| Summary: | Supersymmetric gauge theories are among the most important objects of study in modern theoretical physics. They are considered as more symmetric versions of gauge theories describing the real world and are often dual to string theories in curved backgrounds. In planar limit some supersymmetric gauge theories | N = 4 SYM and ABJM theory among them | manifest integrability, a property which might allow to solve them exactly. In this thesis we discuss application of integrability methods to spectral problems in supersymmetric gauge theories. Our main topic is the Quantum Spectral Curve (QSC) method, the ultimate simpli cation of integrability tools developed over the last decade. We describe the objects of our study, N = 4 SYM and ABJM theories and their dual string theories. Then we review the integrable structures appearing in these theories in the planar limit. A chapter is dedicated to description of QSC in N = 4 SYM and then we solve the QSC equations in various limits, including near-BPS limits of twist operators and of a cusped Wilson line. For the last observable we explore the quasiclassical limit in more detail, nding the matrix model reformulation and the corresponding algebraic curve. We also apply QSC method to BFKL physics, a regime which establishes a more direct connection between N = 4 SYM and QCD. In particular, we nd a new, NNLO, coecient in the weak-coupling expansion of BFKL eigenvalue. We describe an ecient algorithm of numerical solution of QSC equations and use it to explore the relation between the spin and the conformal dimension for wide range of conformal dimension, including complex values. In ABJM we also consider the near-BPS limit of twist operators, calculating the slope function, and extracted from this calculation a conjecture for so-called interpolating function h() | the long-sought-for missing ingredient for the integrability construction in ABJM.
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