Since 2015, the United Nations have called for a global effort to reach Sustainable Development Goals (SDGs). Firms play a vital role in contributing to SDGs. While many empirical approaches were used to map firms' contributions to SDGs, online social networks are an underexplored but promising setting. This paper maps large UK firms' discussions on Twitter, specifically focusing on their SDG-related discussions, with complex network methods from statistical physics. Results show that: 1) SDGs are the topics that tie conversations among major UK firms together; 2) compared to the environmental and economic dimensions, the social dimension is predominant; 3) the attention to different SDGs varies depending on the community and sector firms belong to; 4) the use of retweets on SDGs-related tweets highlights a high stakeholder engagement on global challenges; 5) large UK companies and stakeholders generally behave differently from Italian ones. This paper provides theoretical contributions, combining institutional, stakeholder and legitimacy theories. It also contributes to developing the literature on businesses and SDGs with an interdisciplinary approach. It offers practical implications and a big-data based tool to monitor firms' discussions on SDGs on Twitter. Last, the paper proposes new avenues for further research.