Modular Forms and Anomaly Cancellations in String Theory and M-Theory

String theory is a candidate theory in particle physics for reconciling quantum mechanics and general relativity, i.e. quantum gravity. Moreover, it’s also a candidate of TOE (“theory of everything”), describing all the known fundamental forces and matters in our universe in a mathematically complete system. In conventional theories, elementary particles are mathematical points, whereas, the fundamental objects in string theory are 1-dimensional oscillating lines or loops.