arXiv subjects

Petr Horava

Publications and source records attributed to Petr Horava.

At least 19 recordsLinked to original sources

arXiv2020

Perelman's Ricci Flow in Topological Quantum Gravity

We find the regime of our recently constructed topological nonrelativistic quantum gravity, in which Perelman's Ricci flow equations on Riemannian manifolds appear precisely as the localization equations in the path integral. In this mapping between physics and mathematics, the role of Perelman's dilaton is played by our lapse function. Perelman's local fixed volume condition emerges dynamically as the $\lambda$ parameter in our kinetic term approaches $\lambda\to-\infty$. The DeTurck trick that decouples the metric flow from the dilaton flow is simply a gauge-fixing condition for the gauge symmetry of spatial diffeomorphisms. We show how Perelman's ${\cal F}$ and ${\cal W}$ entropy functionals are related to our superpotential. We explain the origin of Perelman's $\tau$ function, which appears in the ${\cal W}$ entropy functional for shrinking solitons, as the Goldstone mode associated with time translations and spatial rescalings: In fact, in our quantum gravity, Perelman's $\tau$ turns out to play the role of a dilaton for anisotropic scale transformations. The map between Perelman's flow and the localization equations in our topological quantum gravity requires an interesting redefinition of fields, which includes a reframing of the metric. With this embedding of Perelman's equations into topological quantum gravity, a wealth of mathematical results on the Ricci flow can now be imported into physics and reformulated in the language of quantum field theory.

hep-th↗

arXiv2020

String Perturbation Theory on the Schwinger-Keldysh Time Contour

We perform the large-$N$ expansion in the Schwinger-Keldysh formulation of non-equilibrium quantum systems with matrix degrees of freedom, and study universal features of the anticipated dual string theory. We find a rich refinement of the topological genus expansion: In the original formulation, the future time instant where the forward and backward branches of the Schwinger-Keldysh time contour meet is associated with its own worldsheet genus expansion. After the Keldysh rotation, the worldsheets decompose into a classical and quantum part.

hep-th↗

arXiv2020

The Geometry of Time in Topological Quantum Gravity of the Ricci Flow

We continue the study of topological nonrelativistic quantum gravity associated with a family of Ricci flow equations on Riemannian manifolds. This topological gravity is of the cohomological type, and it exhibits an ${\cal N}=2$ extended BRST symmetry. In our previous work, we constructed this theory in a two-step procedure in the appropriate nonrelativistic ${\cal N}=2$ superspace, first presenting a topological theory of the spatial metric $g_{ij}$, and then adding the superspace versions of the lapse and shift variables $n$ and $n^i$ while gauging the symmetries of foliation-preserving spacetime diffeomorphisms. In the relation to Perelman's theory of the Ricci flow, the role of Perelman's dilaton is played by our nonprojectable lapse. Here we demonstrate that this construction is equivalent to a standard one-step BRST gauge-fixing of a theory whose fields are $g_{ij}$, $n^i$ and $n$, and whose gauge symmetries consist of (i) the topological deformations of $g_{ij}$, and (ii) the ultralocal nonrelativistic limit of spacetime diffeomorphisms. The supercharge $Q$ of our superspace construction plays the role of the BRST charge. The spacetime diffeomorphism symmetries appear in an interestingly "shifted" form, which may be of broader interest for nonrelativistic quantum gravity outside of the present topological context. In contrast to the foliation-preserving spacetime diffeomorphisms, the gauge symmetries identified in this paper act nonprojectably on time, making it clear that this theory has no local propagating degrees of freedom. We point out an intriguing dual interpretation of the same theory, as a gauge fixing of a dual copy of ultralocal spacetime diffeomorphisms, with the role of ghosts and antighosts interchanged and the second supercharge $\bar Q$ of the ${\cal N}=2$ superalgebra playing the role of the BRST charge in the dual picture.

hep-th↗

arXiv2020

Topological Quantum Gravity of the Ricci Flow

We present a family of topological quantum gravity theories associated with the geometric theory of the Ricci flow on Riemannian manifolds. First we use BRST quantization to construct a "primitive" topological Lifshitz-type theory for only the spatial metric, with spatial diffeomorphism invariance and no gauge symmetry, associated with Hamilton's Ricci flow: Hamilton's flow equation appears as the localization equation of the primitive theory. Then we extend the primitive theory by gauging foliation-preserving spacetime symmetries. Crucially, all our theories are required to exhibit an ${\cal N}=2$ extended BRST symmetry. First, we gauge spatial diffeomorphisms, and show that this gives us access to the mathematical technique known as the DeTurck trick. Finally, we gauge foliation-preserving time reparametrizations, both with the projectable and nonprojectable lapse function. The path integral of the full theory is localized to the solutions of Ricci-type flow equations, generalizing those of Perelman. The role of Perelman's dilaton is played by the nonprojectable lapse function. Perelman's ${\cal F}$-functional appears as the superpotential of our theory. Since there is no spin-statistics theorem in nonrelativistic quantum field theory, the two supercharges of our gravity theory do not have to be interpreted as BRST charges and, after the continuation to real time, the theory can be studied as a candidate for nonrelativistic quantum gravity with propagating bosonic and fermionic degrees of freedom.

hep-th↗

arXiv2020

Keldysh Rotation in the Large-N Expansion and String Theory Out of Equilibrium

We extend our study of the large-$N$ expansion of general non-equilibrium many-body systems with matrix degrees of freedom $M$, and its dual description as a sum over surface topologies in a dual string theory, to the Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh rotation trades the original fields $M_\pm$ -- defined as the values of $M$ on the forward and backward segments of the closed time contour -- for their linear combinations $M_{\textrm{cl}}$ and $M_{\textrm{qu}}$, known as the "classical" and "quantum" fields. First we develop a novel "signpost" notation for non-equilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each worldsheet surface $Σ$ in the dual string theory expansion was found to exhibit a triple decomposition into the parts $Σ^\pm$ corresponding to the forward and backward segments of the closed time contour, and $Σ^\wedge$ which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the worldsheet surface $Σ$ of the dual string theory undergoes a very different natural decomposition: $Σ$ consists of a "classical" part $Σ^{\textrm{cl}}$, and a "quantum embellishment" part $Σ^{\textrm{qu}}$. We show that both parts of $Σ$ carry their own independent genus expansion. The non-equilibrium sum over worldsheet topologies is naturally refined into a sum over the double decomposition of each $Σ$ into its classical and quantum part. We apply this picture to the classical limits of the quantum non-equilibrium system (with or without interactions with a thermal bath), and find that in these limits, the dual string perturbation theory expansion reduces to its appropriately defined classical limit.

hep-th↗

arXiv2020

Large-N Expansion and String Theory Out of Equilibrium

We analyze the large-$N$ expansion of general non-equilibrium systems with fluctuating matrix degrees of freedom and $SU(N)$ symmetry, using the Schwinger-Keldysh formalism and its closed real-time contour with a forward and backward component. In equilibrium, the large-$N$ expansion of such systems leads to a sum over topologies of two-dimensional surfaces of increasing topological complexity, predicting the possibility of a dual description in terms of string theory. We extend this argument away from equilibrium, and study the universal features of the topological expansion in the dual string theory. We conclude that in non-equilibrium string perturbation theory, the sum over worldsheet topologies is further refined: Each worldsheet surface $Σ$ undergoes a triple decomposition into the part $Σ^+$ corresponding to the forward branch of the time contour, the part $Σ^-$ on the backward branch, and the part $Σ^\wedge$ that corresponds to the instant in the far future where the two branches of the time contour meet. The sum over topologies becomes a sum over the triple decompositions. We generalize our findings to the Kadanoff-Baym time contour relevant for systems at finite temperature, and to the case of closed and open, oriented or unoriented strings. Our results are universal, and follow solely from the features of the large-$N$ expansion without any assumptions about the worldsheet dynamics.

hep-th↗

arXiv2016

Nonrelativistic Short-Distance Completions of a Naturally Light Higgs

Nonrelativistic scalar field theories can exhibit a natural cascading hierarchy of scales, protected by a hierarchy of polynomial shift symmetries. Using a simple model, we argue that a high-energy cross-over to such nonrelativistic behavior naturally leads to light scalars, and thus represents a useful ingredient for technically natural resolutions of scalar mass hierarchies, perhaps even the Higgs mass hierarchy puzzle.

hep-ph↗

arXiv2016

Surprises with Nonrelativistic Naturalness

We explore the landscape of technical naturalness for nonrelativistic systems, finding surprises which challenge and enrich our relativistic intuition already in the simplest case of a single scalar field. While the immediate applications are expected in condensed matter and perhaps in cosmology, the study is motivated by the leading puzzles of fundamental physics involving gravity: The cosmological constant problem and the Higgs mass hierarchy problem.

hep-th↗

arXiv2015

Scalar Field Theories with Polynomial Shift Symmetries

We continue our study of naturalness in nonrelativistic QFTs of the Lifshitz type, focusing on scalar fields that can play the role of Nambu-Goldstone (NG) modes associated with spontaneous symmetry breaking. Such systems allow for an extension of the constant shift symmetry to a shift by a polynomial of degree $P$ in spatial coordinates. These "polynomial shift symmetries" in turn protect the technical naturalness of modes with a higher-order dispersion relation, and lead to a refinement of the proposed classification of infrared Gaussian fixed points available to describe NG modes in nonrelativistic theories. Generic interactions in such theories break the polynomial shift symmetry explicitly to the constant shift. It is thus natural to ask: Given a Gaussian fixed point with polynomial shift symmetry of degree $P$, what are the lowest-dimension operators that preserve this symmetry, and deform the theory into a self-interacting scalar field theory with the shift symmetry of degree $P$? To answer this (essentially cohomological) question, we develop a new graph-theoretical technique, and use it to prove several classification theorems. First, in the special case of $P=1$ (essentially equivalent to Galileons), we reproduce the known Galileon $N$-point invariants, and find their novel interpretation in terms of graph theory, as an equal-weight sum over all labeled trees with $N$ vertices. Then we extend the classification to $P>1$ and find a whole host of new invariants, including those that represent the most relevant (or least irrelevant) deformations of the corresponding Gaussian fixed points, and we study their uniqueness.

hep-th↗

arXiv2015

Cascading Multicriticality in Nonrelativistic Spontaneous Symmetry Breaking

Without Lorentz invariance, spontaneous global symmetry breaking can lead to multicritical Nambu-Goldstone modes with a higher-order low-energy dispersion $ω\sim k^n$ ($n=2,3,\ldots$), whose naturalness is protected by polynomial shift symmetries. Here we investigate the role of infrared divergences and the nonrelativistic generalization of the Coleman-Hohenberg-Mermin-Wagner (CHMW) theorem. We find novel cascading phenomena with large hierarchies between the scales at which the value of $n$ changes, leading to an evasion of the "no-go" consequences of the relativistic CHMW theorem.

hep-th↗

arXiv2014

GR 20 Parallel Session A3: Modified Gravity

This is the contribution representing Parallel Session A3, on Modified Gravity, in the Proceedings of the GR 20 Conference (July 2013, Warszawa, Poland). It consists of three invited chapters, selected by the Session Chair (P.H.) to represent the broad spectrum of topics discussed in the Session, which ranged from theoretical and phenomenological, to experimental, observational and numerical aspects of gravity. The three chapters are "Einstein-Aether Theory: Thermodynamics of Universal Horizons" by Arif Mohd, "The Curious Case of Conformal Anomalies in Horava-Lifshitz Gravity" by Charles M. Melby-Thompson, and "Detectability of Scalar Gravitational-Wave Bursts with LIGO and Virgo" by Peter Shawhan.

gr-qc↗

arXiv2013

Multicritical Symmetry Breaking and Naturalness of Slow Nambu-Goldstone Bosons

We investigate spontaneous global symmetry breaking in the absence of Lorentz invariance, and study technical Naturalness of Nambu-Goldstone (NG) modes whose dispersion relation exhibits a hierarchy of multicritical phenomena with Lifshitz scaling and dynamical exponents $z>1$. For example, we find NG modes with a technically natural quadratic dispersion relation which do not break time reversal symmetry and are associated with a single broken symmetry generator, not a pair. The mechanism is protected by an enhanced `polynomial shift' symmetry in the free-field limit.

hep-th↗

arXiv2012

Lifshitz Gravity for Lifshitz Holography

We argue that Horava-Lifshitz (HL) gravity provides the minimal holographic dual for Lifshitz-type field theories with anisotropic scaling and dynamical exponent z. First we show that Lifshitz spacetimes are vacuum solutions of HL gravity, without need for additional matter. Then we perform holographic renormalization of HL gravity, and show how it reproduces the full structure of the z=2 anisotropic Weyl anomaly in dual field theories in 2+1 dimensions, while its minimal relativistic gravity counterpart yields only one of two independent central charges in the anomaly.

hep-th↗

arXiv2012

Conformal Lifshitz Gravity from Holography

We show that holographic renormalization of relativistic gravity in asymptotically Lifshitz spacetimes naturally reproduces the structure of gravity with anisotropic scaling: The holographic counterterms induced near anisotropic infinity take the form of the action for gravity at a Lifshitz point, with the appropriate value of the dynamical critical exponent $z$. In the particular case of 3+1 bulk dimensions and $z=2$ asymptotic scaling near infinity, we find a logarithmic counterterm, related to anisotropic Weyl anomaly of the dual CFT, and show that this counterterm reproduces precisely the action of conformal gravity at a $z=2$ Lifshitz point in 2+1 dimensions, which enjoys anisotropic local Weyl invariance and satisfies the detailed balance condition. We explain how the detailed balance is a consequence of relations among holographic counterterms, and point out that a similar relation holds in the relativistic case of holography in $AdS_5$. Upon analytic continuation, analogous to the relativistic case studied recently by Maldacena, the action of conformal gravity at the $z=2$ Lifshitz point features in the ground-state wavefunction of a gravitational system with an interesting type of spatial anisotropy.

hep-th↗

arXiv2011

Quantizing Horava-Lifshitz Gravity via Causal Dynamical Triangulations

We extend the discrete Regge action of causal dynamical triangulations to include discrete versions of the curvature squared terms appearing in the continuum action of (2+1)-dimensional projectable Horava-Lifshitz gravity. Focusing on an ensemble of spacetimes whose spacelike hypersurfaces are 2-spheres, we employ Markov chain Monte Carlo simulations to study the path integral defined by this extended discrete action. We demonstrate the existence of known and novel macroscopic phases of spacetime geometry, and we present preliminary evidence for the consistency of these phases with solutions to the equations of motion of classical Horava-Lifshitz gravity. Apparently, the phase diagram contains a phase transition between a time-dependent de Sitter-like phase and a time-independent phase. We speculate that this phase transition may be understood in terms of deconfinement of the global gravitational Hamiltonian integrated over a spatial 2-sphere.

hep-th↗

arXiv2011

General Covariance in Gravity at a Lifshitz Point

This paper is based on the invited talks delivered by the author at GR 19: the 19th International Conference on General Relativity and Gravitation, Ciudad de México, México, July 2010. In Part 1, we briefly review some of the main features of quantum gravity with anisotropic scaling, and comment on its possible relation to the causal dynamical triangulations (CDT) approach to lattice quantum gravity. Part 2 explains the construction of gravity with anisotropic scaling with an extended gauge symmetry -- essentially a nonrelativistic version of general covariance. This extra symmetry eliminates the scalar graviton polarization, and thus brings the theory closer to general relativity at long distances.

hep-th↗

arXiv2010

General Covariance in Quantum Gravity at a Lifshitz Point

In the minimal formulation of gravity with Lifshitz-type anisotropic scaling, the gauge symmetries of the system are foliation-preserving diffeomorphisms of spacetime. Consequently, compared to general relativity, the spectrum contains an extra scalar graviton polarization. Here we investigate the possibility of extending the gauge group by a local U(1) symmetry to "nonrelativistic general covariance." This extended gauge symmetry eliminates the scalar graviton, and forces the coupling constant $λ$ in the kinetic term of the minimal formulation to take its relativistic value, $λ=1$. The resulting theory exhibits anisotropic scaling at short distances, and reproduces many features of general relativity at long distances.

hep-th↗

arXiv2010

Emergent Gravity at a Lifshitz Point from a Bose Liquid on the Lattice

We propose a model with quantum bosons on the fcc lattice, which has a stable algebraic Bose liquid phase at low energy. We show that this phase is described by emergent quantum gravity at the Gaussian z = 3 Lifshitz fixed point in 3+1 dimensions. The stability of this algebraic Bose liquid phase is guaranteed by the gauge symmetry of gravitons and self-duality of the low energy field theory. By tuning one parameter in the lattice boson model we can drive a phase transition between the z = 3 Lifshitz gravity and another algebraic Bose liquid phase, described by gravity at the z = 2 Lifshitz point.

hep-th↗