# Blockchain Interoperability Landscape

Inwon Kang  Department of Computer Science
Rensselaer Polytechnic Institute
Troy, NY, USA
<EMAIL_ADDRESS>Aparna Gupta  Lally School of Management
Rensselaer Polytechnic Institute
Troy, NY, USA
<EMAIL_ADDRESS>Oshani Seneviratne  Department of Computer Science
Rensselaer Polytechnic Institute
Troy, NY, USA
<EMAIL_ADDRESS>
###### Abstract

Blockchain has become a popular emergent technology in many industries. It is
suitable for a broad range of applications, from its base role as an immutable
distributed ledger to the deployment of distributed applications. Many
organizations are adopting the technology, but choosing a specific blockchain
implementation in an emerging field exposes them to significant technology
risk. Selecting the wrong implementation could expose an organization to
security vulnerabilities, reduce access to its target audience, or cause
issues in the future when switching to a more mature protocol. Blockchain
interoperability aims to solve this adaptability problem by increasing the
extensibility of blockchain, enabling the addition of new use cases and
features without sacrificing the performance of the original blockchain.
However, most existing blockchain platforms need to be designed for
interoperability, and simple operations like sending assets across platforms
create problems. Cryptographic protocols that are secure in isolation may
become insecure when several different (individually secure) protocols are
composed. Similarly, utilizing trusted custodians may undercut most of the
benefits of decentralization offered by blockchain-based systems. Even though
there is some research and development in the field of blockchain
interoperability, a characterization of the interoperability solutions for
various infrastructure options is lacking. This paper presents a methodology
for characterizing blockchain interoperability solutions that will help focus
on new developments and evaluate existing and future solutions in this space.

###### Index Terms:

blockchain interoperability, survey, solution characterization

††publicationid: pubid: 978-1-6654-8045-1/22/$31.00 ©2022 IEEE

##  1 Introduction

Since the introduction of Bitcoin [15], blockchain technology has continued to
evolve and be applied to fields beyond digital asset management. The
introduction of Ethereum and its programmability [26] has supported impressive
research and development in the blockchain space toward new mechanisms and
projects to extend its use cases. However, many blockchains to date suffer
from a lack of interoperability, which limits their scope of use cases. As
projects mature, their blockchain interoperability emerges as an increasingly
important feature. Without standardized protocols, projects seeking blockchain
interoperability resort to their unique architecture and mechanisms for a
specific purpose, and designing a universally interoperable framework requires
understanding the goals of past projects and how they sought to achieve
interoperability. This paper characterizes the blockchain interoperability
landscape by examining several existing projects and comparing their
similarities and differences.

###  1.1 What is Blockchain Interoperability?

We define blockchain interoperability as the ability of blockchains to work
together rather than being limited to their isolated environments. In this
regard, interoperability for blockchain-based systems can mean different
things, such as supporting asset exchange, asset transfer, or data exchange.
Bitcoin, the first cryptocurrency to gain widespread attention, was not
designed for interoperability or programmability. However, the emergence of
Ethereum, followed by many other blockchains, has introduced many new
constructs to the blockchain world. For instance, Ethereum introduced
programmable smart contracts and the ability to send arbitrary data.

The ability of blockchain to handle more than just transactions has extended
the possibilities of this technology. Having different blockchains, each with
its niche, bring several benefits. However, it also raises questions about
their ability to work together. Therefore, blockchain interoperability is a
challenging goal to achieve in the current landscape. Blockchain technologies
are inherently “trustless,” meaning parties without previously established
relationships can interact reliably. However, no system works in isolation,
and these blockchain primitives must be extended to the data coming from
beyond a single blockchain ecosystem. This challenge is known as the “oracle
problem” [5]. In other words, a blockchain must devise a verification approach
for the data received from outside its ecosystem.

Another question faced is how new systems may join an interoperating system.
Ethereum’s layer 2 networks [21] can be considered a set of interoperable
chains. However, chains outside the Ethereum network cannot natively
communicate with these layer 2 chains, thus limiting the scope of
interoperability. The solution must account for the possibility of unforeseen
networks joining the system to offer global interoperability.

The National Institute of Standards and Technology (NIST) defines _blockchain
interoperability_ as: “a composition of distinguishable blockchain systems,
each representing a unique distributed data ledger, where atomic transaction
execution may span multiple heterogeneous blockchain systems, and where data
recorded in one blockchain are reachable, verifiable, and referable by another
foreign transaction in a semantically compatible manner” [27]. In our
definition of blockchain interoperability, we consider the various features
essential for a blockchain to interoperate with another blockchain, or another
legacy solution, especially when taken across multiple application domains.
Therefore, we define blockchain interoperability as the ability for a given
blockchain to join multiple blockchain networks and other software systems in
a scalable manner that extends the NIST definition. Essentially, a blockchain
interoperability solution must enable a blockchain to conduct transactions
utilizing information housed in another information system in a verifiable
manner. The above definition encompasses common interoperability use cases,
such as a token exchange and many other decentralized applications. It also
contains uses such as reading stock market data from a non-decentralized
system in a verifiable way. We consider this definition because it is the
broadest definition that retains the properties of a host blockchain and
enables its extensibility.

###  1.2 Blockchain Interoperability Use Cases

As blockchain technology evolves, there is a constant stream of new projects
with new promises. However, the current siloed nature of blockchain systems
may interfere with the projects’ ability to scale. Blockchain interoperability
can help isolated ecosystems trust and work with each other to maximize their
utility. The types of information available in a blockchain network can be
classified into two broad categories: assets (cryptocurrencies, tokens) and
data. This section examines how these two domains can benefit from blockchain
interoperability.

####  1.2.1 Asset Movement

The ability to mine cryptocurrencies and trade digital assets that are not
regulated by a central authority constitutes the main reason for the
popularity of blockchains. However, with an increasing number of projects in
the ecosystem, the ability to transfer these siloed assets between different
networks will become ever more critical. Pillai et al. [16] define three types
of use cases for assets in a cross-chain environment. These are (1) asset
exchange, (2) asset transfer, and (3) asset migration. Asset exchange is a
two-way operation in which entities on different blockchains can exchange an
agreed-upon amount of their respective assets. Asset transfer is an operation
in which one entity moves its asset from one blockchain to another. Asset
migration is when an entity permanently moves its asset between two
blockchains. Although the last two use cases sound similar, they are, in fact,
different in the protocols used for each case. Asset movement across
blockchains can offer advantages that a single blockchain cannot–for example,
when an application developer wishes to develop a decentralized application
(DApp) without making its users bear the burden of the high network fee of
Ethereum. An interoperability solution can allow the developer and the
potential users to seamlessly convert their assets onto a different network
with lower gas fees. The developer can then attract a more extensive user base
while the users benefit from a more affordable service.

####  1.2.2 Web3 Applications

Web3 provides an application substrate for a new generation of Web
applications, bringing decentralization and tokenomics to the current Web.
Instead of traditional databases or servers, blockchain(s) can act as the
database for the Web3 applications and program a set of constructs for the
applications to interface with it. A significant obstacle to achieving a fully
connected Web that lets users read, write and own assets using blockchains is
that most blockchains are siloed ecosystems. A well-thought-out
interoperability solution must be in place for an application to seamlessly
use multiple blockchains. For example, one could build a DApp for selling
customized shirts, and this application could connect different vendors to the
end user to deliver the final product. If every vendor, shipper, shirt
manufacturer, shirt printer, etc., has a different blockchain system running
for its operations, such an application would need to be able to communicate
with every party through their respective blockchain. If there is an
interoperability solution between blockchains, the application will utilize it
to deliver the final product to the user. This example is simple, but it shows
a glimpse into the possible future of an Internet economy powered by
decentralized solutions. While blockchains possess several appealing
properties, their gamut of possibilities is limited without interoperability
since all stakeholders cannot be expected to use the same blockchain platform
in a truly decentralized environment.

####  1.2.3 Secure and Accountable Information Exchange

Handling sensitive information in an accountable manner is another critical
task that can be driven by blockchain technology, which is made possible by
two aspects of blockchain systems. First is unique identifiability across the
entire network: in the case of blockchain, the interconnected network. Each
device has a unique ‘name’ in the network, allowing applications such as
search engines or routing protocols to optimize their operations. For example,
this can be the assignment of a unique identifier to each wallet in the
specific network. Identifiability inside each network is a part of the
blockchain mechanism. However, sharing this identity information outside each
blockchain’s silo is an unsolved problem. If a unique identity across the
entire blockchain space is established, it can lead to more efficient data
routing and more security for cross-chain operations.

Second is the ability to verify the off-chain identities of individuals in a
permissioned manner. Know Your Customer (KYC), and Anti Money Laundering (AML)
are essential properties in traditional finance to prevent crime and
misconduct, primarily when the trade partners are not known or are untrusted.
These properties are more difficult to enforce in the blockchain space due to
the pseudo-anonymous nature of the underlying technology. Some centralized
exchanges, such as Coinbase, require KYC for customers to create an account.
However, in the space of Decentralized Exchanges (DEX), KYC is not a
requirement. Managing such sensitive information in a decentralized
environment is a tricky task. Chainlink [3] is an example of a blockchain
solution that addresses this problem by providing verified external data
through its network. Some enterprises are also evaluating permissioned
blockchains to manage such sensitive data. However, for a blockchain to use
the data provided by such solutions, it must have some degree of
interoperability, which can be addressed by establishing communication between
oracle providers and the consumer blockchain. Furthermore, permissioned
sharing of information can be enabled if the chains can verify the identity –
thus trustworthiness – of the entities across different blockchains. The
problem of identity is not limited to financial services. Other industries,
such as healthcare or logistics and supply chains, can also benefit from
sharing this type of sensitive data, potentially leading to innovations that
were, heretofore, unavailable.

###  1.3 Our Contribution

This work provides an overview of the current landscape of blockchain
interoperability by categorizing nine existing projects. While past research
has focused on a theoretical categorization of possible interoperability
approaches and pointing out some real-life examples, we approach the problem
with a bottom-up approach by examining the existing solutions and analyzing
and categorizing them. We use and expand upon the definitions provided in
previous literature and apply them to the existing interoperability solutions
to understand their underlying enabling mechanisms better.

##  2 Previous Work

Siris et al. [20] presented a literature survey of interoperability
approaches. The authors analyze techniques such as cross-chain transactions,
bridges, sidechains, and protocols like the W3C Interledger Protocol (ILP) [9]
and classify them by their supported assets, scalability, trust, and cost.
Kannengiesser et al. [12] characterized approaches toward blockchain
interoperability and their strengths and weaknesses. The authors collected
literature related to cross-chain transactions, extracted common design
patterns used in the literature, and assessed the strengths and weaknesses of
their administration, flexibility, performance, and security properties.

Belchior et al. [2] conducted a systematic review of blockchain
interoperability by collecting and evaluating the related literature. The
authors reviewed the existing mechanisms by classifying them by use case,
payload, and degree of decentralization of each framework. Furthermore, they
outlined the case for blockchain interoperability research and the need for
adaptability driving the growth of both new blockchains and blockchain
interoperability. They also explored various definitions and constructed a
concept map for blockchain interoperability. Overall, their work highlighted
the fragmented nature of blockchain interoperability research and the need for
further classification of solutions. The authors highlighted several risks for
interoperability solutions, namely, what if (1) security vulnerabilities are
uncovered?, (2) the chosen blockchain does not support new use cases?, (3) the
rest of the industry adopted an improved version of the technology?, or (4) an
organization’s users exist on another solution? Mitigating such risks requires
adaptability by either improving the existing blockchain to make it adequate
for transacting with other blockchains or integrating it with an existing
blockchain, thus making the blockchain extensible.

Lohachab et al. [14] explore the generations of blockchain technology and
introduce the need for blockchain interoperability. They describe potential
interoperability architectures and taxonomy to compare the state-of-the-art
interoperability research, concluding with the challenges for future research
directions to achieve blockchain interoperability. Pillai et al. [16]
characterize the different layers of a blockchain interoperability solution,
providing a guideline for building blockchain interoperability solutions
depending on the use case, such as data relay or a crypto exchange, along with
the different mechanisms that can be used to achieve these goals. Twelve
properties that should be considered when evaluating a blockchain
interoperability solution are described by Koens et al. [13] in analyzing two
existing solutions, Polkadot [25] and Cosmos [10]. Based on these metrics, the
authors outline the similarities and differences in terms of the 12
properties. Finally, Singh et al. [19] focus on sidechain mechanisms to
achieve interoperability. They outline the different sidechain designs, their
use cases, and their strengths.

##  3 Layers of Blockchain Interoperability

The blockchain ecosystem is made complete by the different layers. The
fundamental building blocks of a blockchain are classified as the consensus,
networking, data, and hardware layers. With the introduction of Decentralized
Applications (DApps), applications can now consume information from the
consensus layer of blockchains. In addition to these layers, modern blockchain
developments have introduced another abstraction layer. This layer handles the
operations between the application and consensus layers, allowing the data to
flow between different blockchains for a particular application in a
blockchain-agnostic manner. Interoperability between blockchains can happen on
separate layers. This work examines applications that either facilitate data
flow between blockchains through the messaging layer or applications that
provide the tools for building new blockchains.

###  3.1 Hardware Layer

Also known as the infrastructure layer, this layer refers to the hardware that
runs the node client software for the blockchain. This node’s security depends
on the blockchain’s permission level. In a public network, anyone may maintain
a node. However, in a permissioned network, only trusted entities are allowed
to run a node.

###  3.2 Transaction Layer

This layer refers to the transactions and blocks in a blockchain. A block is
formed by grouping together different transactions, and it must be verified by
a fraction of nodes greater than a certain threshold to be accepted into the
ledger. In this layer, the nodes listen for transaction information and batch
them together into a block. The transaction layer is where differences in most
blockchain projects start to appear. Different protocols specify the ordering
mechanism of transactions and the size of each block, preventing them from
being interoperable. The encryption and hashing of data also happen in this
layer. Each blockchain can use a different algorithm to encrypt its payload,
resulting in varying degrees of difficulty and security guarantees. This
inconsistency adds to the interoperability challenge.

###  3.3 Networking Layer

The networking layer consists of node clients that use peer-to-peer networking
to finalize the input to the ledger. It involves nodes submitting their work
to each other and checking it, where the data protocol may vary for each
network, thus requiring different fields or routing protocols.

###  3.4 Consensus Layer

The consensus layer is where the nodes check each other’s work and agree on a
result to add to the ledger. Blockchains can have different consensus
mechanisms, such as Bitcoin’s Proof of Work (PoW), Ethereum’s Proof of Stake
(PoS), or Solana’s Proof of History (PoH). Even if some blockchains share the
mechanism, the protocol can differ by their validation threshold. The
differences in this layer can also impact the security threshold of individual
blockchains, making it difficult to achieve safe interoperability.

###  3.5 Messaging Layer

In a traditional blockchain system, the messaging layer would be a part of the
application layer. However, as the need for blockchain interoperability grows,
some projects, such as Overledger [24] or Hyperledger Cactus [4], have added
this layer. Such an abstraction layer, which is often located between the
consensus and application layer, can be an entry point between multiple
blockchains working with a single multi-DApp. Therefore, a solution must
choose a standard protocol for sharing consensus results between numerous
blockchains. Overledger, for example, solves this problem by writing the
message payload to the ‘comment’ field of different blockchains.

###  3.6 Application Layer

This layer interfaces with the end users of DApps, and delivers the
information passed from the blockchain to its DApps. It can also involve
application-supporting features, such as smart contracts.

Figure 1: Blockchain System Layers

##  4 Variations in Blockchain Implementations

Computational efficiency and interoperability are key issues addressed by
these new technologies. As the blockchain space becomes increasingly complex,
it is necessary to separate layers within blockchains based on their roles. In
our survey of interoperability solutions, there are three types of mechanisms,
which are sidechains, layer-2 blockchains, and layer-0 blockchains. A
simplified diagrams of these categories are shown in Figures 2(a), 2(b) and
2(c).

(a) Layer 2

(b) Layer 0

(c) Sidechain

###  4.1 Sidechains

Sidechains, one of the earliest approaches towards interoperability, were
first introduced by Back et al. [1]. This mechanism uses the existing
blockchain network (the parent chain) and allows assets to move between
various networks. A sidechain is a separate blockchain and can be connected to
the parent chain to allow asset movement. A two-way peg mechanism [17] makes
the asset movement possible, which enables the chain-specific asset to be
wrapped into the supported asset on the destination chain. This mechanism is
characterized by the fact that they are connected to their parent chains and
run parallel to them.

###  4.2 Layer-2 Blockchains

Layer-2 blockchain is another form of blockchain that runs on top of a parent
(or layer-1) blockchain. It shares many similarities with the sidechain
mechanism in that it runs parallel to its parent chain and is connected to it.
The difference is that while sidechains may maintain their state and security
protocol separately, layer-2 blockchains rely on the state of the layer-1
blockchain. The main goal of layer-2 blockchains is to extend the
functionality of layer-1, as well as enable faster transaction speed by
relying on layer-1’s infrastructures. Layer-2 can be implemented using
mechanisms other than a two-way peg [17], such as rollups [22], or bridges
[12]. Sometimes the terms sidechain and layer-2 are used interchangeably in
the literature. Our interoperability landscape characterization uses the above
definitions (dependence on the layer-1 chain) to avoid confusion between these
two mechanisms.

###  4.3 Layer-0 Blockchains

Some interoperability projects, such as Polkadot and Cosmos, use this term to
describe their implementations. This idea is an extension of the relationship
between layer-1 and layer-2 blockchains. Similar to how the layer-1 network
supports the layer-2 network by providing consensus and security, layer-0
chains provide functionalities that the layer-1 chain can use. Thus, layer-0
blockchains are meta-chains providing infrastructure and interoperability
between the depending layer-1 chains.

Connection Method | Payload | Network Permission | Projects
---|---|---|---
On Chain | layer-0 | Crypto, Data | All | Polkadot, Cosmos,
layer-2 | Crypto | All | Loom
Crypto | Public | Hybrix
Off-Chain | Gateway | Crypto | Public | Poly Network
Data | All | Overledger, Cactus, Firefly
Oracle | Data | All | Chainlink

TABLE I: Blockchain Interoperability Project Categorization.
The connection method contains the on-chain and off-chain characteristics and
further breaks that down to how it is achieved (layer-0, layer-2, gateway, or
oracle). The payload column refers to the type of data the framework handles,
which includes crypto-assets (crypto) and arbitrary data (data). The network
permission column describes the permission level the framework can handle,
such as public or private or all (the framework supports both types of
permission).

##  5 Characterization of Existing Blockchain Interoperability Approaches

Blockchain interoperability is a relatively new field, and advances are
happening rapidly. There is a variety of solutions with different approaches
for achieving interoperability. We analyze a range of blockchain
interoperability projects and present an analysis of nine projects with
exciting features introduced in Section 5.1.

We then compare the differences in these approaches, namely, what they
promise, how their goals are achieved, how they can be extended in the future,
and their strengths and weaknesses. Table II provides an overall summary of
our analysis.

###  5.1 Summary of Solutions

Blockchain interoperability presents many new paradigms, ranging from simply
transferring assets to being able to share sensitive data in an accountable
manner over the Internet. Although there are some overlaps in the goals of
each framework discussed, each framework incorporates a unique goal and a
method of achieving it.

Overledger [24]: Overledger’s official purpose is to provide a ‘messaging
layer’ for blockchain-based applications, which is agnostic to each
blockchain’s consensus mechanism. Instead, it allows the different chains to
send messages to each other through a standardized format. Quant Network, the
company behind Overledger, describes the project as an “operating system of
blockchains”, as it can serve as the entry point for existing payment systems
and blockchains to be abstracted for the application layer.

Hyperledger Cactus [4]: Hyperledger Cactus is an enterprise-grade application
that allows for seamless cross-chain transactions. On top of achieving
interoperability for assets, it also looks into the interoperability of data
between blockchains, such as sharing inventory data between permissioned
enterprise networks.

Hyperledger Firefly [8]: Hyperledger Firefly is an API abstraction layer for
applications based on the blockchain, similar to Overledger. It aims to allow
for permissioned data passing among different chains. Compared to Hyperledger
Cactus, this project focuses more on the application development side, as it
seeks to provide a software development kit for building DApps based on
multiple chains.

Polkadot [25]: Polkadot is a layer-0 solution for blockchain interoperability.
It enables the communication between different blockchains through what is
called “parachains” and the “relay chains” [18]. It looks to address
scalability in computing and networking while working with systems with
diverse needs and guarantees in the blockchain space.

Cosmos [10]: Cosmos shares a similar goal as Polkadot, providing a layer-0
solution for interoperability. It aims to address the drawbacks of many
current blockchain networks, such as energy inefficiency and low performance,
and bring all networks together through a parallelized structure called
“zones.”

Loom [7]: The Loom Network is an Ethereum sidechain powered by the native LOOM
token, a native ERC-20 token. It strives to achieve interoperability by
connecting different blockchains to the ‘basechain’ using a gateway mechanism.
Loom tokens can be burned and minted on the source and destination chain using
these gateways, allowing liquidity flow across various blockchains.

Hybrix [11]: Hybrix is a peer-to-peer (P2P) protocol to achieve
interoperability between blockchains. A node on a Hybrix system runs software
called hybrixd, which includes connection mechanisms for supported
blockchains, a data abstraction mechanism, and a secure signing mechanism to
enable a connection between interoperating blockchains.

Chainlink [3]: Chainlink is a network of Decentralized Oracle Networks (DONs).
It aims to provide reliable data – both from on-chain and off-chain – for
other blockchains to consume. With the announcement of Chainlink v2 [3], the
project has introduced a new term called hybrid smart contracts, which is an
abstraction of on-chain computation using off-chain data from DONs.

Poly Network [23]: Finally, Poly Network is another project that follows the
hub and spoke architecture (Figure 3(a)) to provide interoperability. Poly
Network itself is not a blockchain, but it provides the protocol for nodes of
blockchains participating in Poly Network to create a consortium blockchain
that has shared storage.

###  5.2 Integration Mode

(a) Hub and Spoke

(b) Decentralized Relay

(c) Decentralized Oracle

Pillai et al. [16] introduced the term “cross-blockchain” to define the method
through which the blockchains are connected. This property is essential
because it affects many other properties of an interoperability solution. If a
solution introduces a new blockchain to solve the problem, it may be able to
guarantee more robust security or governance. However, it presents a new issue
of scaling the said blockchain to the ever-evolving blockchain space. If a
solution uses an off-chain mechanism to connect, it may have a lower security
guarantee, but it can also benefit from a reduced scaling overhead. In
general, we see three different methods of approaching this problem, namely
Hub and Spoke, Decentralized Relays, and Decentralized Oracles.

Hub and Spoke: The hub and spoke approach is a commonly used architecture for
constructing large networks. Figure 3(a) shows a simple architecture diagram.
This approach connects individual ‘nodes’ – blockchains in this case – to a
central hub through which they can interoperate. Some projects implementing
this architecture are Polkadot, Cosmos, Loom, and Poly Network.

In Polkadot, the parachains are analogous to the spokes and the relay chain to
the hub. For Cosmos, the Zones is equivalent to the spokes and the Hubs to the
hub. Poly Network also uses this approach to enable the interoperability of
cryptocurrencies of connected networks. The relayer, as it is known in Poly
Network, is in charge of converting the currency and syncing with the main
network. Loom implements this design by using an Ethereum sidechain and
allowing its tokens to be interoperable in other supported sidechains.

Decentralized Relay: The decentralized relay architecture is a design in which
networks can communicate through ‘gateways.’ A blockchain gateway is an entry
point to the underlying ledger with read/write access. In this design, each
gateway is assigned a particular blockchain to connect to the main network or
handle connections between the main network and all other blockchains. Figure
3(b) shows a simple diagram describing this architecture. For projects such as
Overledger, Hyperledger Cactus, and Firefly, the software must run on a
trusted machine – not necessarily an active network node – and use connectors
for each blockchain. Hybrix also uses this architecture to relay messages from
one blockchain to another through a P2P gateway mechanism. At a glance, this
architecture can appear similar to the Hub and Spoke architecture, as it
achieves a similar network scheme of connecting different chains through a
designated entry point. To avoid confusion, we differentiate these by their
usage of additional blockchains. Then, the Hub and Spoke approach is an
architecture in which both the hub and spoke are different blockchains that
sync with the target chains. On the other hand, a decentralized relay is a
piece of software that connects blockchains without introducing additional
blockchains, meaning that it does not necessarily have to deal with the state
of the networks as long as it forwards data between blockchains.

Decentralized Oracles: An oracle in the blockchain space refers to an entity
that allows a blockchain to access out-of-chain information. This information
can be both on-chain from another blockchain, such as the amount of liquidity
present in the network, or off-chain, such as foreign exchange rates. If a
blockchain depends on information from a centralized oracle service, the
validity of the entire blockchain ledger can be compromised by compromising
the oracle’s data source. Decentralized oracles try to tackle this problem by
adding redundancy at the oracle level by having multiple nodes serve the same
data only if it can be agreed upon by the validators and verified. Previous
works in analyzing blockchain interoperability have classified the Oracle
approach as a mechanism used to achieve interoperability, similar to that of
other mechanisms such as Hashed Time Lock Contracts [6] or Burn-and-Mint [2].
However, unlike those mechanisms, oracles are not necessarily crypto-centric.
It can convey information related to transactions, but it can also communicate
any other kind of data. In this work, we view decentralized oracles as an
interoperability solution because of the wide range of data they can transfer
between blockchains and the traditional world. Chainlink [3] implements this
approach by connecting various Decentralized Oracle Networks (DONs) to create
a meta-chain of oracles that can provide data feeds to its target blockchains.

###  5.3 Validation

Blockchains can have different security assumptions that need to be validated.
Existing solutions take different approaches towards this problem:

Native Validation: Native validation is defined as validation happening inside
the interoperability solution. This approach applies to solutions implemented
in their chains, which can have their state validated. Several projects, such
as Polkadot, Cosmos, Loom, Chainlink, and Poly Network, use this approach.

Polkadot: Polkadot’s relay chain state is maintained by its validator nodes.
To prevent malicious behavior from validators, Polkadot has a unique role
called fishermen who are in charge of checking the blocks signed by the
validators. If it turns out that the fishermen did indeed detect misbehavior,
they are rewarded by the network. The reward amount scales with the number of
validators who signed off on the illegal block, as it implies higher risk in
the network [25]. In each parachain, collators are the nodes responsible for
working with the validators. As the name suggests, the collators combine the
transactions by building a block for the validators.

Cosmos: In Cosmos, each zone updates its parent hub with the state of its
blockchain. The zones also update their status based on the hub, allowing for
bi-directional information flow [10]. When a zone needs to send or receive
information from another zone, it posts a Merkle tree proof of the sender on
the hub, which the destination zone receives. Similarly, the destination zone
also publishes evidence that the information was received. This mechanism is
similar to how sidechains communicate by establishing a bi-directional channel
for proof-of-existence transactions.

Loom: The Loom base chain uses Delegated Proof of Stake (DPoS), an improvement
over the commonly used Proof of Stake (PoS) mechanism, to validate the data
coming from outside. Unlike PoS, in which the validator is chosen with
randomness proportional to each user’s staked amount, DPoS allows users to
vote for new validators, each vote worth the amount they have staked.

Chainlink: As an oracle framework, Chainlink must verify that it serves the
correct data, whether off-chain or on-chain. Before the data is relayed to the
requesting blockchain, validation is done by selecting random nodes of each
DON to compare the data against its own.

Poly Network: Validation happens in Poly Network by recording the block header
of the relayed data. Using the Merkle root inside the block’s header, the Poly
Chain (the main chain) re-verifies the block’s validity. Once this block is
verified and recorded in the relay chain, the data can be sent to other
chains, completing the cross-chain operation.

Third-party Validation: Third-party validation refers to a validation
mechanism in which the task of validating is delegated to a third party who is
not a part of the interoperability solution. A federation of validating nodes
is commonly used to provide a robust security guarantee. While convenient
because different validators can be switched in and out, this approach can
also introduce a security bottleneck if it needs to be decentralized enough
compared to the connected blockchains. Projects that use this approach are
Hyperledger Cactus and Firefly, both of which make use of ledger plugins
(called blockchain plugins in Firefly). The choice of validator service is
left up to the user, meaning that the security of these projects’ instances
depends on the validator service.

Hybrid Validation: Hybrid validation is when verification happens through a
different mechanism than extending the connected chains’ state. This mechanism
is applicable when the data being passed in the interoperability protocol are
not assets but pieces of arbitrary data. A typical implementation of this
mechanism uses the message field in transaction data. By setting a protocol
for reading and writing to this field, the interoperability solution can
ensure that the data it receives is original. Two projects that use this
approach are Overledger and Hybrix. Overledger uses a proprietary algorithm
known as TrustTag to generate and verify hashes for messages. When a
transaction needs to be sent off-chain, the user’s identity and transaction
details are encrypted using the application’s public key and added to the
metadata field. The digest is verified using the application’s private key
when this message arrives at the application layer. This mechanism allows the
app to confirm that the message has not been tampered with and provides the
ability to assign unique IDs for each user. This technology is closed source
and is patented by Quant, the company behind Overledger. Like Overledger’s
message verification mechanism, Hybrix also uses the message header field to
validate the originality of the incoming data. To achieve this, empty
transactions that carry data are composed and pushed to each blockchain. The
first block contains a genesis recipe, which establishes the interoperability
protocol. The cross-chain transactions sent after this genesis recipe is
checked using its hash value.

Poly Network allows each chains to validate each other by checking the
transaction’s block header in the source chain. This is made possible by the
Poly Chain’s synchronized header mechanism which synchronizes with every
participating chain’s block height and records the information.

###  5.4 Upgradability

Upgradability measures how the framework can stay flexible to keep up with the
evolving technologies in the blockchain space. A good step of upgradeability
is that the framework should continuously upgrade itself to keep up with the
evolving technologies and provide a reliable solution. One good example of a
measure for this property is the possibility of updates causing a hard fork.
For instance, if the community cannot agree on the proposal leading to a hard
fork, it can cause a divide. Such an event can cause the interoperability
solution to have fewer interoperating blockchains and cause further
fragmentation in the blockchain space. This property is also closely related
to the security of the framework, as there can be unexpected bugs in the
software. Upgradability in interoperability solutions can be broadly
categorized into two groups: Software-based and Blockchain-based.

Software-based: This category applies to solutions that provide the software
to construct or connect to the network, such as Overledger, Cactus, Firefly,
and Hybrix. These solutions implement a microservice-based architecture,
meaning that the connector service for each network can be upgraded separately
without impacting the other functionalities.

Blockchain-based: This category applies to solutions that leverage a new
blockchain to achieve interoperability, including projects such as Polkadot,
Cosmos, Loom, Chainlink, and Poly Network. Upgradability is an essential
feature in this category of solutions because they can be susceptible to a
hard fork if the community does not unanimously agree with a substantial
update. Polkadot addresses this problem with its protocol. Its governance is a
meta-protocol, meaning that the governance can define upgrades without
requiring a hard fork. Chainlink addresses this problem by providing a service
called contract-update to account for unforeseen changes. This service allows
users to modify the existing smart contract, publish a new one, and update the
DON connections to point to the revised contract. Cosmos, Loom, and Poly
Network do not specify plans for unforeseen changes to the network. Thus, with
the exceptions of Polkadot and Chainlink, blockchain-based solutions may face
a hard fork when there are essential requirements to upgrade.

###  5.5 Adoptability

Adoptability is how new networks can be adopted into the existing framework,
allowing it to scale further in the blockchain space. The distinction between
upgradability and adoptability can be characterized as follows. Upgradability
refers to the innate ability of the framework, while adopotibility refers to
how external networks can join the framework. There are many variables to
consider when considering a framework’s adoptability, ranging from the data
protocol to the network structure. An interoperability framework with higher
adoptability is more likely to gain a larger community, which is crucial in
the current blockchain environment.

Hub and Spoke: Projects that implemented the hub and spoke architecture can
have more complicated requirements for new networks to join the
interoperability network. These requirements arise from the hub’s blockchain
need to stay secure. Depending on the implementation of the main chain’s
validation and consensus, each project has a slightly different requirement
for new blockchains.

Polkadot: Anyone can build a parachain for Polkadot [25]. For a parachain to
become a part of the Polkadot ecosystem, it must have validators. To maintain
the integrity of the main network when adding new networks, Polkadot requires
that the existing stakeholders of the network be incentivized by some means –
whether that be paying a fee or providing a new feature. For connecting a new
chain that does not conform to Polkadot’s governance, such as Bitcoin,
Polkadot provides a “bridge chain.” Additionally, Polkadot runs a test network
known as “Kusama” to provide a sandbox for developers to test their
blockchains.

Cosmos: Anyone can build a zone for Cosmos. Cosmos separates the governance
between the hub and zones, allowing its zones’ failures to be isolated from
the entire network. Compared to Polkadot, Cosmos has a lower entry condition
as the zone creator can choose their validators for the zone instead of being
designated validators by the main network.

Loom: Loom network’s main chain exists as a layer-2 Ethereum blockchain and is
supported by the Loom ERC-20 token. Loom has achieved interoperability by
allowing different ‘versions’ of this token to be transferred between various
blockchains using smart chain mechanisms such as burning and minting. Although
Loom makes no specific requirements for a blockchain to be supported, it can
be assumed that the new blockchain must support some form of ERC-20 compatible
tokens and some form of smart contracts.

Poly Network: The relayer function needs to be implemented for a new
blockchain to join the Poly Network environment. The new network must also
support light client verification mechanisms such as Simple Payment
Verification (SPV) or Light Ethereum Subprotocol (LES) to implement the
validation step in the relay chain.

Decentralized Relays: Decentralized relays such as Overledger, Hyperledger
Cactus, Hyperledger Firefly, and Hybrix use the comment field to facilitate
interoperability. Thus, a critical requirement for a new network to join these
systems is that it must support a comment field in the transaction data. For
Hybrix, the new blockchain must also be public for validation purposes.

Decentralized Oracles: Chainlink does not have any requirements because its
role is simply to serve the data. However, some security requirements exist
for new DONs to join its meta-network [3].

###  5.6 Governance

The governance of a decentralized system is crucial in ensuring a bright
future for the system by making the best decisions. When putting together
different networks, this can become an interesting problem. Should the sub-
networks (lower-level blockchains) have a say in the governance of the overall
interoperability framework? Or should this framework have a separate community
that makes the decisions? We show two significant ways of dealing with this
problem. Overledger, Cactus, Firefly, and Hybrix take the off-chain governance
approach, in which connected blockchains’ governance does not affect the
framework. On the other hand, Polkadot, Cosmos, Loom, Chainlink, and Poly
Network implement a governance protocol made up of the validators in the
system.

On-chain: On-chain governance is applicable for solutions that maintain the
main blockchain network that other networks connect through.

Polkadot: Polkadot governance is composed of a ‘council’ and a ‘technical
committee.’ A council is a group of elected stakeholders, of which there are
currently 13 members. The council is responsible for: 1. managing treasury, 2.
proposing referenda, 3. canceling dangerous referenda, and finally, 4. voting
for the technical committee members. Only teams or individuals who have added
successful implementation to Polkadot can be a part of the technical
committee. The technical committee can submit emergency proposals and veto a
proposal.

Cosmos: Cosmos’ governance is responsible for managing updates to the network,
changing protocol parameters, and using community pool funds. Each validator
needs to deposit a minimum amount of ATOM tokens. A validator may be
deactivated for a certain amount of time if a validator does not vote. On top
of the Hub’s governance, each zone may also have its governance. Nevertheless,
the Hub’s governance will reign superior, acting as a constitution for zones
inside the Cosmos ecosystem.

Loom: Loom Network’s governance is based on the native LOOM token. Validators
and delegators stake the tokens to secure the network. Application developers
who use Loom’s base chain also pay for their service in LOOM, even though the
individual users do not need to pay. The token can also be bonded when a
developer adds a new chain to the network to discourage malicious behavior.

Chainlink: With the announcement for v2 of Chainlink, no specific governance
protocol has been announced for the network. However, the governance model
will be evolutionary [3], meaning that the governance mechanism itself can be
subject to change. Each DON may also have its governance mechanism to account
for DON-specific changes.

Poly Network: Governance happens on its main chain, called the Poly Chain
[23]. However, Poly Network does not issue a token, so it remains to be seen
how the network members can partake in the governance of the main chain.

Off-chain: Interoperability solutions that provide the software rather than
another blockchain do not have on-chain governance. These solutions are
Hyperledger Cactus, Hyperledger Firefly, and Hybrix. They are designed to be
used by a consortium of entities to run their nodes and form an off-chain
network. Overledger is an exception because it requires the Quant network’s
QNT token to be licensed. However, governance within the consortium network is
still dependent on the user group. The provided software can also include
functionality for the consortium to make decisions, as Hyperledger Firefly
does.

###  5.7 Open Source

Out of the nine projects examined, all but one are open source, and their code
can be audited by anyone online. Overledger is the exception. The client part
of the API is documented publicly, but the internal implementation of message
hashing, known as TrustTag, is patented by Quant and is closed-source. While
having the core part of the solution be closed-source allows Quant to gain an
advantage in the current market and prevent possible exploits, it also comes
with the drawback of possibly slowing down the advancement of their
interoperability solution to be compatible with other blockchain technologies.

###  5.8 Trust

_Trust_ in a given blockchain is the ability to engage in transactions (e.g.,
transfer of assets) with an entity that a user has not interacted with before
or does not necessarily trust. If there is a guarantee that the entity will
behave correctly, then the system is trustless. However, if one needs to trust
the entity to behave correctly – with no provable guarantee – then the system
is trusted. This property is closely linked with how the verification happens
on the system. In the traditional blockchain architecture, trust is
established by having the nodes check each other’s work and agree.
Interoperability solutions that leverage a blockchain (Hub and Spoke,
Decentralized Oracle) are trustless because the blockchain maintains the
system’s security. However, decentralized relays need to be trusted because
they are designed to be operated by a permissioned consortium of entities.

| Summary | Integration Mode | Validation | Upgradability | Adoptability | Governance | Open Source | Trust
---|---|---|---|---|---|---|---|---
Overledger |  | Message layer for
---
multi-chain applications
Decentralized Relay | Hybrid | Through plugins | Allow metadata field | Off-chain | FALSE | Trusted
Hyperledger Cactus |  | Integration framework
---
for multi-chain transactions
Decentralized Relay | Third Party | Through plugins | Implement plugin | Off-chain | TRUE | Trusted
Hyperledger Firefly |  | API layer for
---
multi-chain applications
Decentralized Relay | Third Party | Through plugins | Implement plugin | Off-chain | TRUE | Trusted
Polkadot |  | Enable value and data to be
---
sent across multiple chains
Hub and Spoke | Native | No hard fork |  | Implement protocol/
---
build in Substrate
On-chain | TRUE | Trustless
Cosmos |  | Enables transfer of
---
value between chains
Hub and Spoke | Native | Possible hard fork |  | Implement protocol/
---
build in Tendermint
On-chain | TRUE | Trustless
Loom |  | Enable interoperability
---
for Etherum DApps
Hub and Spoke | Native | Possible hard fork |  | Loom sidechain must
---
be implemented
On-chain | TRUE | Trustless
Hybrix |  | Blockchain-agnostic
---
interoperability protocol
Decentralized Relay | Hybrid | Through P2P daemon | Allow metadata field | Off-chain | TRUE | Trusted
Chainlink |  | Network of Decentralized
---
Oracle Networks (DON)
Oracle | Native | Contract can self-generate | No requirement | Onchain | TRUE | Trustless
Poly Network |  | Interoperability protocol for
---
heterogenous blockchains
Hub and Spoke | Hybrid | Possible hard fork |  | Implement
---
Smart contract & Relayer
On-chain | TRUE | Trustless

TABLE II: Overview of Existing Blockchain Solutions.
The cells are colored to denote the security of the properties. Green: Good,
Yellow: OK, Red: Can be improved

##  6 Conclusion

We examined nine different implementations for blockchain interoperability.
These projects are compared based on properties we consider essential for such
a framework. In particular, upgradability and adoptability can directly lead
to the widespread adoption of an interoperability solution. Among the
discussed solutions, three broad integration modes can be seen in these
projects: decentralized relays, hub and spoke, and decentralized oracles. The
decentralized relay architecture is well suited for dealing with arbitrary
data and can be easier to configure. However, depending on the third-party
validation service, it can also be less secure. The hub and spoke architecture
can provide better security and validation by using secondary blockchains to
implement the system. However, achieving complete upgradeability can be more
difficult, requiring more work for developers to join new networks to the
system. The decentralized oracle approach is a tool for blockchains to use
off-chain data. Although uni-directional, it can also facilitate
interoperability by allowing different blockchains to read data from others.
Chainlink provides a novel solution to upgradability by enabling smart
contracts to generate updated copies. While it is still too early to make any
statement about which approach is better, we have highlighted some differences
in these implementations to provide a bottom-up view of the current
interoperability landscape in the blockchain space.

## Acknowledgement

We acknowledge the support from National Science Foundation
Industry–University Cooperative Research Centers (NSF IUCRC) Center for
Research toward Advancing Financial Technologies (CRAFT) research grant for
this research.

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curvature of the string will be very small, allowing the use of a finite-term
effective action.

In the case of a fundamental critical string, the worldsheet theory is
conformal. Therefore, the worldsheet theory is known at arbitrarily high
energies or extrinsic curvatures. This fact is crucial for a successful
quantization of the critical string. For an effective string, the theory is
not necessarily conformal. However, the worldsheet theory flows to a CFT in
the infrared, i.e. for long wavelengths. Therefore, we can assume that in the
regime of the validity of the effective string theory (which is long
wavelengths) the theory is almost conformal.

Let us focus on the worldsheet action in equation (86). The full action is
model-dependent but there is a universal term imposed by the tension of the
string—the Nambu–Goto term,

$S_{\text{String}}=-\mathcal{T}\int
d^{2}\sigma\sqrt{-g_{\text{ind}}}+\ldots\,,$ (88)

where $g_{\text{ind}}$ is the 2d metric induced on the string worldsheet

$\displaystyle(g_{\rm ind})_{ij}=\frac{\partial
X^{\mu}}{\partial\sigma^{i}}\frac{\partial
X^{\nu}}{\partial\sigma^{j}}g_{\mu\nu}\,.$ (89)

This action can be expressed in different coordinate systems on the worldsheet
due to the diffeomorphism gauge symmetry. We can go to the light-cone gauge
($\sigma^{+},\sigma^{-}$) to partially fix the gauge

$\displaystyle(g_{\rm{ind}})_{++}=(g_{\rm{ind}})_{--}=0\,.$ (90)

In this gauge, the string action takes the form

$\displaystyle S=2\mathcal{T}\int
d\sigma^{+}d\sigma^{-}\partial_{+}X^{\mu}\partial_{-}X_{\mu}+\ldots\,,$ (91)

and the remnant reparametrization symmetry is given by

$\displaystyle\sigma^{+}\rightarrow f^{+}(\sigma^{+})\,,$
$\displaystyle\sigma^{-}\rightarrow f^{-}(\sigma^{-})\,.$ (92)

Polchinski and Strominger found the appropriate term induced by the
Fadeev–Popov determinant which ensures the theory’s invariance under the above
reparametrization at the quantum level Polchinski:1991ax . The action receives
a series of corrections that can be organized in terms of the extrinsic
curvature of the worldsheet in units of the string tension $\mathcal{T}$. The
leading contribution takes the form

$\displaystyle S_{\rm PS}=\int
d\sigma^{+}d\sigma^{-}2\mathcal{T}\partial_{+}X^{\mu}\partial_{-}X_{\mu}+\frac{d-26}{4\pi}\frac{(\partial^{2}_{+}X^{\mu}\partial_{-}X_{\mu})(\partial_{+}X^{\nu}\partial^{2}_{-}X_{\nu})}{(\partial_{+}X^{\rho}\partial_{-}X_{\rho})^{2}}+\mathcal{O}\left((\mathcal{T}L^{2})^{-1}\right)+\ldots\,,$
(93)

where $d$ is the dimension of the spacetime and $L\sim\partial X$ is the order
of magnitude of the length scale associated with the worldsheet curvature.
Note that the above action closely resembles Polyakov’s non-critical string
action Polyakov:1981rd . The action (93) can be transformed into the Polyakov
action by using the first-order on-shell equation
$\partial_{+}\partial_{-}X^{\mu}=0$ in addition to replacing
$\partial_{+}X^{\mu}\partial_{-}X_{\mu}$ with $\exp(\phi)$.

$\displaystyle S_{\text{Polyakov}}=\int
d\sigma^{+}d\sigma^{-}\mathcal{T}\partial_{+}X^{\mu}\partial_{-}X_{\mu}+\frac{d-26}{4\pi}\partial_{+}\phi\partial_{-}\phi.$
(94)

In this theory $\phi$ is an additional scalar. This is due to the fact that
for the Polyakov string, the worldsheet metric is auxiliary and $\phi$ is the
Goldstone mode of the conformal symmetry on the worldsheet. For the
Polchinski–Strominger string, however, everything is formulated in terms of
the physical induced 2d metric.

Note that the second term in (93) is the extrinsic curvature of the string
expressed in terms of the spacetime coordinates. The coefficient of the
extrinsic curvature can change depending on the rest of the 2d action. For
example, in the presence of worldsheet fermions in the neglected ($\ldots$)
part of the action, the coefficient changes. However, we can draw the
following general conclusion: If the extrinsic curvature is much smaller than
the string tension, the leading contribution to the effective action of a
weakly coupled string comes from the Nambu–Goto area term.

The weak coupling condition is important to ensure that the self-interaction
of the string due to its coupling with bulk modes can be ignored. For example,
if the string couples to a 2-form gauge potential as in (87), it will source a
$B$-filed background. If the gauge coupling of the $B$-field is significant,
the interaction of the string with the $B$-field sourced by the other part of
it would not be negligible. Therefore, it is only for weakly coupled strings
for which we can ignore their self-interactions through the bulk.

##  Appendix B Gross–Mende saddles

In this section we review the Gross–Mende saddles Gross:1987ar , of the
Polyakov action which play an important role in the contribution of strings to
the high-energy scattering amplitudes. Since the Polyakov action and the
Nambu–Goto action coincide at the classical level, and the saddle point is a
classical configuration, finding a saddle of the Polyakov action also amounts
to finding a saddle of the Nambu–Goto action. We will elaborate on this point
after reviewing the saddles at the end of the Appendix.

Let us start by summarizing some special covering spaces of the sphere which
are discussed at length in Hamidi:1986vh  and were used by Gross and Mende
Gross:1987ar  to find worldsheet saddles. Suppose
$\\{z_{1},...,z_{L}\\}\subset\mathbb{C}$ are $L$ points on this sphere. Then

$y^{N}=\prod_{i=1}^{L}(z-z_{i})^{L_{i}}$ (95)

describes a covering of the sphere embedded in $(\mathbb{C}\cup\infty)^{2}$
provided the integers $L_{1\leq i\leq L}$ satisfy

$\sum_{i=1}^{L}L_{i}\equiv 0\mod N\,.$ (96)

Since the Polyakov action is invariant under Weyl rescaling, we will work with
the flat coordinates $ds^{2}=dzd\bar{z}$ that cover all but one point on
$\mathbb{CP}^{1}$. For $z\notin\\{z_{i}|1\leq i\leq N\\}$, the above
polynomial equation has $N$ complex solutions for $y$. Therefore the space
described by (95) is an $N$-cover of the sphere with $L$ branch points at
$z=z_{i}$ of order $N-1$ (assuming $L_{i}$’s and $N$ are coprime). The genus
of this space is given by

$\displaystyle G=\frac{(N-1)(L-2)}{2}\,.$ (97)

One can verify this using the Gauss-Bonnet theorem. The Euler characteristic
of this punctured surface has two contributions, one from the integral of the
Ricci scalar over the interior of the surface, and the other from the
contribution of the branch points which have angular deficits. Therefore, we
find

$\displaystyle\chi=-\sum_{\text{Punctures}}(\text{Ord}_{i}+1)+\frac{1}{4\pi}\int_{\text{interior}}\sqrt{g}\mathcal{R}\,.$
(98)

Here, $\text{Ord}_{i}$ represents the order of the branch point at $z=z_{i}$.
The contribution of the first terms is $-LN$ given that there are $L$ branch
points each with degree $N-1$. Since the space is an $N$-cover of the sphere,
the contribution from integrating the 2d Ricci scalar $N$-times over the
smooth component of the surface is $N$ times that of the sphere. The
contribution of the second term is $N\cdot\chi_{\text{sphere}}=2N$. In total
this leads to

$\chi=-L(N-1)+2N-L.$ (99)

To find the genus associated with this surface in the perturbative expansion
of string amplitudes, we use

$2-2G-L=\chi=-(N-1)L+2N-L,$ (100)

which results in

$G=\frac{(N-1)(L-2)}{2}.$ (101)

In the main text we are interested in computing $2\rightarrow 2$ amplitudes
describing scattering processes with four external legs. Since external legs
correspond to branch points we are mainly interested in covering spaces with
$L=4$ branch points. For these covering spaces we have

$G_{2\rightarrow 2}(N)=N-1.$ (102)

Having reviewed the covering spaces of the sphere, we are ready to describe
the Gross–Mende saddles. Suppose we are interested in a string amplitude
involving 4 external legs. Then the amplitude for the $2\rightarrow 2$ process
can be determined via the Polyakov path integral in the Euclidean signature,
which takes the form

$\mathcal{A}=\sum_{\text{topologies}}\int\prod_{a}\left(dt^{a}f(t^{a})\right)\,\mathcal{D}X^{\mu}\,e^{-S_{\text{Polyakov}}}\prod_{i=1}^{4}\left(dz_{i}\right):e^{ip_{i}\cdot
X(z_{i})}:\mathcal{V}_{i}(z_{i})\,.$ (103)

Let us explain the ingredients in the above path integral:

  * •

The operators $\mathcal{V}_{i}$ are the vertex operators that identify the
species of the external legs. Note that for every pair of conformal Killing
vectors, there is a vertex operator $\mathcal{V}_{i}$ that contains a
$c$-field ghost insertions multiplied by $\delta(z-z^{\prime})$ that fixes the
location of $z_{i}$.

  * •

The vectors $p_{i}$ are the momenta of the external legs and $::$ represents
worldsheet normal ordering to regulate the worldsheet OPEs.

  * •

The integration variables $t^{a}$ are the moduli of the Riemann surface.

  * •

The integration variables $z_{i}$ are complex worldsheet coordinates
corresponding to the location of the insertions.

  * •

The factor $f(t^{a})$ is the integration measure over the moduli space which
is determined by the Fadeev–Popov determinant and $b$-field ghost insertions.

Note that here we neglected the fermionic worldsheet fields for simplicity as
their inclusion does not affect our final conclusions. At high energies
($p_{i}\gg M_{s}$), the behavior of this path integral is not sensitive to the
choice of the species of the in/out-going particles. Therefore, the amplitude
can be approximated by the 4-point function of the $:e^{ip_{i}\cdot
X(z_{i})}:$ operators which is given by

$\mathcal{A}\sim\delta^{d}\left(\sum_{i}p_{i}\right)\int\prod_{j}dz_{j}\exp\left(\frac{1}{2}\sum_{1\leq
i<j\leq 4}^{4}p_{i}\cdot p_{j}G(z_{i},z_{j})\right)\,,$ (104)

where $G(z_{i},z_{j})$ is the propagator on the surface. Here we use
$\mathcal{T}=1/(2\pi\alpha^{\prime})$ and our convention for the signature of
the spacetime metric is $(-,+,\ldots,+)$. Gross and Mende approximated this
integral using the saddle point approximation. For that, we need to find the
saddle point of $\mathcal{E}=\frac{1}{2}\sum_{i,j}(-p_{i}\cdot
p_{j})G(z_{i},z_{j})$. Fortunately, this is a familiar problem since
$G(z_{i},z_{j})$ is the solution to

$\partial\bar{\partial}G(z,z_{0})=-\left(\frac{\alpha^{\prime}\pi}{2}\right)\delta^{2}(z-z_{0})\,,$

which also describes the electrostatic potential of a charged particle with
unit charge at $z=z_{0}$ in two Euclidean dimensions. Therefore, $\mathcal{E}$
is the electrostatic energy of a configuration of particles with charges
$p_{i}$ at points $z_{i}$.

The mimimum energy configuration depends on the topology of the saddle. Let us
start with the simplest case of the sphere. On the sphere, the propagator is
$G(z_{i},z_{j})=-(\alpha^{\prime}/4)\log|z_{i}-z_{j}|^{2}$ and the energy is
given by

$\displaystyle\mathcal{E}=\frac{\alpha^{\prime}}{4}\sum_{1\leq i<j\leq
4}p_{i}\cdot p_{j}\log|z_{i}-z_{j}|\,,$ (105)

which can be expressed in terms of the cross ratio

$\lambda=\frac{(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{1}-z_{2})(z_{3}-z_{4})}\,,$

as

$\mathcal{E}=-\frac{\alpha^{\prime}}{8}(t\log|\lambda|+u\log|1-\lambda|)\,,$
(106)

where $s$, $t$, and $u$ are the usual Mandelstam variables (see (119)). In
writing the above equation, we have used $s\simeq-t-u$ which is a good
approximation at high energies $E\gg M_{s}$ where the external legs can be
treated as massless relativistic particles. The extremum of the above action
is reached for

$\displaystyle\lambda_{\rm saddle}=-t/s\simeq 1+u/s\,.$ (107)

For this critical $\lambda$, the amplitude is given by

$\displaystyle\mathcal{A}_{\text{tree}}\sim\exp\left(-\frac{\alpha^{\prime}}{8}(s\log
s+u\log u+t\log t)\right)\,,$ (108)

which agrees with the high-energy behavior of the Virasoro–Shapiro amplitude.
Note that there are also polynomial factors multiplying the exponential term
that result from integrating around the saddle (107).

To study surfaces with higher genus, we can repeat the same analysis for the
covering spaces of the sphere which we discussed earlier. Consider an
$N$-cover of the sphere. The energy function $\mathcal{E}$ is suppressed by a
factor $1/N$. This is because on every sheet the electric charges, and the
corresponding electric field $E$, are divided by $N$. On the other hand, the
energy $\mathcal{E}$, is proportional to the integral of $E^{2}$ over $N$
sheets which leads to a $N\times(1/N^{2})=1/N$ suppression compared to the
tree-level diagram. The rest of the analysis gives the same cross-ratio
$\lambda$ for the positions $z_{i}$. Therefore, Gross and Mende found

$\displaystyle\mathcal{A}_{N}\sim
g_{s}^{2G-2}\exp\left(-\frac{\alpha^{\prime}}{8N}(s\log s+u\log u+t\log
t)\right)\,,$ (109)

where $g_{s}$ is the string coupling. Using (102), we can express $G$ in terms
of $N$.

$\displaystyle\mathcal{A}_{N}\sim
g_{s}^{2N-4}\exp\left(-\frac{\alpha^{\prime}}{8N}(s\log s+u\log u+t\log
t)\right)\,,$ (110)

Every $N$ corresponds to a different saddle. However, one can see that the
amplitude is maximized for

$N_{\rm max}\simeq\sqrt{\frac{\alpha^{\prime}\left(s\log s+u\log u+t\log
t\right)}{16\left(-\log g_{s}\right)}}\,,$ (111)

for which we have

$\mathcal{A}\sim\exp(-\sqrt{\left(-\log(g_{s})\right)\alpha^{\prime}\left(s\log
s+u\log u+t\log t\right)})\,,$ (112)

up to multiplication by factors polynomial in energy. Therefore, we conclude
that the dominant contribution from the Gross-Mende saddles goes like

$\log(\mathcal{A})\sim-\sqrt{s}\cdot f(s,\theta)\,,$ (113)

where $\theta$ is the scattering angle and $f(s,\theta)$ is a function with
subpolynomial (logarithmic) growth/decay in $\sqrt{s}$. Note that the
polynomial prefactors, which we neglected in (109), contribute to
$f(s,\theta)$

It is also instructive to review the embedding of these saddles in spacetime.
Around each external leg, the position is given by
$X^{\mu}\simeq\frac{ip_{j}^{\mu}}{4\pi\mathcal{T}}\log|z-z_{j}|$, where
$\mathcal{T}=1/(2\pi\alpha^{\prime})$ is the string tension. Therefore, we
have

$X^{\mu}=\sum_{j=1}^{4}\frac{ip_{j}^{\mu}}{4\pi
N\mathcal{T}}\log|z-z_{j}|+\mathcal{O}\left(\frac{\sqrt{\mathcal{T}}}{s}\right)\,.$
(114)

From the above expression, one can see that the length scale associate with
the extrinsic curvature ($L\sim\partial X^{\mu}$) behaves like $L\sim
E/(N\mathcal{T})$ where $E$ is the center of mass energy. For the dominant
saddle, using (111) this length scale evaluates to $L\sim l_{s}$ up to
multiplicative logarithmic corrections in energy. Note that for $1\ll N\sim
N_{\rm max}(E/M_{s})^{-\epsilon}\ll N_{\rm max}$, where $\epsilon>0$ and $E\gg
M_{s}$, we have

$L\gg l_{s}\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
\log(\mathcal{A})\sim-\frac{E}{\sqrt{\mathcal{T}}}\widetilde{\mathcal{O}}\left(L\sqrt{\mathcal{T}}\right)\,.$
(115)

In the above discussion, we used the Polyakov action to find the saddle.
Saddles satisfy the classical equations of motion, and both Polyakov and
Nambu–Goto action lead to the following equation of motion for the embedding
of the worldsheet

$\displaystyle\Box X^{\mu}=0\,,$ (116)

where $\Box$ is the d’Alembertian with respect to the induced metric
$h_{ij}=\partial_{i}X^{\mu}\partial_{j}X_{\mu}$. However, one might be
concerned whether the boundary conditions imposed at the external legs are the
same in the two pictures. For each external leg, we have chosen a spacetime
momentum which is the conserved charge under translations in $X^{\mu}$. In the
Polyakov action this conserved quantity is given by

$\displaystyle p^{\mu}=\frac{\mathcal{T}}{2}\int
d\sigma\sqrt{h}\,h^{0i}{\partial_{i}}X^{\mu}\,,$ (117)

while in the Nambu–Goto action, this conserved charge is

$\displaystyle p^{\mu}=\frac{\mathcal{T}}{2}\int
d\sigma\sqrt{g_{\text{ind}}}\,g_{\text{ind}}^{0i}{\partial_{i}}X^{\mu}\,,$
(118)

where $g_{\text{ind}}$ is the induced metric on the worldsheet. Therefore,
after plugging in the equations of motion $h\propto g_{\text{ind}}$, we see
that the two definitions of momenta match. In other words, the embedding (114)
which is a Polyakov saddle, is also a solution to the Nambu-Goto equations of
motion with external legs that carry spacetime momenta $p_{i}^{\mu}$.
Therefore, the same embeddings are also saddles of the Nambu–Goto action.

##  Appendix C Amplitude toolbox

In this Appendix we review some results about scattering amplitudes and how
they apply to high-energy amplitudes in quantum gravity. We focus on
$2\rightarrow 2$ scattering.

Let us start by defining our conventions and notations. The in-going spacetime
momenta are $k_{1}$ and $k_{1}$ and the out-going spacetime momenta are
$-k_{3}$ and $-k_{4}$. Our convention for the signature of the spacetime
metric is $(-,+,\ldots,+)$. The conservation of energy-momentum implies
$\sum_{i}k_{i}^{\mu}=0$. The Mandelstam variables are defined as

$\displaystyle s$ $\displaystyle=-(k_{1}+k_{2})^{\mu}(k_{1}+k_{2})_{\mu}\,,$
(119) $\displaystyle t$
$\displaystyle=-(k_{1}+k_{3})^{\mu}(k_{1}+k_{3})_{\mu}\,,$ $\displaystyle u$
$\displaystyle=-(k_{1}+k_{4})^{\mu}(k_{1}+k_{4})_{\mu}\,.$

The scattering angle $\theta$ can be expressed in terms of $k_{1}$ and $k_{3}$
as

$\cos(\theta)=\frac{\sum_{\alpha\neq
0}k_{1}^{\alpha}{k_{3}}_{\alpha}}{\sqrt{(\sum_{\beta\neq
0}k_{1}^{\beta}{k_{1}}_{\beta})(\sum_{\gamma\neq
0}k_{3}^{\gamma}{k_{3}}_{\gamma})}}\,.$ (120)

Suppose the in-going and out-going particles are on-shell and have physical
mass $m$. We can express the scattering angle in terms of the Mandelstam
variables as

$\displaystyle\cos(\theta)=1+\frac{2t}{s-4m^{2}}\,.$ (121)

Since $|\cos(\theta)|\leq 1$, the variable $t$ is bounded to lie in the
interval $[-s+4m^{2},0]$. This regime is referred to as the physical regime
for $t$.

###  C.1 Regge behavior

An interesting limit in the scattering processes is obtained by keeping $t$
fixed while $s$ is taken to infinity. In this limit, the scattering angle
$\theta$ goes to 0. Therefore, this limit corresponds to two very high energy
particles that barely change each other’s trajectory while exchanging a fixed
amount of spacetime momentum $t$. The dependence of the t-channel amplitude on
$s$ for the exchange of a spin-$L$ particle can be determined in the following
way. A spin-$L$ particle is represented by a traceless symmetric field with
$L$ indices. Suppose a vertex has two scalar legs and one spin-$L$ leg. Due to
Lorentz invariance, the vertex must contain $L$ derivatives. Therefore, each
one of the two vertices in the t-channel yields a contribution proportional to
$(i\sqrt{s})^{L}$ in the high energy limit. Therefore, the overall amplitude
is proportional to $(-s)^{L}$.

The contribution of the exchange of such a particle to the t-channel amplitude
depends on $t$ as well. As $\sqrt{{-t}}$ approaches the mass of a particle,
the amplitude diverges. However, for a fixed value of $t$, at high center of
mass energies, the contribution of this particle exchange grows polynomially
as $(-s)^{L}$. Note that this is not the full t-channel amplitude since there
could be more contributions to the t-channel. However, for fixed $t$, the
number of weakly coupled particles that can participate in the t-channel
exchange will be finite. Therefore, the polynomial growth of the amplitude is
expected to continue as long as we are at a generic value of $t$ which does
not coincide with the mass of the exchanged particles. This behavior is called
the Regge behavior. To be more precise, for every fixed $t$ and $s\gg|t|$ we
can parameterize the amplitude as in Caron-Huot:2016icg  in terms of a
continuous function $j(t)$ and a meromorphic function $F(t)$ as

$\displaystyle\mathcal{A}\simeq F(t)(-s)^{j(t)}\,.$ (122)

In tree-level string theory $j(t)$ is a linear function in $t$. In Caron-
Huot:2016icg , the authors showed that in the presence of higher-spin weakly
coupled (tree-level) particles, linearity of $j(t)$ at high energies is
universal. As we discussed above, a single spin-$L$ particle with mass $m$
will result in a non-negative integer value $j(t)=L$ and a pole for $F(t)$ at
$-m^{2}$. Therefore, we expect $j(t)$ to be integer at poles of $F(t)$ and
vice versa (see A.2 in Caron-Huot:2016icg  for an argument).

###  C.2 Locality bound

In this subsection we briefly review the locality bounds on amplitudes and
discuss how (and if) they apply to quantum gravity.

The most well-known bound on high-energy scattering amplitudes based on
locality is the Cerulus–Martin bound Cerulus:1964cjb  which is a lower bound
for the amplitude of a hard scattering process at high energies. The
Cerulus–Martin bound is obtained under certain assumptions on the amplitude.
In particular, the amplitude is assumed to have the following three
properties:

  1. 1.

The amplitude is analytic as a function of $s$ and $z=\cos(\theta)$ in a
complicated sub-region of the cut $z$-plane which is described in
Cerulus:1964cjb  (see section 3.2 in Buoninfante:2023dyd  for a more detailed
explanation).

  2. 2.

The amplitude is polynomially bounded at high energies for a fixed range of
scattering angles, $z\in[-a,a]$ dor some fixed $a$,

$\displaystyle|\mathcal{A}(s,z)|<A\left(\frac{s}{s_{0}}\right)^{N},$ (123)

where $\sqrt{s_{0}}$ is some reference energy and $A$ and $N$ are constants.
This assumption is often understood to follow from locality since amplitudes
that are reproduced by a local effective field theory with finitely many terms
are polynomially bounded.

  3. 3.

The forward scattering amplitude is polynomially bounded from below

$\displaystyle|\mathcal{A}(s,1)|>\frac{B}{(\frac{s}{s_{0}})^{\beta}}\,,$ (124)

for some positive constants $B$ and $\beta$ at high energies (i.e. $s\gg
s_{0}$).

Given an amplitude that satisfied these three assumption for a fixed range of
scattering angles $z\in[-a,a]$, the Cerulus–Martin bound states that the
amplitude is bounded as

$\displaystyle|\mathcal{A}(s,z)|>\mathcal{N}(s)\,e^{-f(a)\sqrt{s}\log(s/s_{0})}\,,$
(125)

where $\mathcal{N}(s)$ is a subdominant function of $s$ and $f(a)$ is
independent from $s$.

Given this bound, we can now ask whether and under what conditions we expect
it to be satisfied in theories of quantum gravity. We first notice that the
Cerulus–Martin bound is violated in tree-level string theory since tree-level
amplitudes violate polynomial boundedness Gross:1987kza . Therefore, the tree-
level string theory does not exhibit the local properties of an effective
field theory. As a consequence high-energy tree-level amplitudes in string
theory only satisfy a bound that is weaker than the Cerulus–Martin bound (cf.
(108))

$\displaystyle|\mathcal{A}_{\rm
tree}(s,z)|>\mathcal{N}(s)e^{-f(a)\,s\log(s/s_{0})}\,.$ (126)

In Buoninfante:2023dyd  the Cerulus-Martin bound was generalized to
exponentially bounded amplitudes and the application of the generalized lower
bound to tree-level string theory yields the lower bound

$\displaystyle|\mathcal{A}_{\rm tree}(s,z)|>\mathcal{N}(s)e^{-f(a)s^{3/2}}\,,$
(127)

which is satisfied. Interestingly, when the higher-genus contributions from
the Gross–Mende saddles, discussed in appendix B, are included, the overall
amplitude can be resummed to Mende:1989wt

$\displaystyle|\mathcal{A}(s,z)|>\mathcal{N}(s)e^{-f(a)\sqrt{s}\log(s/s_{0})^{q}},$
(128)

where $q$ is a negative number. The above amplitude satisfies the
Cerulus–Martin bound. It thus seems that including higher-genus corrections
restores some notion of locality. Nonetheless, the exact connection to
locality is not well-understood (see section 6.4 in Buoninfante:2023dyd ).

Still, under certain assumptions, we can motivate the Cerulus–Martin bound for
high-energy amplitudes in theories of gravity. However, instead of showing the
bound in the fixed angle scattering, we first argue that it holds for any
fixed impact parameter. The inequality then follows for the fixed angle
scattering amplitude since it can be estimated using a saddle point
approximation to be of the order of the fixed impact parameter that forms a
saddle Amati:1988tn .

In section 5.2, we argued that the density of one-particle states at energies
above the quantum gravity cutoff $\Lambda$ must grow exponentially in energy.
As reviewed in section 3.1, we assume that the density of states is related to
the high-energy/low-impact parameter scattering amplitude as
$\log\rho(\sqrt{s})\sim-\log(|\mathcal{A}(s,b)|^{2})$. The intuition behind
this assumption is that at high-energies the in-going particles form bound
states/saddles that exhibit thermal properties with $e^{S}$ microstates such
that $S\sim\log(\rho)$. Therefore, the probability of having that thermal
state decay into a 2-particle state is $e^{S}$. This assumption is correct in
string theory Bedroya:2022twb  even though a bottom-up argument justifying
this assumption is missing. Based on this assumption, we find that the high-
energy amplitude for energies $\Lambda\ll\sqrt{s}<M_{\text{min}}$ satisfies
the Cerulus–Martin bound as

$\displaystyle\log(|\mathcal{A}(s,b)|^{2})\gtrsim-\log\rho(\sqrt{s})=-f(s)\sqrt{s},$
(129)

where $f(s)$ has logarithmic growth. The above inequality is saturated at low
impact parameters, but not at large impact parameters where less intermediate
channels are available.

###  C.3 Unphysical amplitudes ($s,t\gg\Lambda$)

As we explained in the beginning of this Appendix, the Mandelstam variable $t$
can only take negative values for on-shell particles. However, in this section
we are interested in the analytic continuation of the amplitude to the
unphysical regime in which $t$ takes large, positive values. As we will
explain, unlike the physical fixed angle amplitudes, the amplitude in this
regime is very sensitive to the aforementioned suppression at small impact
parameters.

Let us start by explaining the relationship between scattering amplitudes in
terms of the impact parameter and Mandelstam variables. The impact parameter
$b$ is the relative position between the two wave-packets while
$q=p_{1}+p_{3}$ is the momentum exchange between the two particles. The
component, $K_{\perp}$, of the spatial part of $q$ perpendicular to $p_{1}$ is
conjugate to $b$ such that the amplitude $\mathcal{A}(s,b)$ can be obtained
from $\mathcal{A}(s,t)$ via

$\displaystyle\mathcal{A}(s,b)=\int
d^{d-2}K_{\perp}e^{i\vec{K}_{\perp}\cdot\vec{b}}\mathcal{A}(s,t)\,.$ (130)

Since $t$ and $b$ are conjugate to each other they cannot both be known. As
explained in the previous subsection, at energies $\sqrt{s}\gg\Lambda$, we
expect the amplitude $\mathcal{A}(s,b)$ to be exponentially suppressed at
sufficiently small impact parameters. For example, if the center of mass
energy $\sqrt{s}$ is higher than the black hole formation threshold
$m_{\text{min}}$, the amplitude satisfies

$\displaystyle\log(\mathcal{A}(s,b\ll r_{s}(M)))\sim-S(M)\,,$ (131)

where $r_{s}(M)$ and $S(M)$ are respectively the radius and entropy of a black
hole with mass $M$. This exponential suppression cannot be easily seen in
$\mathcal{A}(s,t)$ because $\mathcal{A}(s,t)$ is the Fourier transform of
$\mathcal{A}(s,b)$ and receives contributions from all impact parameter.
Therefore, the exponentially small contribution of low-impact parameters is
overshadowed by the polynomially large contributions of the amplitudes at high
impact parameters. On the other hand, the exponential suppression at small $b$
leaves an exponential phase imprint in the amplitude
$\sim\exp(2\sqrt{t}\,r_{s}(\sqrt{s}))$. This term becomes dominant in the
unphysical regime $t>0$ (see Giddings:2007qq  for details) such that this
regime is a good probe for the exponential suppression of the amplitudes at
small impact parameters.

More generally, the same argument also applies to non-black hole bound states
as we show in the following. Suppose the amplitude $\mathcal{A}(s,b)$ is
exponentially suppressed for impact parameters much smaller than some critical
value $b_{c}(\sqrt{s})$. Further assume that it is polynomial for impact
parameters $b\gg b_{c}$. By modeling the scattering process with a black disk
that blocks low-impact parameter amplitudes one can find that the amplitude in
the unphysical regime behaves as

$\displaystyle\mathcal{A}\left(s,t\gg
b_{c}(\sqrt{s})^{-2}\right)=\mathcal{N}(s,t)\exp(\sqrt{t}\,\mathcal{O}(b_{c}(\sqrt{s}))\,,$
(132)

where $\mathcal{N}(s,t)$ is a subdominant function. This statement was proven
in Giddings:2007qq  for black holes but the same argument applies to any
amplitude which is exponentially suppressed at low impact parameters. The
reason is that the diffraction off of the edge of the black disc will lead to
contributions proportional to
$\sim\sin(\mathcal{O}(E\theta\,b_{c}(\sqrt{s})))$ to the amplitude. If we
express the scattering angle in terms of Mandelstam variables
$-t/s\simeq\theta^{2}/4$ and analytically continue to positive $t$ we find
(132).

In section C.2 we argued that the low-impact parameter amplitude in quantum
gravity should be exponentially suppressed based on the identification between
the density of one-particle states and the scattering amplitude. Assuming that
this exponential suppression is realized for $b\ll b_{c}(\sqrt{s})$ we find
that the unphysical high-energy amplitude below the black hole threshold is
given by (132). Moreover, we know that this amplitude must match with that of
the black hole at the black hole threshold $\sqrt{s}\sim M_{\text{min}}$.
Above the black hole threshold, $b_{c}$ is given by the radius of the black
hole. Therefore, we find

$\displaystyle b_{c}(M_{\text{min}})=r_{\min}=\Lambda_{\rm min}^{-1}\,.$ (133)

While this relation determines $b_{c}(\sqrt{s})$ at energies corresponding to
the black hole threshold, it does not yet fix the energy dependence of $b_{c}$
at energies below $M_{\rm min}$. We can provide a bound on the energy
dependence of $b_{c}$ below the black hole threshold by using the Regge
behavior reviewed in section C.1 which is expected to hold below the black
hole threshold. From (132), we conclude that if $b_{c}(\sqrt{s})$ depended
polynomially on $\sqrt{s}$ for $\sqrt{s}<M_{\rm min}$, the off-shell amplitude
$\mathcal{A}(s,t>0)$ would have an exponential dependence on $s$. By (122)
such an exponential dependence is, however, inconsistent with the Regge
behavior which forces $b_{c}(\sqrt{s})$ to have at best a logarithmic
dependence on $\sqrt{s}$ at high energies above the cutoff $\Lambda_{\rm
min}$. Still, the Regge behavior is motivated for a perturbative description
of the amplitudes which we expect to be violated at the black hole threshold
$\sqrt{s}=M_{\rm min}$. Therefore, we find that for energies $\Lambda_{\rm
min}\ll\sqrt{s}\ll M_{\rm min}$,

$\displaystyle b_{c}(\sqrt{s})^{-1}=\widetilde{\mathcal{O}}\left(\Lambda_{\rm
min}\right)\,.$ (134)

Plugging equation (134) back into (132) leads to

$\displaystyle\mathcal{A}(s,t\gg\Lambda_{\rm
min}^{2})\sim\mathcal{N}(s,t)\exp\left[\widetilde{\mathcal{O}}(\sqrt{t}/\Lambda_{\rm
min})\right]\,.$ (135)

This result for the amplitude is the main ingredient for our argument in favor
of Proposition 2 in section 4.1.

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