Loads of people simply walk out of their jobs every day, without any regard for notice periods. In the vast majority of cases, employers do nothing about it, other than immediately ceasing to pay the employee and refusing to provide a reference.
However, because those (ex-)employees are in breach of their contracts, their (former) employers could sue them for any losses sustained by their company through their unauthorised absence. For example, if a senior employee walked out and the company lost a multi-million pound contract because of it, the employer could sue for full compensation. Similarly, if an employer has to pay other staff overtime rates or engage expensive agency staff, because an employee has walked out, the employer can sue for compensation.
Chris