| 總結: | <p>In the first part of this thesis, we discuss some aspects of the four-dimensional <em>N</em> = 2 superconformal index of theories of class <em>S</em>. We first consider a generalized index on <em>S<sup>1</sup></em> × <em>S<sup>3</sup></em>/ℤ<sub><em>r</em></sub>, and prove S-duality in a particular fugacity slice. We then go on to study the (round) superconformal index in the presence of surface defects. We develop a systematic prescription to compute surface defects labeled by arbitrary irreducible representations of the gauge group and subject those defects to various tests in several different limits. Each of these limits is interesting in its own right, and we go on to explore them in some depth.</p> <p>In the second part of this thesis, we construct the gravity duals of large <em>N</em> supersymmetric gauge theories defined on squashed five-spheres with <em>SU</em>(3) × <em>U</em>(1) symmetry. The gravity duals are constructed in Euclidean Romans <em>F</em>(4) gauged supergravity in six- dimensions, and uplift to massive type IIA supergravity. We compute the partition function and Wilson loop in the large N limit of the gauge theory and compare them to their corresponding supergravity dual quantities. As expected from AdS/CFT, both sides agree perfectly. Based on these results, we conjecture a general formula for the partition function and Wilson loop on any five-sphere background, which for fixed gauge theory depends only on a certain supersymmetric Killing vector. We then go on to construct rigid supersymmetric gauge theories on more general Riemannian five-manifolds. We follow a holographic approach, realizing the manifold as the conformal boundary of the six-dimensional bulk supergravity solution. This leads to a systematic classification of five-dimensional supersymmetric backgrounds with gravity duals.</p>
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