scholarly journals Partition functions of p-forms from Harish-Chandra characters

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Justin R. David ◽  
Jyotirmoy Mukherjee
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Integral Representation ◽  
Integral Transform ◽  
Entanglement Entropy ◽  
Integral Transforms ◽  
Partition Functions ◽  
De Sitter ◽  
Symmetric Tensors ◽  
Hyperbolic Cylinders

Abstract We show that the determinant of the co-exact p-form on spheres and anti-de Sitter spaces can be written as an integral transform of bulk and edge Harish-Chandra characters. The edge character of a co-exact p-form contains characters of anti-symmetric tensors of rank lower to p all the way to the zero-form. Using this result we evaluate the partition function of p-forms and demonstrate that they obey known properties under Hodge duality. We show that the partition function of conformal forms in even d + 1 dimensions, on hyperbolic cylinders can be written as integral transforms involving only the bulk characters. This supports earlier observations that entanglement entropy evaluated using partition functions on hyperbolic cylinders do not contain contributions from the edge modes. For conformal coupled scalars we demonstrate that the character integral representation of the free energy on hyperbolic cylinders and branched spheres coincide. Finally we propose a character integral representation for the partition function of p-forms on branched spheres.

scholarly journals AdS one-loop partition functions from bulk and edge characters

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Zimo Sun
Keyword(s):  
Partition Function ◽  
Integral Representation ◽  
Integral Transform ◽  
Integral Formula ◽  
Partition Functions ◽  
De Sitter Spacetime ◽  
De Sitter ◽  
Sitter Spacetime ◽  
Higher Spin ◽  
The One

Abstract We show that the one-loop partition function of any higher spin field in (d + 1)-dimensional Anti-de Sitter spacetime can be expressed as an integral transform of an SO(2, d) bulk character and an SO(2, d − 2) edge character. We apply this character integral formula to various higher-spin Vasiliev gravities and find miraculous (almost) cancellations between bulk and edge characters that lead to agreement with the predictions of HS/CFT holography. We also discuss the relation between the character integral representation and the Rindler-AdS thermal partition function.

scholarly journals Partition functions of higher derivative conformal fields on conformally related spaces

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Jyotirmoy Mukherjee
Keyword(s):  
Partition Function ◽  
Integral Representation ◽  
Gauge Field ◽  
Partition Functions ◽  
Conformal Vector ◽  
Higher Derivative ◽  
Conformal Fields ◽  
Vector Gauge ◽  
Bulk Part ◽  
Hyperbolic Cylinders

Abstract The character integral representation of one loop partition functions is useful to establish the relation between partition functions of conformal fields on Weyl equivalent spaces. The Euclidean space Sa × AdSb can be mapped to Sa+b provided Sa and AdSb are of the same radius. As an example, to begin with, we show that the partition function in the character integral representation of conformally coupled free scalars and fermions are identical on Sa × AdSb and Sa+b. We then demonstrate that the partition function of higher derivative conformal scalars and fermions are also the same on hyperbolic cylinders and branched spheres. The partition function of the four-derivative conformal vector gauge field on the branched sphere in d = 6 dimension can be expressed as an integral over ‘naive’ bulk and ‘naive’ edge characters. However, the partition function of the conformal vector gauge field on $$ {S}_q^1 $$ S q 1 × AdS5 contains only the ‘naive’ bulk part of the partition function. This follows the same pattern which was observed for the partition of conformal p-form fields on hyperbolic cylinders. We use the partition function of higher derivative conformal fields on hyperbolic cylinders to obtain a linear relationship between the Hofman-Maldacena variables which enables us to show that these theories are non-unitary.

scholarly journals Hyperbolic cylinders and entanglement entropy: gravitons, higher spins, p-forms

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Justin R. David ◽  
Jyotirmoy Mukherjee
Keyword(s):  
Stress Tensor ◽  
Entanglement Entropy ◽  
Partition Functions ◽  
Point Function ◽  
Expectation Value ◽  
Massive Spin ◽  
Twist Operator ◽  
Higher Spins ◽  
Hyperbolic Cylinders ◽  
Kaluza Klein

Abstract We show that the entanglement entropy of D = 4 linearized gravitons across a sphere recently computed by Benedetti and Casini coincides with that obtained using the Kaluza-Klein tower of traceless transverse massive spin-2 fields on S1× AdS3. The mass of the constant mode on S1 saturates the Brietenholer-Freedman bound in AdS3. This condition also ensures that the entanglement entropy of higher spins determined from partition functions on the hyperbolic cylinder coincides with their recent conjecture. Starting from the action of the 2-form on S1× AdS5 and fixing gauge, we evaluate the entanglement entropy across a sphere as well as the dimensions of the corresponding twist operator. We demonstrate that the conformal dimensions of the corresponding twist operator agrees with that obtained using the expectation value of the stress tensor on the replica cone. For conformal p-forms in even dimensions it obeys the expected relations with the coefficients determining the 3-point function of the stress tensor of these fields.

scholarly journals Deformations of JT gravity via topological gravity and applications

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Stefan Förste ◽  
Hans Jockers ◽  
Joshua Kames-King ◽  
Alexandros Kanargias
Keyword(s):  
Phase Transition ◽  
Partition Function ◽  
Form Factors ◽  
Temperature Limit ◽  
Partition Functions ◽  
De Sitter ◽  
Spectral Form ◽  
Kdv Hierarchy ◽  
Topological Gravity ◽  
Perturbative Corrections

Abstract We study the duality between JT gravity and the double-scaled matrix model including their respective deformations. For these deformed theories we relate the thermal partition function to the generating function of topological gravity correlators that are determined as solutions to the KdV hierarchy. We specialise to those deformations of JT gravity coupled to a gas of defects, which conforms with known results in the literature. We express the (asymptotic) thermal partition functions in a low temperature limit, in which non-perturbative corrections are suppressed and the thermal partition function becomes exact. In this limit we demonstrate that there is a Hawking-Page phase transition between connected and disconnected surfaces for this instance of JT gravity with a transition temperature affected by the presence of defects. Furthermore, the calculated spectral form factors show the qualitative behaviour expected for a Hawking-Page phase transition. The considered deformations cause the ramp to be shifted along the real time axis. Finally, we comment on recent results related to conical Weil-Petersson volumes and the analytic continuation to two-dimensional de Sitter space.

scholarly journals Superstrings in thermal anti-de Sitter space

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Sujay K. Ashok ◽  
Jan Troost
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Isometry Group ◽  
Three Dimensional ◽  
Open Interval ◽  
Energy Calculation ◽  
De Sitter ◽  
Half Open Interval ◽  
Sitter Spacetime ◽  
Discrete Representations

Abstract We revisit the calculation of the thermal free energy for string theory in three-dimensional anti-de Sitter spacetime with Neveu-Schwarz-Neveu-Schwarz flux. The path integral calculation is exploited to confirm the off-shell Hilbert space and we find that the Casimir of the discrete representations of the isometry group takes values in a half-open interval. We extend the free energy calculation to the case of superstrings, calculate the boundary toroidal twisted partition function in the Ramond-Ramond sector, and prove lower bounds on the boundary conformal dimension from the bulk perspective. We classify Ramond-Ramond ground states and construct their second quantized partition function. The partition function exhibits intriguing modular properties.

scholarly journals The Bakerian lecture - The anomalous specific heats of crystals, with special reference to the contribution of molecular rotations

1935 ◽  
Vol 151 (872) ◽  
pp. 1-22 ◽  
Author(s):  
Ralph Howard Fowler
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Specific Heat ◽  
Energy Content ◽  
Partition Functions ◽  
Free Energy Function ◽  
General Survey ◽  
Final State ◽  
Specific Heats ◽  
Molecular Rotations

The primary purpose of this lecture is to describe a tentative theory, which I have recently tried to improve, of the contribution that molecular rotations may make to the specific heats and in general to the equilibrium properties of crystals. The occasion of the Bakerian lecture, however, seems a fitting one for a general survey rather than for the detailed exposition of a particular theory, which at best is still far from a final state. I shall spend most of my time therefore in such a general survey of the theory of the specific heats of crystalline solids. Strictly speaking, the theory is not so much a theory of the specific heat as a theoretical construction of the free-energy function of the solid in terms of an assumed model for the atoms or molecules constituting its crystal lattice. From this free energy, the total energy content, specific heat, equation of state and other related equilibrium properties of the solid can of course be deduced by differentiation, and specific heats provide the most striking property easily submitted to comparison with experiment. When we set out to construct theoretical free-energy functions in this way, using the methods of statistical mechanics, the function that we actually construct is the partition function, the quantum mechanical generalization of Gibb’s phase integral ∫ e -E/ k T d Ω over the phase space of solid or other system. The free energy is merely - k T log (partition function). My first object is therefore to describe the present state of the theory of the partition functions of solids and their success or failure in describing the observed facts. This will lead us on naturally to the discussion of various types of anomaly in the specific heat curves not provided for by the simpler versions of the theory, and so to a description of attempts that have been made to explain them, among which the theory of molecular rotations in solids finds its natural place.

scholarly journals Quantum holographic entanglement entropy to all orders in 1/N expansion

2020 ◽  
Vol 2020 (4) ◽  
Author(s):  
Shinji Hirano
Keyword(s):  
Partition Function ◽  
Minimal Surface ◽  
Path Integrals ◽  
Entanglement Entropy ◽  
Saddle Points ◽  
De Sitter ◽  
Three Sphere ◽  
Holographic Entanglement Entropy ◽  
Bulk Volume ◽  
Minimal Surface Area

Abstract We study holographic entanglement entropy in four-dimensional quantum gravity with negative cosmological constant. By using the replica trick and evaluating path integrals in the minisuperspace approximation, in conjunction with the Wheeler–DeWitt equation, we compute quantum corrections to the holographic entanglement entropy for a circular entangling surface on the boundary three-sphere. Similarly to our previous work on the sphere partition function, the path integrals are dominated by a replica version of asymptotically anti-de Sitter conic geometries at saddle points. As expected from a general conformal field theory argument, the final result is minus the free energy on the three-sphere, which agrees with the logarithm of the Airy partition function for the Aharony–Bergman–Jafferis–Maldacena theory that sums up all perturbative $1/N$ corrections despite the absence of supersymmetries. The all-order holographic entanglement entropy cleanly splits into two parts, (1) the $1/N$-corrected Ryu–Takayanagi minimal surface area and (2) the bulk entanglement entropy across the minimal surface, as suggested in the earlier literature. It is explicitly shown that the former comes from the localized conical singularity of the replica geometries and the latter from the replication of the bulk volume.

A canonical ensemble derivation of the McMillan-Mayer solution theory

1983 ◽  
Vol 48 (10) ◽  
pp. 2888-2892 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Partition Function ◽  
Energy Function ◽  
Special Function ◽  
Helmholtz Free Energy ◽  
Canonical Ensemble ◽  
Free Energy Function ◽  
Solution Theory ◽  
Pure Solvent ◽  
The Common

A special free energy function is defined for a solution in the osmotic equilibrium with pure solvent. The partition function of the solution is derived at the McMillan-Mayer level and it is related to this special function in the same manner as the common partition function of the system to its Helmholtz free energy.

scholarly journals Free partition functions and an averaged holographic duality

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.

scholarly journals Gauge theories on compact toric manifolds

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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