A company is planning to set up a REST API application on AWS. The application team wants to set up a new identity store on AWS. The IT team does not want to maintain any infrastructure or servers for

This option includes deploying EC2 instances to run Active Directory for authentication, which directly contradicts the requirement that the IT team "does not want to maintain any infrastructure or servers" for the deployment.

AWS AppSync is primarily designed for GraphQL APIs, not specifically for REST APIs as requested. More critically, it also suggests deploying EC2 instances for Active Directory authentication, violating the "no infrastructure or servers to maintain" requirement.

Deploying the application logic as AWS Lambda functions and exposing them via Amazon API Gateway provides a fully serverless solution for the REST API, eliminating server management. Integrating Amazon Cognito as the identity store further enhances operational efficiency by providing a fully managed authentication and authorization service without requiring any server or infrastructure maintenance.

Amazon EKS (Elastic Kubernetes Service) requires managing Kubernetes worker nodes (unless using Fargate, which still has container management overhead) and is not as operationally efficient or serverless as a Lambda/API Gateway architecture. Furthermore, it also includes EC2 instances for Active Directory, violating the "no infrastructure or servers to maintain" requirement.