scholarly journals Partition functions of 𝒩 = 1 gauge theories on S2 × ℝ𝜀2 and duality

2020 â—˝  
Vol 35 (33) â—˝  
pp. 2050207
Author(s):  
Taro Kimura â—˝  
Jun Nian â—˝  
Peng Zhao
Keyword(s):  
Path Integral â—˝  
Gauge Theories â—˝  
Building Blocks â—˝  
Partition Functions â—˝  
Seiberg Duality â—˝  
Nekrasov Partition Functions

We compute the partition functions of [Formula: see text] gauge theories on [Formula: see text] using supersymmetric localization. The path integral reduces to a sum over vortices at the poles of [Formula: see text] and at the origin of [Formula: see text]. The exact partition functions allow us to test Seiberg duality beyond the supersymmetric index. We propose the [Formula: see text] partition functions on the [Formula: see text]-background, and show that the Nekrasov partition functions can be recovered from these building blocks.

scholarly journals Tests of Seiberg-like dualities in three dimensions

2020 â—˝  
Vol 2020 (8) â—˝  
Author(s):  
Anton Kapustin â—˝  
Brian Willett â—˝  
Itamar Yaakov
Keyword(s):  
Gauge Theories â—˝  
Three Dimensions â—˝  
Simons Theory â—˝  
Partition Functions â—˝  
Seiberg Duality â—˝  
Monopole Operators â—˝  
Real Mass â—˝  
Supersymmetric Gauge â—˝  
The Relationship â—˝  
Chern Simons

Abstract We use localization techniques to study several duality proposals for supersymmetric gauge theories in three dimensions reminiscent of Seiberg duality. We compare the partition functions of dual theories deformed by real mass terms and FI parameters. We find that Seiberg-like duality for $$ \mathcal{N} $$ N = 3 Chern-Simons gauge theories proposed by Giveon and Kutasov holds on the level of partition functions and is closely related to level-rank duality in pure Chern-Simons theory. We also clarify the relationship between the Giveon-Kutasov duality and a duality in theories of fractional M2 branes and propose a generalization of the latter. Our analysis also confirms previously known results concerning decoupled free sectors in $$ \mathcal{N} $$ N = 4 gauge theories realized by monopole operators.

scholarly journals More on topological vertex formalism for 5-brane webs with O5-plane

2021 â—˝  
Vol 2021 (4) â—˝  
Author(s):  
Hirotaka Hayashi â—˝  
Rui-Dong Zhu
Keyword(s):  
Gauge Theory â—˝  
Partition Function â—˝  
Vertex Operator â—˝  
Vertex Function â—˝  
Gauge Theories â—˝  
Partition Functions â—˝  
Topological Vertex â—˝  
Concrete Form â—˝  
Nekrasov Partition Functions â—˝  
Chern Simons

Abstract We propose a concrete form of a vertex function, which we call O-vertex, for the intersection between an O5-plane and a 5-brane in the topological vertex formalism, as an extension of the work of [1]. Using the O-vertex it is possible to compute the Nekrasov partition functions of 5d theories realized on any 5-brane web diagrams with O5-planes. We apply our proposal to 5-brane webs with an O5-plane and compute the partition functions of pure SO(N) gauge theories and the pure G2 gauge theory. The obtained results agree with the results known in the literature. We also compute the partition function of the pure SU(3) gauge theory with the Chern-Simons level 9. At the end we rewrite the O-vertex in a form of a vertex operator.

scholarly journals 6d/5d exceptional gauge theories from web diagrams

2021 â—˝  
Vol 2021 (7) â—˝  
Author(s):  
Hirotaka Hayashi â—˝  
Hee-Cheol Kim â—˝  
Kantaro Ohmori
Keyword(s):  
Gauge Theory â—˝  
Gauge Theories â—˝  
Partition Functions â—˝  
Topological Vertex â—˝  
Nekrasov Partition Functions â—˝  
Chern Simons â—˝  
The Web â—˝  
Kaluza Klein

Abstract We construct novel web diagrams with a trivalent or quadrivalent gluing for various 6d/5d theories from certain Higgsings of 6d conformal matter theories on a circle. The theories realized on the web diagrams include 5d Kaluza-Klein theories from circle compactifications of the 6d G2 gauge theory with 4 flavors, the 6d F4 gauge theory with 3 flavors, the 6d E6 gauge theory with 4 flavors and the 6d E7 gauge theory with 3 flavors. The Higgsings also give rise to 5d Kaluza-Klein theories from twisted compactifications of 6d theories including the 5d pure SU(3) gauge theory with the Chern-Simons level 9 and the 5d pure SU(4) gauge theory with the Chern-Simons level 8. We also compute the Nekrasov partition functions of the theories by applying the topological vertex formalism to the newly obtained web diagrams.

scholarly journals Boundaries, Vermas and factorisation

2021 â—˝  
Vol 2021 (4) â—˝  
Author(s):  
Mathew Bullimore â—˝  
Samuel Crew â—˝  
Daniel Zhang
Keyword(s):  
Partition Function â—˝  
Boundary Conditions â—˝  
Gauge Theories â—˝  
Building Blocks â—˝  
Partition Functions â—˝  
Verma Modules â—˝  
Coulomb Branch â—˝  
Superconformal Index â—˝  
Chiral Rings â—˝  
Real Mass

Abstract We revisit the factorisation of supersymmetric partition functions of 3d $$ \mathcal{N} $$ N = 4 gauge theories. The building blocks are hemisphere partition functions of a class of UV $$ \mathcal{N} $$ N = (2, 2) boundary conditions that mimic the presence of isolated vacua at infinity in the presence of real mass and FI parameters. These building blocks can be unambiguously defined and computed using supersymmetric localisation. We show that certain limits of these hemisphere partition functions coincide with characters of lowest weight Verma mod- ules over the quantised Higgs and Coulomb branch chiral rings. This leads to expressions for the superconformal index, twisted index and S3 partition function in terms of such characters. On the way we uncover new connections between boundary ’t Hooft anomalies, hemisphere partition functions and lowest weights of Verma modules.

Path integral for gauge theories with fermions

Physical Review D â—˝  
1980 â—˝  
Vol 21 (10) â—˝  
pp. 2848-2858 â—˝  
Author(s):  
Kazuo Fujikawa
Keyword(s):  
Path Integral â—˝  
Gauge Theories

scholarly journals Cwebs beyond three loops in multiparton amplitudes

2021 â—˝  
Vol 2021 (3) â—˝  
Author(s):  
Neelima Agarwal â—˝  
Lorenzo Magnea â—˝  
Sourav Pal â—˝  
Anurag Tripathi
Keyword(s):  
Gauge Theories â—˝  
Anomalous Dimension â—˝  
Wilson Line â—˝  
Building Blocks â—˝  
Feynman Diagrams â—˝  
Loop Order â—˝  
Abelian Gauge â—˝  
Sum Rule â—˝  
Combinatorial Properties â—˝  
Low Dimensional

Abstract Correlators of Wilson-line operators in non-abelian gauge theories are known to exponentiate, and their logarithms can be organised in terms of collections of Feynman diagrams called webs. In [1] we introduced the concept of Cweb, or correlator web, which is a set of skeleton diagrams built with connected gluon correlators, and we computed the mixing matrices for all Cwebs connecting four or five Wilson lines at four loops. Here we complete the evaluation of four-loop mixing matrices, presenting the results for all Cwebs connecting two and three Wilson lines. We observe that the conjuctured column sum rule is obeyed by all the mixing matrices that appear at four-loops. We also show how low-dimensional mixing matrices can be uniquely determined from their known combinatorial properties, and provide some all-order results for selected classes of mixing matrices. Our results complete the required colour building blocks for the calculation of the soft anomalous dimension matrix at four-loop order.

scholarly journals Gauge theories on compact toric manifolds

2021 â—˝  
Vol 111 (3) â—˝  
Author(s):  
Giulio Bonelli â—˝  
Francesco Fucito â—˝  
Jose Francisco Morales â—˝  
Massimiliano Ronzani â—˝  
Ekaterina Sysoeva â—˝  
...  
Keyword(s):  
Partition Function â—˝  
Gauge Theories â—˝  
Blow Up â—˝  
Gauge Coupling â—˝  
Piecewise Constant Function â—˝  
Partition Functions â—˝  
Piecewise Constant â—˝  
Toric Manifolds â—˝  
Holomorphic Anomaly â—˝  
The One

AbstractWe compute the $$\mathcal{N}=2$$ N = 2 supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the Kähler form with jumps along the walls where the gauge symmetry gets enhanced. The partition function on such manifolds is written as a sum over the residues of a product of partition functions on $$\mathbb {C}^2$$ C 2 . The evaluation of these residues is greatly simplified by using an “abstruse duality” that relates the residues at the poles of the one-loop and instanton parts of the $$\mathbb {C}^2$$ C 2 partition function. As particular cases, our formulae compute the SU(2) and SU(3) equivariant Donaldson invariants of $$\mathbb {P}^2$$ P 2 and $$\mathbb {F}_n$$ F n and in the non-equivariant limit reproduce the results obtained via wall-crossing and blow up methods in the SU(2) case. Finally, we show that the U(1) self-dual connections induce an anomalous dependence on the gauge coupling, which turns out to satisfy a $$\mathcal {N}=2$$ N = 2 analog of the $$\mathcal {N}=4$$ N = 4 holomorphic anomaly equations.

scholarly journals Three-dimensional đť’© = 2 supersymmetric gauge theories and partition functions on Seifert manifolds: A review

2019 â—˝  
Vol 34 (23) â—˝  
pp. 1930011 â—˝  
Author(s):  
Cyril Closset â—˝  
Heeyeon Kim
Keyword(s):  
Correlation Functions â—˝  
Gauge Theories â—˝  
Three Dimensional â—˝  
Partition Functions â—˝  
Exact Results â—˝  
Strongly Coupled â—˝  
Seifert Manifolds â—˝  
Recent Developments â—˝  
Supersymmetric Gauge â—˝  
Chern Simons

We give a pedagogical introduction to the study of supersymmetric partition functions of 3D [Formula: see text] supersymmetric Chern–Simons-matter theories (with an [Formula: see text]-symmetry) on half-BPS closed three-manifolds — including [Formula: see text], [Formula: see text], and any Seifert three-manifold. Three-dimensional gauge theories can flow to nontrivial fixed points in the infrared. In the presence of 3D [Formula: see text] supersymmetry, many exact results are known about the strongly-coupled infrared, due in good part to powerful localization techniques. We review some of these techniques and emphasize some more recent developments, which provide a simple and comprehensive formalism for the exact computation of half-BPS observables on closed three-manifolds (partition functions and correlation functions of line operators). Along the way, we also review simple examples of 3D infrared dualities. The computation of supersymmetric partition functions provides exceedingly precise tests of these dualities.

scholarly journals AN ALGORITHMIC APPROACH TO QUANTUM FIELD THEORY

2006 â—˝  
Vol 21 (03) â—˝  
pp. 405-447 â—˝  
Author(s):  
MASSIMO DI PIERRO
Keyword(s):  
Field Theory â—˝  
Quantum Field Theory â—˝  
Path Integral â—˝  
Gauge Theories â—˝  
Quantum Field â—˝  
Algorithmic Approach â—˝  
Main Emphasis â—˝  
Lattice Computation â—˝  
Theoretical Foundations â—˝  
Minimal Residual

The lattice formulation provides a way to regularize, define and compute the Path Integral in a Quantum Field Theory. In this paper, we review the theoretical foundations and the most basic algorithms required to implement a typical lattice computation, including the Metropolis, the Gibbs sampling, the Minimal Residual, and the Stabilized Biconjugate inverters. The main emphasis is on gauge theories with fermions such as QCD. We also provide examples of typical results from lattice QCD computations for quantities of phenomenological interest.

TI/PIMC METHOD WITH THE TAKAHASHI–IMADA APPROXIMATION FOR THE EQUILIBRIUM CONSTANT OF THE O + HCl ⇌ OH + Cl REACTION

2013 â—˝  
Vol 12 (04) â—˝  
pp. 1350026 â—˝  
Author(s):  
MARCIN BUCHOWIECKI
Keyword(s):  
Monte Carlo â—˝  
Temperature Dependence â—˝  
Equilibrium Constant â—˝  
Path Integral â—˝  
Thermodynamic Integration â—˝  
Partition Functions â—˝  
Integration Path â—˝  
Path Integral Monte Carlo

The thermodynamic integration/path integral Monte Carlo (TI/PIMC) method of calculating the temperature dependence of the equilibrium constant quantum mechanically is applied to O + HCl ⇌ OH + Cl reaction. The method is based upon PIMC simulations for energies of the reactants and the products and subsequently on thermodynamic integration for the ratios of partition functions. PIMC calculations are performed with the primitive approximation (PA) and the Takahashi–Imada approximation (TIA).

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