The Sachdev-Ye-Kitaev (SYK) model is a quantum mechanical model of N Majorana fermions with random interactions. It is tractable in the large N limit, where the classical variable is a bilocal fermion bilinear. The model becomes strongly interacting at low energies, developing an emergent conformal symmetry. The model exhibits a conformal limit where the symmetry is spontaneously broken to SL(2,R), leading to zero modes. These zero modes are lifted by a small residual explicit breaking, producing an enhanced contribution to the four-point function. This contribution displays a maximal Lyapunov exponent in the chaos region, suggesting universal properties of large N quantum mechanics systems with emergent reparametrization symmetry. The model's two-point functions are studied, showing that the conformal limit leads to a classical bilocal field. The four-point functions are analyzed, revealing a conformal limit where the four-point function is infinite due to Nambu-Goldstone bosons. However, the explicit breaking of the symmetry lifts these modes, leading to a finite four-point function. The enhanced contribution to the four-point function saturates the chaos bound, indicating a universal feature of large N systems with emergent conformal symmetry. The model's effective theory of reparametrizations is discussed, showing that the conformal symmetry is broken to SL(2,R). The density of states and free energy are analyzed, revealing a zero-temperature entropy of order N and a finite-temperature entropy linear in temperature. The model's connection to black holes and holography is explored, with the SYK model's four-point function matching the expected chaotic dynamics in a gravity theory. The model's four-point function is derived, showing that the conformal limit leads to an infinite four-point function. However, the explicit breaking of the symmetry lifts the Nambu-Goldstone modes, leading to a finite four-point function. The enhanced contribution to the four-point function saturates the chaos bound, indicating a universal feature of large N systems with emergent conformal symmetry. The model's connection to black holes and holography is explored, with the SYK model's four-point function matching the expected chaotic dynamics in a gravity theory. The model's effective theory of reparametrizations is discussed, showing that the conformal symmetry is broken to SL(2,R). The density of states and free energy are analyzed, revealing a zero-temperature entropy of order N and a finite-temperature entropy linear in temperature. The model's connection to black holes and holography is explored, with the SYK model's four-point function matching the expected chaotic dynamics in a gravity theory.The Sachdev-Ye-Kitaev (SYK) model is a quantum mechanical model of N Majorana fermions with random interactions. It is tractable in the large N limit, where the classical variable is a bilocal fermion bilinear. The model becomes strongly interacting at low energies, developing an emergent conformal symmetry. The model exhibits a conformal limit where the symmetry is spontaneously broken to SL(2,R), leading to zero modes. These zero modes are lifted by a small residual explicit breaking, producing an enhanced contribution to the four-point function. This contribution displays a maximal Lyapunov exponent in the chaos region, suggesting universal properties of large N quantum mechanics systems with emergent reparametrization symmetry. The model's two-point functions are studied, showing that the conformal limit leads to a classical bilocal field. The four-point functions are analyzed, revealing a conformal limit where the four-point function is infinite due to Nambu-Goldstone bosons. However, the explicit breaking of the symmetry lifts these modes, leading to a finite four-point function. The enhanced contribution to the four-point function saturates the chaos bound, indicating a universal feature of large N systems with emergent conformal symmetry. The model's effective theory of reparametrizations is discussed, showing that the conformal symmetry is broken to SL(2,R). The density of states and free energy are analyzed, revealing a zero-temperature entropy of order N and a finite-temperature entropy linear in temperature. The model's connection to black holes and holography is explored, with the SYK model's four-point function matching the expected chaotic dynamics in a gravity theory. The model's four-point function is derived, showing that the conformal limit leads to an infinite four-point function. However, the explicit breaking of the symmetry lifts the Nambu-Goldstone modes, leading to a finite four-point function. The enhanced contribution to the four-point function saturates the chaos bound, indicating a universal feature of large N systems with emergent conformal symmetry. The model's connection to black holes and holography is explored, with the SYK model's four-point function matching the expected chaotic dynamics in a gravity theory. The model's effective theory of reparametrizations is discussed, showing that the conformal symmetry is broken to SL(2,R). The density of states and free energy are analyzed, revealing a zero-temperature entropy of order N and a finite-temperature entropy linear in temperature. The model's connection to black holes and holography is explored, with the SYK model's four-point function matching the expected chaotic dynamics in a gravity theory.