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Partition Function of Semichiral Fields on Sphere and General Kähler

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Partition Function of Semichiral Fields on Sphere and General Kähler SSStttooonnnyyy BBBrrrooooookkk UUUnnniiivvveeerrrsssiiitttyyy The official electronic file of this thesis or dissertation is maintained by the University Libraries on behalf of The Graduate School at Stony Brook University. ©©© AAAllllll RRRiiiggghhhtttsss RRReeessseeerrrvvveeeddd bbbyyy AAAuuuttthhhooorrr... Some Studies on Partition Functions in Quantum Field Theory and Statistical Mechanics A Dissertation presented by Jun Nian to The Graduate School in Partial Fulllment of the Requirements for the Degree of Doctor of Philosophy in Physics Stony Brook University August 2015 Copyright by Jun Nian 2015 Stony Brook University The Graduate School Jun Nian We, the dissertation committe for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation. Martin Ro£ek - Dissertation Advisor Professor, C. N. Yang Institute for Theoretical Physics Peter van Nieuwenhuizen - Chairperson of Defense Distinguished Professor, C. N. Yang Institute for Theoretical Physics Alexander Abanov Associate Professor, Simons Center for Geometry and Physics Dominik Schneble Associate Professor, Department of Physics and Astronomy Leon Takhtajan Professor, Department of Mathematics This dissertation is accepted by the Graduate School Charles Taber Dean of the Graduate School ii Abstract of the Dissertation Some Studies on Partition Functions in Quantum Field Theory and Statistical Mechanics by Jun Nian Doctor of Philosophy in Physics Stony Brook University 2015 Quantum eld theory and statistical mechanics share many common features in their formulations. A very important example is the similar form of the partition function. Exact computations of the partition function can help us understand non-perturbative physics and calculate many physical quantities. In this thesis, we study a number of examples in both quantum eld theory and statistical mechanics, in which one can compute the partition function exactly. Recently, people have found a consistent way of dening supersymmetric theories on curved backgrounds. This gives interesting deformations to the original supersymmetric theories de- ned on the at spacetime. Using the supersymmetric localization method one is able to calculate the exact partition functions of some supersymmetric gauge theories on compact manifolds, which provides us with many more rigorous checks of dualities and many geomet- rical properties of the models. We discuss the localization of supersymmetric gauge theories on squashed S3, round S2 and T 2. Entanglement entropy and Rényi entropy are key concepts in some branches of condensed matter physics, for instance, quantum phase transition, topological phase and quantum com- putation. They also play an increasingly important role in high energy physics and black-hole physics. A generalized and related concept is the supersymmetric Rényi entropy. In the the- iii sis we review these concepts and their relation with the partition function on a sphere. We also consider the thermal correction to the Rényi entropy at nite temperature. Partition functions can also be used to relate two apparently dierent theories. One example discussed in the thesis is the Gross-Pitaevskii equation and a string-like nonlinear sigma model. The Gross-Pitaevskii equation is known as a mean-eld description of Bose- Einstein condensates. It has some nontrivial solutions like the vortex line and the dark soliton. We give a eld theoretic derivation of these solutions, and discuss recent developments of the relation between the Gross-Pitaevskii equation and the Kardar-Parisi-Zhang equation. Moreover, a (2+1)-dimensional system consisting of only vortex solutions can be described by a statistical model called point-vortex model. We evaluate its partition function exactly, and nd a phase transtition at negative temperature. The order parameter, the critical exponent and the correlation function are also discussed. iv To the people I love v Contents 1 Introduction1 1.1 Historical Developments (before 1980) . .1 1.1.1 Quantum Theory . .1 1.1.2 Statistical Mechanics . .5 1.2 Recent Developments (after 1980) . .6 1.2.1 Quantum Theory . .7 1.2.2 Statistical Mechanics . .8 1.3 Organization of the Thesis . 10 2 Supersymmetric Localization 12 2.1 Introduction to Supersymmetric Localization . 12 2.2 Mathematical Aspects . 13 2.3 S3 Localization . 18 2.3.1 Review of Squashed S3 .......................... 18 2.3.2 Some Relevant Formulae . 20 2.3.3 Review of 3D N = 2 Supersymmetry . 23 2.3.4 Squashed S3 with U(1) × U(1) Isometry . 25 2.3.5 Squashed S3 with SU(2) × U(1) Isometry . 35 2.4 S2 Localization . 60 2.4.1 2D N = (2; 2) Supersymmetry . 60 2.4.2 Semichiral Fields on S2 .......................... 66 2.4.3 Localization on the Coulomb Branch . 71 2.5 T 2 Localization . 74 2.5.1 Hamiltonian formalism . 75 2.5.2 Path integral formalism . 78 2.5.3 Eguchi-Hanson space . 79 2.5.4 Taub-NUT space . 82 3 Entanglement Entropy 84 3.1 Introduction to Entanglement Entropy . 84 vi 3.2 Thermal Corrections to Rényi Entropy . 89 3.2.1 The Free Scalar Case . 92 3.2.2 Odd Dimensions and Contour Integrals . 94 3.3 Thermal Corrections to Entanglement Entropy . 97 3.4 Numerical Check . 98 3.5 Discussion . 101 4 BEC, String Theory and KPZ Equation 103 4.1 Gross-Pitaevskii Equation . 104 4.2 BEC and String Theory . 106 4.2.1 Derivation of the String Action . 106 4.2.2 Derrick's Theorem, 1D and 2D Solutions . 110 4.3 KPZ Equation . 113 4.3.1 Review of the KPZ Equation . 113 4.3.2 KPZ Equation and GP Equation . 114 4.3.3 KPZ Equation and String Theory . 116 5 2-Dimensional Point-Vortex Model 117 5.1 Review of the Model . 118 5.2 System in a Disc . 120 5.2.1 Review of the Approach by Smith and O'Neil . 120 5.2.2 Results for a System in Disc . 132 5.3 System in a Box . 135 5.3.1 Review of the Approach by Edwards and Taylor . 135 5.3.2 Results for a System in a Box . 141 5.4 Discussion . 145 6 Conclusion and Prospect 147 A Convention in S3 Localization 150 B S3 as an SU(2)-Group Manifold 152 C Dierent Metrics of Squashed S3 154 D BPS Solutions in S3 Localization 158 E Some Important Relations in S3 Localization 161 F 2-Dimensional N = (2; 2) Superspace 164 G Gauged Linear Sigma Model with Semichiral Superelds in Components 165 vii H Semichiral Stückelberg Field 170 I Jerey-Kirwan Residue 172 J Two-Point Functions in d = (4 + 1) Dimensions 174 K Examples of Thermal Corrections to Rényi Entropies 176 Bibliography 178 viii List of Figures 1:1 The illustration of dierent paths from classical mechanics to quantum eld theory . 3 3:1 The contour we choose to evaluate the contour integral (3.2.30).............. 95 3:2 The numerical results for the thermal correction to the Rényi entropy with n = 3 in (2+1) dimensions . 100 3:3 The numerical results for the thermal correction to the Rényi entropy with n = 3 in (3+1) dimensions . 100 3:4 The numerical results for the thermal correction to the Rényi entropy with n = 3 in (4+1) dimensions . 101 3:5 The numerical results for the thermal correction to the Rényi entropy with n = 3 in (5+1) dimensions . 101 5:1 The β-E curve for the 2-dimensional point-vortex system in a disc . 124 5:2 The S-E curve for the 2-dimensional point-vortex system in a disc . 132 5:3 The prole of n+(r) for the 2-dimensional point-vortex system in a disc at the bifurcation point . 133 5:4 The prole of n−(r) for the 2-dimensional point-vortex system in a disc at the bifurcation point . 133 5:5 The prole of σ(r) for the 2-dimensional point-vortex system in a disc at the bifurcation point . 133 5:6 The prole of n+(r) + n−(r) for the 2-dimensional point-vortex system in a disc at the bifurcation point . 133 5:7 The solution perserving the U(1) rotational symmetry with the energy above the bifurcation point . 134 5:8 The solution breaking the U(1) rotational symmetry with the energy above the bifurcation point . 134 5:9 The hDi-E curve for the 2-dimensional point-vortex system in a disc . 135 5:10 The contour we choose to evaluate the contour integral (5.3.16).............. 137 5:11 The contour we choose to evaluate the contour integral (5.3.16) after the changes of variable . 137 5:12 The contour we choose to evaluate the contour integral (5.3.26).............. 139 5:13 The S-E curve for the 2-dimensional point-vortex system in a box . 142 5:14 The β-E curve for the 2-dimensional point-vortex system in a box . 142 5:15 The correlation function hσ(~r) σ(~r + ~s)i as a function of j~sj for E = 3:2 E0 ....... 143 5:16 The correlation function hσ(~r) σ(~r + ~s)i as a function of j~sj for E = −0:4 E0 ...... 144 5:17 The βhD2i-E curve for the 2-dimensional point-vortex system in a box . 145 ix List of Tables 2:1 The Weyl charge, vector R-charge and axial R-charge assignment for the component elds in the semichiral multiplet . 68 2:2 Charge assignments of the components of the semichiral multiplets . 78 2:3 Charges of the components of the semichiral multiplets under the R-symmetry . 80 2:4 Charges of the components of the eld strength superelds under the R-symmetry . 80 2:5 Charges of the components of the Stückelberg eld under the R-symmetry . 83 x Acknowledgements First, I would like to thank my advisor Prof.
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