A Symphony of Supersymmetry and Geometry: Invariants, Dualities and Chiral Rings

• Main Author: Ye, Ke
• Format: Others
• Published: 2018
• Online Access: https://thesis.library.caltech.edu/10844/1/Ye_Ke_2018_thesis.pdf; Ye, Ke (2018) A Symphony of Supersymmetry and Geometry: Invariants, Dualities and Chiral Rings. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/K2K2-9790. https://resolver.caltech.edu/CaltechTHESIS:05032018-153255038 <https://resolver.caltech.edu/CaltechTHESIS:05032018-153255038>

<p>The present dissertation discusses aspects of supersymmetric quantum field theory, whose main themes are two-folded. First, we explore connections between superconformal theories in various dimensions and geometric invariants. Such correspondence arises from compactification of string theory or M-theory, which encodes geometric quantities into physical observables. Second, we study in detail the chiral rings and their quantum corrections in certain supersymmetric gauge theory. The goal is to shed some light on the hitherto mysterious electric-magnetic dualities.</p> <p>We first consider M5 brane on the product manifold <i>L</i>(<i>k</i>, 1) × <i>M</i>₃, where <i>M</i>₃ = <i>L</i>(<i>p</i>, 1). Compactification on <i>L</i>(<i>p</i>, 1) gives rise to three dimensional theory <i>T</i>[<i>L</i>(<i>p</i>, 1)] whose partition function, according to 3d-3d correspondence, is equivalent to Chern-Simons invariants with complex gauge group on <i>L</i>(<i>p</i>, 1). We test the statement in Chapter 2 by taking <i>k</i> = 0 and calculating the supersymmetric index. We find a full agreement between two seemingly distinct quantities. In particular, when <i>p</i> = 1, we see the familiar <i>S</i>³ partition function of Chern-Simons theory arises from the index of a free theory.</p> <p>We then move on in Chapter 3 to consider <i>M</i>₃ = <i>S</i>¹ × Σ, and twisted compactification on general Riemann surface Σ with tame punctures. The twisted partition function of lens space theory <i>T</i>[<i>L</i>(<i>k</i>, 1)] on <i>S</i>¹ × Σ computes the graded dimension of the Hilbert space after geometrically quantizing Hitchin moduli space <i>M</i>_H, dubbed as "tame Hitchin characters" or "equivariant Verlinde formula". We show that this quantity can be computed from the "Coulomb branch index" of the class <i>S</i> theory <i>T</i>[Σ] on <i>L</i>(<i>k</i>, 1) × <i>S</i>¹. The gauge groups on two sides of the equivalence are naturally <i>G</i> and the Langlands dual group ^<i>L</i><i>G</i>. We check explicitly the relation for <i>G</i> = <i>SU</i>(2) or <i>SO</i>(3). We also consider more general case where <i>G</i> is <i>SU</i>(<i>N</i>) or <i>PSU</i>(<i>N</i>) and show that the <i>SU</i>(<i>N</i>) equivariant Verlinde formula can be derived using field theory via (generalized) Argyres-Seiberg duality.</p> <p>As a further application, in Chapter 4 we use Coulomb branch indices of Argyres-Douglas theories on <i>S</i>¹ × <i>L</i>(<i>k</i>, 1) to quantize moduli spaces <i>M</i>_H of wild/irregular Hitchin systems. We obtain the "wild Hitchin characters", and observe that the characters can always be written as a sum over fixed points in <i>M</i>_H under the <i>U</i>(1) Hitchin action, and a limit of them can be identified with matrix elements of the vii modular transform <i>ST^kS</i> in certain vertex operator algebras. The appearance of vertex operator algebras, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising.</p> <p>The BPS spectrum of superconformal theories probe the geometry of Hitchin moduli space. Conversely, physical data of superconformal theories can be read off from Hitchin moduli space as well. We study this dictionary in Chapter 5 for general Argyres-Douglas theories and obtain a refined classification. We also discuss the S-duality of these theories, and find that the weakly coupled descriptions are given by the degeneration limit of auxiliary Riemann sphere with marked points.</p> <p>Finally, in Chapter 6, we analyze classical and quantum chiral ring relations of four dimensional <i>N</i> = 1 adjoint SQCD with superpotential turned on for the adjoint field. In particular, for the mass deformed theory we obtain the complete on-shell vacuum expectation value for various gauge invariant chiral operators and find non trivial gaugino condensations. We argue that the solution of the chiral ring is in one-to-one correspondence with supersymmetric vacua, provided that an additional Konishi anomaly equation is included.</p>