CaltechTHESIS
A Caltech Library Service

# Knots, Trees, and Fields: Common Ground Between Physics and Mathematics

## Citation

Saberi, Ingmar Akira (2017) Knots, Trees, and Fields: Common Ground Between Physics and Mathematics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9VX0DHZ. http://resolver.caltech.edu/CaltechTHESIS:07282016-144651775

## Abstract

One main theme of this thesis is a connection between mathematical physics (in particular, the three-dimensional topological quantum field theory known as Chern-Simons theory) and three-dimensional topology. This connection arises because the partition function of Chern-Simons theory provides an invariant of three-manifolds, and the Wilson-loop observables in the theory define invariants of knots. In the first chapter, we review this connection, as well as more recent work that studies the classical limit of quantum Chern-Simons theory, leading to relations to another knot invariant known as the $A$-polynomial. (Roughly speaking, this invariant can be thought of as the moduli space of flat $SL\left(2,C\right)$ connections on the knot complement.) In fact, the connection can be deepened: through an embedding into string theory, categorifications of polynomial knot invariants can be understood as spaces of BPS states.

We go on to study these homological knot invariants, and interpret spectral sequences that relate them to one another in terms of perturbations of supersymmetric theories. Our point is more general than the application to knots; in general, when one perturbs any modulus of a supersymmetric theory and breaks a symmetry, one should expect a spectral sequence to relate the BPS states of the unperturbed and perturbed theories. We consider several diverse instances of this general lesson. In another chapter, we consider connections between supersymmetric quantum mechanics and the de Rham version of homotopy theory developed by Sullivan; this leads to a new interpretation of Sullivan's minimal models, and of Massey products as vacuum states which are entangled between different degrees of freedom in these models.

We then turn to consider a discrete model of holography: a Gaussian lattice model defined on an infinite tree of uniform valence. Despite being discrete, the matching of bulk isometries and boundary conformal symmetries takes place as usual; the relevant group is $PGL\left(2,Q$p), and all of the formulas developed for holography in the context of scalar fields on fixed backgrounds have natural analogues in this setting. The key observation underlying this generalization is that the geometry underlying AdS3/CFT2 can be understood algebraically, and the base field can therefore be changed while maintaining much of the structure. Finally, we give some analysis of $A$-polynomials under change of base (to finite fields), bringing things full circle.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Topological quantum field theory; knots and links; holography; BPS states.
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Physics
Thesis Availability:Public (worldwide access)
• Gukov, Sergei
Group:Caltech Theory
Thesis Committee:
• Gukov, Sergei (chair)
• Schwarz, John H.
• Marcolli, Matilde
• Spiropulu, Maria
Defense Date:16 June 2016
Funders:
Funding AgencyGrant Number
U.S. Department of EnergyDE-SC-0011632
U.S. Department of EnergyDE-FG03-92-ER40701FG-02
National Science FoundationPHY-0757647
Record Number:CaltechTHESIS:07282016-144651775
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:07282016-144651775
DOI:10.7907/Z9VX0DHZ
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/1512.07883arXivPreprint; excerpted for Chap. 3
http://theory.caltech.edu/~saberiAuthorPersonal webpage
http://dx.doi.org/10.1007/JHEP03(2016)004DOIPublished version of article excerpted in Chap. 3
http://dx.doi.org/10.1090/conm/613/12235DOIPublished version of article constituting Chap. 1
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9896
Collection:CaltechTHESIS
Deposited By: Ingmar Saberi
Deposited On:22 Aug 2016 22:45