextualization and explanatory
evidence. Unlike previous systems, our method infers final answers from
multiple observations, supports semantic qualifiers for the counts, and
provides evidence by enumerating representative instances. Experiments with a
wide variety of queries show the benefits of our method. To promote further
research on this underexplored topic, we release an annotated dataset of 5k
queries with 200k relevant text spans.

Question Answering, Count Queries, Explainable AI

††journalyear: 2022††copyright: rightsretained††conference: Proceedings of the
45th International ACM SIGIR Conference on Research and Development in
Information Retrieval; July 11–15, 2022; Madrid, Spain.††booktitle:
Proceedings of the 45th Int’l ACM SIGIR Conference on Research and Development
in Information Retrieval (SIGIR ’22), July 11–15, 2022, Madrid, Spain††doi:
10.1145/3477495.3531870††isbn: 978-1-4503-8732-3/22/07††ccs: Information
systems Question answering

##  1\. Introduction

### Motivation and Problem

Question answering (QA) and web search with telegraphic queries have been
greatly advanced over the last decade (Balog, 2018; Diefenbach et al., 2018;
Huang et al., 2020; Usbeck et al., 2019). Nevertheless, queries that can have
multiple correct answers due to variance in semantic qualifiers (“top 10
albums”, “singles albums”, “remastered albums”) and alternative
representations through instances remain underexplored and pose open
challenges. This paper addresses the class of count queries, to return the
number of instances that have a certain property. Examples are:

  * •

How many songs did John Lennon write for the Beatles?

  * •

How many languages are spoken in Indonesia?

  * •

How many unicorn companies are there?

Count queries are frequent in search engine logs as well as QA benchmarks
(Rajpurkar et al., 2016; Kwiatkowski et al., 2019; Dubey et al., 2019;
Voorhees, 2001). If the required data is in a structured knowledge base (KB)
such as Wikidata (Vrandečić and Krötzsch, 2014), then answering is relatively
straightforward. However, KBs are limited not only by their sparsity, but also
by the lack of direct links between instances and explicit counts when both
are present. Besides, evaluating the additional condition “for the Beatles”
(i.e., a subset of his songs) is beyond their scope.

Search engines handle popular cases reasonably well, but also fail on
semantically refined requests (e.g., “for the Beatles”), merely returning
either a number without explanatory evidence or multiple candidate answers
with high variance. Answering count queries from web contents thus poses
several challenges:

  1. 1.

Aggregation and inference: Returning just a single number from the highest-
ranked page can easily go wrong. Instead, joint inference over a set of
candidates, with an awareness of the distribution and other signals, is
necessary for a high-confidence answer.

  2. 2.

Contextualization: Counts in texts often come with contexts on the relevant
instance set. For example, John Lennon co-wrote about 180 songs for the
Beatles, 150 as a solo artist, etc. For correct answers it is crucial to
capture context from the underlying web pages and properly evaluate these
kinds of semantic qualifiers.

  3. 3.

Explanatory Evidence: A numeric answer alone, such as “180” for the Beatles
songs by Lennon, is often unsatisfactory. The user may even perceive this as
non-credible, and think that it is too high as she may have only popular songs
in mind. It is, therefore, crucial to provide users with explanatory evidence.

### Approach and Contribution

This paper presents CoQEx, Count Question answering with Explanatory evidence,
which answers count queries via three components: i) answer inference ii)
answer contextualization and, iii) answer explanation.

Given a full-fledged question or telegraphic query and relevant text passages,
CoQEx applies joint inference to compute a high-confidence answer for the
count itself. It provides contextualization of the returned count answer,
through semantic qualifiers into equivalent or subclass categories, and
extracts a set of representative instances as explanatory evidence,
exemplifying the returned number for enhanced credibility and user
comprehension.

Novel contributions of this work are:

  1. 1.

introducing the problem of count query answering with explanatory evidence;

  2. 2.

developing a method for inferring high-confidence counts from noisy candidate
sets;

  3. 3.

developing techniques to provide answer contextualization and explanations;

  4. 4.

evaluating CoQEx against state-of-the-art baselines on a variety of test
queries;

  5. 5.

releasing an annotated data resource with 5k count queries and 200k text
passages, with preliminary access for reviewing at
https://github.com/ghoshs/CoQEx.

##  2\. Related Work

Where structured data is available in KBs, structured QA is the method of
choice. However, for many topics, no relevant count information can be found
in KBs. For example, Wikidata contains 217 songs attributed to John
Lennon111https://w.wiki/4XVq,

but is incomplete in indicating whether these written for the Beatles or
otherwise. Previous analyses found that 5-10% of queries in popular QA
datasets are count queries (Mirza et al., 2018). In the KB-QA domain, systems
like QAnswer (Diefenbach et al., 2019) tackle count queries by aggregating
instances using the SPARQL count modifier. This is liable to incorrect
answers, when instance relations are incomplete. Recent research has also
explored QA over web tables, with a focus on table retrieval and span
prediction (Herzig et al., 2021). Another challenging aspect of KB-QA is
reducing the semantic gap between natural language query and the SPARQL query
formulation (Berant et al., 2013), which would in turn lead to better count
query answering.

State-of-the-art systems typically approach QA via the machine reading
paradigm (Karpukhin et al., 2020; Joshi et al., 2020; Sanh et al., 2019; Chen
et al., 2017; Dua et al., 2019), where the systems find the best answer in a
given passage. The retriever-reader approach in open-domain QA uses several
text segments to return either a single best answer (Chen et al., 2017; Wang
et al., 2018) or a ranked list of documents with the best answer per document
(Karpukhin et al., 2020). The DPR system (Karpukhin et al.,
2020)222http://qa.cs.washington.edu:2020 returns “approximately 180” from its
rank-1 text passage to both, the simple John Lennon query, and the refined
variant with “…for the Beatles”. The other top-10 snippets include false
results such as “five” and contradictory information such as “180 jointly
credited” (as if Lennon had not written any songs alone). Thus, QA systems are
not robust (yet) and lack explanatory evidence beyond merely returning the
top-ranked text snippet.

Attempts have also been made to improve recall by hybrid QA over text and KB,
yet without specific consideration of counts (Lu et al., 2019; Xu et al.,
2016). Search engines can answer simple count queries from their underlying
KBs, if present, a trait which we exploit to create our CoQuAD dataset
(Section. 4). But more often they return informative text snippets, similar to
QA-over-text systems. The basic Lennon query has a highest-ranked Google
snippet with “more than 150” when given the telegraphic input “number of songs
by John Lennon” and “almost 200” when given the full-fledged question “how
many songs did John Lennon write”. For the latter case, the top-ranked snippet
talks about the composer duo “John Lennon and Paul McCartney”. When refining
the query by qualifiers, this already puzzling situation becomes even more
complex with “84.55 of 209 songs” being ranked first followed by varying
counts such as “18 Beatles songs” (co-written with McCartney) and “61”
(written separately). Because of the lack of consolidation, the onus is on the
user to decide whether there are multiple correct answers across text
segments.

It is recognized that just literally answering questions is often not
sufficient for use cases. One line of work tackles this by returning
comprehensive answer in full sentences, using templates (Kacupaj et al.,
2020). Another line concerns long form question answering, where the QA model
retrieves multiple relevant documents to generate a whole answer paragraph
(Krishna et al., 2021). The ELI5 dataset (Fan et al., 2019) contains diverse
open-ended queries with supporting information from relevant web sources.
While the setting is related, long form QA is concerned with generating
textual answers evidenced on multiple documents, while we focus on answering
count queries by consolidating counts and grounding them in instances.

Figure 1. System overview of CoQEx.

##  3\. Methodology

We approach count question answering by a combination of per-document answer
span prediction, context extraction, and consolidation of counts and instances
across documents. Fig. 1 gives the overview of CoQEx. We consider as input a
query that asks for the count of named entities that stand in relation with a
subject, for instance full queries like “How many songs did John Lennon write
for the Beatles”, or a keyword query like “songs by lennon”.

We further assume that relevant documents or passages are given. This could be
the result of a standard keyword/neural embedding-based IR procedure over a
larger (locally indexed) background corpus, like Wikipedia or the Web. We
focus on extracting counts and instances (entity-mentions) from the text
segments so as to i) consolidate the counts to present the best answer, ii)
present contextualization as a means to semantically qualifying the predicted
count, and iii) ground the count in instances.

### Answer Inference

For obtaining answer candidates, we use the popular SpanBERT model (Joshi et
al., 2020), trained on the CoQuAD train split for candidate generation. Span
prediction models return general text spans, which may contain worded answers
(five children, $\textit{Conf}=0.8$), modifier words and other context (17
regional languages, $\textit{Conf}=0.75$). These answer spans have two
components - the count itself and qualifiers, which we separate with the help
of the CogComp Quantifier (Roy et al., 2015). To consolidate the resulting
candidate counts, we compare four methods:

  * 1.

Most confident: The candidate given the highest confidence by the neural
model. This is commonly used in textual QA (Chen et al., 2017; Wang et al.,
2018).

  * 2.

Most frequent: A natural alternative is to rank answers by frequency, and
prefer the ones returned most often.

While most confident may be susceptible to single outliers, most frequent
breaks down in cases where there are few answer candidates. But unlike textual
answers, numbers allow further statistical aggregation:

  * 3.

Median: The midpoint in the ordered list of candidates.

  * 4.

Weighted Median: The median can be further adapted by weighing each candidate
with the model’s score.

### Answer Contextualization

The answer candidates from the previous module often contain nouns with
phrasal modifiers, such as 17 regional languages. We call these count-modified
noun phrases (CNPs). These CNPs stand in some relation with the predicted
count from the answer inference module. The representative CNP,
CNP${}_{\text{rep}}$, which best accompanies the predicted count is first
chosen and then compared with the remaining CNPs. Since answer inference uses
a consolidation strategy, we select the CNP with count within $\pm\alpha$ of
the predicted count having the highest confidence as CNP${}_{\text{rep}}$,
where $\alpha$ is between 0 and 100%, 0 being most restrictive. The remaining
CNPs are categorized as follows:

  1. 1.

Subgroups: CNPs which are semantically more specific than
CNP${}_{\textit{rep}}$, and are expected to count only a subset of the
instances counted by CNP${}_{\textit{rep}}$.

  2. 2.

Synonyms: CNPs, whose meaning is highly similar to CNP${}_{\textit{rep}}$.

  3. 3.

Incomparables: CNPs which count instances of a completely different type.

We assign these categories based on (textual) semantic relatedness of the
phrasal modifier, and numeric proximity of the count. For example, regional
languages is likely a subgroup of 700 languages, especially if it occurs with
counts $\langle 23,17,42\rangle$. tongue is likely a synonym, especially if it
occurs with counts $\langle 530,810,600\rangle$. Speakers is most likely
incomparable, especially if it co-occurs with counts in the millions.

CNPs with embedding-cosine similarity (Reimers and Gurevych, 2019) less than
zero are categorized as incomparable, while from the remainder, those with a
count within $\pm\alpha$ are considered synonyms, lower count CNPs are
categorized as subgroups, and higher count CNPs as incomparable.

For instance, for the query “How many languages are spoken in Indonesia”, with
a prediction 700, estimated 700 languages would be the
$\textit{CNP}_{\textit{rep}}$, {700 languages, 750 dialects} would be
classified as synonyms, {27 major regional languages, 5 official languages} as
subgroups and {2000 ethnic groups, 85 million native speakers} as
incomparables.

### Answer Explanation

Beyond classifying count answer contexts, showing relevant sample instances is
an important step towards explainability. To this end, we aim to identify
entities that are among the ones counted in the query.

We again use the SpanBERT model to obtain candidates, this time with a
modified query, replacing ”how many” in the query with ”which” (or adding it),
so as to not confuse the model on the answer type. We extract named entities
from the answer spans and rank them using the following alternative
approaches:

  1. 1.

QA w/o Consolidation. In the spirit of conventional QA, where results come
from a single document, we return instances from the document with the most
confident answer span.

  2. 2.

QA + Context Frequency. The instances are ranked by their frequency.

  3. 3.

QA + Summed Confidence. We rank the instances based on the summed confidence
of all answer spans that contain them.

  4. 4.

QA + Type Compatibility. Here instances are ranked by their compatibility with
the query’s answer type, extracted via the dependency parse tree. We form a
hypothesis “(instance) is a (answer type)” and use the probability of its
entailment from the parent sentence in the context from which the instance was
extracted to measure type compatibility. We use (Liu et al., 2019) to obtain
entailment scores, which are again summed over all containing answer spans.

##  4\. The CoQuAD Dataset

### Dataset construction

Existing QA datasets only incidentally contain count queries; we leverage
search engine autocomplete suggestions to automatically compile count queries
that reflect real user queries (Sullivan, 2020). We provide the Google search
engine with iterative query prefixes of the form “How many X”, where
$X\in\\{a,aa,\ldots,zzz\\}$, similar to the candidate generation from patterns
used in (Romero et al., 2019), and collect all autocomplete suggestions via
SERP API333https://serpapi.com. We keep those with at least one named-entity
(to avoid too general queries) and no measurement term (to avoid non-entity
answer types). This gives us 11.3k count queries.

We automatically obtain count ground truth by collecting structured answers
from the same search engine. Executing each query on Google, we scrape
knowledge graph (KG) answers and featured snippets, using an off-the-shelf QA
extraction model (Sanh et al., 2019) to obtain best answers from the latter.
This gives us KG ground truth for 131 queries, and ground truth from featured
snippets for 6.7k queries. We again discard queries whose automated ground
truth answer contains a measurement term, and manually annotate 100 queries
from those without automated ground truth.

We next scrape the top-50 snippets per query from Bing, and obtain text
segment ground truth by labelling answer spans returned by the count extractor
(Roy et al., 2015) as positive when the count lies within $\pm 10\%$ from the
ground truth. There are around 800 queries with no positive snippets, which we
do not discard, so the system is not forced to generate an answer. In the end
we have $5162$ count queries with automated ground truth, and an average of
$40$ annotated text segments per query. We use 80% of the count queries with
automated ground truth for training. The test set contains 50 count queries
with KG ground truth, 100 with ground truths from featured snippets, and 100
with manually annotated ground truth for quantitative and qualitative
analysis. We also manually annotate 75 queries with at least top-5 prominent
instances for evaluating answer explanations.

### Dataset Characteristics

Queries in CoQuAD cover a range of topics, notably entertainment (27%), social
topics (20%), organizations (12%), technology (8%) and politics (7%). We find
that 20% of query results are fully stable (a company’s founders, casts in
produced movies), 55% are low-volatile (lakes in a region, band members of an
established but active band), 25% are near-continuous (employment numbers,
COVID cases). Most queries count entities in a simple relation to one named
entity, i.e., the avg. query length is 6.40 words, with an average of 1.08
named entities per query.

Research access to the data is at https://tinyurl.com/countqueryappendix.

##  5\. Evaluation

### Implementation Details

The transformer model for answer inference candidate generation is trained for
2 epochs, at a learning rate of $3e^{-5}$. An input datapoint consists of a
query, a text segment and a text span containing the count answer (empty if no
answer). We train over 3 seeds and report the average score. For getting the
instances from the answer spans, we use the pre-trained SpaCy NER
model444https://spacy.io on the en_core_web_sm model..

### Component analysis of CoQEx

We evaluate the CoQEx components to determine the best configurations for
answer inference, consolidation and explanation. While we can use regular IR
metrics of precision and recall for evaluating answer explanations
(Precision@k, Recall@k, Hit@k and MRR) and accuracy of classification for
evaluating count context categories, we need a new metric for counts. We use
Relaxed Precision (RP), which is the fraction of queries where the prediction
lies within $\pm 10\%$ of the gold answer. We also report Coverage, which
measures the fraction of queries that a systems returns an answer for.

We test the candidate generator for count spans on SpanBERT (Joshi et al.,
2020) finetuned on i) CoQuAD and, ii) the popular general QA dataset SQuAD
(Rajpurkar et al., 2016). Fine-tuning on SQuAD gives slightly higher precision
scores (29.8% vs. 27.4% RP), but CoQuAD gives a higher coverage (82% vs.
73.8%) resulting in overall more correctly answered queries.

For answer inference, the frequent and weighted median consolidation schemes
outperform the others, with weighted median achieving 27.4% RP just 0.6% ahead
of frequent. Thus, for queries backed by less variant data, frequent is good
enough, but to have an edge in more variant data weighted median is the way to
go.

We assess the classification accuracy of CNPs for a manually labelled sample
of 294 CNPs for 64 queries. While a strict threshold of 10% ensures a high
accuracy of 90% for Synonyms, which only decreases with increasing $\alpha$,
the accuracy of Subgroups and Incomparables is initially low (at or less than
50%), and peaks only at a much higher $\alpha$. A weighted optimum is reached
at $\alpha=30\%$, where the accuracy of the Synonyms does not degrade much
(89%), and the accuracy of Subgroups and Incomparables is both above 60% (61%
and 62% respectively).

In answer explanation, QA + context frequency consolidation performs
consistently well. While QA w/o consolidation has highest P (11.5%) and R
(2.3%) at rank 1, its performance decreases subsequently to only 2.7% P and
5.1% R at rank 10, compared to 10.8% P and 18.2% R of QA + context frequency.
This indicates that the QA model is tailored to the typical setting QA of a
single correct answer, and that consolidation helps beyond that.

### Comparing CoEx with baselines

We compare our proposed system with two complementary paradigms.

  1. 1.

Knowledge-base question answering: QAnswer (Diefenbach et al., 2019).

  2. 2.

Commercial search engine QA: Google Search Direct Answers (GoogleSDA).

For fairness to QAnswer, we post-processed the results to extract count and
instances. For evaluating instances by GoogleSDA, we post-processed knowledge
graph and featured snippet of the search engine result page, keeping items
from list-like structures as instances ranked in their order of appearance.

Table 1. Comparing answer inference results (in percentages).

System CoQuAD LCQuAD${}_{\textit{count}}$ Stresstest RP Cov RP Cov RP Cov
QAnswer (Diefenbach et al., 2019) 6.6 95.6 29.8 50.0 3.0 45.0 GoogleSDA 84.6
60.0 5.7 10.5 22.0 37.0 CoQEx 27.4 82.0 6.7 45.2 38.6 88.3

Table 2. Precision@k (P@k), Recall@k (R@k), Hit@10 and MRR for the answer
explanations of CoQEx and baselines.

System P@1 P@5 P@10 R@1 R@5 R@10 Hit@10 MRR QAnswer (Diefenbach et al., 2019)
4.7 4.5 3.7 2.1 0.8 1.1 12.8 0.064 GoogleSDA 7.7 14.2 21.7 1.3 2.2 3.2 12.8
0.089 CoQEx 6.4 12.8 10.8 1.3 11.7 18.2 55.1 0.218

For answer inference, Table 1 compares the three systems on the 250 annotated
test CoQuAD queries. We also present the performances on 100 count queries
from an existing dataset LCQuAD (Dubey et al., 2019), and a manually curated
dataset of 100 challenging count queries called Stresstest. While GoogleSDA
has a high precision on CoQuAD (consisting 150 KG and snippet answerable
queries), CoQEx not only provides high coverage but a decent RP. On LCQuAD, a
dataset designed specifically for KG queries, CoQEx loses to LCQuAD while
still maintaining a better coverage and RP score compared to GoogleSDA. The
results indicate that reliance on structured KBs (QAnswer) is not sufficient
for general queries, and robust consolidation from crisper text segments is
necessary.

For answer explanation, the comparison results on the 75 instance-annotated
CoQuAD queries are in Table 2. CoQEx provides more R than the baselines, with
competitive P at rank 1, and better at ranks 5 and 10. CoQEx performs
consistently better at hits@k and MRR, losing only slightly in precision at
ranks 1 and 5. Both baselines answer less than 25 queries at rank 5, and at
rank 10 less than 20 queries, and QAnswer performs poorly in the returned
answers. GoogleSDA operates extremely conservative, thus maintaining precision
at lower ranks, at the cost of tiny recall.

Table 3. Extrinsic user study (precision in percent).

Class Only Count +Instances +CNPs +Snippet All Correct 73 63 78 75 88
Incorrect 28 45 40 53 45 Both 55 56 63 66 71

### Qualitative Comparison

We looked further into the 250 queries (50 KG answerable, 100 snippet
answerable and 100 with ranked snippets) for understanding how baselines
tackle progressively difficult queries. As explained in the CoQuAD dataset
creation, we encountered three ways in which Google answers count queries.
Around 1% of the answers came from the Google KG, 59% through featured
snippets, 40% through page results. Besides these high-level categories,
answers can be categorized by the following aspects:

  1. 1.

Listing only instances (e.g., for “mayors of New York”),

  2. 2.

Listing only counts (e.g., for “employees of NHS”),

  3. 3.

Listing both instances and counts, or

  4. 4.

Listing counts refined with semantic qualifiers (e.g., “7 official languages
and 30 regional languagues”).

On the above 250 queries, QAnswer returns mostly counts (95.6%) and rarely
instances (2%) or no answers (2.4%). Among the KG-based answers, GoogleSDA
returns both counts and instances for 90% of the queries, and only counts for
the remaining 10%. Among the featured-snippets-based answers, 85% contain only
counts, and just 15% contain both counts and instances. Semantic qualifiers
are common in featured snippets, coming up for 73% of queries. While semantic
qualifiers can be expressed in KG answers (“volcanic islands in
Hawaii”$\Rightarrow$islands$\rightarrow$Hawaii$\rightarrow$volcanoes), this
rarely shows up, unless the queried entity and the qualifier are extremely
popular.

### User Studies

We asked 120 MTurk users for pairwise preferences between answer pages that
reported bare counts, and counts enhanced by either of the explanation types.
50% of participants preferred interfaces with CNPs, 80% with a snippet, 73%
with instances, 63% preferred an interface with all three enabled (remaining
percentage: prefer bare count answer/same/cannot decide). While snippets are
already in use in search engines, the results indicate that CNPs and instances
are considered valuable, too.

We also validated the merit of explanations extrinsically. We took 5 queries
with correct count results, 5 with incorrect results, and presented the system
output under the 5 explanation settings to 500 users. The users’ task was to
judge the count as correct or not based on the explanations present. The
measured precision scores are in Table 3. All explanation had a positive
effect on overall annotator precision, especially for incorrect counts.

##  6\. Conclusion

In this work we highlighted the distinct challenges of count query answering,
in particular, that count answer candidates form numeric distributions, that
answer contexts stand in hierarchical relations, and that counts coexist with
instances, that ideally should be returned as well. We introduced an approach
for count query answering that contains these three components.

We construct the CoQEx dataset of count queries that reflect real user needs.
Through CoQEx, we show how to provide a better trade-off between precision and
coverage than the current deployed systems, on multiple challenging datasets.
User studies show that explanatory evidence improves user comprehension,
especially through CNPs when there exist similar competing answers attributed
to specific qualifiers. To foster further research, we release all
datasets555https://github.com/ghoshs/CoQEx.

###### Acknowledgements.

We would like to thank the anonymous reviewers for their valuable feedback and
suggestions.

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# Jacob’s ladders and new families of $\zeta$-kindred real continuous
functions.

Jan Moser  Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
<EMAIL_ADDRESS>
###### Abstract.

In this paper we obtain, by our method of crossbreeding in certain set of
$\zeta$-factorization formulas, the corresponding complete hybrid formulas.
These are playing the role of criterion for selection of new families of
$\zeta$-kindred real continuous functions.

###### Key words and phrases:

Riemann zeta-function

##  1\. Introduction

### 1.1.

In this paper we use the following notions:

  * (A)

crossbreeding in the set of $\zeta$-factorization formulas;

  * (B)

complete hybrid formula,

  * (C)

definition of $\zeta$-kindred elements in the set of real continuous
functions,

that we have introduced in the paper [11].

Let us present a short survey of these notions.

  * (a)

We begin with the set of functions

(1.1) $\begin{split}&f_{m}(t)\in\tilde{C}_{0}[T,T+U],\ U=o\left(\frac{T}{\ln
T}\right),\ T\to\infty\\\ &m=1,\dots,M,\ M\in\mathbb{N},\end{split}$

where $M$ is arbitrary and fixed.

  * (b)

Next we obtain, by application of the operator $\hat{H}$ (introduced in the
paper [8], (3.6)), the vector-valued functions

(1.2)
$\begin{split}&\hat{H}f_{m}(t)=(\alpha_{0}^{m,k_{m}},\dots,\alpha_{k_{m}}^{m,k_{m}},\beta_{1}^{k_{m}},\dots,\beta_{k_{m}}^{k_{m}}),\\\
&m=1,\dots,M,,\ 1\leq k_{m}\leq k_{0},\ k_{0}\in\mathbb{N},\end{split}$

where $k_{0}$ is arbitrary and fixed. Simultaneously, we obtain by our
algorithm (for the short survey of this one see [8], (3.1) – (3.11)), also the
following set of $\zeta$-factorization formulas

(1.3)
$\begin{split}&\prod_{r=1}^{k_{m}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{m,k_{m}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{m}}\right)}\right|^{2}\sim
E_{m}(U,T)F_{m}[f_{m}(\alpha_{0}^{m,k_{m}})],\\\ &m=1,\dots,M,\
L\to\infty.\end{split}$

  * (c)

Further, we will suppose that we have obtained the following complete hybrid
formula

(1.4)
$\begin{split}&\mathcal{F}\left\\{\prod_{r=1}^{k_{1}}(\cdots),\prod_{r=1}^{k_{2}}(\cdots),\dots,\prod_{r=1}^{k_{M}}(\cdots),F_{1}[f_{1}(\alpha_{0}^{1,k_{1}})],\dots,F_{M}[f_{m}(\alpha_{0}^{M,k_{M}})]\right\\}\\\
&=1+\mathcal{O}\left(\frac{\ln\ln T}{\ln T}\right)\sim 1,\
T\to\infty\end{split}$

after the finite number of stages of crossbreeding (every memeber of (1.3) is
the participant on this process of crossbreeding) in the set (1.3) - that is:
after the finite number of eliminations of the external functions

(1.5) $E_{m}(U,T),\ m=1,\dots,M$

from the set (1.3).

  * (d)

Now we see that the complete hybrid formula (1.4) expresses the functional
dependence of the set of vector-valued functions (1.2). Consequently, the
back-projection of this functional dependence of the set (1.2) into the
generating set (1.1) leaves us with the following (see [1])

######  Definition.

We will call the subset

(1.6) $\\{f_{1}(t),\dots,f_{M}(t)\\},\ t\in[T,T+U]$

of the real continuous functions (comp. (1.1)), for which there is the
complete hybrid formula (1.4), as the family of $\zeta$-kindred functions.

### 1.2.

In this paper we obtain the following new families of $\zeta$-kindred real
continuous functions:

(1.7) $\begin{split}&\left\\{\frac{1}{\cos^{2}t},\frac{\sin^{2}t}{\cos^{4}t},\
(t-\pi L)^{\Delta}\right\\},\\\ &t\in[\pi L,\pi L+U],\
U\in(0,\pi/2-\epsilon],\Delta>0,\ L\to\infty,\end{split}$

and

(1.8) $\begin{split}&\left\\{\frac{1}{\cos^{2}t},\cos t,\ \cos^{2}t,\
\cos^{3}t,\ \sin^{2}t,\ (t-2\pi L)^{\Delta}\right\\},\\\ &t\in[2\pi L,2\pi
L+U],\ U\in(0,\pi/2-\epsilon],\Delta>0,\ L\to\infty.\end{split}$

Let us remind we have introduced in the papers [1] – [11] new notions in the
theory of the Riemann zeta-function based on Jacob’s ladders (see [1]). The
present paper contains new results in this direction.

##  2\. The first class of lemmas

By making use of our algorithm for generating $\zeta$-factorization formulas
(see [8], (3.1) – (3.11)) we obtain the following results.

######  Lemma 1.

For the function

(2.1) $f_{1}(t)=\frac{1}{\cos^{2}t}\in\tilde{C}_{0}[\pi L,\pi L+U],\
U\in(0,\pi/2-\epsilon]$

there are the vector-valued functions

(2.2)
$\begin{split}&(\alpha_{0}^{1,k_{1}},\dots,\alpha_{k_{1}}^{1,k_{1}},\beta_{1}^{k_{1}},\dots,\beta_{k_{1}}^{k_{1}}),\\\
&1\leq k_{1}\leq k_{0},\ k_{0}\in\mathbb{N}\end{split}$

such that the following factorization formula

(2.3)
$\prod_{r=1}^{k_{1}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{1,k_{1}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{1}}\right)}\right|^{2}\sim\frac{\tan
U}{U}\cos^{2}(\alpha_{0}^{1,k_{1}}),\ L\to\infty$

holds true, where

(2.4) $\begin{split}&\alpha_{r}^{1,k_{1}}=\alpha_{r}(U,L,k_{1};f_{1}),\
r=0,1,\dots,k_{1},\\\ &\beta_{r}^{k_{1}}=\beta_{r}(U,L,k_{1}),\
r=1,\dots,k_{1},\\\ &\pi L<\alpha_{0}^{1,k_{1}}<2\pi L+U\ \Rightarrow\
0<\alpha_{0}^{1,k_{1}}-\pi L<U.\end{split}$

######  Lemma 2.

For the function

(2.5) $f_{2}(t)=\frac{\sin^{2}t}{\cos^{4}t}\in\tilde{C}_{0}[\pi L,\pi L+U],\
U\in(0,\pi/2-\epsilon]$ (2.6)
$\begin{split}&(\alpha_{0}^{2,k_{2}},\dots,\alpha_{k_{2}}^{2,k_{2}},\beta_{1}^{k_{2}},\dots,\beta_{k_{2}}^{k_{2}}),\\\
&1\leq k_{2}\leq k_{0},\ k_{0}\in\mathbb{N}\end{split}$

such that the following factorization formula

(2.7)
$\prod_{r=1}^{k_{2}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{2,k_{2}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{2}}\right)}\right|^{2}\sim\frac{1}{3}\frac{\tan^{3}U}{U}\frac{\cos^{4}(\alpha_{0}^{2,k_{2}})}{\sin^{2}(\alpha_{0}^{2,k_{2}})},\
L\to\infty$

holds true, where

(2.8) $\begin{split}&\alpha_{r}^{2,k_{2}}=\alpha_{r}(U,L,k_{2};f_{2}),\
r=0,1,\dots,k_{2},\\\ &\beta_{r}^{k_{2}}=\beta_{r}(U,L,k_{2}),\
r=1,\dots,k_{2},\\\ &0<\alpha_{0}^{2,k_{2}}-\pi L<U.\end{split}$

######  Lemma 3.

For the function

(2.9) $\bar{f}_{\Delta}(t,L)=\bar{f}_{\Delta}(t)=(t-\pi
L)^{\Delta}\in\tilde{C}_{0}[\pi L,\pi L+U],\ U\in(0,\pi/2),\ \Delta>0$

there are vector-valued functions

(2.10)
$\begin{split}&(\bar{\alpha}_{0}^{\Delta,\bar{k}_{\Delta}},\dots,\alpha_{\bar{k}_{\Delta}}^{\Delta,\bar{k}_{\Delta}},\beta_{1}^{\bar{k}_{\Delta}},\dots,\beta_{\bar{k}_{\Delta}}^{\bar{k}_{\Delta}}),\\\
&1\leq\bar{k}_{\Delta}\leq k_{0},\ k_{0}\in\mathbb{N}\end{split}$

such that the following factorization formula

(2.11)
$\prod_{r=1}^{\bar{k}_{\Delta}}\left|\frac{\zeta\left(\frac{1}{2}+i\bar{\alpha}_{r}^{\Delta,\bar{k}_{\Delta}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{\bar{k}_{\Delta}}\right)}\right|^{2}\sim\frac{1}{1+\Delta}\left(\frac{U}{\bar{\alpha}_{0}^{\Delta,\bar{k}_{\Delta}}-\pi
L}\right)^{\Delta},\ L\to\infty$

holds true, where

(2.12)
$\begin{split}&\alpha_{r}^{\Delta,\bar{k}_{\Delta}}=\alpha_{r}(U,L,\bar{k}_{\Delta};\bar{f}_{\Delta}),\
r=0,1,\dots,\bar{k}_{\Delta},\\\
&\beta_{r}^{\bar{k}_{\Delta}}=\beta_{r}(U,L,\bar{k}_{\Delta}),\
r=1,\dots,\bar{k}_{\Delta},\\\
&0<\bar{\alpha}_{0}^{\Delta,\bar{k}_{\Delta}}-\pi L<U.\end{split}$

(see [9], (2.1) – (2.5), $L\rightarrow\ \pi L$).

##  3\. Theorem 1

###  3.1. The first stage of the crossbreeding

It is the result of the crossbreeding between the $\zeta$-factorization
formula (2.3) and (2.7):

(3.1)
$\begin{split}&\prod_{r=1}^{k_{1}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{1,k_{1}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{1}}\right)}\right|^{2}\sim\\\
&\sim\frac{3}{U^{2}}\frac{\cos^{6}(\alpha_{0}^{1,k_{1}})\sin^{2}(\alpha_{0}^{2,k_{2}})}{\cos^{4}(\alpha_{0}^{2,k_{2}})}\prod_{r=1}^{k_{2}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{2,k_{2}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{2}}\right)}\right|^{2},\
L\to\infty.\end{split}$

###  3.2. The second stage of the crossbreeding

Now, the crossbreeding between the formula (3.1) and the formula (see (2.11))

(3.2) $U^{\Delta}\sim(1+\Delta)(\bar{\alpha}_{0}^{\Delta,\bar{k}_{\Delta}}-\pi
L)^{\Delta}\prod_{r=1}^{\bar{k}_{\Delta}}\left|\frac{\zeta\left(\frac{1}{2}+i\bar{\alpha}_{r}^{\Delta,\bar{k}_{\Delta}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{\bar{k}_{\Delta}}\right)}\right|^{2}$

gives the following

######  Complete Hybrid Formula 1.

(3.3)
$\begin{split}&\prod_{r=1}^{\bar{k}_{\Delta}}\left|\frac{\zeta\left(\frac{1}{2}+i\bar{\alpha}_{r}^{\Delta,\bar{k}_{\Delta}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{\bar{k}_{\Delta}}\right)}\right|^{2}\sim\\\
&\sim\frac{3^{\Delta/2}}{1+\Delta}\left[\frac{|\cos^{3}(\alpha_{0}^{1,k_{1}})||\sin(\alpha_{0}^{2,k_{2}})|}{(\bar{\alpha}_{0}^{\Delta,\bar{k}_{\Delta}}-\pi
L)\cos^{2}(\alpha_{0}^{2,k_{2}})}\right]^{\Delta}\times\\\
&\times\left\\{\prod_{r=1}^{k_{1}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{1,k_{1}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{1}}\right)}\right|^{2}\right\\}^{-3\Delta/2}\times\left\\{\prod_{r=1}^{k_{2}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{2,k_{2}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{2}}\right)}\right|^{2}\right\\}^{\Delta/2},\\\
&1\leq\bar{k}_{\Delta},k_{1},k_{2}\leq k_{0},\ \Delta>0,\
L\to\infty.\end{split}$

### 3.3.

Now, we obtain from (3.3) by Definition the following

######  Theorem 1.

The subset

(3.4) $\begin{split}&\left\\{\frac{1}{\cos^{2}t},\frac{\sin^{2}t}{\cos^{4}t},\
(t-\pi L)^{\Delta}\right\\},\\\ &t\in[\pi L,\pi L+U],\
U\in(0,\pi/2-\epsilon],\Delta>0,\ L\to\infty\end{split}$

is the family of $\zeta$-kindred elements in the class of real continuous
functions.

##  4\. The second class of lemmas

### 4.1.

If we put in [8], (4.1) – (4.10)

$L\rightarrow 2L,\ \mu=0$

then we obtain the following result.

######  Lemma 4.

For the function

(4.1) $f_{3}(t)=\sin^{2}t\in\tilde{C}_{0}[2\pi L,2\pi L+U],\ U\in(0,\pi/2)$

there are vector-valued functions

(4.2)
$(\alpha_{0}^{3,k},\dots,\alpha_{k_{3}}^{3,k_{3}},\beta_{1}^{k_{3}},\dots,\beta_{k_{3}}^{k_{3}}),\
1\leq k_{3}\leq k_{0}$

such that the following factorization formula

(4.3)
$\begin{split}&\prod_{r=1}^{k_{3}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{3,k_{3}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{3}}\right)}\right|^{2}\sim\\\
&\sim\left\\{\frac{1}{2}-\frac{1}{2}\frac{\sin U}{U}\cos
U\right\\}\frac{1}{\sin^{2}(\alpha_{0}^{3,k_{3}})},\ L\to\infty\end{split}$

holds true, where

(4.4) $\begin{split}&\alpha_{r}^{3,k_{3}}=\alpha_{r}(U,2L,k_{3};f_{3}),\
r=0,1,\dots,k_{3},\\\ &\beta_{r}^{k_{3}}=\beta_{r}(U,2L,k_{3}),\
r=1,\dots,k_{3},\\\ &0<\alpha_{0}^{3,k_{3}}-2\pi L<U.\end{split}$

######  Lemma 5.

For the function

(4.5) $f_{4}(t)=\cos^{2}t\in\tilde{C}_{0}[2\pi L,2\pi L+U],\ U\in(0,\pi/2)$

there are vector-valued functions

(4.6)
$(\alpha_{0}^{4,k_{4}},\dots,\alpha_{k_{4}}^{4,k_{4}},\beta_{1}^{k_{4}},\dots,\beta_{k_{4}}^{k_{4}}),\
1\leq k_{4}\leq k_{0}$

such that the following factorization formula

(4.7)
$\begin{split}&\prod_{r=1}^{k_{4}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{4,k_{4}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{4}}\right)}\right|^{2}\sim\\\
&\sim\left\\{\frac{1}{2}+\frac{1}{2}\frac{\sin U}{U}\cos
U\right\\}\frac{1}{\cos^{2}(\alpha_{0}^{4,k_{4}})},\ L\to\infty\end{split}$

holds true, where

(4.8) $\begin{split}&\alpha_{r}^{4,k_{4}}=\alpha_{r}(U,2L,k_{4};f_{4}),\
r=0,1,\dots,k_{4},\\\ &\beta_{r}^{k_{4}}=\beta_{r}(U,2L,k_{4}),\
r=1,\dots,k_{4},\\\ &0<\alpha_{0}^{4,k_{4}}-2\pi L<U.\end{split}$

######  Lemma 6.

For the function

(4.9) $f_{5}(t)=\frac{1}{\cos^{2}t}\in\tilde{C}_{0}[2\pi L,2\pi L+U],\
U\in(0,\pi/2-\epsilon]$

there are vector-valued functions

(4.10)
$(\alpha_{0}^{5,k_{5}},\dots,\alpha_{k_{5}}^{5,k_{5}},\beta_{1}^{k_{5}},\dots,\beta_{k_{5}}^{k_{5}}),\
1\leq k_{5}\leq k_{0}$

such that the following factorization formula

(4.11)
$\begin{split}&\prod_{r=1}^{k_{5}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{5,k_{5}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{5}}\right)}\right|^{2}\sim\\\
&\sim\frac{\sin U}{U}\frac{\cos^{2}(\alpha_{0}^{5,k_{5}})}{\cos U},\
L\to\infty\end{split}$

holds true, where

(4.12) $\begin{split}&\alpha_{r}^{5,k_{5}}=\alpha_{r}(U,2L,k_{5};f_{5}),\
r=0,1,\dots,k_{5},\\\ &\beta_{r}^{k_{5}}=\beta_{r}(U,2L,k_{5}),\
r=1,\dots,k_{5},\\\ &0<\alpha_{0}^{5,k_{5}}-2\pi L<U,\end{split}$

(comp. (2.1) – (2.4) at $L\to\infty$).

### 4.2.

Next, we have the following results (see [10], (2.1) – (2.6)).

######  Lemma 7.

For the function

(4.13) $f_{6}(t)=\cos^{3}t\in\tilde{C}_{0}[2\pi L,2\pi L+U],\ U\in(0,\pi/2)$

there are vector-valued functions

(4.14)
$(\alpha_{0}^{6,k_{6}},\dots,\alpha_{k_{6}}^{6,k_{6}},\beta_{1}^{k_{6}},\dots,\beta_{k_{6}}^{k_{6}}),\
1\leq k_{6}\leq k_{0}$

such that the following factorization formula

(4.15)
$\begin{split}&\prod_{r=1}^{k_{6}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{6,k_{6}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{6}}\right)}\right|^{2}\sim\\\
&\sim\left\\{\frac{\sin U}{U}-\frac{U^{2}}{3}\left(\frac{\sin
U}{U}\right)^{3}\right\\}\frac{1}{\cos^{3}(\alpha_{0}^{6,k_{6}})},\
L\to\infty\end{split}$

holds true, where

(4.16) $\begin{split}&\alpha_{r}^{6,k_{6}}=\alpha_{r}(U,L,k_{6};f_{6}),\
r=0,1,\dots,k_{6},\\\ &\beta_{r}^{k_{6}}=\beta_{r}(U,L,k_{6}),\
r=1,\dots,k_{6},\\\ &0<\alpha_{0}^{6,k_{6}}-\pi L<U.\end{split}$

######  Lemma 8.

For the function

(4.17) $f_{7}(t)=\cos t\in\tilde{C}_{0}[2\pi L,2\pi L+U],\ U\in(0,\pi/2)$

there are vector-valued functions

(4.18)
$(\alpha_{0}^{7,k_{7}},\dots,\alpha_{k_{7}}^{7,k_{7}},\beta_{1}^{k_{7}},\dots,\beta_{k_{7}}^{k_{7}}),\
1\leq k_{7}\leq k_{0}$

such that the following factorization formula

(4.19)
$\begin{split}&\prod_{r=1}^{k_{7}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{7,k_{7}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{7}}\right)}\right|^{2}\sim\\\
&\sim\frac{\sin U}{U}\frac{1}{\cos(\alpha_{0}^{7,k_{7}})},\
L\to\infty\end{split}$

holds true, where

(4.20) $\begin{split}&\alpha_{r}^{7,k_{7}}=\alpha_{r}(U,L,k_{7};f_{7}),\
r=0,1,\dots,k_{7},\\\ &\beta_{r}^{k_{7}}=\beta_{r}(U,L,k_{7}),\
r=1,\dots,k_{7},\\\ &0<\alpha_{0}^{7,k_{7}}-2\pi L<U.\end{split}$

##  5\. Theorem 2

###  5.1. The first stage of the crossbreeding

The crossbreeding between the formula (4.3) and (4.7) gives the following
formula

(5.1) $\begin{split}&\frac{\cos^{2}(\alpha_{0}^{4,k_{4}})}{\cos
U}\prod_{r=1}^{k_{4}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{4,k_{4}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{4}}\right)}\right|^{2}-\frac{\sin^{2}(\alpha_{0}^{3,k_{3}})}{\cos
U}\prod_{r=1}^{k_{3}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{3,k_{3}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{3}}\right)}\right|^{2}\\\
&\sim\frac{\sin U}{U},\ L\to\infty.\end{split}$

###  5.2. The second stage of the crossbreeding

The crossbreeding between the formula (4.15) and (4.19) gives the following
formula

(5.2)
$\begin{split}&\prod_{r=1}^{k_{6}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{6,k_{6}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{6}}\right)}\right|^{2}\sim\\\
&\sim\frac{\cos(\alpha_{0}^{7,k_{7}})}{\cos^{3}(\alpha_{0}^{6,k_{6}})}\prod_{r=1}^{k_{7}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{7,k_{7}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{7}}\right)}\right|^{2}-\\\
&-\frac{U^{2}}{3\cos^{3}U\cos^{3}(\alpha_{0}^{6,k_{6}})}\times\\\
&\left\\{\cos^{2}(\alpha_{0}^{4,k_{4}})\prod_{r=1}^{k_{4}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{4,k_{4}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{4}}\right)}\right|^{2}-\sin^{2}(\alpha_{0}^{3,k_{3}})\prod_{r=1}^{k_{3}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{3,k_{3}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{3}}\right)}\right|^{2}\right\\}^{3},\\\
&L\to\infty.\end{split}$

###  5.3. The third stage of the crossbreeding

The crossbreeding between the formula (4.11) and (5.1) gives the following
formula

(5.3)
$\begin{split}&\frac{1}{\cos^{2}U}\sim\frac{1}{\cos^{2}(\alpha_{0}^{5,k_{5}})}\prod_{r=1}^{k_{5}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{5,k_{5}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{5}}\right)}\right|^{2}\times\\\
&\left\\{\cos^{2}(\alpha_{0}^{4,k_{4}})\prod_{r=1}^{k_{4}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{4,k_{4}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{4}}\right)}\right|^{2}-\sin^{2}(\alpha_{0}^{3,k_{3}})\prod_{r=1}^{k_{3}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{3,k_{3}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{3}}\right)}\right|^{2}\right\\}^{-1},\\\
&L\to\infty.\end{split}$

In the next stage of the crossbreeding we use the following variant of the
formula (5.3).

(5.4)
$\begin{split}&\frac{1}{\cos^{3}U}\sim\frac{1}{\cos^{3}(\alpha_{0}^{5,k_{5}})}\left\\{\prod_{r=1}^{k_{5}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{5,k_{5}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{5}}\right)}\right|^{2}\right\\}^{3/2}\times\\\
&\left\\{\cos^{2}(\alpha_{0}^{4,k_{4}})\prod_{r=1}^{k_{4}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{4,k_{4}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{4}}\right)}\right|^{2}-\sin^{2}(\alpha_{0}^{3,k_{3}})\prod_{r=1}^{k_{3}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{3,k_{3}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{3}}\right)}\right|^{2}\right\\}^{-3/2},\\\
&L\to\infty.\end{split}$

###  5.4. The fourth stage of the crossbreeding

The crossbreeding between the formula (5.2) and (5.4) gives the following
formula

(5.5)
$\begin{split}&\prod_{r=1}^{k_{6}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{6,k_{6}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{6}}\right)}\right|^{2}\sim\\\
&\sim\frac{\cos(\alpha_{0}^{7,k_{7}})}{\cos^{3}(\alpha_{0}^{6,k_{6}})}\prod_{r=1}^{k_{7}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{7,k_{7}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{7}}\right)}\right|^{2}-\\\
&-\frac{U^{2}}{3}\frac{1}{\cos^{3}(\alpha_{0}^{5,k_{5}})\cos^{3}(\alpha_{0}^{6,k_{6}})}\left\\{\prod_{r=1}^{k_{5}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{5,k_{5}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{5}}\right)}\right|^{2}\right\\}^{3/2}\times\\\
&\left\\{\cos^{2}(\alpha_{0}^{4,k_{4}})\prod_{r=1}^{k_{4}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{4,k_{4}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{4}}\right)}\right|^{2}-\sin^{2}(\alpha_{0}^{3,k_{3}})\prod_{r=1}^{k_{3}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{3,k_{3}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{3}}\right)}\right|^{2}\right\\}^{3/2},\\\
&L\to\infty.\end{split}$

###  5.5. The fifth stage of the crossbreeding

In this stage we use the formula (2.11) in the case

$L\rightarrow 2L;\ \pi L\rightarrow\pi 2L,$

i.e. we use the following formula

(5.6)
$\begin{split}&U^{2}\sim(1+\Delta)^{2/\Delta}(\overset{*}{\bar{\alpha}}_{0}^{\Delta,\bar{k}_{\Delta}}-2\pi
L)^{2}\left\\{\prod_{r=1}^{\bar{k}_{\Delta}}\left|\frac{\zeta\left(\frac{1}{2}+i\overset{*}{\bar{\alpha}}_{r}^{\Delta,\bar{k}_{\Delta}}\right)}{\zeta\left(\frac{1}{2}+i\overset{*}{\bar{\beta}}_{r}^{\bar{k}_{\Delta}}\right)}\right|^{2},\right\\}^{2/\Delta},\\\
&\Delta>0,\ L\to\infty,\end{split}$

where (comp. (2.12))

(5.7)
$\begin{split}&\overset{*}{\bar{\alpha}}_{0}^{\Delta,\bar{k}_{\Delta}}=\alpha_{r}(U,2L,\bar{k}_{\Delta};\overset{*}{\bar{f}}_{\Delta}),\
r=0,1,\dots,\bar{k}_{\Delta},\\\
&\overset{*}{\bar{\beta}}_{r}^{\bar{k}_{\Delta}}=\beta_{r}(U,2L,\bar{k}_{\Delta}),\
r=1,\dots,\bar{k}_{\Delta},\\\ &\overset{*}{\bar{f}}_{\Delta}=(t-2\pi
L)^{\Delta}\in\tilde{C}_{0}[2\pi L,2\pi L+U].\end{split}$

Finally, we obtain by the crossbreeding between the formulas (5.5) and (5.6)
the following

######  Complete Hybrid Formula 2.

(5.8)
$\begin{split}&\prod_{r=1}^{k_{6}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{6,k_{6}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{6}}\right)}\right|^{2}\sim\frac{\cos(\alpha_{0}^{7,k_{7}})}{\cos^{3}(\alpha_{0}^{6,k_{6}})}\prod_{r=1}^{k_{7}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{7,k_{7}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{7}}\right)}\right|^{2}-\\\
&-\frac{(1+\Delta)^{2/\Delta}}{3}\frac{(\overset{*}{\bar{\alpha}}_{0}^{\Delta,\bar{k}_{\Delta}}-2\pi
L)^{2}}{\cos^{3}(\alpha_{0}^{5,k_{5}})\cos^{3}(\alpha_{0}^{6,k_{6}})}\left\\{\prod_{r=1}^{\bar{k}_{\Delta}}\left|\frac{\zeta\left(\frac{1}{2}+i\overset{*}{\bar{\alpha}}_{r}^{\Delta,\bar{k}_{\Delta}}\right)}{\zeta\left(\frac{1}{2}+i\overset{*}{\bar{\beta}}_{r}^{\bar{k}_{\Delta}}\right)}\right|^{2}\right\\}^{2/\Delta}\times\\\
&\times\left\\{\prod_{r=1}^{k_{5}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{5,k_{5}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{5}}\right)}\right|^{2}\right\\}^{3/2}\times\\\
&\left\\{\cos^{2}(\alpha_{0}^{4,k_{4}})\prod_{r=1}^{k_{4}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{4,k_{4}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{4}}\right)}\right|^{2}-\sin^{2}(\alpha_{0}^{3,k_{3}})\prod_{r=1}^{k_{3}}\left|\frac{\zeta\left(\frac{1}{2}+i\alpha_{r}^{3,k_{3}}\right)}{\zeta\left(\frac{1}{2}+i\beta_{r}^{k_{3}}\right)}\right|^{2}\right\\}^{3/2},\\\
&1\leq k_{3},k_{4},k_{5},k_{6},k_{7},\bar{k}_{\Delta}\leq k_{0},\ \Delta>0,\
L\to\infty.\end{split}$

### 5.6.

Now, we obtain from (5.8) by Definition the following

######  Theorem 2.

The set of functions

(5.9)
$\begin{split}&\left\\{\sin^{2}t,\cos^{2}t,\frac{1}{\cos^{2}t},\cos^{3}t,\cos
t,(1-2\pi L)^{\Delta}\right\\},\\\ &t\in[2\pi L,2\pi L+U],\
U\in(0,\pi/2-\epsilon],\ \Delta>0,\ L\to\infty\end{split}$

is the family of $\zeta$-kindred elements in the class of real continuous
functions.

I would like to thank Michal Demetrian for their moral support of my study of
Jacob’s ladders.

## References

  * [1] J. Moser, ‘Jacob’s ladders and almost exact asymptotic representation of the Hardy-Littlewood integral‘, Math. Notes 88, (2010), 414-422, arXiv: 0901.3937.
  * [2] J. Moser, ‘Jacob’s ladders, structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations‘, Proc. Steklov Inst. 276 (2011), 208-221, arXiv: 1103.0359.
  * [3] J. Moser, ‘Jacob’s ladders, reverse iterations and new infinite set of $L_{2}$-orthogonal systems generated by the Riemann zeta-function, arXiv: 1402.2098.
  * [4] J. Moser, ‘Jacob’s ladders, $\zeta$-factorization and infinite set of metamorphoses of a multiform‘, arXiv: 1501.07705v2.
  * [5] J. Moser, ‘Jacob’s ladders, Riemann’s oscillators, quotient of the oscillating multiforms and set of metamorphoses of this system‘, arXiv: 1506.00442.
  * [6] J. Moser, ‘Jacob’s ladders, factorization and metamorphoses as an appendix to the Riemann functional equation for $\zeta(s)$ on the critical line‘, Proc. Steklov Inst. 296 (2017), pp. 92-102, arXiv: 1506.00442v1.
  * [7] J. Moser, ’Jacob’s ladders, $\mathcal{Z}_{\zeta,Q^{2}}$-transformation of real elementary functions and telegraphic signals generated by the power functions’, arXiv: 1602.04994.
  * [8] J. Moser, ’Jacob’s ladders, interactions between $\zeta$-oscillating systems and $\zeta$-analogue of an elementary trigonometric identity’, arXiv: 1609.09293v1.
  * [9] J. Mosser, ‘Jacob’s ladders and some unbounded decomposition of the $\zeta$-oscillating systems on products of other $\zeta$-oscillating systems as an $\zeta$-analogue of the prime decomposition‘, arXiv:1701.08979v1.
  * [10] J. Moser, ‘Jacob ladders, nonlinear interactions between $\zeta$-oscillating systems and corresponding constraints‘, arXiv: 1702.0417v1.
  * [11] J. Moser, ‘Jacob’s ladders, crossbreeding in the set of $\zeta$-factorization formulas and the selection of families of $\zeta$-kindred real continuous functions‘, arXiv: 1710.04428 .


# An infinite family of $w_{1+\infty}$ invariant theories on the celestial
sphere

Shamik Banerjee  National Institute of Science Education and Research (NISER),
Bhubaneswar 752050, Odisha, India
and
Homi Bhabha National Institute, Anushakti Nagar, Mumbai, India-400085
<EMAIL_ADDRESS>Harshal Kulkarni  Department of Physical
Sciences, IISER Kolkata, Mohanpur, West Bengal 741246, India
<EMAIL_ADDRESS>Partha Paul  Centre for High Energy Physics, Indian
Institute of Science, C.V. Raman Avenue, Bangalore 560012, India.
<EMAIL_ADDRESS>
###### Abstract

In this note we determine the graviton-graviton OPE and the null states in any
$w_{1+\infty}$ symmetric theory on the celestial sphere. Our analysis shows
that there exists a discrete infinite family of such theories. The MHV-sector
and the quantum self dual gravity are two members of this infinite family.
Although the Bulk Lagrangian description of this family of theories is not
currently known to us, the graviton scattering amplitudes in these theories
are heavily constrained due to the existence of null states. Presumably they
are exactly solvable in the same way as the minimal models of $2$-D CFT.

###### Contents

  1. I Introduction
  2. II General structure of $w$-invariant OPE
    1. II.1 What is theory $A$?
  3. III Null states in the MHV sector
    1. III.1 $N^{0,0}_{k}$
    2. III.2 $N^{0,1}_{k}$
  4. IV Organization of the OPE at every order
    1. IV.1 Action of $sl_{2}(R)_{V}$ on the null states
      1. IV.1.1 $N^{0,0}_{k}$
      2. IV.1.2 $N^{0,1}_{k}$
  5. V w invariance
    1. V.1 $w$ invariance at $\mathcal{O}(z^{0}\bar{z}^{0})$
    2. V.2 $w$ invariance at $\mathcal{O}(z^{0}\bar{z}^{1})$
  6. VI Finiteness and an infinite family of theories
  7. VII $w$ invariant OPE coefficients
  8. VIII Knizhnik-Zamolodchikov type null states
  9. IX Acknowledgements

##  I Introduction

Celestial holography Strominger:2017zoo  is an attempt to formulate a dual
theory of quantum gravity in asymptotically flat space time. When the bulk is
$(3+1)$ dimensional, the dual theory is conjectured to be two dimensional and
it lives on the celestial sphere. The correlation functions of this theory are
given by the Mellin transform of the bulk $S$-matrix elements
Pasterski:2016qvg ; Pasterski:2017kqt ; Banerjee:2018gce  and an important
question is how to compute them in the dual formulation.

Now in the absence of a Lagrangian description one can resort to the Bootstrap
technique and the job is sometimes simplified if the theory has a large number
of symmetries. For example, one can solve the minimal models in $2$-D CFT by
using the representation theory of Virasoro and other current algebras even
when the Lagrangian description is not known. In celestial holography the two
dimensional dual theory also has various (infinite dimensional) current
algebra symmetries Sachs:1962zza ; Strominger:2013jfa ; He ; Barnich:2009se ;
Barnich:2011ct ; Kapec:2016jld ; Kapec:2014opa ; He:2017fsb ; Banerjee:2020zlg
; Guevara:2021abz ; Stieberger:2018onx ; Banerjee:2022wht ; Banerjee:2021cly ;
Strominger:2021lvk ; Donnay:2020guq ; Himwich:2021dau ; Ball:2021tmb ;
Adamo:2021lrv ; Costello:2022wso ; Gupta:2021cwo ; Freidel:2021dfs ;
Freidel:2021ytz  and it is expected that the bootstrap approach should
simplify considerably in this case111Please see Atanasov:2021cje  for a
discussion of conformal bootstrap approach to celestial holography. Also
recently celestial holography has been used to shed light on $S$-matrix
bootstrap in Ghosh:2022net ..

In this paper we consider dual theories which are invariant under the
$w_{1+\infty}$ symmetry222To be more precise the wedge subalgebra of
$w_{1+\infty}$.Guevara:2021abz ; Strominger:2021lvk ; Himwich:2021dau ;
Ball:2021tmb ; Adamo:2021lrv ; Costello:2022wso . We determine the graviton-
graviton OPE and the null states in such theories. So far two examples of such
theories were known. One of them is the MHV-sector Banerjee:2020zlg ;
Banerjee:2021cly  and the other one is the quantum self dual gravity
Ball:2021tmb . However, we find that there is a discrete infinite family of
such theories and the MHV-sector and the quantum self dual gravity are two
members of this family. Bulk description of this family of theories is not
known to us and we leave this to future work.

##  II General structure of $w$-invariant OPE

The general structure of the $w$-invariant OPE between two positive helicity
outgoing graviton primaries with weights $\Delta_{1}$ and $\Delta_{2}$ can be
written as,

$\begin{gathered}G^{+}_{\Delta_{1}}(z_{1},\bar{z}_{1})G^{+}_{\Delta_{2}}(z_{2},\bar{z}_{2})=-\frac{\bar{z}_{12}}{z_{12}}\sum_{n=0}^{\infty}B\left(\Delta_{1}-1+n,\Delta_{2}-1\right)\frac{{\bar{z}_{12}}^{n}}{n!}\bar{\partial}^{n}G^{+}_{\Delta_{1}+\Delta_{2}}(z_{2},\bar{z}_{2})\\\
+\sum_{p,q=0}^{\infty}z_{12}^{p}\bar{z}_{12}^{q}\sum_{k=1}^{\tilde{n}_{p,q}}\tilde{C}^{k}_{p,q}(\Delta_{1},\Delta_{2})\tilde{O}^{p,q}_{k}(\Delta_{1},\Delta_{2},z_{2},\bar{z}_{2})\end{gathered}$
(1)

where the first line contains the universal terms Guevara:2021abz  having
simple pole in $z_{12}$. The second line contains the non-singular terms in
$z_{12}$ and $\bar{z}_{12}$ and they are linear combinations of the $w$
descendants of a positive helicity graviton primary, denoted by
$\tilde{O}^{p,q}_{i}$. The positive integer $\tilde{n}_{p,q}$ may be finite or
infinite. Our goal is to determine the operators $\tilde{O}^{p,q}_{i}$ and the
OPE coefficients $\tilde{C}^{i}_{p,q}(\Delta_{1},\Delta_{2})$ which can appear
in a $w$-invariant theory. Now before we proceed let us briefly describe the
symmetry algebra that follows from the universal singular terms of the OPE
(1).

So we start by defining the tower of conformally soft Donnay:2018neh
gravitons Guevara:2021abz

$H^{k}(z,\bar{z})=\lim_{\Delta\rightarrow
k}(\Delta-k)G^{+}_{\Delta}(z,\bar{z}),\ k=1,0,-1,-2,...$ (2)

with weights $\left(\frac{k+2}{2},\frac{k-2}{2}\right)$. It follows from the
structure of the singular terms in (1) that we can introduce the following
truncated mode expansion

$H^{k}(z,\bar{z})=\sum_{m=\frac{k-2}{2}}^{\frac{2-k}{2}}\frac{H^{k}_{m}(z)}{\bar{z}^{m+\frac{k-2}{2}}}$
(3)

and the modes $H^{k}_{m}(z)$ are the conserved holomorphic currents. The
currents $H^{k}_{m}(z)$ can be further mode expanded

$H^{k}_{m}(z)=\sum_{\alpha\in\mathbb{Z}-\frac{k+2}{2}}\frac{H^{k}_{\alpha,m}}{z^{\alpha+\frac{k+2}{2}}}$
(4)

and one can show Guevara:2021abz  that the modes $H^{k}_{\alpha,m}$ satisfy
the algebra333Here we are assuming that $\kappa=\sqrt{32\pi G_{N}}=2$.

$\begin{gathered}\left[H^{k}_{\alpha,m},H^{l}_{\beta,n}\right]\\\
=-\left[n(2-k)-m(2-l)\right]\frac{\left(\frac{2-k}{2}-m+\frac{2-l}{2}-n-1\right)!}{\left(\frac{2-k}{2}-m\right)!\left(\frac{2-l}{2}-n\right)!}\frac{\left(\frac{2-k}{2}+m+\frac{2-l}{2}+n-1\right)!}{\left(\frac{2-k}{2}+m\right)!\left(\frac{2-l}{2}+n\right)!}H^{k+l}_{\alpha+\beta,m+n}\end{gathered}$
(5)

This is called the Holographic Symmetry Algebra (HSA). Now if we make the
following redefinition (or discrete light transformation)Strominger:2021lvk

$w^{p}_{\alpha,m}=\frac{1}{2}\left(p-m-1\right)!\left(p+m-1\right)!H^{-2p+4}_{\alpha,m}$
(6)

then (5) turns into the $w_{1+\infty}$ algebra444This is the wedge subalgebra
of $w_{1+\infty}$.

$\left[w^{p}_{\alpha,m},w^{q}_{\beta,n}\right]=\left[m(q-1)-n(p-1)\right]w^{p+q-2}_{\alpha+\beta,m+n}$
(7)

For our purpose it is more convenient to work with the HSA (5) rather than the
$w_{1+\infty}$ algebra. However, we continue to refer to the HSA as the $w$
algebra.

Now suppose we have two different $w$ invariant theories which we call $A$ and
$B$. $A$ and $B$ have identical symmetry algebras (5) because there are no
central charges and moreover, their operator contents are also the same.
Therefore if the OPE is completely determined by $w$ covariance then there
must exist a universal form of the OPE which holds in both theory $A$ and
theory $B$. But how is this possible if the graviton scattering amplitudes in
theory $A$ and theory $B$ are different ? This is still possible if the
following holds:

Let us denote the universal OPE by $\text{OPE}_{w}$ and the OPEs obtained from
the graviton scattering amplitudes of theory A and theory B by
$\text{OPE}_{A}$ and $\text{OPE}_{B}$, respectively. Note that
$\text{OPE}_{w}$ holds in both theory $A$ and theory $B$. Therefore it must be
true that

$\begin{gathered}\text{OPE}_{w}=\text{OPE}_{A}+\text{Linear combination of
Null-States of theory A}\\\ \text{OPE}_{w}=\text{OPE}_{B}+\text{Linear
combination of Null-States of theory B}\end{gathered}$ (8)

This is a major simplification if we know the OPE and null states of either
theory $A$ or theory $B$.

###  II.1 What is theory $A$?

Our task will be simplified if we can find a theory $A$ such that the null
states of theory $B$ can be written as a linear combination of the null states
of theory $A$. This does not mean that the theory $A$ and theory $B$ have the
same null states but, this implies that the null states of theory $A$ form a
basis in terms of which the null states of theory $B$ can be expanded. Once we
find such a theory $A$ we can write, as a consequence of (8),

$\boxed{\text{OPE}_{B}=\text{OPE}_{A}+\text{Linear combination of Null-States
of theory A}}$ (9)

Now in our case we take the theory $A$ to be the MHV-sector. This is the
simplest theory in the sense that although this is $w$ invariant, the
graviton-graviton OPE can be written as Banerjee:2020zlg  a linear combination
of supertranslation and $\overline{sl_{2}}$ current algebra555Supertranslation
and $\overline{sl_{2}}$ current algebras are subalgebras of the $w_{1+\infty}$
algebra. They are generated by the soft gravitons $H^{1}(z,\bar{z})$ and
$H^{0}(z,\bar{z})$. descendants only. Therefore MHV sector has the maximum
number of null states, i.e, maximally constrained among all the $w$ invariant
theories. Now the OPE and the null states in the MHV-sector are known
Banerjee:2020zlg ; Banerjee:2021cly  and we will use them to construct the
general $w$ invariant OPE.

##  III Null states in the MHV sector

According to our discussion in the previous section, we can write

$\begin{gathered}G^{+}_{\Delta_{1}}(z_{1},\bar{z}_{1})G^{+}_{\Delta_{2}}(z_{2},\bar{z}_{2})|_{B}=G^{+}_{\Delta_{1}}(z_{1},\bar{z}_{1})G^{+}_{\Delta_{2}}(z_{2},\bar{z}_{2})|_{\text{MHV}}\\\
+\sum_{p,q=0}^{\infty}z_{12}^{p}\bar{z}_{12}^{q}\sum_{k=1}^{n_{p,q}}C^{k}_{p,q}(\Delta_{1},\Delta_{2})N^{p,q}_{k}(z_{2},\bar{z}_{2})\end{gathered}$
(10)

where $N^{p,q}_{k}$ are the null states of the MHV sector which can appear at
$\mathcal{O}(z^{p}\bar{z}^{q})$ and the integers $n_{p,q}$ can be finite or
infnite. Since the MHV-sector OPE contains the universal singular part of
$\text{OPE}_{B}$, the second line of (10) contains only the non-singular
terms.

We now discuss the MHV null states at
$\mathcal{O}(z_{12}^{0}\bar{z}_{12}^{0})$ and at
$\mathcal{O}(z_{12}^{0}\bar{z}_{12}^{1})$. This can be done at any order
however, for the sake of simplicity, we focus on these two orders only.
Moreover, the $\mathcal{O}(z_{12}^{0}\bar{z}_{12}^{1})$ terms in the
$\text{OPE}_{B}$ give rise to interesting constraints on the graviton
scattering amplitudes Banerjee:2020zlg  in theory $B$. So it will be
interesting to determine them.

###  III.1 $N^{0,0}_{k}$

The MHV null states that can appear at
$\mathcal{O}({z_{12}^{0}\bar{z}_{12}^{0}})$ are given by Banerjee:2020zlg ;
Banerjee:2021cly 666The null states are obtained by starting from the
graviton-graviton OPE in the MHV sector and then making one of the gravitons
conformally soft. This is discussed in detail in Banerjee:2020zlg ;
Banerjee:2021cly .,

$\Phi_{k}(\Delta)=\left[H^{1-k}_{\frac{k-3}{2},\frac{k+1}{2}}\left(-H^{1}_{-\frac{1}{2},-\frac{1}{2}}\right)^{k}-\frac{(-1)^{k}}{k!}\frac{\Gamma(\Delta+k-2)}{\Gamma(\Delta-2)}H^{1}_{-\frac{3}{2},\frac{1}{2}}\right]G^{+}_{\Delta-1}$
(11)

where $k=1,2,3,\cdots,\infty$. However, it is more convenient to work with the
new basis defined by

$\begin{gathered}\Omega_{k}(\Delta)=\sum_{n=1}^{k}\frac{1}{(k-n)!}\frac{\Gamma(\Delta+k-2)}{\Gamma(\Delta+n-2)}\Phi_{n}(\Delta)\end{gathered}$
(12)

The physical significance of this basis will become clear in the next section.

###  III.2 $N^{0,1}_{k}$

The MHV null states that can appear at
$\mathcal{O}({z_{12}^{0}\bar{z}_{12}^{1}})$ are given by Banerjee:2020zlg ;
Banerjee:2021cly ,

$\begin{gathered}\Psi_{k}(\Delta)=\bigg{[}H^{-k}_{\frac{k-2}{2},\frac{k}{2}}\left(-H^{1}_{-\frac{1}{2},-\frac{1}{2}}\right)^{k+1}-\frac{(-1)^{k}}{k!}\frac{\Gamma(\Delta+k-2)}{\Gamma(\Delta-2)}H^{0}_{-1,0}\left(-H^{1}_{-\frac{1}{2},-\frac{1}{2}}\right)\\\
-\frac{(-1)^{k}k}{(k+1)!}\frac{\Gamma(\Delta+k-2)}{\Gamma(\Delta-3)}H^{1}_{-\frac{3}{2},-\frac{1}{2}}\bigg{]}G^{+}_{\Delta-2}\end{gathered}$
(13)

where $k=1,2,3\cdots,\infty$. Again we introduce the new basis

$\Pi_{k}(\Delta)=\sum_{n=1}^{k}\frac{1}{(k-n)!}\frac{\Gamma(\Delta+k-2)}{\Gamma(\Delta+n-2)}\Psi_{n}(\Delta)$
(14)

which is more convenient for our purpose.

There is a second set of null states at this order which involve the
descendant $L_{-1}G^{+}_{\Delta}$. These are the null states of the Knizhnik-
Zamolodchikov(KZ) type and their decoupling gives rise to differential
equations for the scattering amplitudes Banerjee:2020zlg ; Banerjee:2020vnt
which are the generalization of the standard KZ equation to Celestial
holography. However, we do not need the KZ-type null states to write down the
$\text{OPE}_{B}$ because the generator $L_{-1}$777$\bar{L}_{-1}$ is a
generator of the $w$ algebra. does not belong to the $w$ algebra. We will give
the explicit form of these null states at a later part of the paper.

##  IV Organization of the OPE at every order

We have argued that the OPE in the theory $B$ can be written as,

$\begin{gathered}G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0,0)|_{B}=G^{+}_{\Delta_{1}}(z_{1},\bar{z}_{1})G^{+}_{\Delta_{2}}(0,0)|_{\text{MHV}}\\\
+\sum_{p,q=0}^{\infty}z^{p}\bar{z}^{q}\sum_{k=1}^{n_{p,q}}C^{k}_{p,q}(\Delta_{1},\Delta_{2})N^{p,q}_{k}(0,0)\end{gathered}$
(15)

where $N^{p,q}_{k}$ are the MHV null states. Now, an infinite number of null
states can potentially appear at every order of the $\text{OPE}_{B}$. So we
need some guiding principle which tells us which null states can appear at a
given order and hopefully, we have only a finite number of them.

The guiding principle is nothing but the $w$-covariance of the OPE. In fact,
if we want to classify the states which can appear at a given order of the
$\text{OPE}_{B}$, then we can work with the $sl_{2}(R)$ subalgebra of the $w$
algebra generated by the operators $\\{H^{1}_{-1/2,-1/2},\ H^{0}_{0,0},\
H^{-1}_{1/2,1/2}\\}$,

$\begin{gathered}\left[H^{0}_{0,0},H^{1}_{-1/2,-1/2}\right]=H^{1}_{-1/2,-1/2}\\\
\left[H^{0}_{0,0},H^{-1}_{1/2,1/2}\right]=-H^{-1}_{1/2,1/2}\\\
\left[H^{1}_{-1/2,-1/2},H^{-1}_{1/2,1/2}\right]=-H^{0}_{0,0}\end{gathered}$
(16)

We denote this by $sl_{2}(R)_{V}$888Here V stands for vertical. Please see
Fig.1 for an explanation. .

Now using the relations

$\begin{gathered}H^{0}_{0,0}G^{+}_{\Delta}(z,\bar{z})=2\bar{h}G^{+}_{\Delta}(z,\bar{z})=\left(\Delta-2\right)G^{+}_{\Delta}(z,\bar{z})\\\
H^{1}_{-1/2,-1/2}G^{+}_{\Delta}(z,\bar{z})=-G^{+}_{\Delta+1}(z,\bar{z})\\\
H^{-1}_{1/2,1/2}G^{+}_{\Delta}(z,\bar{z})=-\frac{1}{2}\hskip
0.85355pt\left[(\Delta-2)(\Delta-3)+4(\Delta-2)\bar{z}\bar{\partial}+3\bar{z}^{2}\bar{\partial}^{2}\right]G^{+}_{\Delta-1}(z,\bar{z})\end{gathered}$
(17)

and the $w$-covariance of the OPE, i.e, the fact that both sides of the OPE
transform in the same way under $w$ transformations, one can easily check
that:

The MHV null states that can appear at $\mathcal{O}(z^{p}\bar{z}^{q})$ of the
OPE, transform in a representation of the $sl_{2}(R)_{V}$ subalgebra.
Moreover, this is the unique $sl_{2}$ subalgebra with this property.

We now verify this explicitly.

###  IV.1 Action of $sl_{2}(R)_{V}$ on the null states

####  IV.1.1 $N^{0,0}_{k}$

Let us consider the action of the $sl_{2}(R)_{V}$ on the MHV null states
$\Omega_{k}(\Delta)$ that can appear at $\mathcal{O}(z^{0}\bar{z}^{0})$. It is
given by,

$H^{1}_{-\frac{1}{2},-\frac{1}{2}}\Omega_{k}(\Delta)=-\Omega_{k}(\Delta+1)$
(18)
$\begin{gathered}H^{-1}_{\frac{1}{2},\frac{1}{2}}\Omega_{k}(\Delta)=\frac{1}{2}(k+1)(k+2)\Omega_{k}(\Delta-1)-\frac{1}{2}(\Delta-4)(\Delta-5)\Omega_{k}(\Delta-1)\\\
-\frac{1}{2}(k+1)(k+2)\Omega_{k+1}(\Delta-1)\end{gathered}$ (19)

and $H^{0}_{0,0}=2\bar{L}_{0}$ is obviously diagonal on these states. So we
can see that the null states $\Omega_{k}(\Delta)$ indeed form a representation
of $sl_{2}(R)_{V}$. However, this representation is reducible. For let us
consider the subspace spanned by the (infinite) set of states
$\\{\Omega_{n+1}(\Delta),\Omega_{n+2}(\Delta),\Omega_{n+3}(\Delta),\cdots\\}$
where $n\geq 0$. It is easy to see from (18) and (19) that this set is closed
under the action of the $sl_{2}(R)_{V}$. Therefore the representation carried
by the states $\\{\Omega_{k}(\Delta)\\}_{k=1,2,...}$ is reducible. As a
consequence we can set the states

$\Omega_{k+1}(\Delta)=0,\ k\geq n\geq 0$ (20)

without violating the $sl_{2}(R)_{V}$ symmetry. After doing this we are left
with $n$ null states $\\{\Omega_{1}(\Delta),...,\Omega_{n}(\Delta)\\}$ which,
as a consequence of (19) and (20), again form a representation of
$sl_{2}(R)_{V}$. The crucial point is that the value of the integer $n$ is not
fixed by the $sl_{2}(R)_{V}$ symmetry.

Now for a given $n$, there is another set of
$\lceil{\frac{n}{2}}\rceil$999$\lceil{\frac{n}{2}}\rceil=\text{Smallest
integer}$ $\geq\frac{n}{2}$. nontrivial101010There are of course the $n$
states $\\{\Omega_{1}(\Delta),...,\Omega_{n}(\Delta)\\}$ which transform in a
representation of $sl_{2}(R)_{V}$ but, we cannot set them to zero because that
will lead us again to the MHV sector. states
$\\{\chi^{1}_{n}(\Delta),...,\chi_{n}^{\lceil{n/2}\rceil}(\Delta)\\}$ defined
as

$\begin{gathered}\chi^{1}_{n}(\Delta)=\sum_{p=1}^{n}\Omega_{p}(\Delta)\\\
\chi^{i}_{n}(\Delta)=\sum_{p=i}^{n}\prod_{q=i}^{2i-2}(p-q)\Omega_{p}(\Delta),\
i=2,3,...,\lceil{\frac{n}{2}}\rceil\end{gathered}$ (21)

which transform in a representation of the $sl_{2}(R)_{V}$ as a consequence of
(20). We can also set these states to zero

$\chi^{i}_{n}(\Delta)=0$ (22)

without violating the $sl_{2}(R)_{V}$ symmetry.

####  IV.1.2 $N^{0,1}_{k}$

The action of the $sl_{2}(R)_{V}$ on the MHV null states $\Pi_{k}(\Delta)$
which can appear at $\mathcal{O}(z^{0}\bar{z}^{1})$ is given by,

$H^{1}_{-\frac{1}{2},-\frac{1}{2}}\Pi_{k}(\Delta)=-\Pi_{k+1}(\Delta+1)$ (23)

and

$\begin{gathered}H^{-1}_{\frac{1}{2},\frac{1}{2}}\Pi_{k}(\Delta)=\frac{1}{2}k(k+1)\Pi_{k}(\Delta-1)-\frac{1}{2}(\Delta-4)(\Delta-5)\Pi_{k}(\Delta-1)\\\
-\frac{1}{2}(k-1)(k+2)\Pi_{k+1}(\Delta-1)\end{gathered}$ (24)

Therefore we can see the states $\Pi_{k}(\Delta)$ form a representation of the
$sl_{2}(R)_{V}$. This representation is reducible because the subspace spanned
by the states
$\\{\Pi_{n+1}(\Delta),\Pi_{n+2}(\Delta),\Pi_{n+3}(\Delta),\cdots\\}$ form a
representation of the $sl_{2}(R)_{V}$. We can get a smaller representation
spanned by the states $\\{\Pi_{1}(\Delta),\cdots,\Pi_{n}(\Delta)\\}$ if we set

$\Pi_{k+1}(\Delta)=0,\ k\geq n\geq 0$ (25)

There is no other nontrivial representation of $sl_{2}(R)_{V}$ in the subspace
spanned by the states $\\{\Pi_{1}(\Delta),...,\Pi_{n}(\Delta)\\}$.

##  V w invariance

So far we have discussed the invariance of the OPE under $sl_{2}(R)_{V}$.
There is another $sl_{2}(R)$ subalgebra which is generated by the global
(Lorentz) conformal transformations
$\\{H^{0}_{0,1},H^{0}_{0,0},H^{0}_{0,-1}\\}$. We call this
$\overline{sl_{2}(R)}$ because this acts only on the $\bar{z}$ coordinate. Now
the $w$ symmetry is generated by the infinite number of soft currents
$\\{H^{k}_{p}(z)\\}$ where $k=1,0,-1,-2,...$ is the dimension $(\Delta)$ of
the soft operator and $\frac{k-2}{2}\leq p\leq-\frac{k-2}{2}$. For a fixed
$k$, the soft currents
$\\{H^{k}_{-\frac{k-2}{2}}(z),...,H^{k}_{\frac{k-2}{2}}(z)\\}$ transform in a
spin $\frac{2-k}{2}$ representation of the $\overline{sl_{2}(R)}$.

Figure 1: The figure shows the soft currents. The rows and the columns are
indexed by the $\overline{sl_{2}(R)}$ weights and the dimension
$(\Delta=k=1,0,-1,-2,...)$ of the conformally soft graviton
$H^{k}(z,\bar{z})$, which generates the currents sitting in a row,
respectively. $\overline{sl_{2}(R)}$ acts horizontally along a row and
$sl_{2}(R)_{V}$ acts vertically along a column. In this way they generate the
whole symmetry algebra starting from the current $H^{1}_{\frac{1}{2}}(z)$ on
the top left corner.

Now let us consider the currents
$\\{H^{1}_{\frac{1}{2}},H^{0}_{1},...,H^{k}_{\frac{2-k}{2}},...\\}$ with the
lowest $\overline{sl_{2}(R)}$ weights. These currents transform in an
irreducible highest weight representation of the $sl_{2}(R)_{V}$. This can be
seen from the following commutation relations following from (5),

$\begin{gathered}\left[H^{-1}_{\frac{1}{2},\frac{1}{2}},H^{k}_{\alpha,\frac{2-k}{2}}\right]=-\frac{1}{2}(k-2)(k-3)H^{k-1}_{\alpha+\frac{1}{2},\frac{2-(k-1)}{2}}\\\
\left[H^{0}_{0,0},H^{k}_{\alpha,\frac{2-k}{2}}\right]=(k-2)H^{k}_{\alpha,\frac{2-k}{2}}\\\
\left[H^{1}_{-\frac{1}{2},-\frac{1}{2}},H^{k}_{\frac{2-k}{2}}\right]=-H^{k+1}_{\alpha-\frac{1}{2},\frac{2-(k+1)}{2}}\end{gathered}$
(26)

Therefore, starting from the current $H^{1}_{\frac{1}{2}}(z)$ we can generate
any other $w$ current by the combined action of the $\overline{sl_{2}(R)}$ and
$sl_{2}(R)_{V}$ (Fig.1).

Now, the way it is reflected in the structure of the OPE is the following.
Suppose we consider the OPE of two positive helicity gravitons. The OPE is $w$
invariant if the singular terms have the following universal structure

$G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0,0)\supset-\frac{\bar{z}}{z}\sum_{n=0}^{\infty}B\left(\Delta_{1}-1+n,\Delta_{2}-1\right)\frac{{\bar{z}}^{n}}{n!}\bar{\partial}^{n}G^{+}_{\Delta_{1}+\Delta_{2}}(0,0)$
(27)

Now it has been shown in Pate:2019lpp  that the leading term in $\bar{z}$ is
uniquely determined by the $sl_{2}(R)_{V}$ invariance111111It is assumed that
the leading term is of $\mathcal{O}(\frac{\bar{z}}{z})$. This follows from the
structure of the universal leading term in the collinear limit of two positive
helicity gravitons.. Once we know the leading term, the subleading terms in
$\bar{z}$ of $\mathcal{O}(\frac{\bar{z}^{q}}{z}),q\geq 2$ are determined by
the $\overline{sl_{2}(R)}$ invariance121212Here the assumption is that the
$\overline{\text{Vir}}$ is not part of the symmetry algebra.This is further
discussed in Banerjee:2022wht .. Therefore if we make sure that the
$sl_{2}(R)_{V}$ and the $\overline{sl_{2}(R)}$ symmetries are not broken then
we are bound to have $w$ invariance. We now discuss the consequences of this.

###  V.1 $w$ invariance at $\mathcal{O}(z^{0}\bar{z}^{0})$

In section-(IV) we have shown that the equations

$\Omega_{k+1}(\Delta)=0,\ k\geq n\geq 0$ (28)

are $sl_{2}(R)_{V}$ invariant. Now one can easily check that

$H^{0}_{0,1}\Omega_{k}(\Delta)=0$ (29)

where $H^{0}_{0,1}$ is, upto normalization, $\bar{L}_{1}$. So the MHV null
states $\Omega_{k}(\Delta)$ are $\overline{sl_{2}(R)}$ primaries and as a
result the equations (28) are $\overline{sl_{2}(R)}$ invariant. This implies
that (28) is also $w$ invariant.

For the same reason, the additional null state relations (22)

$\chi^{i}_{n}(\Delta)=0,\ i=1,2,...,\lceil{\frac{n}{2}}\rceil$ (30)

at $\mathcal{O}(z^{0}\bar{z}^{0})$ are also $w$ invariant.

###  V.2 $w$ invariance at $\mathcal{O}(z^{0}\bar{z}^{1})$

Similarly, we have shown in section-(IV) that the equations

$\Pi_{k+1}(\Delta)=0,\ k\geq n^{\prime}\geq 0$ (31)

are $sl_{2}(R)_{V}$ invariant. Now the action of $H^{0}_{0,1}$ on
$\Pi_{k}(\Delta)$ is given by,

$\begin{gathered}H^{0}_{0,1}\Pi_{k+1}(\Delta)=-(\Delta+k-2)\Omega_{k+1}(\Delta-1)+(k+3)\Omega_{k+2}(\Delta-1)\end{gathered}$
(32)

So if we want to impose (31) without breaking the $\overline{sl_{2}(R)}$
invariance then we also have to impose

$\Omega_{k+1}(\Delta)=0,\ k\geq n^{\prime}\geq 0$ (33)

Therefore, (31) is $w$ invariant if (33) holds and we know from our previous
analysis that (33) is indeed $w$ invariant. This also shows that
$n_{0,0}=n_{0,1}$ in (10) as a consequence of $w$ invariance.

##  VI Finiteness and an infinite family of theories

Our analysis in the last section shows that $w$ symmetry remains unbroken if
we set

$\Omega_{k+1}(\Delta)=0,\ k\geq n\geq 0$ (34)

and

$\Pi_{k+1}(\Delta)=0,\ k\geq n\geq 0$ (35)

simultaneously. Therefore we can construct a $w$ invariant OPE at
$\mathcal{O}(z^{0}\bar{z}^{0})$ and $\mathcal{O}(z^{0}\bar{z}^{1})$ by keeping
only the finite131313There are still an infinite number of states
$\\{\Omega_{p}(\Delta+\mathbb{Z})\\}_{1\leq p\leq n}$ required by the global
time translation and the global subsubleading
($H^{-1}_{\frac{1}{2},\frac{1}{2}}$) invariance. number of MHV null states
$\\{\Omega_{1}(\Delta),...,\Omega_{n}(\Delta)\\}$ and
$\\{\Pi_{1}(\Delta),\cdots,\Pi_{n}(\Delta)\\}$, respectively. The crucial
point here is that the value of the integer n, is not fixed by $w$ invariance.
So we get different $w$ invariant OPEs for different choices of the integer
$n$. For example, $n=0$ gives the MHV-sector. Similarly, direct calculation of
graviton-graviton OPE using the scattering amplitudes shows that $n=4$ gives
the OPE of the quantum self-dual gravity theory which is known to be $w$
invariant Ball:2021tmb . The existence of this discrete infinite family of $w$
invariant OPEs is the main outcome of our analysis.

##  VII $w$ invariant OPE coefficients

In this section we write down the $w$ invariant OPE for a given value of $n$,
at $\mathcal{O}(z^{0}\bar{z}^{0})$ and $\mathcal{O}(z^{0}\bar{z}^{1})$, using
the states $\\{\Omega_{1}(\Delta),...,\Omega_{n}(\Delta)\\}$ and
$\\{\Pi_{1}(\Delta),\cdots,\Pi_{n}(\Delta)\\}$. At
$\mathcal{O}(z^{0}\bar{z}^{0})$ we have

$\begin{gathered}G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0,0)\big{|}_{\mathcal{O}(z^{0}\bar{z}^{0})}=G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0,0)\big{|}_{\text{MHV
at}\ \mathcal{O}(z^{0}\bar{z}^{0})}\\\
+\sum_{p=1}^{n}B(\Delta_{1}-1+p,\Delta_{2}-1)\
\Omega_{p}(\Delta_{1}+\Delta_{2})\end{gathered}$ (36)

and at $\mathcal{O}(z^{0}\bar{z}^{1})$

$\begin{gathered}G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0,0)\big{|}_{\mathcal{O}(z^{0}\bar{z}^{1})}=G^{+}_{\Delta_{1}}(z,\bar{z})G^{+}_{\Delta_{2}}(0,0)\big{|}_{\text{MHV
at}\ \mathcal{O}(z^{0}\bar{z}^{1})}\\\
+\bar{z}\sum_{p=1}^{n}B(\Delta_{1}+p,\Delta_{2}-1)\
\Pi_{p}(\Delta_{1}+\Delta_{2}+1)\end{gathered}$ (37)

The MHV sector OPE at $\mathcal{O}(z^{0}\bar{z}^{0})$ and
$\mathcal{O}(z^{0}\bar{z}^{1})$ are given in Banerjee:2020zlg .

As we have already discussed the states
$\\{\Omega_{1}(\Delta),...,\Omega_{n}(\Delta)\\}$ are not all independent
because of the existence of the additional null state relations (22)

$\chi^{i}_{n}(\Delta)=0,\ i=1,2,...,\lceil{\frac{n}{2}}\rceil$ (38)

We can further simplify (36) by using these null state relations. However, we
choose to leave it in this form because the OPE coefficients are particularly
nice when written in terms of these linearly dependent states. Now different
values of $n$ give OPEs in different $w$ invariant theories. We now discuss
the KZ type null states of these theories.

##  VIII Knizhnik-Zamolodchikov type null states

KZ type null states occur at $\mathcal{O}(z^{0}\bar{z}^{1})$ of the OPE. The
simplest way to derive it is to demand the commutativity of the OPE, i.e,

$G^{+}_{\Delta_{1}}(z_{1},\bar{z}_{1})G^{+}_{\Delta_{2}}(z_{2},\bar{z}_{2})=G^{+}_{\Delta_{2}}(z_{2},\bar{z}_{2})G^{+}_{\Delta_{1}}(z_{1},\bar{z}_{1})$
(39)

If we use the consistency condition (39) and the OPE given in (36) and (37) we
get the following KZ type null state

$\boxed{\Xi_{n}(\Delta)=\xi(\Delta)+\sum_{k=1}^{n}\Pi_{k}(\Delta+1)=0}$ (40)

where $\xi(\Delta)$ is the KZ type null state in the MHV sector given by
Banerjee:2020zlg

$\xi(\Delta)=\boxed{L_{-1}}G^{+}_{\Delta}+H^{0}_{0,-1}H^{1}_{-\frac{3}{2},\frac{1}{2}}G^{+}_{\Delta-1}+H^{0}_{-1,0}G^{+}_{\Delta}+(\Delta-1)\,H^{1}_{-\frac{3}{2},-\frac{1}{2}}G^{+}_{\Delta-1}$
(41)

We have used that $\chi^{1}_{n}(\Delta)$ is a null state in this theory to
arrive at the form (40).

As consistency checks we can compute the actions of $\overline{sl_{2}(R)}$ and
$sl_{2}(R)_{V}$ generators on $\Xi_{n}(\Delta)$. For example,

$H^{0}_{0,1}\Xi_{n}(\Delta)=(n+2)\Omega_{n+1}(\Delta)-(\Delta-3)\chi^{1}_{n}(\Delta)=0$
(42)

because $\Omega_{n+1}(\Delta)$ and $\chi^{1}_{n}(\Delta)$ are both null states
in this theory. Therefore $\Xi_{n}(\Delta)$ is a $\overline{sl_{2}(R)}$
primary.

Similarly, we have

$\begin{gathered}H^{-1}_{\frac{1}{2},\frac{1}{2}}\Xi_{n}(\Delta)=-\frac{1}{2}(\Delta-2)(\Delta-3)\Xi_{n}(\Delta-1)\\\
-\frac{1}{2}(n+2)(n-1)\Pi_{n+1}(\Delta)-H^{-1}_{\frac{1}{2},-\frac{1}{2}}\chi^{1}_{n}(\Delta)-H^{0}_{0,-1}\left((\Delta-1)\chi^{1}_{n}(\Delta-1)+\chi^{2}_{n}(\Delta-1)\right)\\\
\end{gathered}$ (43)

But, since $\Pi_{n+1}(\Delta),\chi^{1}_{n}(\Delta)$ and $\chi^{2}_{n}(\Delta)$
are null states in the theory, we get

$H^{-1}_{\frac{1}{2},\frac{1}{2}}\Xi_{n}(\Delta)=-\frac{1}{2}(\Delta-2)(\Delta-3)\Xi_{n}(\Delta-1)$
(44)

Therefore $\Xi_{n}(\Delta)$ transforms under a representation of the
$sl_{2}(R)_{V}$ and we can consistently set it to zero without violating the
$sl_{2}(R)_{V}$ symmetry. Therefore, (40) is indeed $w$ invariant.

Decoupling of null states gives rise to differential equations which the
graviton scattering amplitudes in this theory have to satisfy. So in principle
we know a lot about these theories however, solving these equations may not be
easy in practice. We hope to return to this in near future.

##  IX Acknowledgements

We would like to thank Andrew Strominger and Sabrina Pasterski for useful
discussion. The work of SB is partially supported by the Swarnajayanti
Fellowship (File No- SB/SJF/2021-22/14) of the Department of Science and
Technology and SERB, India and by SERB grant MTR/2019/000937 (Soft-Theorems,
S-matrix and Flat-Space Holography). The work of HK is partially supported by
the KVPY fellowship of the Department of Science and Technology (DST),
Government of India and the Summer Students’ Visiting Program at the Institute
of Physics, Bhubaneswar, India. The work of PP is supported by an IOE endowed
postdoctoral position at IISc, Bengaluru, India.

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