Chiral algebras and partition functions

This chapter discusses an approximate approach—transition-state theory—to the calculation of rate constants for bimolecular reactions. A reaction coordinate is identified from a normal-mode coordinate analysis of the activated complex, that is, the supermolecule on the saddle-point of the potential energy surface. Motion along this coordinate is treated by classical mechanics and recrossings of the saddle point from the product to the reactant side are neglected, leading to the result of conventional transition-state theory expressed in terms of relevant partition functions. Various alternative derivations are presented. Corrections that incorporate quantum mechanical tunnelling along the reaction coordinate are described. Tunnelling through an Eckart barrier is discussed and the approximate Wigner tunnelling correction factor is derived in the limit of a small degree of tunnelling. It concludes with applications of transition-state theory to, for example, the F + H2 reaction, and comparisons with results based on quasi-classical mechanics as well as exact quantum mechanics.

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Hamilton-Jacobi Theory

10.1093/oso/9780198822370.003.0019 ◽

2018 ◽

Author(s):

Peter Mann

Keyword(s):

Phase Space ◽

Bbgky Hierarchy ◽

Distribution Functions ◽

Symplectic Integrators ◽

Partition Functions ◽

Von Neumann ◽

Equipartition Theorem ◽

Recurrence Theorem ◽

Phase Amplitude ◽

Canonical Ensembles

This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

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BMS field theories and Weyl anomaly

Journal of High Energy Physics ◽

10.1007/jhep07(2021)101 ◽

2021 ◽

Vol 2021 (7) ◽

Author(s):

Arjun Bagchi ◽

Sudipta Dutta ◽

Kedar S. Kolekar ◽

Punit Sharma

Keyword(s):

Three Dimensions ◽

Field Theories ◽

Partition Functions ◽

Two Dimensional ◽

Speed Of Light ◽

Asymptotically Flat ◽

Weyl Invariance ◽

Flat Spacetimes ◽

Liouville Action ◽

Null Surfaces

Abstract Two dimensional field theories with Bondi-Metzner-Sachs symmetry have been proposed as duals to asymptotically flat spacetimes in three dimensions. These field theories are naturally defined on null surfaces and hence are conformal cousins of Carrollian theories, where the speed of light goes to zero. In this paper, we initiate an investigation of anomalies in these field theories. Specifically, we focus on the BMS equivalent of Weyl invariance and its breakdown in these field theories and derive an expression for Weyl anomaly. Considering the transformation of partition functions under this symmetry, we derive a Carrollian Liouville action different from ones obtained in the literature earlier.

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Estimating reciprocal partition functions to enable design space sampling

The Journal of Chemical Physics ◽

10.1063/5.0025358 ◽

2020 ◽

Vol 153 (20) ◽

pp. 204102

Author(s):

Alex Albaugh ◽

Todd R. Gingrich

Keyword(s):

Design Space ◽

Partition Functions

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Parafermionization, bosonization, and critical parafermionic theories

Journal of High Energy Physics ◽

10.1007/jhep04(2021)285 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

Yuan Yao ◽

Akira Furusaki

Keyword(s):

Field Theory ◽

Conformal Field Theory ◽

Degrees Of Freedom ◽

Lattice Models ◽

Partition Functions ◽

Minimal Models ◽

Fermionic Model ◽

Conformal Field ◽

Critical Theories ◽

Wigner Transformation

AbstractWe formulate a ℤk-parafermionization/bosonization scheme for one-dimensional lattice models and field theories on a torus, starting from a generalized Jordan-Wigner transformation on a lattice, which extends the Majorana-Ising duality atk= 2. The ℤk-parafermionization enables us to investigate the critical theories of parafermionic chains whose fundamental degrees of freedom are parafermionic, and we find that their criticality cannot be described by any existing conformal field theory. The modular transformations of these parafermionic low-energy critical theories as general consistency conditions are found to be unconventional in that their partition functions on a torus transform differently from any conformal field theory whenk >2. Explicit forms of partition functions are obtained by the developed parafermionization for a large class of critical ℤk-parafermionic chains, whose operator contents are intrinsically distinct from any bosonic or fermionic model in terms of conformal spins and statistics. We also use the parafermionization to exhaust all the ℤk-parafermionic minimal models, complementing earlier works on fermionic cases.

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Free partition functions and an averaged holographic duality

Journal of High Energy Physics ◽

10.1007/jhep01(2021)130 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Nima Afkhami-Jeddi ◽

Henry Cohn ◽

Thomas Hartman ◽

Amirhossein Tajdini

Keyword(s):

Partition Function ◽

Spectral Gap ◽

Central Charge ◽

Three Dimensional ◽

Two Dimensions ◽

Three Dimensions ◽

Partition Functions ◽

General Constraints ◽

Dimensional Gravity ◽

Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

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Fermi gas approach to general rank theories and quantum curves

Journal of High Energy Physics ◽

10.1007/jhep10(2020)158 ◽

2020 ◽

Vol 2020 (10) ◽

Author(s):

Naotaka Kubo

Keyword(s):

Matrix Models ◽

Critical Role ◽

Three Dimensional ◽

Fermi Gas ◽

Fermi Gases ◽

Partition Functions ◽

Density Matrices ◽

General Rank ◽

Chern Simons ◽

The Ideal

Abstract It is known that matrix models computing the partition functions of three-dimensional $$ \mathcal{N} $$ N = 4 superconformal Chern-Simons theories described by circular quiver diagrams can be written as the partition functions of ideal Fermi gases when all the nodes have equal ranks. We extend this approach to rank deformed theories. The resulting matrix models factorize into factors depending only on the relative ranks in addition to the Fermi gas factors. We find that this factorization plays a critical role in showing the equality of the partition functions of dual theories related by the Hanany-Witten transition. Furthermore, we show that the inverses of the density matrices of the ideal Fermi gases can be simplified and regarded as quantum curves as in the case without rank deformations. We also comment on four nodes theories using our results.

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Gauge theories on compact toric manifolds

Letters in Mathematical Physics ◽

10.1007/s11005-021-01419-9 ◽

2021 ◽

Vol 111 (3) ◽

Author(s):

Giulio Bonelli ◽

Francesco Fucito ◽

Jose Francisco Morales ◽

Massimiliano Ronzani ◽

Ekaterina Sysoeva ◽

...

Keyword(s):

Partition Function ◽

Gauge Theories ◽

Blow Up ◽

Gauge Coupling ◽

Piecewise Constant Function ◽

Partition Functions ◽

Piecewise Constant ◽

Toric Manifolds ◽

Holomorphic Anomaly ◽

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.

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