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Motivated by the prospect of constraining microscopic models, we calculate the exact one-loop corrected de Sitter entropy (the logarithm of the sphere partition function) for every effective field theory of quantum gravity, with particles in arbitrary spin representations. In doing so, we universally relate the sphere partition function to the quotient of a quasi-canonical bulk and a Euclidean edge partition function, given by integrals of characters encoding the bulk and edge spectrum of the observable universe. Expanding the bulk character splits the bulk (entanglement) entropy into quasinormal mode (quasiqubit) contributions. For 3D higher-spin gravity formulated as an sl($n$) Chern-Simons theory, we obtain all-loop exact results. Further to this, we show that the theory has an exponentially large landscape of de Sitter vacua with quantum entropy given by the absolute value squared of a topological string partition function. For generic higher-spin gravity, the formalism succinctly relates dS, AdS$^\pm$ and conformal results. Holography is exhibited in quasi-exact bulk-edge cancelation.

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We study the spectrum of semiclassical rotating strings in de Sitter space and its consistency. Even though a naive extrapolation of the linear Regge trajectory on flat space implies a violation of the Higuchi bound (a unitarity bound on the mass of higher-spin particles in de Sitter space), the curved space effects turn out to modify the trajectory to respect the bound. Interestingly, as a consequence of accelerated expansion, there exists a maximum spin for each Regge trajectory, which is helpful to make the spectrum consistent with the Higuchi bound, but at the same time, it could be an obstruction to stringy UV completion based on an infinite higher-spin tower. By pushing further this observation, we demonstrate that the vacuum energy V inflating the Universe has to be bounded by the string scale Ms as V≲Ms4, if UV completion is achieved with the leading Regge trajectory of higher spin states up to the 4D Planck scale. Its application to inflation in the early Universe implies an upper bound on the tensor-to-scalar ratio, r≲0.01×(Ms/1016 GeV)4, which is within the scope of the near future CMB experiments. We also discuss another possibility that UV completion is achieved by multiple Regge trajectories.

We study aspects of Jackiw-Teitelboim (JT) quantum gravity in two-dimensional nearly de Sitter (dS) spacetime, as well as pure de Sitter quantum gravity in three dimensions. These are each theories of boundary modes, which include a reparameterization field on each connected component of the boundary as well as topological degrees of freedom. In two dimensions, the boundary theory is closely related to the Schwarzian path integral, and in three dimensions to the quantization of coadjoint orbits of the Virasoro group. Using these boundary theories we compute loop corrections to the wavefunction of the universe, and investigate gravitational contributions to scattering.Along the way, we show that JT gravity in dS2 is an analytic continuation of JT gravity in Euclidean AdS2, and that pure gravity in dS3 is a continuation of pure gravity in Euclidean AdS3. We define a genus expansion for de Sitter JT gravity by summing over higher genus generalizations of surfaces used in the Hartle-Hawking construction. Assuming a conjecture regarding the volumes of moduli spaces of such surfaces, we find that the de Sitter genus expansion is the continuation of the recently discovered AdS genus expansion. Then both may be understood as coming from the genus expansion of the same double-scaled matrix model, which would provide a non-perturbative completion of de Sitter JT gravity.A preprint version of the article is available at ArXiv.

We model the back-reaction of a static observer in four-dimensional de Sitter spacetime by means of a singular ℤq quotient. The set of fixed points of the ℤq action consists of a pair of codimension two minimal surfaces given by 2-spheres in the Euclidean geometry. The introduction of an orbifold parameter q > 1 permits the construction of an effective action for the bulk gravity theory with support on each of these minimal surfaces. The effective action corresponds to that of Liouville field theory on a 2-sphere with a finite vacuum expectation value of the Liouville field. The intrinsic Liouville theory description yields a thermal Cardy entropy that we reintrepret as a modular free energy at temperature T = q−1, whereupon the Gibbons-Hawking entropy arises as the corresponding modular entropy. We further observe that in the limit q → ∞ the four-dimensional geometry reduces to that of global dS3 spacetime, where the two original minimal surfaces can be mapped to the future and past infinities of dS3 by means of a double Wick rotation. In this limit, the Liouville theories on the minimal surfaces become boundary theories at zero temperature whose total central charge equals that computed using the dS3/CFT2 correspondence.A preprint version of the article is available at ArXiv.

In this brief note we consider the interaction between high spin excitations in string theory along the Regge trajectory and the Higuchi bound in de Sitter space. There is always a point along the Regge trajectory where the Higuchi bound is violated. However, this point precisely coincides with a string whose length is of order the de Sitter Hubble scale. String theory therefore manifests no immediate inconsistency as long as the string scale Ms is above the Hubble scale H. However, an implication is that the Regge trajectory must be significantly modified at some ultraviolet scale. Insisting that this modification should occur no earlier than the Planck scale would lead to a bound on the string scale of Ms>HMp.

A bstract
We use the Einstein-Hilbert gravitational path integral to investigate gravita- tional entanglement at leading order O (1 /G ). We argue that semiclassical states prepared by a Euclidean path integral have the property that projecting them onto a subspace in which the Ryu-Takayanagi or Hubeny-Rangamani-Takayanagi surface has definite area gives a state with a flat entanglement spectrum at this order in gravitational perturbation theory. This means that the reduced density matrix can be approximated as proportional to the identity to the extent that its Renyi entropies Sn are independent of n at this order. The n -dependence of Sn in more general states then arises from sums over the RT/HRT- area, which are generally dominated by different values of this area for each n . This provides a simple picture of gravitational entanglement, bolsters the connection between holographic systems and tensor network models, clarifies the bulk interpretation of alge- braic centers which arise in the quantum error-correcting description of holography, and strengthens the connection between bulk and boundary modular Hamiltonians described by Jafferis, Lewkowycz, Maldacena, and Suh.

A bstract
If the graviton is the only high spin particle present during inflation, then the form of the observable tensor three-point function is fixed by de Sitter symmetry at leading order in slow-roll, regardless of the theory, to be a linear combination of two possible shapes. This is because there are only a fixed number of possible on-shell cubic structures through which the graviton can self-interact. If additional massive spin-2 degrees of freedom are present, more cubic interaction structures are possible, including those containing interactions between the new fields and the graviton, and self-interactions of the new fields. We study, in a model-independent way, how these interactions can lead to new shapes for the tensor bispectrum. In general, these shapes cannot be computed analytically, but for the case where the only new field is a partially massless spin-2 field we give simple expressions. It is possible for the contribution from additional spin-2 fields to be larger than the intrinsic Einstein gravity bispectrum and provides a mechanism for enhancing the size of the graviton bispectrum relative to the graviton power spectrum.

A bstract
We study fermionic bulk fields in the dS/CFT dualities relating $$ \mathcal{N} $$ N = 2 su- persymmetric Euclidean vector models with reversed spin-statistics in three dimensions to supersymmetric Vasiliev theories in four-dimensional de Sitter space. These dualities specify the Hartle-Hawking wave function in terms of the partition function of deforma- tions of the vector models. We evaluate this wave function in homogeneous minisuperspace models consisting of supersymmetry-breaking combinations of a half-integer spin field with either a scalar, a pseudoscalar or a metric squashing. The wave function appears to be well-behaved and globally peaked at or near the supersymmetric de Sitter vacuum, with a low amplitude for large deformations. Its behavior in the semiclassical limit qualitatively agrees with earlier bulk computations both for massless and massive fermionic fields.

A bstract
It has been proposed that a certain ℤ N orbifold, analytically continued in N , can be used to describe the thermodynamics of Rindler space in string theory. In this paper, we attempt to implement this idea for the open-string sector. The most interesting result is that, although the orbifold is tachyonic for positive integer N , the tachyon seems to disappear after analytic continuation to the region that is appropriate for computing Tr $$ {\rho}^{\mathcal{N}} $$ ρ N , where ρ is the density matrix of Rindler space and Re $$ \mathcal{N} $$ N > 1. Analytic continuation of the full orbifold conformal field theory remains a challenge, but we find some evidence that if such analytic continuation is possible, the resulting theory is a logarithmic conformal field theory, necessarily nonunitary.

A bstract
The zeta function of an arbitrary field in ( d + 1)-dimensional anti-de Sitter (AdS) spacetime is expressed as an integral transform of the corresponding so (2 , d ) representation character, thereby extending the results of [ arXiv:1603.05387 ] for AdS 4 and AdS 5 to arbitrary dimensions. The integration in the variables associated with the so ( d ) part of the character can be recast into a more explicit form using derivatives. The explicit derivative expressions are presented for AdS d +1 with d = 2 , 3 , 4 , 5 , 6.

A bstract
We compute the one-loop free energies of the type-A ℓ and type-B ℓ higher-spin gravities in ( d + 1)-dimensional anti-de Sitter (AdS d +1 ) spacetime. For large d and ℓ, these theories have a complicated field content, and hence it is difficult to compute their zeta functions using the usual methods. Applying the character integral representation of zeta function developed in the companion paper [ arXiv:1805.05646 ] to these theories, we show how the computation of their zeta function can be shortened considerably. We find that the results previously obtained for the massless theories ( ℓ = 1) generalize to their partially-massless counterparts (arbitrary ℓ ) in arbitrary dimensions.

Entanglement entropy for spatial subregions is difficult to define in string theory because of the extended nature of strings. Here we propose a definition for bosonic open strings using the framework of string field theory. The key difference (compared to ordinary quantum field theory) is that the subregion is chosen inside a Cauchy surface in the “space of open string configurations.” We first present a simple calculation of this entanglement entropy in free light-cone string field theory, ignoring subtleties related to the factorization of the Hilbert space. We reproduce the answer expected from an effective field theory point of view, namely a sum over the one-loop entanglement entropies corresponding to all the particle-excitations of the string, and further show that the full string theory regulates ultraviolet divergences in the entanglement entropy. We then revisit the question of factorization of the Hilbert space by analyzing the covariant phase-space associated with a subregion in Witten’s covariant string field theory. We show that the pure gauge (i.e., BRST exact) modes in the string field become dynamical at the entanglement cut. Thus, a proper definition of the entropy must involve an extended Hilbert space, with new stringy edge modes localized at the entanglement cut.

In this Letter we provide a complete holographic reconstruction of the cubic couplings in the minimal bosonic higher spin theory in (d+1)-dimensional anti– de Sitter space. For this purpose, we also determine the operator-product expansion coefficients of all single-trace conserved currents in the d-dimensional free scalar O(N) vector model, and we compute the tree-level three-point Witten diagram amplitudes for a generic cubic interaction of higher spin gauge fields in the metriclike formulation.

We compute logarithmic corrections to the entropy of supersymmetric extremal black holes in N = 4 and N = 8 supersymmetric string theories and find results in perfect agreement with the microscopic results. In particular these logarithmic corrections vanish for quarter BPS black holes in N = 4 supersymmetric theories, but has a finite coefficient for 1/8 BPS black holes in the N = 8 supersymmetric theory. On the macroscopic side these computations require evaluating the one loop determinant of massless fields around the near horizon geometry, and include, in particular, contributions from dynamical four dimensional gravitons propagating in the loop. Thus our analysis provides a test of one loop quantum gravity corrections to the black hole entropy, or equivalently of the AdS 2 /CF T 1 correspondence. We also extend our analysis to N = 2 supersymmetric STU model and make a prediction for the logarithmic correction to the black hole entropy in that theory.

We consider the universal logarithmic divergent term in the entanglement
entropy of gauge fields in the Minkowski vacuum with an entangling sphere.
Employing the mapping in arXiv:1102.0440, we analyze the corresponding thermal
entropy on open Einstein universe and on the static patch of de Sitter. Using
the heat kernel of the vector Laplacian we resolve a discrepancy between the
free field calculation and the expected Euler conformal anomaly. The resolution
suggests a modification of the well known formulas for the vacuum expectation
value of the spin-1 energy-momentum tensor on conformally flat space-times.

The spectral function (also known as the Plancherel measure), which gives the spectral distribution of the eigenvalues of the Laplace–Beltrami operator, is calculated for a field of arbitrary integer spin (i.e., for a symmetric traceless and divergence-free tensor field) on the N-dimensional real hyperbolic space (HN). In odd dimensions the spectral function μ(λ) is analytic in the complex λ plane, while in even dimensions it is a meromorphic function with simple poles on the imaginary axis, as in the scalar case. For N even a simple relation between the residues of μ(λ) at these poles and the (discrete) degeneracies of the Laplacian on the N sphere (SN) is established. A similar relation between μ(λ) at discrete imaginary values of λ and the degeneracies on SN is found to hold for N odd. These relations are generalizations of known results for the scalar field. The zeta functions for fields of integer spin on HN are written down. Then a relation between the integer-spin zeta functions on HN and SN is obtained. Applications of the zeta functions presented here to quantum field theory of integer spin in anti-de Sitter space–time are pointed out.

Using the approach of Fradkin and Vasiliev (1987), an action is constructed in 2+1 spacetime dimensions describing interacting massless fields of all integer and half-integer spins s>or=3/2. The action is associated with an infinite-dimensional superalgebra, denoted shs(1, 2)(+)shs(1,2). Truncation to the spin 3/2-spin 2 sector gives the (1,1) type anti-de Sitter (AdS) supergravity theory corresponding to osp(1,2; R)(+)osp(1,2; R). Various properties of the D=3 higher-spin theory, and its relevance to the higher-spin problem in four dimensions, are discussed.

In this review we describe the statistical mechanics of quantum systems in the presence of a Killing horizon and compare statistical-mechanical and 1-loop contributions to black-hole entropy. The study of these questions was motivated by attempts to explain the entropy of black holes as a statistical-mechanical entropy of quantum fields propagating near the black-hole horizon. We provide an introduction to this field of research and review its results. In particular, we discuss the relation between the statistical-mechanical entropy of quantum fields and the Bekenstein-Hawking entropy in the standard scheme with renormalization of gravitational coupling constants and in the theories of induced gravity.

We show that there exists a one-parameter family of infinite-dimensional algebras that includes the bosonicd=3 Fradkin-Vasiliev higher-spin algebra and the non-Euclidean version of the algebra of area-preserving diffeomorphisms of the two-sphereS
2 as two distinct members. The non-Euclidean version of the area preserving algebra corresponds to the algebra of area-preserving diffeomorphisms of the hyperbolic spaceS
1,1, and can be rewritten as
limN ® ¥ su(N,N)\mathop {\lim }\limits_{N \to \infty } su(N,N)
. As an application of our results, we formulate a newd=2+1 massless higher-spin field theory as the gauge theory of the area-preserving diffeomorphisms ofS
1,1.

A definition for the entanglement entropy in a gauge theory was given recently in arXiv:1501.02593. Working on a spatial lattice, it involves embedding the physical state in an extended Hilbert space obtained by taking the tensor product of the Hilbert space of states on each link of the lattice. This extended Hilbert space admits a tensor product decomposition by definition and allows a density matrix and entanglement entropy for the set of links of interest to be defined. Here, we continue the study of this extended Hilbert space definition with particular emphasis on the case of Non-Abelian gauge theories. We extend the electric centre definition of Casini, Huerta and Rosabal to the Non-Abelian case and find that it differs in an important term. We also find that the entanglement entropy does not agree with the maximum number of Bell pairs that can be extracted by the processes of entanglement distillation or dilution, and give protocols which achieve the maximum bound. Finally, we compute the topological entanglement entropy which follows from the extended Hilbert space definition and show that it correctly reproduces the total quantum dimension in a class of Toric code models based on Non-Abelian discrete groups.

We revisit noninteracting string partition functions in Rindler space by summing over fields in the spectrum. In field theory, the total partition function splits in a natural way into a piece that does not contain surface terms and a piece consisting of solely the so-called edge states. For open strings, we illustrate that surface contributions to the higher-spin fields correspond to open strings piercing the Rindler origin, unifying the higher-spin surface contributions in string language. For closed strings, we demonstrate that the string partition function is not quite the same as the sum over the partition functions of the fields in the spectrum: an infinite overcounting is present for the latter. Next we study the partition functions obtained by excluding the surface terms. Using recent results of He et al. [J. High Energy Phys. 05 (2015) 106], this construction, first done by Emparan [arXiv:hep-th/9412003], can be put on much firmer ground. We generalize to type II and heterotic superstrings and demonstrate modular invariance. All of these exhibit an IR divergence that can be interpreted as a maximal acceleration close to the black hole horizon. Ultimately, since these partition functions are only part of the full story, divergences here should not be viewed as a failure of string theory: maximal acceleration is a feature of a faulty treatment of the higher-spin fields in the string spectrum. We comment on the relevance of this to Solodukhin’s recent proposal [Phys. Rev. D 91, 084028 (2015)]. A possible link with the firewall paradox is apparent.

We calculate the free energies F for U(1) gauge theories on the d dimensional sphere of radius R. For the theory with free Maxwell action we find the exact result as a function of d; it contains the term d-4/2 log R consistent with the lack of conformal invariance in dimensions other than 4. When the U(1) gauge theory is coupled to a sufficient number N-f of massless four-component fermions, it acquires an interacting conformal phase, which in d < 4 describes the long distance behavior of the model. The conformal phase can be studied using large N-f methods. Generalizing the d = 3 calculation in arXiv:1112.5342, we compute its sphere free energy as a function of d, ignoring the terms of order 1/N-f and higher. For finite N-f, following arXiv:1409.1937 and arXiv:1507.01960, we develop the 4 - epsilon expansion for the sphere free energy of conformal QED(d). Its extrapolation to d = 3 shows very good agreement with the large N-f approximation for N-f > 3. For N-f at or below some critical value N-crit, the SU(2N(f)) symmetric conformal phase of QED(3) is expected to disappear or become unstable. By using the F-theorem and comparing the sphere free energies in the conformal and broken symmetry phases, we show that N-crit <= 4. As another application of our results, we calculate the one loop beta function in conformal QED(6), where the gauge field has a four-derivative kinetic term. We show that this theory coupled to N-f massless fermions is asymptotically free.

Based on the talk given by X.B. at the international conference “Mathematics Days in Sofia” held in July 2014 at Sofia (Bulgaria)

Flux compactifications of string theory exhibiting the possibility of discretely tuning the cosmological constant to small values have been constructed. The highly tuned vacua in this discretuum have curvature radii which scale as large powers of the flux quantum numbers, exponential in the number of cycles in the compactification. By the arguments of Susskind/Witten (in the AdS case) and Gibbons/Hawking (in the dS case), we expect correspondingly large entropies associated with these vacua. If they are to provide a dual description of these vacua on their Coulomb branch, branes traded for the flux need to account for this entropy at the appropriate energy scale. In this note, we argue that simple string junctions and webs ending on the branes can account for this large entropy, obtaining a rough estimate for junction entropy that agrees with the existing rough estimates for the spacing of the discretuum. In particular, the brane entropy can account for the (A)dS entropy far away from string scale correspondence limits.

The correspondence between supergravity (and string theory) on AdS space and boundary conformal field theory relates the thermodynamics of N = 4 super-Yang-Mills theory in four dimensions to the thermodynamics of Schwarzschild black holes in anti-de Sitter space. In this description, quantum phenomena such as the spontaneous breaking of the center of the gauge group, magnetic confinement and the mass gap are coded in classical geometry. The correspondence makes it manifest that the entropy of a very large AdS Schwarzschild black hole must scale "holographically" with the volume of its horizon. By similar methods, one can also make a speculative proposal for the description of large N gauge theories in four dimensions without supersymmetry.

The massive scalar and Dirac fields quantized on a de Sitter background
geometry prove to be exactly soluble models in general-relativistic
field theory. The Feynman Green's function is computed for both the
scalar and Dirac fields. A dimensional regularization procedure applied
in coordinate space facilitates the calculation of their respective
effective Lagrangians, which describe the vacuum corrections due to
closed matter loops. The model is found to be renormalizable. There is
no creation of real particle pairs.

Following an historical introduction, the conventional canonical formulation of general relativity theory is presented. The canonical Lagrangian is expressed in terms of the extrinsic and intrinsic curvatures of the hypersurface x0=constant, and its relation to the asymptotic field energy in an infinite world is noted. The distinction between finite and infinite worlds is emphasized. In the quantum theory the primary and secondary constraints become conditions on the state vector, and in the case of finite worlds these conditions alone govern the dynamics. A resolution of the factor-ordering problem is proposed, and the consistency of the constraints is demonstrated. A 6-dimensional hyperbolic Riemannian manifold is introduced which takes for its metric the coefficient of the momenta in the Hamiltonian constraint. The geodesic incompletability of this manifold, owing to the existence of a frontier of infinite curvature, is demonstrated. The possibility is explored of relating this manifold to an infinite-dimensional manifold of 3-geometries, and of relating the structure of the latter manifold in turn to the dynamical behavior of space-time. The problem is approached through the WKB approximation and Hamilton-Jacobi theory. Einstein's equations are revealed as geodesic equations in the manifold of 3-geometries, modified by the presence of a "force term." The classical phenomenon of gravitational collapse shows that the force term is not powerful enough to prevent the trajectory of space-time from running into the frontier. The as-yet unresolved problem of determining when the collapse phenomenon represents a real barrier to the quantum-state functional is briefly discussed, and a boundary condition at the barrier is proposed. The state functional of a finite world can depend only on the 3-geometry of the hypersurface x0=constant. The label x0 itself is irrelevant, and "time" must be determined intrinsically. A natural definition for the inner product of two such state functionals is introduced which, however, encounters difficulties with negative probabilities owing to the barrier boundary condition. In order to resolve these difficulties, a simplified model, the quantized Friedmann universe, is studied in detail. In order to obtain nonstatic wave functions which resemble a universe evolving, it is necessary to introduce a clock. In order that the combined wave functions of universe-cum-clock be normalizable, it turns out that the periods of universe and clock must be commensurable. Wave packets exhibiting quasiclassical behavior are constructed, and attention is called to the phenomenological character of "time." The innerproduct definition is rescued from its negative-probability difficulties by making use of the fact that probability flows in a closed finite circuit in configuration space. The article ends with some speculations on the uniqueness of the state functional of the actual universe. It is suggested that a viewpoint due to Everett should be adopted in its interpretation.

One can evaluate the action for a gravitational field on a section of
the complexified spacetime which avoids the singularities. In this
manner we obtain finite, purely imaginary values for the actions of the
Kerr-Newman solutions and de Sitter space. One interpretation of these
values is that they give the probabilities for finding such metrics in
the vacuum state. Another interpretation is that they give the
contribution of that metric to the partition function for a grand
canonical ensemble at a certain temperature, angular momentum, and
charge. We use this approach to evaluate the entropy of these metrics
and find that it is always equal to one quarter the area of the event
horizon in fundamental units. This agrees with previous derivations by
completely different methods. In the case of a stationary system such as
a star with no event horizon, the gravitational field has no entropy.

We have formulated the statistical mechanics in terms of the S matrix, which describes the scattering processes taking place in the thermodynamical system of interest. Such a formulation is necessary for studying the systems whose microscopic constituents behave according to the laws of relativistic quantum mechanics. Our result is a simple prescription for calculating the grand canonical potential of any gaseous system given the free-particle energies and S-matrix elements. When applied to a nonrelativistic gas, it gives a simple prescription for calculating all virial coefficients. Simplified relativistic gas models are considered as examples of application. A general form of the Levinson's Theorem for any number of particles follows immediately from our formalism. Its applications in statistical mechanics are briefly discussed.

The one-loop effective potential is investigated in quantum gravity with or without cosmological constant in general D dimensions. It is shown that the partition function at the one-loop level depends only on the laplacian plus the scalar curvature with no terms of the form delta(0) ln g, i.e. Z(1-loop) = det-1/2 (Delta2 + 2R/D) × det1/2 (Delta1 + 2R/D). An explicit calculation is carried out at the one-loop level to demonstrate gauge independence of the partition function in a class of general linear gauges.

Various estimates of the lower bound of the holomorphic invariant α(M), defined in [T], are given here by using branched coverings, potential estimates and Lelong numbers of positive,d-closed (1, 1) currents of certain type, etc. These estimates are then applied to produce Kähler-Einstein metrics on complex surfaces withC
1>0, in particular, we prove that there are Kähler-Einstein structures withC
1>0 on any manifold of differential type\(CP^2 \# \overline {nCP^2 } (3 \leqq n \leqq 8)\).

We calculate the one-loop, off-shell, effective action in O(4) gauged supergravity assuming an (anti) de Sitter metric and constant scalar fields as a background. The problem of the large induced Lambda term (present already for free matter fields) is stressed and the possibility of dynamical breakdown of local supersymmetry is pointed out. We illustrate our techniques and qualitative conclusions on a number of examples, including Ø4 theory and QED scalar potentials on a de Sitter background and an effective action in Einstein theory with a cosmological constant. Possible solutions of the Lambda-term problem are also discussed.

The gravitational saddle point integral which arises in recent work of Hawking and Coleman need not be real. We determine the phase to be (-i)d+2, implying that Coleman's solutions to thecosmological constant problem can work, in its present form, only in dimension d = 2 mod 4. The physical interpretation of this phase is obscure. A.P. Sloan foundation fellow.

This report the following topics: loops and states in conformal field theory; brief review of the Liouville theory; 2D Euclidean quantum gravity I: path integral approach; 2D Euclidean quantum gravity II: canonical approach; states in 2D string theory; matrix model technology I: method of orthogonal polynomials; matrix model technology II: loops on the lattice; matrix model technology III: free fermions from the lattice; loops and states in matrix model quantum gravity; loops and states in the C=1 matrix model; 6V model fermi sea dynamics and collective field theory; and string scattering in two spacetime dimensions.

It is shown that the close connection between event horizons and thermodynamics which has been found in the case of black holes can be extended to cosmological models with a repulsive cosmological constant. An observer in these models will have an event horizon whose area can be interpreted as the entropy or lack of information of the observer about the regions which he cannot see. Associated with the event horizon is a surface gravity κ which enters a classical "first law of event horizons" in a manner similar to that in which temperature occurs in the first law of thermodynamics. It is shown that this similarity is more than an analogy: An observer with a particle detector will indeed observe a background of thermal radiation coming apparently from the cosmological event horizon. If the observer absorbs some of this radiation, he will gain energy and entropy at the expense of the region beyond his ken and the event horizon will shrink. The derivation of these results involves abandoning the idea that particles should be defined in an observer-independent manner. They also suggest that one has to use something like the Everett-Wheeler interpretation of quantum mechanics because the back reaction and hence the spacetime metric itself appear to be observer-dependent, if one assumes, as seems reasonable, that the detection of a particle is accompanied by a change in the gravitational field.

The quantum state of a spatially closed universe can be described by a wave function which is a functional on the geometries of compact three-manifolds and on the values of the matter fields on these manifolds. The wave function obeys the Wheeler-DeWitt second-order functional differential equation. We put forward a proposal for the wave function of the "ground state" or state of minimum excitation: the ground-state amplitude for a three-geometry is given by a path integral over all compact positive-definite four-geometries which have the three-geometry as a boundary. The requirement that the Hamiltonian be Hermitian then defines the boundary conditions for the Wheeler-DeWitt equation and the spectrum of possible excited states. To illustrate the above, we calculate the ground and excited states in a simple minisuperspace model in which the scale factor is the only gravitational degree of freedom, a conformally invariant scalar field is the only matter degree of freedom and Λ>0. The ground state corresponds to de Sitter space in the classical limit. There are excited states which represent universes which expand from zero volume, reach a maximum size, and then recollapse but which have a finite (though very small) probability of tunneling through a potential barrier to a de Sitter-type state of continual expansion. The path-integral approach allows us to handle situations in which the topology of the three-manifold changes. We estimate the probability that the ground state in our minisuperspace model contains more than one connected component of the spacelike surface.

The effective Lagrangian and vacuum energy-momentum tensor 〈Tμν〉 due to a scalar field in a de Sitter-space background are calculated using the dimensional-regularization method. For generality the scalar field equation is chosen in the form (□2+ξR+m2)ϕ=0. If ξ=1/6 and m=0, the renormalized 〈Tμν〉 equals gμν(960π2a4)-1, where a is the radius of de Sitter space. More formally, a general zeta-function method is developed. It yields the renormalized effective Lagrangian as the derivative of the zeta function on the curved space. This method is shown to be virtually identical to a method of dimensional regularization applicable to any Riemann space.

Symmetric transverse traceless tensor harmonics of arbitrary rank are constructed on spheres Sn of dimensionality n≥3, and the associated eigenvalues of the Laplacian are computed. It is shown that these tensor harmonics span the space of symmetric transverse traceless tensors on Sn and are eigenfunctions of the quadratic Casimir operator of the group O(n+1). The dimensionalities of the eigenspaces of the Laplacian are computed for harmonics of rank 1 and rank 2.

We propose a dual non-perturbative description for maximally extended Schwarzschild Anti-de-Sitter spacetimes. The description involves two copies of the conformal field theory associated to the AdS spacetime and an initial entangled state. In this context we also discuss a version of the information loss paradox and its resolution.

U(1) gauge theory onR
4 is known to possess an electric-magnetic duality symmetry that inverts the coupling constant and extends to an action ofSL(2,Z). In this paper, the duality is studied on a general four-manifold and it is shown that the partition function is not a modular-invariant function but transforms as a modular form. This result plays an essential role in determining a new low-energy interaction that arises whenN=2 supersymmetric Yang-Mills theory is formulated on a four-manifold; the determination of this interaction gives a new test of the solution of the model and would enter in computations of the Donaldson invariants of four-manifolds with b
2
+
1. Certain other aspects of abelian duality, relevant to matters such as the dependence of Donaldson invariants on the second Stieffel-Whitney class, are also analyzed.

The remarkable representations of the 3+2 de Sitter group, discovered by Dirac, later called singleton representations and here denoted Di and Rac, are shown to possess the following truly remarkable property: Each of the direct products Di Di, Di Rac, and Rac Rac decomposes into a direct sum of unitary, irreducible representations, each of which admits an extension to a unitary, irreducible representation of the conformal group SO(4, 2). Therefore, in de Sitter space, every state of a free, massless particle may be interpreted as a state of two free singletons — and vice versa. The term massless is associated with a set of particle-like representations of SO(3, 2) that, besides the noted conformal extension, exhibit other phenomena typical of masslessness, especially gauge invariance.