Notes on massless scalar field partition functions, modular invariance and Eisenstein series

The partition function of a massless scalar field on a Euclidean spacetime manifold ℝd−1 × 𝕋2 and with momentum operator in the compact spatial dimension coupled through a purely imaginary chemical potential is computed. It is modular covariant and admits a simple expression in terms of a real analytic SL(2, ℤ) Eisenstein series with s = (d + 1)/2. Different techniques for computing the partition function illustrate complementary aspects of the Eisenstein series: the functional approach gives its series representation, the operator approach yields its Fourier series, while the proper time/heat kernel/world-line approach shows that it is the Mellin transform of a Riemann theta function. High/low temperature duality is generalized to the case of a non-vanishing chemical potential. By clarifying the dependence of the partition function on the geometry of the torus, we discuss how modular covariance is a consequence of full SL(2, ℤ) invariance. When the spacetime manifold is ℝp × 𝕋q+1, the partition function is given in terms of a SL(q + 1, ℤ) Eisenstein series again with s = (d + 1)/2. In this case, we obtain the high/low temperature duality through a suitably adapted dual parametrization of the lattice defining the torus. On 𝕋d+1, the computation is more subtle. An additional divergence leads to an harmonic anomaly.

Effective action of string theory at order $$\alpha '$$ in the presence of boundary

Recently, using the assumption that the string theory effective action at the critical dimension is background independent, the classical on-shell effective action of the bosonic string theory at order $$\alpha '$$ α ′ in a spacetime manifold without boundary has been reproduced, up to an overall parameter, by imposing the O(1, 1) symmetry when the background has a circle. In the presence of the boundary, we consider a background which has boundary and a circle such that the unit normal vector of the boundary is independent of the circle. Then the O(1, 1) symmetry can fix the bulk action without using the lowest order equation of motion. Moreover, the above constraints and the constraint from the principle of the least action in the presence of boundary can fix the boundary action, up to five boundary parameters. In the least action principle, we assume that not only the values of the massless fields but also the values of their first derivatives are arbitrary on the boundary. We have also observed that the cosmological reduction of the leading order action in the presence of the Hawking–Gibbons boundary term, produces zero cosmological boundary action. Imposing this as another constraint on the boundary couplings at order $$\alpha '$$ α ′ , we find the boundary action up to two parameters. For a specific value for these two parameters, the gravity couplings in the boundary become the Chern–Simons gravity plus another term which has the Laplacian of the extrinsic curvature.

Hubble Expansion as an Einstein Curvature

Hubble expansion may be considered as a velocity per photon travel time rather than as velocity or redshift per distance. Dimensionally, this is an acceleration and will have an associated curvature of space under general relativity. This paper explores this theoretical curvature as an extension to the spacetime manifold of general relativity, generating a modified solution with three additional non-zero Christoffel symbols, and a reformulated Ricci tensor and curvature. The observational consequences of this reformulation were compared with the ΛCDM model for luminosity distance using the extensive type Ia supernovae (SNe Ia) data with redshift corrected to the CMB, and for angular diameter distance with the recent baryonic acoustic oscillation (BAO) data. For the SNe Ia data, the modified GR and ΛCDM models differed by −0.15+0.11μB mag. over zcmb=0.01−1.3, with overall weighted RMS errors of ±0.136μB mag for modified GR and ±0.151μB mag for ΛCDM espectively. The BAO measures spanned a range z=0.106−2.36, with weighted RMS errors of ±0.034 Mpc with H0=67.6±0.25 for the modified GR model, and ±0.085 Mpc with H0=70.0±0.25 for the ΛCDM model. The derived GR metric for this new solution describes both the SNe Ia and the BAO observations with comparable accuracy to ΛCDM without requiring the inclusion of dark matter or w’-corrected dark energy.

Metamaterial Acoustics on the (2 + 1)D Einstein Cylinder

The Einstein cylinder is the first cosmological model for our universe in modern history. Its geometry not only describes a static universe—a universe being invariant under time reversal—but it is also the prototype for a maximally symmetric spacetime with constant positive curvature. As such, it is still of crucial importance in numerous areas of physics and engineering, offering a fruitful playground for simulations and new theories. Here, we focus on the implementation and simulation of acoustic wave propagation on the Einstein cylinder. Engineering such an extraordinary device is the territory of metamaterial science, and we will propose an appropriate tuning of the relevant acoustic parameters in such a way as to mimic the geometric properties of this spacetime in acoustic space. Moreover, for probing such a space, we derive the corresponding wave equation from a variational principle for the underlying curved spacetime manifold and examine some of its solutions. In particular, fully analytical results are obtained for concentric wave propagation. We present predictions for this case and thereby investigate the most significant features of this spacetime. Finally, we produce simulation results for a more sophisticated test model which can only be tackled numerically.

Spherical space in the Newtonian limit: The cosmological constant

We compute the cosmological constant of a spherical space in the limit of weak gravity. To this end, we use a duality developed by the present authors in a previous work. This duality allows one to treat the Newtonian cosmological fluid as the probability fluid of a single particle in nonrelativistic quantum mechanics. We apply this duality to the case when the spacetime manifold on which this quantum mechanics is defined is given by [Formula: see text]. Here, [Formula: see text] stands for the time axis and [Formula: see text] is a 3-dimensional sphere endowed with the standard round metric. A quantum operator [Formula: see text] satisfying all the requirements of a cosmological constant is identified, and the matrix representing [Formula: see text] within the Hilbert space [Formula: see text] of quantum states is obtained. Numerical values for the expectation value of the operator [Formula: see text] in certain quantum states are obtained, which are in good agreement with the experimentally measured cosmological constant.

The Mystery of the Lorentz Transform: A Reconstruction and Its Implications for Einstein’s Theories of Relativity and cosmology.

The Lorentz Transformation (LT) is an arbitrary and poorly conceived mathematical tool designed to make Maxwell’s electromagnetism conform to Galilean relativity, which formed the basis of classical mechanics and physics. A strange combination of this transform with an axiomatic assumption by Albert Einstein that the velocity of light c is an absolute and universal constant has led to an idealist, geometrical and phenomenological view of the universe, that is at variance with objective reality. This conundrum that has lasted for more than hundred years has led to rampant mysticism and has impaired the development of positive knowledge of the universe. The present reconstruction of LT shows that the gamma term, which fueled mysticism in physics and cosmology is, on the contrary, a natural outcome of the subjective geometrical rendition of the speed of light and the idealist unification of abstract space and time into a 4D “spacetime” manifold; by Minkowski and Einstein. Only a materialist dialectical perspective of space and time can rid physics of all mysticism arising out of LT; from the quantum to the cosmic.

Is There Change on the B-Theory of Time?

The purpose of this paper is to explore the connection between change and the B-theory of time, sometimes also called the Scientific view of time, according to which reality is a four-dimensional spacetime manifold, where past, present and future things equally exist, and the present time and non-present times are metaphysically the same. I argue in favour of a novel response to the much-vexed question of whether there is change on the B-theory or not. In fact, B-theorists are often said to hold a ‘static’ view of time. But this far from being innocent label: if the B-theory of time presents a model of temporal reality that is static, then there is no change on the B-theory. From this, one can reasonably think as follows: of course, there is change, so the B-theory must be false. What I plan to do in this paper is to argue that in some sense there is change on the B-theory, but in some other sense, there is no change on the B-theory. To do so, I present three instances of change: Existential Change, namely the view that things change with respect to their existence over time; Qualitative Change, the view that things change with respect to how they are over time; Propositional Change, namely the view that things (i.e. propositions) change with respect to truth value over time. I argue that while there is a reading of these three instances of change that is true on the B-theory, and so there is change on the B-theory in this sense, there is a B-theoretical reading of each of them that is not true on the B-theory, and therefore there is no change on the B-theory in this other sense.

Exact solutions of (deformed) Jackiw–Teitelboim gravity

It is well known that Jackiw–Teitelboim (JT) gravity posses the simplest theory on 2-dimensional gravity. The model has been fruitfully studied in recent years. In the present work, we investigate exact solutions for both JT and deformed JT gravity recently proposed in the literature. We revisit exact Euclidean solutions for Jackiw-Teitelboim gravity using all the non-zero components of the dilatonic equations of motion using proper integral transformation over Euclidean time coordinate. More precisely, we study exact solutions for hyperbolic coverage, cusp geometry and another compact sector of the AdS$$_2$$ 2 spacetime manifold.

Nilpotent symmetries as a mechanism for Grand Unification

In the classic Coleman-Mandula no-go theorem which prohibits the unification of internal and spacetime symmetries, the assumption of the existence of a positive definite invariant scalar product on the Lie algebra of the internal group is essential. If one instead allows the scalar product to be positive semi-definite, this opens new possibilities for unification of gauge and spacetime symmetries. It follows from theorems on the structure of Lie algebras, that in the case of unified symmetries, the degenerate directions of the positive semi-definite invariant scalar product have to correspond to local symmetries with nilpotent generators. In this paper we construct a workable minimal toy model making use of this mechanism: it admits unified local symmetries having a compact (U(1)) component, a Lorentz (SL(2, ℂ)) component, and a nilpotent component gluing these together. The construction is such that the full unified symmetry group acts locally and faithfully on the matter field sector, whereas the gauge fields which would correspond to the nilpotent generators can be transformed out from the theory, leaving gauge fields only with compact charges. It is shown that already the ordinary Dirac equation admits an extremely simple prototype example for the above gauge field elimination mechanism: it has a local symmetry with corresponding eliminable gauge field, related to the dilatation group. The outlined symmetry unification mechanism can be used to by-pass the Coleman-Mandula and related no-go theorems in a way that is fundamentally different from supersymmetry. In particular, the mechanism avoids invocation of super-coordinates or extra dimensions for the underlying spacetime manifold.

From $$ \mathcal{N} $$ = 4 Super-Yang-Mills on ℝℙ4 to bosonic Yang-Mills on ℝℙ2

We study the four-dimensional $$ \mathcal{N} $$ N = 4 super-Yang-Mills (SYM) theory on the unorientable spacetime manifold ℝℙ4. Using supersymmetric localization, we find that for a large class of local and extended SYM observables preserving a common supercharge $$ \mathcal{Q} $$ Q , their expectation values are captured by an effective two-dimensional bosonic Yang-Mills (YM) theory on an ℝℙ2 submanifold. This paves the way for understanding $$ \mathcal{N} $$ N = 4 SYM on ℝℙ4 using known results of YM on ℝℙ2. As an illustration, we derive a matrix integral form of the SYM partition function on ℝℙ4 which, when decomposed into discrete holonomy sectors, contains subtle phase factors due to the nontrivial η-invariant of the Dirac operator on ℝℙ4. We also comment on potential applications of our setup for AGT correspondence, integrability and bulk-reconstruction in AdS/CFT that involve cross-cap states on the boundary.