Partition function for quantum gravity in 4 dimensions as a 1/N expansion Godwill Mbiti Kanyolo, Titus Masese

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Godwill Mbiti Kanyolo, Titus Masese. Partition function for quantum gravity in 4 dimensions as a 1/N expansion. 2021. ￿hal-03335930v2￿

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Distributed under a Creative Commons Attribution - NonCommercial - NoDerivatives| 4.0 International License Partition function for quantum gravity in 4 dimensions as a 1/ expansion N Godwill Mbiti Kanyolo1, ∗ and Titus Masese2,3, † 1The University of Electro-Communications, Department of Engineering Science, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585 2Research Institute of Electrochemical Energy (RIECEN), National Institute of Advanced Industrial Science and Technology (AIST), 1-8-31 Midorigaoka, Ikeda, Osaka 563-8577, Japan 3AIST-Kyoto University Chemical Energy Materials Open Innovation Laboratory (ChEM-OIL), Yoshidahonmachi, Sakyo-ku, Kyoto-shi 606-8501, Japan

Quantum gravity is the solution ascribed to rendering the geometric description of classi- cal gravity, in any dimensions, completely consistent with principles of quantum theory. The serendipitous theoretical discovery that black holes are thermodynamical objects that must participate in the second law has led to these purely gravitational objects to be dubbed, ‘the hydrogen atom for quantum gravity’, analogous to the atomic spectrum of hydrogen, effectively used by Neils Bohr and his contemporaries to successfully for- mulate quantum mechanics in the early 20th century. Here, we employ the temperature and entropy formulae describing Schwarzschild black holes to consider the emergence of Einstein Field Equations from a complex-Hermitian structure, [Ricci tensor √ 1 Yang-Mills field strength], where the gravitational degrees of freedom are the SU(± N−) colors with N Z 0 and a condensate comprising color pairs, k = N/2, appropri- ately coupled to∈ the≥ Yang-Mills gauge field. The foundations of our approach reveal the complex-Hermitian structure is analogous to Cayley-Dickson algebras, which aids in formulating the appropriate action principle for our formalism. The SU(N) gauge group is broken into an effective SU(4) SO(4) SO(1, 3) field theory on the tangent space with two terms: the Einstein-Hilbert→ action↔ and a Gauss-Bonnet topological term. Moreover, since topologically classifying all n = 4 (dimensional) Riemannian manifolds is not a clear-cut endeavor, we only consider the Euclidean path integral as the sum over manifolds with distinct topologies, h Z 0 homeomorphic to connected sums of an arbitrary number of n = 4-spheres and∈ h number≥ of n = 4-tori. Consequently, the par- tition function adopts a reminiscent form of the sum of the vacuum Feynman diagrams for a large = exp(βM/2) theory, provided = βM/2 = πN is the Schwarzschild black hole entropy,N β = 8πGM is the inverse temperature,S G is gravitational constant, M is the black hole mass and horizon area, = 2GβM = 4πGN is pixelated in units of 4πG. This leads us to conclude that theA partition function for quantum gravity is equivalent to the vacuum Feynman diagrams of a yet unidentified large theory in n = 4 dimensions. Our approach also sheds new light on the asymptoticN behavior of dark matter-dominated galaxy rotation curves (the empirical baryonic Tully-Fisher re- lation) and emergent gravity in condensed matter systems with defects such as layered materials with cationic vacancies as topological defects.

CONTENTS 2. Defining the central mass 13 3. Constraint 2 14 I. Introduction 2 C. Lagrangian density 16

II. Notation and Nomenclature 4 IV. Results 18 A. Group scalars and vectors (tensors) 4 A. Generating all mass terms 18 B. Introduction to p forms 7 B. Quantum gravity 19 C. Group scalars and vectors (p forms) 10 C. 1/ expansion 19 N D. Topology 20 III. Theory 11 1. Old quantum condition 20 A. Motivation 11 2. Entropy as an adiabatic invariant 20 B. Equations of motion 12 3. Random acceleration 21 1. Constraint 1 12 4. Gauss-Bonnet theorem 22 5. Euler characteristic, χ4 = χ2 22 E. Average energy of manifolds 23

∗ [email protected] V. Ramifications 23 † [email protected] A. Fermion/boson picture 23 2

B. Asymptotic behavior in galaxy rotation curves 25 are not entirely guaranteed to be more than an anal- C. Layered materials with cationic vacancies 27 ogy.(Dougherty and Callender, 2016) Nonetheless, consistency with well-tested ther- VI. Discussion 30 modynamical behavior(Isi et al., 2021), coupled Acknowledgements 32 with experiments using analogue gravitational systems(Jacquet et al., 2020; Steinhauer, 2014) offer a References 32 robust test for the correctness of the results, albeit with major unresolved problems such as the black hole infor- mation paradox(Almheiri et al., 2021; Hawking, 1976b) I. INTRODUCTION related to the apparent violation of information conser- vation in such systems(Hawking, 2005), and the nature The gravitational field in general relativity has of the black hole microstates.(Bekenstein, 2008) In par- a geometric description that, on one hand avails a ticular, according to the no-hair theorem(Thorne et al., myriad of visual cues which aid to build physicists’ 2000), a charged-rotating (Kerr-Newman) black hole intuitions on the nature of space-time(Thorne et al., is completely characterized by its mass (M), charge 2000), but on the other hand, notoriously does not lend (Q) and angular momentum (L), implying that black itself to renormalization.(Doboszewski and Linnemann, holes are the ‘simplest’ objects in the universe, with no 2018; Feynman et al., 2018; Hamber, 2008; Shomer, other internal degrees of freedom. Moreover, Birkhoff’s 2007; Thorne et al., 2000; Zee, 2010) This has led theorem(Israel, 1967; Jebsen, 2005) guarantees that to theoretic descriptions of gravity as an emer- the simplest such black holes are static, uncharged and gent phenomenon garnering some traction(Jacobson, non-rotating (Schwarszchild) black holes. While these 1995; Kleinert, 1987; Konopka et al., 2008; Lee et al., theorems are consistent with classical general relativity, 2013; Markopoulou and Smolin, 2004; Oh et al., 2018; semi-classical approaches to black hole thermodynamics Padmanabhan, 2010; Swingle and Van Raamsdonk, pioneered by Hawking(Hawking, 1974, 1976a, 1975) 2014; Van Raamsdonk, 2010; Verlinde, 2011) over quan- introduce a quantum field theory in a fixed space-time tum field theoretic approaches since emergence appears background, leading to the serendipitous conclusion that to by-pass the need for renormalization.(Burgess, 2004; black holes not only are black bodies with a temperature, Linnemann and Visser, 2017) Arguments for a successful T = κ/2πkB proportional to their surface gravity, κ, quantum theory of gravity range from the need to under- and entropy, = /4G proportional to their surface S A stand the nature of cosmic inflation(Hartle and Hawking, area, , where G is the gravitational constant and kB 1983) and black holes(Hawking, 1976a,b, 2005), to aes- is Boltzmann’sA constant, but also pack the maximum thetic considerations in favor of the unification of gravity amount of information possible within a given space, with gauge symmetries in the Standard Model of particle often referred to as the Bekenstein bound.(Bekenstein, physics.(Ross, 2003) 2005; Bousso, 1999) The Bekenstein bound, coupled Employing a semi-classical approach, Bekenstein and with the no-hair theorem, already implies an apparent his contemporaries showed that black holes are thermo- contradiction, namely that the ‘simplest’ objects in the dynamical objects that must participate in the second universe simultaneously harbor the most amount of law.(Bardeen et al., 1973; Bekenstein, 1973; Hawking, information. 1974, 1975) As a consequence, some proponents of emer- However, the black hole information paradox is a gent gravity have argued that gravity in general relativ- far more insidious problem than the aforementioned ity may not be as fundamental as the other gauge the- apparent contradiction, since black hole entropy and ories after all, since Boltzmann’s investigations demon- temperature require the black hole to eventually evap- strated that the second law is statistical, as it emerges orate by Unruh-Hawking radiation.(Hawking, 1974; from underlying microscopic physics.(Jacobson, 1995; Unruh, 1977) In particular, the radiation is essentially Padmanabhan, 2010; Verlinde, 2011) Nonetheless, these a black body spectrum in a mixed quantum state. considerations have led to black holes being dubbed, ‘the Whence, according to the semi-classical approach, hydrogen atom for quantum gravity’(Corda and Feleppa, quantum objects in a pure quantum state which fall 2019; ’t Hooft, 2016), setting up black hole thermody- into a black hole will eventually escape as Unruh- namics as an indispensable tool, analogous to the atomic Hawking radiation in a mixed state via particle-pair spectrum of hydrogen, effectively used by Neils Bohr production at the event horizon.(Giddings, 2012; and his contemporaries to successfully formulate quan- Mathur, 2009) This appears to violate conservation tum mechanics in the early 20th century. However, unlike of information since unitarity in quantum theory the atomic spectrum of hydrogen that was well-grounded precludes pure states evolving into mixed states and in observations prior to Bohr’s insights, most effects of vice versa. In essence, the black hole evaporation black hole thermodynamics are not only presently be- process in the semi-classical approach is not time- yond experimental reach(Hossenfelder, 2010), but also reversible(Hawking, 2005), since black hole entropy ap- 3 pears not to follow, e.g. the Page curve.(Almheiri et al., the partition function of n = 4 dimensional quantum 2019, 2021, 2020; Hashimoto et al., 2020; Page, gravity, corresponding to a quantum theory of Einstein’s 1980, 1993) Attempts to conclusively address general relativity, is a 1/ expansion for a rank n = 4 the black hole information paradox range from random tensor model/groupN field theory.(Gurau, 2011, proposing black hole remnants(Chamseddine et al., 2012; Gurau and Rivasseau, 2011) 2019), entangled black holes via an Einstein-Rosen Motivated by questions pertaining to the nature of the bridge(Maldacena and Susskind, 2013) to introduc- black hole microstates(Bekenstein, 2008), as well as an ing mechanisms for encoding black hole informa- overall pursuit for a suitable partition function for quan- tion in Unruh-Hawking radiation by circumventing tum gravity in n = 4 dimensions, in this work, we em- the challenges posed by monogamy of quantum ploy the temperature and entropy formulae describing entanglement(Grudka et al., 2018), with no clear Schwarzschild black holes in order to consider the emer- consensus. gence of Einstein Field Equations (EFE) from a complex- A related problem is the nature of black hole Hermitian structure, [Ricci tensor √ 1 Yang-Mills field ± − microstates.(Bekenstein, 2008) In statistical mechanics, strength], where the gravitational degrees of freedom are entropy is a measure of the number of microstates, the SU(N) colors with N Z 0 and a condensate N ∈ ≥ in a thermodynamical system, namely = kB ln . A comprising color pairs, k = N/2, appropriately coupled myriad of approaches conjecture thatS black holeN mi- to the Yang-Mills gauge field. The foundations of our crostates ought to correspond to the degrees of freedom approach reveal the complex-Hermitian structure is anal- associated with quantum gravity.(Becker et al., 2006; ogous to Cayley-Dickson algebras(Schafer, 1954), which Bousso, 2002; Einstein et al., 1935; Einstein and Rosen, aids in formulating the appropriate action principle for 1935; Hawking et al., 1999; Maldacena, 1999; our formalism. A similar construction, [metric tensor Maldacena and Susskind, 2013; Polchinski, 1998a,b; √ 1 U(1) field strength] was considered by Einstein ± − Susskind, 1995; Swingle and Van Raamsdonk, as an attempt to geometrize Maxwell’s theory using a 2014) For instance, it has been conjectured complex-Hermitian metric tensor, albeit with limited suc- that black hole information is entanglement cess.(Einstein, 1945, 1948) entropy(Swingle and Van Raamsdonk, 2014), with Within the formalism, the complex-Hermitian struc- µ † ideas such as the Einstein-Rosen (ER) bridge = ture avails the constraints, DµK ν = Ψ(Dν Ψ) and µν Einstein-Podolski-Rosen (EPR)(Einstein et al., 1935; Dν DµK = 0, where Ψ plays the role of wave func- Einstein and Rosen, 1935; Maldacena and Susskind, tion and K = R iF is a complex-Hermitian ten- µν µν − µν 2013) playing a major role in such approaches, with sor constructed using the Ricci tensor, Rµν and a Yang- the most successful proposals centered around the Mills/SU(N) field strength, Fµν . The SU(N) gauge holographic principle(Bousso, 2002; Susskind, 1995) group is broken into an effective SU(4) SO(4) → ↔ and the Anti-de Sitter/Conformal Field Theory SO(1, 3) field theory on the tangent space resulting in two (AdS/CFT) correspondence(Hawking et al., 1999; terms in the effective cation: the Einstein-Hilbert action Hubeny et al., 2007; Maldacena, 1999; Nishioka et al., and a Gauss-Bonnet topological term.(Lovelock, 1971) 2009; Ryu and Takayanagi, 2006a,b) and the re- The Euclidean path integral is considered as the sum liance of extra dimensions in the context of string over manifolds with distinct topologies, h Z 0 topo- ∈ ≥ theory.(Becker et al., 2006; Polchinski, 1998a,b) logically equivalent (homeomorphic) to connected sums Fairly recent developments have highlighted random of any number of n =4-spheres and h number of n =4- tensor models/group field theories of rank n as prime can- tori. Consequently, the partition function for quantum didates for quantum gravity.(Freidel, 2005; Gielen et al., gravity in n = 4 dimensions takes the form,

2013; Gurau, 2011, 2012; Gurau and Rivasseau, 2011; E E E ±χ4 (M ) Maldacena, 1999; Oriti, 2006; Thorn, 1994) This follows ZQG = exp( IME (λ)) , (1a) E ∓ N from the observation that their partition functions can MX∈h be Taylor expanded using 1/N as a small parameter, E λ 4 E E IME (λ)= d x det(gµν )R, (1b) where N is the size of the group and χn( ) is the π ME order of the expansion, which also correspondsM to the Z q Euler characteristic of an emergent compact manifold, where = exp(πN) is the number of microstates, E N E of dimensions n.(Aharony et al., 2000; ’t Hooft, 1993) IME (λ) is the Einstein-Hilbert action, with gµν the met- M 2 For instance, for random matrix models/group field the- ric tensor in Euclidean signature, λ = 1/16G = mP/16, ories (tensors of rank n = 2), the emergent manifold m = 1/√G the Planck mass and χ ( E) the Euler P 4 M is (n = 2 dimensional, with χ2( )=2 2h, where characteristic of a 4 dimensional Riemannian manifold, h Z 0 is a positive integer countingM the −sum of holes E, related to the 1, 3 pseudo-Riemannian space-time and∈ handles≥ (genus) in the manifold. Such matrix theo- manifoldM in Einstein’s general relativity by a Wick ro- ries have been argued to be dual to 2 dimensional quan- tation, t = itE. This result requires that the hori- tum gravity.(Gross, 1992) Thus, it is conjectured that zon area, =± 2GβM = 4πGN is pixelated in units of A 4

4πG.(Bekenstein and Mukhanov, 1998, 1995; Mukhanov, Throughout this paper, we use Einstein’s summation 1986; Vaz and Witten, 1999) convention(Thorne et al., 2000) for all repeated Greek Moreover, since it is not straightforward to topo- and Roman indices, and natural units where Planck’s logically classify all n = 4 dimensional Riemannian constant, speed of light in vacuum and Boltzmann con- manifolds,(Freedman, 1982) the Euclidean path integral stant, respectively are set to unity (~ = c = kB = 1). We is considered as the sum over manifolds with distinct also assume a torsion-free 1 + 3 dimensional space-time topologies, h Z 0 homeomorphic to connected sums manifold(Hehl et al., 1976), where is the metric com- ∇µ of an arbitrary∈ number≥ of n = 4-spheres and h number patible covariant derivative, g = 0 and g = g ∇σ µν µν νµ of n = 4-tori, thus obtaining the Euler characteristic, is the metric tensor in Lorenzian signature. We shall χ4 = 2 2h. Consequently, eq. (1a) coincidentally cor- indicate reduced quantities with an overhead bar. For responds− to the generic form of the 1/ expansion of instance β¯ = β/2π is the reduced inverse temperature N the free energy of a large theory.(Aharony et al., 2000; andm ¯ P = mP/√8π is the reduced Planck mass. Gurau, 2012; ’t Hooft, 1993N ) This leads to the conclusion a¯ Moreover, in the tetrad formalism, (ωµ) ¯b = that the quantum gravity partition function in n = 4 di- ω a¯ a¯ α a¯ α a¯ α β ( µ)¯b = e α µe ¯b = e α∂µe ¯b + e αΓ µβe ¯b is the mensions is equivalent to the sum of vacuum Feynman − ∇ α α 1 αβ ν spin connection, Γ µν = Γ νµ = 2 g (∂gµβ/∂x + diagrams of a yet unidentified large theory taking the µ β N ∂gβν/∂x ∂gµν /∂x ) are the torsion-free Christoffel form, − a¯ symbols/affine connection, eµ are the tetrad fields, γµ = exp(ZE ), (2a) are the gamma matrices in curved space-time satisfy- Z QG ¯ ∗T 0 −1 † 0 −1 ± ing γµγν + γν γµ = 2gµν , ψ = (ψ) (γ ) = ψ (γ ) µ µ a¯ a¯ = [ϕ]exp N Tr(γ(ϕ)) , (2b) is the Dirac adjoint spinor, γ = e a¯γ with γ the Z D − λ± Z   Dirac matrices in the tangent Minkowski space-time a¯ µ ± manifold satisfying γa¯γ¯b + γ¯bγa¯ = 2ηa¯¯b, and e µ,e a¯ are where S(ϕ) = N Tr(γ(ϕ)) is the action for uniden- a¯ µ λ tetrad fields satisfying eµeaν¯ = gµν and e a¯eµ¯b = ηa¯¯b tified field tensors, ϕ of rank n = 4, transforming ap- i.e.
with ηa¯¯b the Minkowski metric tensor diag(η¯ab¯) = propriately under some unidentified group, and γ is the (1, 1, 1, 1).(De Felice and Clarke, 1992) unidentified function of ϕ defining the large theory. − − − ¯ N The overhead bar ina, ¯ b, c¯ is used to distinguish the Thus, the vacuum Feynman diagrams pave n = 4 (di- Minkowski tangent manifold indices from the curved mensional) manifolds(Gurau, 2012), contrary to random space-time indices and group field theory indices, e.g. matrix large N theories where ϕ would be a tensor of 0¯ 0 a¯ a¯ in the tetrad fields, e 0 = e 0¯ and (ωµ) ¯b = ωµa(λa) ¯b, rank 2 (matrix), with Feynman diagrams defining n =2 where (λ )a¯ = (λ ) a¯ are6 n(n 1)/2 = 6 (with n = 4) E a ¯b a ¯b dimensional Riemannian manifolds, with a boundary, anti-symmetric− matrices corresponding− to the generators A E with Euler characteristic given by χ2( h)=2 2h, of rotation(Ba¸skal et al., 2021) in the tangent Euclidean where h = g + b/2 with g Z 0 the genus,A ∈ Z a positive− ∈ ≥ space-time manifold and related to generators of SO(1, 3) integer and b the boundary contribution.(Aharony et al., after Wick rotation. 2000; Gurau, 2012; ’t Hooft, 1993) Finally, we consider the possibility of defining E E χ4( ) in terms of χ2( ), where the matrix large A. Group scalars and vectors (tensors) Mgauge theories takingA the form of eq. (2), can Nbe identified with quantum gravity in n = 4 di- mensions. Our approach also sheds new light on Since we will be interested in Yang-Mills theory, where t ~t are the SU(N) Hermitian matrices, we shall adopt the asymptotic behavior of dark matter-dominated a ≡ galaxy rotation curves (the empirical baryonic the following notation for objects which transform as Tully-Fisher relation)(Eisenstein and Loeb, 1997; scalars and vectors respectively under SU(N), Famaey and McGaugh, 2012; Kanyolo and Masese, T γ···δ T γ···δI , (3a) 2021; Keeton, 2001; Martel and Shapiro, 2003; 0α···β ≡ α···β N McGaugh, 2012; McGaugh et al., 2000; Persic et al., γ···δ ~ γ···δ Taα···β Tα···β , (3b) 1996) and emergent gravity in condensed matter ≡ f A T γ···δ A~ T~ γ···δ, (3c) systems with defects(Holz, 1988; Kleinert, 1987; abc aµ bα···β ≡ µ × α···β Kleinert and Zaanen, 2004; Ver¸cin, 1990; Zaanen et al., where I is the N N identity matrix and, 2004) such as layered materials(Kanyolo et al., 2021; N × Masese et al., 2021b, 2018) with cationic vacancies as T γ···δ T γ···δ = T γ···δI + iT γ···δ, (3d) topological defects.(Kanyolo and Masese, 2020) kα···β ≡ α···β α···β N α···β is the most general object of any space-time tensor that II. NOTATION AND NOMENCLATURE transforms as a scalar (k = 0) or a vector (k = a) under SU(N) with the boldface indicating the vectors corre- 5 sponding to, not a scalar, but must transform as a vector,

γ···δ γ···δ Aµ = A~ µ ~t Aaµta, (3e) D (T )= D (T~ ~t) · ≡ µ α···β µ α···β · γ···δ γ···δ T T~ ~t, (3f) γ···δ 1 γ···δ α···β α···β = ( T~ + A~ T~ ) ~t ≡ · ∇µ α···β 2 µ × α···β · 1 in order for IN and i~t in eq. (3d) to be interpreted as γ···δ γ···δ = µTα···β i[Aµ, Tα···β ], (6d) N N basis matrices. This also implies that T γ···δ ∇ − 2 × α···β γ···δ ~ and Tα···β must have the same rank in space-time where we ought to have Dµt = 0. Moreover, the product indices. of a scalar and a vector, Since the identity matrix is already implied for the σ···ρ γ···δ γ···ρ scalars, we shall refrain from explicitly writing it unless Tµ···ν Tα···β = Jα···ν , (6e) for clarity. In this notation, the Yang-Mills field strength must be vector. This can be verified e.g. by noting that can be written as, differentiating by parts behaves as desired,

··· Fµν = ∂µAν ∂ν Aµ i[Aµ, Aν ] D J γ···ρ D σ···ρT γ δ − − η α···ν = η(Tµ···ν α···β ) ~ ~ ~ ~ ~ ··· ··· = (∂µAν ∂ν Aµ + Aµ Aν ) t, (4) T γ δD σ···ρ σ···ρD T γ δ − × · = α···β ηTµ···ν + Tµ···ν η α···β γ···δ σ···ρ σ···ρ γ···δ where we choose the normalization of the SU(N) matri- = T ηT ··· + T ··· ηT α···β ∇ µ ν µ ν ∇ α···β ces, ta as, 1 T σ···ρ i[A , T γ···δ] − µ···ν 2 η α···β N 1 Tr(tatb)= δab, (5a) = J γ···ρ i[A , J γ···ρ]. (6f) 4 ∇η α···ν − 2 η α···ν [ta,tb]= ifabctc, (5b) γ···δ Since there exist complex objects like Tα···β that are with Tr the trace, fabc the structure constants and δab neither scalars nor vectors but a combination of both, we the Kronecker delta. This normalization is chosen for note that the derivative, later convenience, and it corresponds to a re-scaling, D = iA , (7a) ta = N/2 Ta and fabc = N/2 Fabc of the customary µ ∇µ − µ † SU(N) matrices, Ta and the structure constants, Fabc D = + iA , (7b) p p µ µ µ of particle physics.(Zee, 2010) The bold-face and the ∇ overhead arrow ( ) unambiguously serve as the disam- acting on a scalar forms such a complex object, where → biguation of group vectors from scalars. necessarily Dµ = Dµ. Proceeding, we can construct a 6 Generically, since scalars commute with the SU(N) complex-valued curvature tensor, Kµν using commuta- gauge field, Aµ but vectors do not, the SU(N) gauge tion with the derivative in eq. (7) and a space-time vec- Vµ covariant derivative, Dµ will act differently on scalars tor, , which is such an object, and vectors to respectively yield scalars and vectors, as µ µ follows, Kµν V [Dµ, Dν ]V ≡ µ µ = Dµ( ν iAν )V Dν ( µ iAµ)V γ···δ γ···δ D ∇ − µ − ∇ − µ µTα···β = µTα···β , (6a) = ( iA )V iA ( iA )V ∇ ∇µ ∇ν − ν − µ ∇ν − ν γ···δ γ···δ 1 γ···δ µ µ ~ ~ ~ ~ ν ( µ iAµ)V + iAν ( µ iAµ)V DµTα···β = µTα···β + Aµ Tα···β . (6b) ∇ 2 × − ∇ ∇µ − µ µ ∇ − µ = [ µ, ν ]V iV µAν + iV ν Aµ [Aµ, Aν ]V ∇ ∇ ρ µ− ∇ ∇ − µ Here, we shall always have the factor of 1/2 in the cross- = Rµρν V i (∂µAν ∂ν Aµ i[Aµ, Aν ]) V product, which guarantees the Yang-Mills field strength − − − = (R i(D A D A )) Vµ can be defined as, µν − µ µ − ν µ = (R iF )Vµ, (8a) µν − µν Fµν DµAν Dν Aµ, (6c) ≡ − where Fµν is the Yang-Mills field strength defined in eq. (6c) and, in order for Dµ to act as a proper covariant derivative, by observing that the SU(N) U(1) exchange corresponds ρ ⇔ [ µ, ν ]Vσ = Rρσµν V , (8b) to Fµν Fµν and Aµ Aµ, where Aµ and Fµν are ∇ ∇ ⇔ ⇔ ρ scalars under SU(N). This is consistent with Dµ µ. with Rρσµν and Rµν = Rµρν the Riemann and Ricci Thus, the gauge covariant derivative transforms⇔ scalars ∇ tensors. Thus, we find, to scalars and vectors to vectors. Consequently, a dot- ~ Kµν = Rµν iFµν , (8c) product of a vector with the SU(N) matrix vector, t is − 6 is the object in question. In our notation, since K discrepancy arises from the fact that t and t SU(3) µν ≡ 8 3 Kkµν , we discover that, K0µν = Rµν and Kaµν = matrices are both proportional to the single octonion F~ νµ. Consequently, Kµν has the property that Kµν = element, e8 hence reducing the octonion components by T∗ ∗T ∗ † (Kµν ) = (Kµν ) = (Kνµ) = (Kµν ) , i.e it is one.(Shrestha et al., 2012) Hermitian with respect to both the space-time and group On the other hand, the SU(4) object has 42 = 16 in- indices. dependent tensors (including the identity) which exactly An intriguing observation is that, when the gauge corresponds to 2 8 = 16, the number of components of × group is SU(2), these objects are space-time dependent a sedenion. Since we can write, 16 = (2 2 [2 2] ), ×{ × × } quaternion tensors. In particular, a typical quaternion, we are always able to construct the SU(4) object from Q corresponds to a space-time scalar represented as, four quaternion sub-algebras,

Q = Q + iQ = Q + iQ~ ~t = Q + iQ t Kµν = Rµν e0 + e1fµν1 + e2 [fµν2 e1fµν3] · a a − = Q + e Q /2+ e Q /2+ e Q /2, (9a) + e4 fµν4 e1fµν5 e2 [fµν6 e1fµν7] 1 1 2 2 3 3 { − − − } + √2e (f e f e [f e f ] where, 8 µν8 − 1 µν9 − 2 µν10 − 1 µν11 e f e f e [f e f ] ) − 4 { µν12 − 1 µν13 − 2 µν14 − 1 µν15 } e1e2 = e2e1 = e3 = iσ3, (9b) = R + e f = R iF , (10b) − µν a µνa µν − µν e2e3 = e3e2 = e1 = iσ1, (9c) − as long as e2e1 = e3, e4e1 = e5, e4e2 = e6, e3e1 = e1e3 = e2 = iσ2, (9d) − − − − e4e3 = e7, e2e1 = e3, e8e1 = e9, e8e2 = e10, e e = − e , e e =− e , e e −= e , e −e = defines the quaternion algebra, Q,Q are real numbers 8 3 11 8 4 12 8 5 13 8 6 a e and−e e = e −is satisfied. In− the last line, and Q = Q~ ~t = Q t with t = σ /2 and σ the Pauli 14 8 7 15 a a a a a e− f = F follows− from setting e = I and K as- matrices.(Meglicki· , 2008) Thus, a dot-product of a vector a µνa µν 0 4 µν sumed Hermitian, since any 4 4 Hermitian matrix can with the SU(2) matrices can be treated as the imaginary always be decomposed into components× with SU(4) basis part of a quaternion, whose real part transforms as a matrices. scalar under SU(2). Evidently, our notation in eq. (3) Finally, introducing Ψ = Ψ U, we note that U = merely exploits the fact that in SU(2), the expressions exp(iS) and its inverse U−1 =| exp(| iS) are N N ma- with f reduce to actual cross-products since the struc- abc trices, while Ψ is a scalar under SU(− N), where,× ture constants, fabc = εabc are proportional to the Levi- | | Civita symbol, εabc. In SU(3), the objects are analogous UU−1 = U−1U = I , (11a) to octonions/Cayley numbers.(Dixon, 2013) N In particular, SU(2) objects can be related to the U(1) −1 is the N N identity matrix which satisfies Dµ(U U)= objects by a Cayley-Dickson construction(Schafer, 1954), −1 × −1 U DµU + UDµU = 0. Consequently, it is straight- where the number of generators (including the identity forward to show, e.g. in SU(2), that Ψ, U and U−1 are matrix) is doubled in the construction. We shall refer to quaternions by recognizing the product property given in γ···δ all such objects, Tα···β with their analogous Cayley- eq. (6e) and calculating, Dickson algebra names, namely complex, quaternion, oc- tonion, sedenion etc., corresponding to complex-valued exp(iS) = exp(iS1σ1/2) exp(iS2σ2/2) exp(iS3σ3/2) tensor objects with their imaginary parts transforming 1 as vectors under SU(N), i.e. U(1), SU(2), SU(3), SU(4) = (iS σ /2)j (iS σ /2)k(iS σ /2)l j!k!l! 1 1 2 1 3 3 etc. respectively. For instance, the Cayley-Dickson con- j,k,l X struction allows one to factorize a quaternion curvature = f(S1,S2,S3)I2 + if(S1,S2,S3), (11b) tensor (eq. (8c)) into two complex tensors, where we have used, σaσb = δab + iεabcσc and the Taylor K = R iF = R iF t expansion, exp(x) = xk/k!. Thus, eq. (11a) implies µν µν − µν µν − µν · k that f 2 + f 2 + f 2 + f 2 = 1, where ~f = (f ,f ,f ). This = Rµν + e1Fµν1/2+ e2Fµν2/2+ e3Fµν3/2, 1 2 P3 1 2 3 property can be generalized for the other SU(N) alge- = Rµν + e1Fµν1/2+ e2(Fµν2 e1Fµν3)/2, (10a) − bras, with ~f fa and S θ a continuous infinitesimal ≡ 2 ≡ where e1, e2, e3 are given in eq. (9b) and the factor of variable, where θ =0 vanishes in order for, 1/2 arises from the normalization choice of the SU(N) U = exp(iθ)= I + θ~ ~t, (11c) matrices given in eq. (5). However, the Cayley-Dickson N · −1 construction is obfuscated as the number of colors, N be- U = exp(iθ)= IN θ~ ~t. (11d) comes large. For instance, SU(3) objects contain 32 =9 − · real-valued tensors, whereas octonion tensors would only Thus, using det(U) = det(U−1) = 1 and U−1U = have 2 4 = 8 real-valued tensors. Nonetheless, this UU−1 = I , it follows that U, U−1 are unitary. × N 7

In this formalism, a gauge transformation corresponds (iv) The Levi-Civita symbol does not transform appro- to, priately as a tensor. To see this, on a manifold, of n dimensions with a metric tensor, g , we canM ′ −1 −1 µν Aµ = U AµU + U DµU. (12a) attempt to lower its indices by,

µν···σρ This requires that the field strength must transform in a ε gµαgνβ gσγ gρδ = det(gµν )εαβ···γδ, (16a) gauge invariant manner as, ··· − where the right-hand side follows from the definition ′ −1 Fµν = U Fµν U. (12b) of determinant of an n n matrix and the minus (-) sign originates from× the Lorenzian signature of the space-time manifold (For a Euclidean signature B. Introduction to p forms E metric, gµν , we would have the determinant multi- plied by a plus (+) sign instead). Thus, one divides A notation usually preferred for its compactness is both sides by the measure det(gµν ) to yield the differential forms.(Demailly, 1997; Tu, 2017) We intro- tensors, − duce several straightforward guidelines on differential p forms particularly helpful to follow the discussion in the vαβ···σρ = det(gµν )εαβ···σρ, (16b) manuscript: − q and, (i) Any p form can be locally written as the contracted form of a space-time tensor of rank p with an equal αβ···σρ 1 αβ···σρ µ v = ε , (16c) number of space-time basis vectors, dx p such that, det(g ) − µν 1 µ1 µp whose raising andp lowering operations by the metric A = Aµ ···µ dx dx , (13a) p! 1 p ∧ · · · tensor are appropriately given by eq. (16a). We can explicitly check each transforms as a tensor using, where, we can device a useful short-hand to write, ′µ ′ν µ µ µ ···µ 1 p ∂x ∂x dx 1 dx p = ε 1 p dx dx , (13b) det (g (x)) = det g′ (x′) ∧ · · · ··· σρ µν ∂xσ ∂xρ µ1···µp   with ε the totally anti-symmetric Levi-Civita ∂x′ 2 symbol normalized as ε1,2,3,···p = 1 and the fac- = det g′ (x′) , (16d) µν ∂x torial, p! introduced for later convenience. This
makes the p form the complete anti-symmetrization and,

of Aµ1···µp . ∂xσ1 ∂xσp dxσ1 dxσp = dx′µ1 dx′µp (ii) The wedge product, is associative (i.e. (A B) ′µ1 ′µp ∧ ∧ ∧ ∧ · · · ∂x ∧ · · · ∂x C = A (B C)), and corresponds to the multipli- σ1 σp ∂x ∂x µ1···µ ′1 p cation ∧ ∧ = ε p dx dx operation for p forms, The wedge product of ∂x′µ1 ··· ∂x′µp ··· a p and a q form yields a p + q form, ∂x = εµ1···µp dx′1 dx′p 1 ∂x′ ··· A B = A B dxµ1 dxµp+q , (14) ∧ p!q! [µ1··· ···µp+q] ∧ · · · ∂x = dx′σ1 dx′σp , (16e) ∂x′ ∧ · · · where [ ] indicates anti-symmetrization of the in- ··· dices. Thus, using this, one can show that, where we recognize,

A B = ( 1)pqB A. (15) ∂x ∂x′ −1 ∧ − ∧ = . (16f) ∂x′ ∂x A B A B This means that = 0 when = A = = B ∧ is the determinant of the Jacobian matrix. are group scalars and p = q is odd. However, when A = A = B = B are group vectors (i.e. can be (v) For an n manifold, the volume element is a p = n decomposed into components multiplied by group form built from eq.M (16b) as, basis matrices which do not commute), eq. (15)

need not vanish, even for odd p = q. Finally, we 1 α1 αp=n dV = vα1···αp=n dx dx , (17a) can determine the wedge product, A1 A2 An (p = n)! ∧ · · · for n number of p forms in a similar manner.∧ ··· where dV is the volume element. Using eq. (13), (iii) Since space-time indices are always hidden in the and the property of the Levi-Civita symbol, notation, only p forms with equal rank can be added α1···αn or subtracted. εα1···αn ε = n!, (17b) 8

the volume of the n = p manifold, becomes, Applying the exterior derivative operation twice an- M nihilates the p form, dV = dx1 dxp=n det(g ) µν 2 1 µ1 µp+2 M M ··· − d A = ∂µ +2 ∂µ +1 Aµ1···µ dx dx =0, Z Z q (p + 2)! p p p ∧ · · · n = d x det(gµν ). (17c) M − where we have used the fact that partial derivatives Z q commute, [∂µp+2 , ∂µp+1 ] = 0. We now recognize the need for the factorial, p! intro- duced in (i). For our purposes, all manifolds shall The expression translates into the Bianchi identities be Riemannian or pseudo-Riemannian, and hence in gauge and gravity theories, and can be familiar- ized as the differential form version of the identities, orientable (meaning det(gµν ) does not change sign ~ (~ A~) = 0 and ~ ~ φ = 0 for spatial vector anywhere on the manifold, to prevent the square- ∇ · ∇× ∇× ∇ root from turning imaginary). and scalar fields in n = 3 dimensions. This is in- structive, since the proof of these identities in vector (vi) The condition p = n, is referred to, in literature, calculus also utilizes the fact that partial derivatives as the condition that the p form is a top form. We commute. note that the volume element requires this condi- For the wedge product of a p and a q form, A and tion. This condition is vital, since the Lagrangian B respectively, the Leibniz rule is given by, density of any field theory must be a top form. Also, due to eq. (13), the relevant Levi-Civita symbol d(A B)= dA B + ( 1)pA dB. (19b) ∧ ∧ − ∧ strictly must have p = n indices in order for eq. (17a) to be a top form. For any p = n 1 form, A, on a non-compact mani- fold (i.e. a manifold− with a boundary, ∂ ), we can (vii) Any p n form, A, has a dual (n p) form, ⋆A elegantly write Stokes’ theorem as, M known≤ as the Hodge dual such that, − A = dA. (19c) 1 α1···αp αp+1 αn ⋆A= v ··· A dx dx . (18a) ∂M M p!(n p)! α1 αn ∧· · · Z Z − (ix) Since the wedge product of an (n p) form with a The appearance of v , defined in eq. (16b), − α1···αn p form is a top form, one can exploit this property guarantees the component of the Hodge dual, to define an inner product for two p forms, A and B, as the integral of their wedge product over the 1 α1···αp (⋆A)αp+1···αn = vα1···αn A , (18b) manifold, , p! M and its raised counterpart, A, B = A ⋆B. (20a) h i ∧ ZM αp+1···αn 1 α1···αn (⋆A) = v Aα1···αp , (18c) There is a sense in which the exterior derivative is p! a raising operator acting on forms since it takes a appropriately transform as tensors. Consequently, p form to a p + 1 form. One can define a lowering operator, d† by introducing a p 1 form C related the Hodge operation done twice yields back the p − form, to A in eq. (20a) by dC = A, and differentiate by parts and drop the boundary term (eq. (19c)), p(n−p) α1 αp C B C B ⋆⋆ A = ( 1) A ··· dx dx , (18d) M d( ⋆ )= ∂M ⋆ = 0, to yield, − α1 αp ∧ · · · ∧ ∧ R R where we have used, dC, B = dC ⋆B = C d⋆ B h i M ∧ − M ∧ µ1···µpαp+1···αn n−p µ1···µp Z Z v vα1···αpαp+1···αn = ( 1) (n p)!δα1···αp , − − = C ⋆( 1)p(n−p) ⋆d⋆ B − ∧ − µ1···µp ZM with δα1···αp a generalized Kronecker delta with the µ ···µ † † property, δ 1 p Aα1···αp = ( 1)pp!Aµ1···µp . = C ⋆d B = C, d B , (20b) α1···αp ∧ h i − ZM (viii) The exterior derivative operation, d on any p form where the lowering operator is given by d† = is given by, ( 1)p(n−p)+1 ⋆ d⋆. − 1 µ1 µp+1 (x) For any p+1= n form, B on the manifold, B is said dA = ∂µp+1 Aµ1···µp dx dx . (19a) (p + 1)! ∧ · · · to be closed if dB = 0. Employing eq. (19), one 9

could suspect that we can always write, B = dA, mathematics, the monopole number is referred with A defined the same way everywhere on ∂ . to as the first Chern number.(Chern, 1946) Gen- It turns out that the statement is false for topo-M erally, Stokes’ theorem (eq. (19c)) directly re- logically relevant manifolds. This stems from de lates such numbers to a class of non-exact but Rham cohomology(Demailly, 1997), a tool expos- otherwise closed p forms known as characteristic ing the extent to which the fundamental theorem of classes.(Milnor and Stasheff, 2016) calculus fails on topological manifolds. Particularly useful for this paper, is the Euler char- In particular, if dA = B, where A is defined the acteristic of an n = 4 dimensional compact Rieman- same way everywhere on ∂ , then B is said to be nian manifold, E (with Euclidean signature, de- M exact. Thus, every exact form is closed, but the noted by the superscript,M E) given by the integral, converse fails due to the non-trivial topology of . Thus, the topology of the manifold can be exploitedM 1 1 to avail an avenue to apply Stokes’ theorem on com- R R j χ4 = 2 Tr( )= CS , (23a) pact manifolds (i.e. manifolds without a boundary). 8π ME ∧ πN ∂ME Z j=± Z j In particular, the boundary operation ∂ applied to X N 2 a compact manifold vanishes, ∂ = 0. However, CSj = Tr ωj dωj + ωj ωj ωj , (23b) M 8π ∧ 3 ∧ ∧ suppose can be divided into two patches, +   and Mdue to topological reasons (which willM be- − where E = E, R = dω + ω ω is the comeM apparent), then we can write the condition j=± j curvatureM p = 2 formM which transforms∧ as R′ = that is compact as, − P ′ − M ΛRΛ 1 under SO(4) rotations, ω = Λ 1ωΛ + −1 j ∂ ± = , (21a) Λ dΛ, ω is the spin connection and CS is the M ±B Chern-Simons p = 3 form(Chern and Simons, 1974) = + + −, (21b) M M M with, where is the boundary shared by + and −, albeit withB opposite orientation ( ),M and mustM be a N ± k = Z 0, (23c) compact manifold, ∂ = 0. Since is n 1 dimen- 2 ∈ ≥ sional when is nBdimensional,B and− both man- ifolds are compactM , this procedure can be carried a positive integer known as the Chern-Simons out successively n 1 times, introducing non-trivial level.(Jackiw and Pi, 2003) This further restricts topologies at each− stage. the number of colors, N to even positive integers. For instance, in eq. (19c), when A = A is the The integrand in eq. (23a) is the second Euler class U(1) gauge field (p = 1 form) and = S2 is the and is equivalent to(Lovelock, 1971), n = 2-sphere, a magnetic monopoleM at the center 2 1 of the S sphere implies that A cannot be defined Tr(R R) 2 E the same way on the northern (∂ + = S+) and det(gµν ) ∧ 2 M southern (∂ − = S ) hemispheres, each bounded µνσρ µν 2 − p = (R Rµνσρ 4R Rµν + R ), (24a) at the equatorM by the n = 2-sphere, = S1 with − B opposite orientation. Nonetheless, the respective where Rµνσρ is the Riemann tensor, Rµν and R are gauge fields are related by a gauge transformation, the Ricci tensor and scalar respectively and we have + − A = A + dθ1, where θ1 is a p = 0 form. Since θ1 used, is the azimuthal angle, defined modulo 2πν on S1, where ν Z is an integer, Stokes’ theorem requires ∈ R = dω + ω ω that, ∧ a¯ a¯ a¯ c¯ = R ¯b = dω ¯b + ω c¯ ω ¯b 1 1 1 ∧ 1 a¯ a¯ a¯ c¯ µ ν dA = dA + dA = ∂µω ¯ ∂ν ω ¯ + ω ω ¯ dx dx 2π 2 2π 2 2π 2 ν b µ b µ c¯ ν b ZS ZS+ ZS− 2 − ∧ ∧  1  1 + 1 − a¯ µ ν = A + A = Rµν ¯bdx dx , (24b) 2π 2 2π 2 2 ∧ Z∂S+ Z∂S− σ a¯ σ ¯b 1 + 1 − with R = R e e the Riemann tensor. = A A ρµν µν ¯b a¯ ρ 2π 1 − 2π 1 ZS ZS E E 1 A Wick rotation, t it, where t is the Eu- = dθ = ν, (22) clidean time, corresponds→ ± to the transformation, 2π 1 ZS1 SO(4) SO(1, 3), where the rotations, Λ now where ν is the monopole number in Maxwell’s correspond→ to local Lorentz transformations (three theory.(Dirac, 1931; Konishi, 2007; Zee, 2010) In boosts and three rotations) on the tangent manifold. 10

2 Thus, the Chern-Simons p = 3 form transforms such where A = AI , A = A~ ~t, A~ = (A , , A 2 ), N 1 N · 1 ··· N −1 − that the wave function, is the number of SU(N) generators and A, A1, , AN 2−1 are real-valued space-time functions. Defining··· anti- + + λ~ ψ = exp i CS Hermitian basis matrices, from the Hermitian ones, CS ~t and their commutation relations as follows,  ZB  = exp i CS− exp(iπNχ ) λ~ = i~t, (29b) 4 −  B  Z − [λa, λb]= fabcλc, (29c) = ψCS exp(iπNχ4) , (25) we recognize that eq. (29a) takes the desired object form, is single-valued under SO(1, 3), where we have used albeit with anti-Hermitian basis vectors, λ~ in place of the ∂ = j with j = from eq. (21). In fact, since j Hermitian vectors, ~t, theMbold-faceB indicates± that ω is a group vector, we can borrow a leaf out of eq. (22), and take the two A = A + iA. (29d) Chern-Simons p = 3 forms to be related to each other by local Lorentz transformations/Euclidean In particular, taking A to be a p = 1 form, the object rotations, form of the Yang-Mills p = 2 form can be defined com- pactly as, ω± = Λ−1ω∓Λ + Λ−1dΛ, (26) DA = F = dA + iA A = F + iF, (30a) which corresponds to, ∧ where d is the exterior derivative, F = dA is the U(1) CS± = CS∓ + Ndθ, (27a) field strength, while F = dA + A A is the Yang-Mills ∧ 1 field strength, which in component form can be written dθ = Tr(Λ−1dΛ)3, (27b) 24π as, Fµν = Fµν + iFµν . For instance, the field equations correspond to, where θ is a p = 2 form, defined modulo 2πν on E 1 corresponding to the second Chern (winding) D†F = J = J + iJ, (30b) Bnumber and, 2π 1 which are equivalent to, Λ−1 Λ 3 χ4 = 2 Tr( d ) =2 dθ = 2ν. (28) 24π BE BE ± 1 µ Z Z DµF ν = Jν = Jν + iJν , (30c) However, eq. (28) is an atypical result since it some- 2π what purports the equivalence between the second where D†F = d†F and D†F = d†F + ⋆(A F), in con- Euler and Chern classes. This is only valid if one formity with eq. (6a), and a factor of 2π introduced∧ for can treat ω as a gauge field, A, which shall have later convenience. Moreover, since d2 = d†2 = 0 vanishes, great utility in the work herein. D2 = D†2 = 0 is also an identity of the covariant deriva- In particular, eq. (28) exploits the fact that SU(4) tives and likewise vanishes. Thus, we can retain our nota- will be broken into SO(4) (F R), which would tion for Yang-Mills objects, namely the complex objects permit the second Chern class integrated→ over S4 to (T), the scalars (T ) and the vectors (T). The bold-face thus indicates that we are dealing with N N matrices, be identified as the second Euler class on the four- × dimensional compact Riemannian manifold, E, whose basis matrices are the anti-Hermitian generators provided (S4,S3) are homeomorphic to ( E,ME) of the SU(N) gauge group. Consequently, gauge trans- under a symmetry breaking U Λ, whereM B the formations correspond to, → object U under SU(4) transforms as a vector un- ′ A = A + dθ, (31a) der SO(4), reflecting the apparent replacement of ′ −1 −1 the complex SU(4) tangent manifold with the real A = U AU + U dU, (31b) SO(4) tangent manifold. This is explored in Section which implies the respective components of the field III.A. strength object transform as a scalar, F ′ = dA′ = F = d(A + dθ)= dA, (31c) C. Group scalars and vectors (p forms) and a vector, Any N N Hermitian matrix, A = A† can be expanded into basis× SU(N) matrices, t including the identity ma- F′ = d(U−1AU + U−1dU) trix, IN as, + (U−1AU + U−1dU) (U−1AU + U−1dU) ∧ A = A A, (29a) = U−1(dA + A A)U = U−1FU, (31d) − ∧ 11

2 a¯ a¯ under SU(N) gauge group, where we have used d θ = where the spin connection, ω ¯b = ω¯b is a 4 4 anti- d2U = d2U−1 = 0 and U−1dU = UdU−1 in eq. (31d). symmetric matrix, and the components− a ¯ running× from Finally, it is interesting to check− that Maxwell’s theory 0¯ to 3,¯ correspond to the four coordinates of the tangent in special relativity takes the form, manifold. Thus, one can expand ω into its basis matrices in order to find, j 0 ∂j F =2πJ , (32a)

jkl j ∂ j iε ∂kFl = 2πJ F , (32b) − − ∂t ω = ωaλa = ω2λ2 + ω5λ5 + ω10λ10 where Ej and Bj are the spatial components of the elec- + ω14λ14 + ω12λ12 + ω7λ7, (35) tric and magnetic fields respectively, εjkl is the n = 3 dimensional totally anti-symmetric Levi-Civita symbol normalized as ε123 =1, Fj = Ej + iBj is a complex vec- where λa are the anti-Hermitian generators of tor constructed from the two U(1) group scalars, Ej ,Bj SU(4)(Sbaih et al., 2013) and we require all the µ µ µ SU(4) generators which do not appear in eq. (35) to be and J = Je + iJm is the complex current density con- µ µ set to nil. Consequently, this strictly breaks SU(4) gauge structed from two U(1) scalars, Je and Jm, correspond- ing to the electric and magnetic charge densities. Thus, symmetry in favor of SO(4), thus capturing the local since Maxwell’s theory with monopoles satisfies S-duality, rotation group of the tangent space in n = 4 dimensions. i.e. Maxwell’s equations are invariant under the ex- To further exploit the formalism, we can always cal- change of electric quantities with their magnetic coun- culate with the full SU(4) symmetry and subsequently terparts(Zee, 2010), eq. (32) can be written in a met- break it to SO(4) afterwards, by setting the appropriate ric independent manner using the language introduced values of A to nil, followed by a Wick rotation, SO(4) above as eq. (30), where the gauge group is U(1) instead SO(1, 3) in order to yield the local Lorentz group on of SU(N) (A A′). Consequently, → → the tangent space replacing the Euclidean rotation group. This corresponds to making the identifications, A = A + iA′, (32c) 1 1 ν = dA, ν′ = dA′, (32d) 2π S2 2π S2 R F, ω A, (36a) Z Z ⇔ ⇔ ′ Λ U, η ¯ δ ¯, (36b) where ν and ν respectively are the monopole and elec- ⇔ a¯b ⇔ a¯b tron numbers and S2 is the n = 2-sphere.

where U = exp(iθ) = exp(iθ~ ~t) = exp(θ~ λ~ ) = exp(θE)= · · III. THEORY Λ are local Lorentz transformations on the tangent man- ifold, and A. Motivation

To gain insights into the scalar and vector structure we ~ E E E ~ E ~ λ = (K2 ,K5 ,K10,L14,L12,L7) = (K , L), (37a) have considered above, we can refrain from suppressing the matrix components, and explicitly write the Yang- Mills vector field strength and gauge transformations as, corresponds to the local generators of Euclidean boost, ~ E E E E ~ K = (K1¯ ,K2¯ ,K3¯ ) and angular momentum L = (L1¯,L2¯,L3¯) on the tangent space. In other words, we Fa¯ = dAa¯ + Aa¯ Ac¯ , (33a) ¯b ¯b c¯ ∧ ¯b shall have, ′a¯ −1 a¯ c¯ d¯ F ¯b = (U ) c¯F d¯(U) ¯b, (33b) ′a¯ −1 a¯ c¯ d¯ −1 a¯ c¯ A ¯b = (U ) c¯A d¯(U) ¯b + (U ) c¯d(U) ¯b, (33c) j a¯ j¯b µ Λ = exp λ ¯bωµ a¯dx where Aa¯ = A∗ a¯ is a space-time dependent N N ¯b ¯b Z  Hermitian matrix. It is evident that, in the tetrad× for- = exp Tr(λωj ) , (37b) malism, the curvature p = 2 form takes a similar form, Z 

a¯ a¯ a¯ c¯ a¯ ′ R ¯b = dω ¯b + ω c¯ ω ¯b, (34a) where j = and λ = λ λ is expanded as a matrix ∧ ¯b a a ′a¯ −1 a¯ c¯ d¯ ± ⊗ R ¯ = (Λ ) R ¯(Λ) ¯, (34b) tensor product into SU(4) according to eq. (35). Requir- b c¯ d b ′ ¯ ing the trace to act on the λa basis matrices, but not ω′a¯ = (Λ−1)a¯ ωc¯ (Λ)d + (Λ−1)a¯ d(Λ)c¯ , (34c) ¯b c¯ d¯ ¯b c¯ ¯b their matrix coefficients, λa, the Lorentz transformation 12 of a topologically relevant manifold is defined as, obtaining an effective action for general relativity, with desirable topological features for quantum gravity. Λ = Λ+(Λ−)−1 We shall use our notation to introduce the gauge co- variant constraint, = exp Tr(λω+) Tr(λω−) − D Kµ = Ψ(D Ψ)†, (39a) Z Z  µ ν ν † = exp λ ω+Tr(λ′ λ′ ) λ′ ω−Tr(λ′ λ′ ) Ψ= √R exp(iS) , Ψ = √R exp( iS) , (39b) a b a b − a b a b −   µν Z Z where R = gµν R is the Ricci scalar, Kµν and Dµ are = exp λ ω+δ λ ω−δ given in eq. (8) and eq. (8c) respectively, and, a b ab − a b ab Z Z  µ µ = exp ω+ ω− = exp dθE S = pµdx + Aµdx , (39c) − ∓ − Z Z   Z  Z Z = exp θE = U∓1, (37c) is an action that transforms as a vector under SU(N), ∓ where, where U+1 U is the unitary object under
SU(4), and we pµ uµ have used eq.≡ (5) with N = 4, eq. (26), eq. (29b) and = κ , (40a) set dθE = Λ−1dΛ. dxµ(xν ) uµ = , (40b) Consequently, 6 of the 10 generators of Poincar´esym- dτ metry can be implemented by gauge/local Lorentz trans- with pµ the n =4-momenta of N particles, tracked by a formations via the vector component of the object, F in central space-time coordinate xν (τ), τ is the proper time eq. (30a). However, to capture the remaining Poincar´e and κ is a mass parameter. We shall also choose the symmetry generators(Zee, 2010) (translations), we can following normalization condition on the n = 4-momenta simply take a linear combination of the Killing vectors and n =4-velocities, and set it equal to the group scalar gauge field, N Tr(pµp )= ~p ~pµ λ, (41) 3 µ 4 µ · ≡ Aµ = c ξµ, (38a) a¯ a¯ where λ is a parameter with dimensions of (mass)2 to be a¯=0 X determined. The central coordinate defines the center of µ where ca¯ are constants with dimensions of mass and ξa¯ mass n =4-velocity, are the 4 generators of space-time translations on the tan- dxµ(τ) gent space. This suggests that the SU(N) scalar current uµ = , (42a) dτ in eq. (30) can be written as, µ u uµ = 1 (42b) ν − Jµ =2Rµν A . (38b) which transforms as a scalar under SU(N). Using eq. (7) and separating the real and imaginary parts of eq. (39) This result is motivated by [ , ]ξ = Rρ ξ and µ ν σa¯ σµν ρa¯ yields, respectively, the Bianchi identity, ξ = ξ , which guarantees∇ ∇ that the scalar cur- ∇ν aµ¯ −∇µ aν¯ rent is conserved. µ 1 µR = ν R, (43) Finally, it will be convenient initially to work with ∇ ν 2∇ SU(N) instead of SU(4), which provides a generalized and the equations of motion for the gauge field, framework for our approach. Comparing Kµν from eq. D Fµ J (8c) with Fµν from eq. (30a), it is evident that, even µ ν =2π ν , (44a) F though µν is also Hermitian under the SU(N) matrix where, indices, unlike Kµν , it is not invariant under complex conjugation followed by transpose of the space-time in- 1 ∗ ∗ R Jν = i(Ψ Dν Ψ ΨDν Ψ )= pν , (44b) dices. We shall refer to Kµν as complex-Hermitian since 4π − −2π it exhibits both levels of hermiticity, while Fµν does not. is the SU(N) current density. Thus, we arrive at familiar expressions in general relativity and Yang-Mills theory. For completeness, we can couple the Yang-Mills field B. Equations of motion strength, Fµν to a Dirac spinor, ψ with N components 1. Constraint 1 using the semi-classical EFE, 1 Rµν Rgµν = 8πGT , (45a) In the preceding work, we exploit the complex- − 2 − µν Hermiticity of K in order to introduce two constraints µν µν µν T = T + T , (45b) analogous to eq. (30). These constraints are essential to µν h Diraci SU(N)   13 where, 2. Defining the central mass

µν 2 µα ν 1 αβ µν However, general relativity does not admit a universal T = Tr F F F Fαβg , (46a) YM Nπ α − 4 et al.   definition of mass in arbitrary space-times.(Thorne , 2000) Typically, one can exploit time translation sym- is the Yang-Mills energy-momentum tensor and, metric (stationary) asymptotically flat space-times such as the Schwarzschild metric to define the mass, M at µν 1 ¯ µ ν ¯ ν µ TDirac = ψγ D ψ + ψγ D ψ spatial infinity, where the curvature vanishes, since such 4i − µ a space-time admits a time-like Killing vector, ξ0¯ as a 1 †µ ¯ ν
†ν ¯ µ µ µ (D ψ)γ ψ + (D ψ)γ ψ , (46b) generator of time-translations, ξ0¯ ∂/∂x = ∂/∂t.This in- 4i troduces the notion of conserved energy, that can appro-
is the energy-momentum tensor of the Dirac equations, priately be compared to expected notions in special rel- µ ativity. Moreover, in addition to ξ¯ , stationary axisym- µ 0 iγ (Dµψ)= Mψ, (47a) metric asymptotic flat space-times such as the Kerr met- i(D† ψ¯)γµ = Mψ,¯ (47b) ric admit an additional unique rotational Killing vector, µ − µ µ µ ξ4¯ as a generator of rotations ξ4¯ ∂/∂x = ∂/∂θ, whose where M is the Dirac mass, Dµψ = ( µ iAµ)ψ, orbits comprise closed curves along the azimuthal angle, † 1 ∇a¯¯b − D ψ¯ = ψ¯ + iψ¯A , ψ = (∂ ω γ γ¯)ψ and 0 θ 2π on the 2-surface of an oblate spheroid. µ µ µ µ µ 4 µ a¯ b ≤ ≤ ∇ 1 a¯¯b ∇ − By definition, the Killing vectors obey the ψ¯ = ∂ ψ¯ + ψ¯( ω γ γ¯). The quantum average, µ µ 4 µ a¯ b et al. ∇is taken only over the Dirac terms in a semi-classicalh· · · i relations(Bardeen , 1973), sense, since fermionic field equations do not have a valid ξ ¯ + ξ ¯ =0, (52a) classical description due to the Pauli-exclusion principle. ∇µ ν0 ∇ν µ0 By contrast, bosonic field equations represent the equa- µξν4¯ + ν ξµ4¯ =0, (52b) ∇ µ∇ ν tions of motion for a large number of bosons occupying µ ν ξ = Rµν ξ¯ , (52c) ∇ ∇ 0¯ 0 the same quantum state, which avails a valid classical µ ν µ ν ξ¯ = Rµν ξ4¯ , (52d) description. ∇ ∇ 4 Consequently, we can define the velocities, uν for the where, Dirac degrees of freedom in the following manner, ξµ ξν = ξµ ξν . (53) 0¯ ∇µ 4¯ 4¯ ∇µ 0¯ ψγ¯ ν tψ uν = h i. (48) µ ψψ¯ Thus, a linear combination of the Killing vectors, ξ = h i µ µ ξ0¯ +Ωhξ4¯ , corresponding to the null vector tangent to µ and proceed to calculate the trace of eq. (45) using eq. the generators at the horizon defined by ξ ξµ = 0, also (47) to yield, satisfies eq. (52),

R = 8πGM ψψ¯ . (49) ξ + ξ =0, (54a) − h i ∇µ ν ∇ν µ ξµ = R ξµ, (54b) Consequently, using eq. (43) to ensure that the right- ∇µ∇ν µν hand side of eq. (45) is divergence-free, we reproduce eq. and the parameter, Ωh is the angular frequency constant (44) with, on the horizon and satisfies, ¯ Jν = ψγν tψ . (50a) ξµ ξν = κξµ, (55) h i ∇µ Plugging in the results in eq. (49) into eq. (44), we find, where κ is the surface gravity we seek to define in terms of the mass M, in a accordance to eq. (51). Since Jν =4GMκ ψψ¯ uν . (50b) h i µξ ¯ = µξ ¯ is anti-symmetric, we can use Stokes’ ∇ ν0 −∇ ν0 Notably, for eq. (50a) to correspond to eq. (50b), we use theorem to transfer the integral of the asymptotically flat space-like hypersurface, S = (tangent to the rotation eq. (48), which requires that, µ V Killing vector, ξ4¯ and intersecting the horizon at a 2- 1 surface ∂B = ), to the boundary, ∂S of S (consisting κ = , (51) A 4GM of ∂B and a 2-surface, ∂S∞ at spatial infinity), and thus motivates identifying κ as the surface µ ν µ ν gravity(Wald, 1984), provided the Dirac mass, M dou- dΣµν ξ¯ + dΣµν ξ¯ ∇ 0 ∇ 0 bles as the central mass, M in the Schwarzschild metric Z∂S∞ Z∂B (for the the non-rotating black-hole solution of eq. (45) µ ν µν = dΣµν ξ0¯ = dΣν ξµ0¯R , (56) with T 0). ∂S ∇ − S µν → Z Z 14 where dΣ = 1 (n ξ n ξ )d and dΣ are the sur- where Φ and Φ are the electric and magnetic potentials µν 2 µ ν − ν µ A µ e m face elements of ∂S and S respectively, while d is the associated with the electric qe and qm monopole charges 3 A area element of the horizon, d = d x det(gµν ) is respectively. Consequently, the exact solutions for the the volume element of S and nV is the other− null vector quantities in eq. (61) are given by(Hawking, 1976a), ν p orthogonal to ∂B, normalized as nµξ = 1. Thus, we µ − 4π 4π 4π define the total mass as measured from infinity (corre- κ = (r r /2) , Φ = q r , Ω = a, (62b) + − S i i + h sponding to the central mass) as, A A A 2 2 2 1 where r± = rS/2 (rS/2) a G i qi is the ra- M = dΣ µξν . (57a) ± − − µν 0¯ dius of the inner( ) or outer(+) horizon, rS = 2GM is 4πG ∂S∞ ∇ − p P Z the Schwarzschild radius, = 4πr+rS and a = Lh/M. In addition, the integral over ∂B yields, Evidently, the surface gravity,A κ is an elaborate function of M and Lh as well as qi, and hence does not corre- µ ν µ ν dΣµν ξ = ξµ ξ nν d spond to eq. (51) unless the solution is Schwarzschild ∂B ∇ − ∂B ∇ A (L 0 and Q 0). This implies that our approach Z Z → → ν will be a good approximation to real systems as the lim- = ξ nν κd = κd = κ , (57b) − A − A − A its, r r and q 0 are satisfied, which correspond Z∂B Z∂B=A + S i to the limits→ ξµ ξ→µ and T µν 0, where the energy- where we have used eq. (55) and the fact that κ is con- 0¯ momentum tensor→is given by eq.→ (45). stant over the horizon.(Bardeen et al., 1973) Likewise, the total angular momentum as measured from infinity and the angular momentum at the horizon defined in a 3. Constraint 2 similar manner as the mass, We shall consider a second constraint, 1 µ ν L = dΣµν ξ4¯ , (58a) 8πG ∂S∞ ∇ µν Z Dν DµK =0, (63) 1 µ ν Lh = dΣµν ξ4¯ , (58b) 8πG ∂B ∇ whose imaginary and real parts respectively correspond Z to, where Stokes’ theorem, using eq. (54), yields, µ DµJ =0, (64a) µ ν µ ν µ dΣµν ξ¯ + dΣµν ξ¯ R =0. (64b) ∇ 4 ∇ 4 ∇µ∇ Z∂S∞ Z∂B µ ν µν The first expression in eq. (64a) is guaranteed by the = dΣ ξ = dΣ ξ ¯R . (59) µν ∇ 4¯ − ν µ4 SU(N) symmetry. However, the second expression is Z∂S ZS novel, since complex-Hermitian objects such as Kµν do Thus, since the EFE in eq. (45) can be transformed into not traditionally appear in the formulation of Einstein’s in n = 4 dimensions, general relativity. 1 It is prudent to define a conserved current density, R = 8πG T g T αβg , (60) µν − µν − 2 αβ µν   1 Jν = ν R, (65) we can plug in the results from eq. (57) and eq. (58) into 2π ∇ eq. (56) to yield, which transforms as a scalar under SU(N) gauge group. κ For the purposes herein, we shall obtain the scalar cur- M = A + 2Ω L +2E, (61a) 4πG h h rent density using, where the energy, E is given by, R = R exp( Φ), (66a) c − µν 1 αβ µν µ E = dΣ ξ ¯(T g T g ), (61b) Φ= pµdx , (66b) µ ν0 − 2 αβ ZS=V Z p = β¯−1u , (66c) where is an n = 3 dimensional space-like hyper-surface. µ µ V For instance, it follows that, for the Kerr-Newmann where R is a constant and β¯ = β/2π is a reduced inverse µν µα ν 1 αβ µν c solution (T = F F α 4 F Fαβg , where Fµν = temperature to be defined. Using the Killing vector ξµ, ∂ A ∂ A is the U(1) field− strength with an appropri- µ ν − ν ν we can exploit the Killing relations in eq. (54) by taking ate choice of gauge for Aµ), the energy becomes the trace to find, ξν = 0 and setting, ∇ν E = qiΦi, (62a) R J = J ξ = p , (67) i∈e,m ν c ν ν X −2π 15 in order to guarantee that the current is conserved, where we have used eq. (71), eq. (42) and, ν ν J = 0. When there is more than one space-time ∇ µν isometry, one should take a linear combination of the R µξν =0, ∇ Killing vectors (such as in eq. (38a)), in order to con- µξν = ν ξµ, µ ∇ −∇ struct a suitable ξ . This ensures that the constraint in 1 ξµ R = ν (R ξµ)= ν ξµ =0, eq. (63) represents a combined space-time and SU(N) 2 ∇µ ∇ µν ∇ ∇µ∇ν gauge symmetry, where the conserved charge is a com- µ plex object given by, from eq. (54), which guarantee that ξ µΦ=0. Thus, we obtain the equation of motion, ∇ 3 0 Q = d x det (gµν ) Tr(J ), (68) µ ¯−1 V − u µuν = β uν + ην , (72a) Z q ∇ − related to this combined symmetry. Here, is the n =3 which takes the form of a Langevin V (dimensional) space-like hyper-surface corresponding to equation(Lemons and Gythiel, 1997), where β¯ plays 1 † spatial volume. Thus, Jµ = 2π Ψ(DµΨ) = Jµ + iJµ is the role of the mean free path between collisions and the conserved current. Nonetheless, the trace, Tr in Q ην the random acceleration (which is nil in eq. (72a)). ensures that Q = 0 vanishes by virtue of the SU(N) Moreover, since, matrices being trace-less. This reduces the number of finite charges to unity, where Q = Q + i0 is purely a β¯−1 exp( Φ)ξ = β¯−1 exp( 2Φ)u − µ − − µ scalar under SU(N). Since Q is the only finite charge, it 1 is reasonable to identify Q = N with the SU(N) colors, = exp( 2Φ) µΦ= µ(exp( 2Φ)) − − ∇ 2∇ − which corresponds to the scalar charge associated with µ 1 ν ν ν the Killing vectors, ξ . Moreover, when N 1, we = µ(ξ ξν )= ξ µξν = ξ ν ξµ, (72b) −2∇ − ∇ ∇ obtain a trivial U(1) gauge symmetry with the→ identity replacing the trace-less matrices, and hence Q remains we obtain eq. (55), where κ = β¯−1 exp( Φ) indeed is the finite satisfying Q 2Q. For U(1), this suggests eq. (39) surface gravity. When ξµ ξµ = ( −1,~0), the Killing → 0¯ and eq. (63) respectively reduce to(Kanyolo and Masese, vector is time-like. Assuming→ a diagonalized− metric ten- 2021), sor, and using the face that Φ is real-valued, we obtain u0 = exp(Φ) 0 and exp( 2Φ) = exp( 2Φ)uµu = µ † ≥ − − − µ µK ν = Ψ(DνΨ) , (69a) ξµξ = ξµξν g = g 0, which means that, ∇ µ µν 00 ≤ Ψ= √R exp(iS), Ψ† = √R exp( iS), (69b) −1 κ µν− β = 0, (73a) µ ν K =0, (69c) 2π g ≥ ∇ ∇ | 00| where, is the Tolmanp relation(Tolman, 1930; Tolman and Ehrenfest, 1930), where κ is given in Dµ = µ iAµ, (70a) µ ∇ − eq. (51), defined at the Killing horizon, ξ ξµ = 0. In K = R iF , (70b) µν µν − µν order to treat Φ as the Newtonian potential, we shall µ µ assume it is small, Φ 1 such that, g00 1 2Φ, S = pµdx + Aµdx . (70c) ≪ ≃ − − implying uµ ξµ, which also lead to, Z Z → κ 1 Proceeding, we employ the definitions in eq. (65) and β−1 = , (73b) eq. (67) to discover that the n =4-velocity is related to ≃ 2π 8πGM the Killing vector by, as expected.(Bekenstein, 2008; Hawking, 1976a; Zee, uν = exp(Φ)ξν , (71) 2010) Consequently, we can introduce the complex ob- − ject, S = S/2 + iS, where S = κ dτ in order for where J = R /β¯ defines relates the constants. Applying Ψ= √ρc exp(S) and ρc is a constant. c c R eq. (71), we obtain equations of motion for the center of Finally, using eq. (73b), eq. (65) and the trace of EFE mass coordinate, given in eq. (49), we find, µ ¯ µ uµ u = exp(Φ)ξµ (exp(Φ)ξ ) J = ψψ u IN . (74) ∇µ ν ∇µ ν h i = exp(2Φ)ξµ ξ + exp(Φ)ξ ξµ exp(Φ) ∇µ ν ν ∇µ In the same spirit as eq. (48), we can define the n = 4- 1 µ velocity as, = exp(2Φ) ν (ξ ξµ) −2 ∇ ¯ 1 ψγν ψ uν = h iIN , (75) = exp(2Φ) ν exp( 2Φ) = ν Φ, ψψ¯ 2 ∇ − −∇ h i 16 where, uν is a scalar under SU(N) and hence an N N C. Lagrangian density matrix. Thus, we proceed to choose the normalization,× To ground the equations of motion on a firmer footing, 3 0 we provide a suitable action principle for our formalism. d x det(gµν ) J (~x) V − We consider the following action on the space-time man- Z q 3 0 ifold, given by, = d x det(gµν ) ψ¯(~x)ψ(~x) u (~x)IN M V − h i 3 Z q 4 3 0 M = d x det(gµν ) Tr ( j ) , (78a) = d x det(gµν ) ψ¯(~x)γ (~x)ψ(~x) IN I − L M j=1 V − h i Z q X Z q ¯ µ ¯ 3 † 1 = iψγ Dµψ Mψψ, (78b) = d x det(gµν ) ψ (~x)ψ(~x) IN = IN , (76a) L − V − h i 1 µν † Z q 2 = Kµν (K ) , (78c) L 2π where we have used (γ0)−1γ0 = 1, in order for J 0(~x) = 1 λ 1 ψ†ψ to have the proper normalization for a probabil- = (D Ψ)(DµΨ)† + Ψ 2 Ψ 4, (78d) 3 2π µ πN 8π ityh densityi . Thus, the scalar charge, Q from eq. (68) L | | − | | becomes, where λ satisfies eq. (41), and Kµν and Ψ are given in eq. (8) and eq. (39) respectively. Note that the Lagrangian Q = Tr(IN )= N, (76b) densities are N N identity matrices, which necessitates the trace, Tr to× be appear in the Lagrangian. as expected. Likewise, we can employ eq. (76a) to obtain Proceeding to set = + + , we derive the the vector charges, Q using, 1 2 3 constraints in eq. (39L) andL eq. (63L ) respectivelyL by, δ δ δ δ d 4x det(g ) Jµ D D µν ν L = L , ν L † = L † , (79a) M − δK δD δ(K ) δ(D ) Z q µν µ µν µ 3 µ dt δ δ = dτ d x det(gµν ) J DµDν L =0, DµDν L =0, (79b) V − dτ δK δ(K )† Z Z q µν µν 3 Jµ 0 † † = dτ d x det(gµν ) u (~x) where Kµν , (Kµν ) and Dµ = µ iAµ, Dµ = µ + V − ∇ − ∇ Z Z q iAµ are treated as independent fields. Moreover, recall µ 3 0 = dτu (τ) d x det(gµν ) J (~x) that the trace, Tr in the action, M will introduce a V − I Z Z q multiplicative factor of N to all SU(N) scalars in the dxµ(τ) = dτuµ(τ)= dτ = ∆xµ(τ), (77a) action. Thus, plugging in eq. (8) into 2 and using dτ F Rµν =0, we can take the trace, Tr( )Lto find, Z Z µν L2 where, = (t, ) is the 1, 3 dimensional pseudo- N µν N µν M V Rµν R F~ µν F~ . (80a) Riemannian manifold. Here, we have applied eq. (48) 2π − 8π · and eq. (50b), and parametrized the vector current Consequently, this corresponds to the Lagrangian density as, Jµ(τ, ~x) = ψ¯(~x)ψ(~x) uµ(τ). Note that, ∆xµ(τ) = of the gravity and gauge fields, gµν and Aµ respectively. xµ(τ) xµ(0) ish the displacementi with xµ(0) the inte- − However, currently, eq. (78) does not yet resemble the gration constant. Thus, eq. (77a) implies that, Einstein-Hilbert action we should obtain in order to be 0 consistent with Einstein’s general relativity. dx 3 0 Tr = d x det(gµν ) Tr(J )=0, (77b) Moreover, the third Lagrangian density, Tr( 3) takes dt V − L   Z q the Ginzburg-Landau form for a Bose-Einstein conden- and thus we shall have, sate of N colors of mass 2λ/πN, whose momenta are given in eq. (40a). Substituting Ψ = √R exp(iS) from Q =0, (77c) p eq. (39) into eq. (78) and taking the trace, Tr ( 3) we obtain, L by virtue of the SU(N) matrices being trace-less, Tr(t)= 0. Consequently, when N 1 i.e. (SU(N) U(1)) in 1 λ N → → Tr( D Ψ 2)+ R R2, (80b) eq. (76b), the trace-less matrices can be replaced by 2π | µ | π − 8π twice the identity matrix, 2I implying that x0 2t, in N where R is the Ricci scalar. Thus, for eq. (78) to repro- order for Q 2Q as required. An approach, consider-→ duce the Einstein-Hilbert action as expected (neglecting ing only this→ limit, has been employed in a preceding the possible leading order terms), we ought to set the paper(Kanyolo and Masese, 2021), which provides the coupling constant, λ to, necessary framework to reproduce asymptotic behavior 2 of galaxy rotation curves, within the context of dark mat- 1 mP λ = , (81) ter (Section V). ≡ 16G 4   17 where mP = 1/√G is the Planck mass. This suggests varying this action with respect to the metric, gµν will that eq. (40a) describes the energy-momentum rela- reproduce eq. (45) where T µν 0, provided we can SU(N) → tion of Planckian particles. Moreover, we can perform find the conditions that allow the kinetic term of Ψ to the symmetry breaking, SU(N) SU(4) SO(4) identically vanish. → → → SO(1, 3), in accordance with eq. (35) and eq. (36), by In n = 4 dimensions, the constraint in eq. (39) is a¯ a¯ µν µν setting (Rµν ) = (Fµν ) , thus obtaining F~ µν F~ = invariant under T 0. This arises from the fact ¯b ¯b · SU(N) → F a¯ F ¯b µνσρ that mass-less SU(N) gauge field theories in n = 4 di- ( µν ) ¯b( µν ) a¯ = RµνσρR , where Rµνσρ is the Rie- mann tensor. In particular, since the tangent space must mensions are conformal field theories, requiring that their have the same dimensions (n = 4) as the manifold, , energy-momentum tensors are mass-less and hence trace- M the symmetry breaking (subsequently employed in eq. less. On the contrary, a finite mass, Mgauge introduces (80a) to arrive at eq. (83)), can be viewed as a Cayley- a preferred length scale comparable to the Compton Dickson (de-)construction of K from SU(N) to SU(4), wavelength, ℓ 1/M .(Francesco et al., 2012) More- µν ∼ gauge in a similar fashion to eq. (10b), setting the irrelevant over, in Ginzburg-Landau theories(Huebener, 2001), one terms to nil and a subsequent Wick rotation to introduce would typically assume Ψ varies slowly, resulting in the the Lorenzian signature. sombrero potential dominating the free energy contribu- Consequently, it is now evident that the coefficients in tion over the kinetic term. Within this paradigm, we eq. (78) and the normalization of the SU(N) matrices can introduce the fluctuations, Ψ 2 Ψ 2 = δ Ψ 2 and | | − h| | i | | in eq. (5) were a priori chosen in order to obtain the S S = δS, and define the Ricci scalar as the fluctu- − h i factors, N/8π, N/2π and N/8π as the coefficients of the ation, R = δ Ψ 2 instead of Ψ 2, where the mean values µν µν 2 2 | | | | terms F~ µν F~ , R Rµν and R respectively, where of Ψ and S are kept independent of space-time coordi- − · − † | | 2 the former two arise from the K (Kµν ) term and the nates ( Ψ /2= Λ and S = Sc). This does not alter µν h| | i − h i latter the Ψ 4 term in the Lagrangian density. In turn, the real and imaginary parts of eq. (39), since the Bianchi this ensures| that| the resultant corrections to the Einstein- identity and Yang-Mills equations, given in eq. (43) and Hilbert action (arising from the Ψ 2 term) correspond to eq. (44) respectively, are invariant under the shift R | | → the Gauss-Bonnet term(Lovelock, 1971), R 2Λ and S S Sc. Thus, we can identify Λ as the cosmological− constant→ − . Consequently, the kinetic term in 1 2 µν µνσρ eq. (83) leads to gauge symmetry breaking by generat- GB = (R 4Rµν R + RµνσρR ), (82a) L −8π − ing the cosmological constant and hence a mass term for the gauge field assuming Λ R, and S δS. How- which is topological in nature (a total derivative). From ≫ c ≫ eq. (23a), ever, since a mass term (spontaneous symmetry break- ing) breaks the aforementioned conformal symmetry by 4 generating an additional energy-momentum tensor pro- dCS = d x det (gµν )Tr( GB). (82b) − − L portional to 1 M 2 Tr(AµAν ) which is not trace-less, q 2 gauge Consequently, we have, this introduces a length scale ℓ Mgauge = 1/√Λ in the gauge field theory.(Zee, 2010)∼On the other hand, in 1 1 order to maintain conformal symmetry, we shall explore χ4( )= Tr(R R)= dCS M 8π2 ∧ πN another approach of identically getting rid of the kinetic ZM ZM 1 1 term of Ψ. = CS+ + CS− πN πN In particular, we exploit the second constraint given in Z∂M+=B Z∂M−=−B 1 1 eq. (63), which serendipitously avails an avenue for the CS+ CS− kinetic term to identically vanish. To see this, we plug in πN − πN ZB ZB Ψ= √R exp(iS), employ eq. (7), eq. (39c), eq. (67) and 1 = dTr(Λ−1dΛ)= 2ν, (82c) eq. (73b), and the normalization conditions in eq. (41) 24π2 ± ZB and eq. (42) to yield, where, we have used eq. (21) and eq. (27). Thus, χ is 4 1 R κ2N the Euler characteristic in n = 4 dimensions, and hence Tr( D Ψ 2)= Tr(pµp )+ uµu , (84a) 2 | µ | 2 µ 4 µ does not contribute to the equations of motion. Finally,   collecting all the terms, we have, which identically vanishes when,

1 2 µ 2 = Tr( D Ψ )+ i ψγ¯ D ψ λ 2 βM 4πr L 2 | µ | h µ i πN = =4πGM = = S = , (84b) πκ¯2 2 4l2 S λ P M ψψ¯ + R + Tr( GB), (83) − h i π L which implies(Kanyolo and Masese, 2021), where we have used the definition, Tr( 1) 1 for the Dirac field Lagrangian density. It isL now≡clear hL ithat βM =2πN, (85a) 18 where we have used eq. (51), eq. (81), the Schwarzschild path integral with respect to Aµ, where the Lagrangian radius, rS =2GM and Planck length, lP = √G =1/mP initially has a term, to identify the entropy, . In fact, eq. (85a) is the al- S 1 4 −1 µν ready proposed black hole area quantization condition by d y det (gµν )Aµ(x)(G ) (x y)Aν (y), several authors(Bekenstein and Mukhanov, 1998, 1995; 2 M − − Z q Mukhanov, 1986; Vaz and Witten, 1999) since N Z with, 0 is the number of colors and hence must be discrete.∈ ≥ Thus, to maintain conformal invariance, Killing horizon α −1 µν Tr(p pα) µν 4 areas must be pixelated in units of 4πG. Another curious (G ) = g δ (x y), (88a) πN det (g ) − observation is, eq. (85a) can be rearranged in order for − µν the mass to take the form of bosonic, N = m or fermionic, the inverse propagatorp , δ 4(x y) the Dirac delta func- N = (2m + 1) Matsubara frequencies(Abrikosov, 1975), 4 4 − α tion normalized as d xδ (x y) = 1, Tr(p pα) = λ given in eq. (41) and gauge field− mass corresponding to ωb = M =2πN/β =2ωf , (85b) R the reduced Planck mass, 2λ/π =m ¯ P =1/√8πG (eq. (81)). Consequently, the propagator, G (x y) is given where m Z 0 is a positive integer. This could have p µν by, − some deep∈ significance≥ for thermal Green functions not explored further in the present work. πN det (g ) Finally, plugging in eq. (84b) into eq. (83), the sum G = − µν g δ 4(x y), (88b) µν pαp µν of the traces in eq. (78) can be transformed into an Tr(p α) − equivalent sum, which satisfies,

3 4 −1 µ α 4 µ Tr( j )= (λ)+ 1 + Tr( GB), (86a) d z (G ) (x z)G (z y)= δ (x y)δ , (88c) L L hL i L α ν ν j=1 − − − X Z µ µα where, (λ)= λR/π corresponds to the Einstein-Hilbert where δ ν = g gαν is the Kronecker delta. A LagrangianL density and, Consequently, performing the path integral, [ µ] obtains the action, D R 4 d x det(gµν )Tr( GB)= χ4( ), (86b) 1 4 4 µ ν M − L −S M d x d y Tr (J (x)J (y)) Gµν (x y). (88d) Z q −2 − ZM ZM . is the Gauss-Bonnet term.(Lovelock, 1971) Thus, we where the Dirac current, Jµ = ψγ¯ µtψ is the Dirac cur- have demonstrated that eq. (78) is a candidate action h i for the formalism earlier introduced, since it reproduces rent in the term, 1 . Moreover, plugging in the equiv- J hL i the Einstein-Hilbert action, albeit with a finite topolog- alent form of µ from eq. (44b) into eq. (88d) obtains, ical term. The significance of this topological term will 2 subsequently be explored within the context of quantum 1 4 R d x det(gµν )πN gravity. − 2 M − 2π Z q   1 = d 4x det(g )Tr Ψ 4 , (89) −8π − µν | | ZM IV. RESULTS q
as earlier remarked. Moreover, the masses of Aµ and Ψ A. Generating all mass terms can effectively be generated by spontaneous symmetry breaking in the following manner, The path integral approach requires that the quan- tum gravity wave function(Gibbons and Hawking, 1977; 1 1 = φ 2 Ψ 2 (D φ)†(Dµφ) V (φ), (90a) Hamber, 2008; Hawking, 1978) be defined by, L4 πN | | | | − 2π µ − λ 1 V (φ)= φ 2 + φ 4, (90b) Ψ = [g , A , ψ,¯ ψ] exp(i M), (87) − 2 | | 4| | QG D µν µ I Z where the sombrero potential V (φ) has its minima at 2 where M is given in eq. (86a). We shall first treat φc = 0, λ, with λ the mass of the φ field as expected I A a¯ a¯ | | the gauge field, ( µ) ¯b and the spin connection, (ωµ) ¯b from eq. (41). Notably, the first term in eq. (90a) yields as different fields an first consider the path integral over the Einstein-Hilbert action, while the kinetic term of φ A . This appears straight-forward to perform since only yields the mass term for the gauge field, φ 2A Aµ/2π, µ | c| µ 1 in eq. (86a) depends on A. It will be clear that we which is integrated out by [Aµ] as discussed in order canhL i effectively generate the Ψ 4/8π term in eq. (78) by a to yield eq. (89). Conversely,D the presence of eq. (89) in | | 19 the Lagrangian density gauge invariance is already spon- Nonetheless, since a Wick rotation, defined by t = E E E taneously broken by the condensate, φ and not Ψ. Like- it and ΨQG ZQG where ZQG is the partition func- wise, recall that the kinetic term of Ψ governs whether ±tion, transforms→ the exponent to a real value, the topolog- the Killing horizon is quantized as in eq. (85a) or cosmo- ical term now contributes in the Euclidean path integral, logical (Λ = 0) in nature, corresponding to either preserv- ing or breaking6 the conformal symmetry of the underly- E E E A ZQG = exp( I E (λ)) exp χ4( ) , (93a) ing gauge theory, µ. Consequently, this can be viewed ∓ M ±S M ME as exploring conformal and non-confomal regimes of the X
theory, corresponding respectively to the length scales, where, ℓ rS =2GM comparable to the black hole radius and ℓ ∼ 1/√Λ comparable to the size of the de Sitter uni- E λ 4 E E ∼ IME (λ)= d x det(gµν )R, (93b) verse. π ME Finally, since ψ no longer couples to Aµ after integrat- Z q ing it out, we recognize that the essence of the above is the Einstein-Hilbert action in Euclidean signature and maneuvers in the case of ℓ r is alreadycaptured in eq. E (β, ) is the n = 4 (dimensional) Riemannian ∼ S M ∈ V (87) by simply setting Aµ = 0. manifold.

B. Quantum gravity C. 1/N expansion

¯ Since is the thermodynamic entropy, it is expected We proceed by considering the path integral, [ψ, ψ]. S We shall perform this integral by the stationaryD to be statistical, arising from quantum gravity degrees of phase/steepest descent approach(Zee, 2010),R where we freedom of some quantum theory. Such a quantum theory simply plug in the classical solutions of motion of ψ, ψ¯ ought to reproduce the partition function given in eq. (93), with the constraint, = ln , where is the black given in eq. (47), which require, 1 = 0. Consequently, S N N the path integral simply acquireshLani unimportant phase, hole/quantum gravity number of microstates. As earlier ¯ remarked, since the path integral measure, [g ] is not ln [ψ, ψ] . Proceeding, we are interested in the path D µν integralD involving the metric tensor, g as the dynamical well-understood, the aspiration is for such a successful
µν field.R However, we face a daunting challenge to appro- quantum theory reproducing eq. (93) to have a well- defined path integral measure, e.g. [ϕ], where ϕ are priately define this path integral since there lacks a con- D sensus in literature on how the measure, [gµν ] ought to dynamical fields, in order for quantum gravity to emerge be defined in order to yield consistent resultsD .(Hamber, from the path integral formulation, hence circumventing 2008) this issue. Nonetheless, we make progress by defining the path in- To make some progress in identifying such a theory, we tegral with the measure as a discrete sum over an ensem- plug into eq. (93) the Boltzmann entropy formula, ble of varied space-time manifolds, i.e. [g ] . D µν → M = ln , (94) Moreover, a neat observation from eq. (25) and eq. (28) S N R P implies, thus obtaining the homestretch,

χ4( )= πNχ4( )= 2πNν, (91) E E E ±χ4 (M ) S M M ± ZQG = exp( IME (λ)) E ∓ N where ν Z is the second Chern (winding) number. Con- MX∈h sequently,∈ this implies that the topological action can = exp dβ E (β) identically be excluded from the exponent. Moreover, we M E − H can rewrite eq. (87) as, MX∈h  Z 

= Tr exp dβ ME (β) , (95) ∗+ − − H ΨQG = exp(iIM(λ))ψ ( )ψ ( )   Z  CS B CS B M X corresponding to eq. (1a), where exp dβ E (β) − HM = exp(iIM(λ)), (92a) takes the form of a Boltzmann factor with the trace, Tr M ER
X going over the ensemble of manifolds, h with dis- tinct topologies characterized by the index,M ∈h to be later where, defined. Note that the classification of all n = 4 dimen-

λ 4 sional Riemannian manifolds into distinct topologies is IM(λ)= d x det (gµν )R, (92b) not a straightforward exercise(Freedman, 1982), requir- π M − Z q ing additional assumptions, tackled in subsequent sec- is the Einstein-Hilbert action (in Lorenzian signature). tions. 20

Nonetheless, we recognize eq. (95) as the 1/ expan- where identifies a world-line in space-time correspond- sion of a large theory with the partition functionN given ing to theC trajectory for the center of mass. Under a wick by, N rotation, t itE and τ is, and closed world-lines → ± →± ± become possible, where s is the arc length for the closed = exp(ZE )= [ϕ]exp N Tr(γ(ϕ)) , (96) trajectory E [0,β]. Z QG D − λ± C ∈ Z   Consequently, the integral over the Langevin eq. (72a) N ± after Wick rotation yields, where S(ϕ)= λ Tr(γ(ϕ)) is the action for unidenti- fied field tensors, ϕ of rank n = 4 transforming appro-
M µ µ priately as vectors under an unknown group, and γ is kg(s) ds + uµ(s)dx (s)= ηµdx , (100a) E N E E the unidentified function of ϕ defining the large the- IC IC IC N ory. In this case, the vacuum Feynman diagrams pave where, n = 4 manifolds(Gurau, 2012), differing from random matrix large N theories, where ϕ is a rank n = 2 tensor β (matrix) with vacuum Feynman diagrams defining n = M ds = βM =2πN =4πk, (100b) 2 manifolds.(Gurau, 2011, 2012; Gurau and Rivasseau, Z0 µ µ µ 2011) u (s)uµ(s) = 1, u (s)= dx /ds and we have used βM = 2πN from eq. (85a) and eq. (23c). 2. Entropy as an adiabatic invariant D. Topology Recalling that ~ = 1, N Z 0 and Muµ has the 1. Old quantum condition right form for momentum, we∈recognize≥ eq. (100b) as the old quantum condition (Wilson–Sommerfeld/Ishiwara Considering the Langevin equation given in eq. (72a), rule)(Ishiwara, 2017; Pauling and Wilson, 2012), which we define the geodesic curvature as, is equivalent to the condition of black hole area quanti- ν µ kg(τ)= u u µuν , (97a) zation, implying that black hole entropy (proportional to ∇ area) is an adiabatic invariant.(Henrard, 1993) motivated by the fact that k (τ) identically vanishes g For instance, considering the constraint equations in as the n = 4-velocity approaches the Killing vector eq. (69) when the gauge field is broken to U(1) U(1), uµ(τ) ξµ(τ) due to the anti-symmetry relation in eq. we can appropriately re-scale the complex function using (54) or→ more favorably when the center of mass equations the trace of EFE, R = β ψψ¯ with β =8πGM in order of motion are geodesics, ¯ − h i M to have,Ψ= ψψ exp(iS), S = 2 ds and, µ ν h i − u µu =0. (97b) ∇ p † R µK = βΨ ∂EΨ, (101a) In particular, when the random acceleration vanishes, ∇ µν − ν η = 0 as in eq. (72a), we can employ uµu = 1 to µ µ where ∂E the partial derivative in Euclidean signature discover the geodesic curvature corresponds to reduced− µ and ∂E iA ∂E. Consequently, the normalization of temperature, k(τ)=1/β¯(τ). Moreover, observe that, µ − µ → µ Ψ= √ρ exp(iS) in Euclidean signature, equivalent to eq. λ (76a), corresponds to, I(λ)= d 4x det(g )R π − µν Z q M 3 ¯ 1= d 3x det(gE )u0(~x(tE)) ψ¯(~x(tE))ψ(~x(tE)) = dt d x det(gµν ) ψψ µν h i − 2 V − h i V Z Z q Z q M 3 0 3 E 0 E † E E = dτ d x det(gµν )Tr(J (~x)) = d x det(gµν )u (~x(t ))Ψ (~x(t ))Ψ(~x(t )) −2N V − ZV Z Z q 2 q M M µ = d 3x det(gE )ξν Ψ†(~x(tE))i∂EΨ(~x(tE)) = dτ = uµ(τ)dx (τ), (98) M µν ν − 2 2 ZV Z Z q2 ∂ where u (xµ(τ)) = dxµ(τ)/dτ, uµ(τ)u (τ) = 1. Note E E µ µ = Ψ(~x(t )) i E Ψ(~x(t )) , (101b) that, we have used eq. (49), eq. (74), eq. (76−) and eq. M h | ∂t | i (81). Thus, eq. (98) guarantees that the Einstein-Hilbert where ξµ = ( 1,~0) is the time-like Killing vector and we action satisfies, have used ξµ−ν R = 1 ξµ∂ER = 0 satisfied by eq. (52) ∇ µν 2 µ M and eq. (101a). Computing the integrals below using eq. IM(λ)= dτ − 2 (101a) and eq. (101b), we find, ZC(M) M µ µ 3 E ν µ βM = uµ(x (τ))dx (τ), (99) i d x det(gµν )ξ Kµν = = , (102a) 2 C(M) − V ∇ 2 S Z Z q 21 and consequently, ψψ¯ = ρc exp( βM¯ Φ(~x)) = ρ the equilibrium Boltz- mannh i density distribution− function which clearly satis- β E −1 E 3 E ν µ fies, ∂ρ(~x)/∂t = 0 and Φ(~x) is the Newtonian poten- iβ dt d x det(gµν )ξ Kµν 2 E − 0 V ∇ tial. Introducing fluctuations by ρ ρ˜ = ρf (t , ~x) Z Z q → β ∂ whereρ ˜ is the fluctuating density distribution, the man- = i dtE Ψ(~x(tE)) Ψ(~x(tE)) = . (102b) ifold will no longer admit a time-like Killing vector, h |∂tE | i S Z0 leading to ∂ρ/∂t˜ E = ρ∂f 2(tE, ~x)/∂tE = 0. Nonethe- 6 Moreover, if Ψ is interpreted as the wave function of less, since the form of the wave function is constrained a quantum gravity system, the entropy, becomes the by the complex charge, Q = Q + i2Q, we must have S ˜ geometric phase/Berry phase.(Berry, 1984) In addition, Ψ=˜ √ρ˜exp(iS)= √ρc exp(S) andρ ˜ = ρc exp(S) where, if a Schwarzschild black hole is described by Ψ and the βM¯ black hole does not undergo quantum transitions as it ac- S(tE, ~x) = ln f(tE, ~x) Φ(~x)+ iS(tE, ~x) cretes/evaporates, the wave function should evolve adi- − 4 abatically. For instance, this requires that the Unruh- = S(tE, ~x)/2+ iS(tE, ~x). (104b) Hawking radiation from the black hole or the accreted It follows from S(tE, ~x) = ln f 2(tE, ~x) βM¯ Φ(~x)/2 that, energy content to be negligible such that the space-time − E can be assumed fairly static, ∂M/∂t 0. Since the ∂ ln f 2 ∂S system at finite temperature is periodic≃ in β, a stan- = , (105) ∂tE ∂tE dard calculation of the adiabatic invariant(Berry, 1984; Cohen et al., 2019) yields, where we can set f(0, ~x) = 1. Applying these assumptions, we proceed to calcu- β late the Kullback-Leibler (KL) divergence, DKL(ρ ρ˜) for E E ∂ E i dt Ψ(~x(t )) Ψ(~x(t )) the two distributions ρ and ρ˜ by(Kullback and Leibler|| , h |∂tE | i Z0 1951), = i d~x Ψ(~x(tE)) ~ Ψ(~x(tE)) ∂AE=CE · h |∇| i ρ Z 3 E DKL(ρ ρ˜)= d x det(gµν ) ρ ln ′E 2 ~ || V ρ˜(t ) = d x ~n Ω= Ω=2πk, (102c) Z q   AE · AE Z Z 3 E 2 ′E = d x det(gµν ) ρ ln f (t ) E − where is the n = 2 manifold enclosed by the Euclidean ZV A ′E q path, ~x(tE), Ω~ = ~ ~ is the Berry curvature(Berry, t ∂ ∇× P = dtE d 3x det(gE ) (ρ ln f 2(tE)) 1984) which can be written as a p = 2 form, µν E − 0 V ∂t Z ′EZ q 1 t ∂S Ω= d = ∂ dxj dxl, (103a) E 3 E j l = dt d x det(gµν )ρ E P 2 P ∧ − 0 V ∂t ′Z Z t E q whereas the Berry connection is given by ~ = † ∂ = dtE d 3x det(gE )Ψ i Ψ ~ P µν E i Ψ(~x) Ψ(~x)) , which can be written as a p = 1 form, 0 V ∂t h |∇| i Z ′ Z t E q l ∂ = i Ψ(~x) ∂l Ψ(~x)) dx , (103b) =i dtE Ψ Ψ =2πk = 0, (106) P h | | i E 0 h |∂t | i S ≥ E Z Z ~n is the unit vector normal to and k = N/2 0 E ′E is the first Chern number(ChernA , 1946), corresponding∈ ≥ where we have used eq. (105), ∂ρ/∂t = 0andset t = β to the Chern-Simons level (eq. (23c)).(Jackiw and Pi, in the last line. Thus, by the positive definite property 2003) of the KL divergence, the black hole entropy is always positive, = πN 0, consistent with N Z 0 as In addition, since the object, Kµν under U(1), cor- S ≥ ∈ ≥ responds to the complex charge, Q = Q + i2Q, the expected. 3. Random acceleration real and imaginary parts of the adiabatic wave function, Ψ= ρ exp(S) capture the same information about the √ c Proceeding to appropriately define the random accel- quantum system. Thus, we write the adiabatic wave eration, η we note that, since the quantum gravity am- function as Ψ = ρ exp(S) where, ν √ c plitude satisfies, βM¯ S(tE, ~x)= Φ(~x)+ iS(tE, ~x) − 4 ΨQG = exp ik p exp( i2πkχ4( )) − M E M C(M) ! = S(0, ~x)/2+ iS(t , ~x), (104a) X Z where S(0, ~x) = βM¯ Φ(~x)/2 is defined in order for = exp ik p , (107) E − M C(M) ! ∂ρ/∂t = 0 to vanish, ρc is a constant distribution with X Z 22 it implies we set, ψ∗+ exp ik p dxµ from eq. (25) where ⋆K is the p = 2 Hodge dual of p = 0 form, K. CS ≡ C µ in order for ΨQG to transform appropriately. This also Consequently, the integral form of the Langevin equation implies that, R
summarizes to, 1 p = CS+, (108a) E E E E kg + ⋆K =2πχ4. (112) C (M ) −k B (M ) E E Z Z Z∂A ZA E E E E where = ∂ ±. Moreover, to find ( ), it is Evidently, this is the Gauss-Bonnet theorem in n = 2 instructiveB to± defineM η such that it generateC Ms the 1/ µ N dimensions(Wu, 2008), where the Euler characteristic of expansion terms in eq. (96), corresponding to the path the n =4 compact manifold, E has been transformed integral contributions. In other words, we ought to set, into the Euler characteristic ofM the n = 2 non-compact E E E 1 − 1 manifold, , i.e. χ4( ) = χ2( ). Consequently, kg(s)ds = CS = ME , (108b) computingAχ ( E) canM be done onA E instead of E, CE(ME) k BE(ME) −k I 4 Z Z suggesting thatM the partition functionA in eq. (95) canM ac- and, tually arise from a random matrix large group theory where ϕ transforms as a vector under a rankN n = 2 tensor η = 2 dθ, (108c) group theory, instead of rank n = 4, as earlier remarked. CE(ME) ± BE(ME) Z Z Thus, these observations take us closer to defining the where θ is given in eq. (27). To make further progress, appropriate large theory for the field variable, ϕ by we consider a series of compact manifolds homeomor- edging us closer toN consistently defining the topology in- phic to the n-sphere, Sn where n = 4, 3, 2, 1, together dex, h in eq. (96) for n = 4 manifolds of interest to our with their non-compact hemispheres (indicated by ) study. ± with boundaries homeomorphic to the n 1 spheres, i.e. Finally, recall that the Gauss-Bonnet the- − ∂Sn = Sn−1. Thus, we employ the definitions, orem in eq. (112) relates the Langevin ± ± equation(Lemons and Gythiel, 1997) (eq. (72a)) to E = E + E , ∂ E = E, (109a) M M+ M− M± ±B the action in the integrand of the path integral in E E E E E = + , ∂ = , (109b) eq. (87). Since varying the action, M leads to the B B+ B− B± ±D I E = E + E , ∂ E = E, (109c) field equations of gravity, we conclude that EFE are D D+ D− D± ±C analogous to the Fokker-Planck equation(Risken, 1996), where E, E, E, E are compact manifolds with di- where the Ricci scalar R is proportional to the proba- mensionsM n B= 4D, 3, 2C, 1 respectiveley. Since E is com- bility density. In diffusion models, the Fokker-Planck pact, we expect that if dθ is non-exact on EB( E), we equation is related to the Langevin equation by Itˆo’s would have, B M lemma, under a non-standard calculus known as Itˆo calculus.(Øksendal, 2003) dθ = θ+ + θ− = dα, (110a) BE BE BE DE Z Z∂ + Z∂ − Z 5. Euler characteristic, χ = χ where θ+ θ− = dα is also assumed and with the p = 4 2 1 form, α−defined modulo 2πν on E. This procedure We shall consider some implications of our approach can be repeated successively underD a similar treatment. by setting, Proceeding, when dα is non-exact on E( E), we have, D M 1 χ2 =2 2g b, (113) dα = α+ + α− = dη, (110b) − − DE ∂DE ∂DE ±2 CE Z Z + Z − Z E 1 where g is the genus of and b = E k(s)ds is A 2π C where α+ α− = 1 dη as before and the random accel- the finite contribution of the geodesic curvature integral 2 R eration, η −is defined± modulo 2πν on E( E). over the boundary, E. Moreover, we shall only consider C M manifolds homeomorphicC to the connected sums of the n = 4-sphere, S4 and the n = 4-torus, T 4 where E is 4. Gauss-Bonnet theorem thus compact. Their respective Euler characteristicsM are given by, We can thus introduce the Gaussian curvature(Wu, 2008), K of the non-compact n = 2 Euclidean manifold, 4 E E E E 4 p 4 bounded by ∂ = ( ), by, χ4(S )= ( 1) bp(S )=2, (114a) − A A C M p=0 X dp = ⋆K, (111a) 4 χ (T 4)= ( 1)pb (T 4)=0, (114b) p = ⋆K, (111b) 4 − p E E p=0 Z∂A ZA X 23 respectively, where, where we have used λ =1/16G fromeq. (81), β =8πGM and = ln = βM/2 = πN from eq. (84b). Mean- n n S N n n while, individual probabilities, PME for a given manifold bp=0(S )= =1, bp=n(S )= =1, (114c) 0 n correspond to,     n n n n! bp=06 ,4(S )=0, bp(T )= = , (114d) E −1 p (n p)!p! P E = Z exp dβ E (β) . (119)   − M QG − HM  Z  and bp is the p-th Betti number.(Bochner and Yano,
2016) The Betti numbers have the property that the Thus, given the above probability distribution for each manifold configuration, the expression for average energy b0 and bn numbers are not additive (remain invariant) under connected sum operations, while the rest of the becomes, Betti numbers are additive.(Zagier, 2017) For instance, 4 ∂ E the connected sum of manifolds homeomorphic to S has = PME ME (β)= ln(ZQG). (120) hHi E H −∂β the Euler characteristic, MX∈h Moreover, when there are no sources in the EFE or χ (S4#S4# #S4) 4 ··· when the energy momentum tensor is trace-less, the Ricci 4 4 4 = b0(S )+ b4(S )= χ4(S ), (115a) scalar vanishes and the stationary phase/steepest descent approach(Zee, 2010) allows the energy of the manifold to since the Betti numbers, b0 = 1 and b4 = 1 are invariant, be transformed into a more palatable expression by plug- while bp=06 ,4 = 0 are additive but do not contribute to ging in R = 0. Thus, the average energy becomes purely additional terms since they are vanishing. Meanwhile, the sum of the topological contributions, the connected sum of manifolds homeomorphic to S4 and T 4 has the Euler characteristic, M = PME χ4(h), (121) hHi 2 E 4 4 4 4 4 4 MX∈h χ4(S #S # #S # T #T #T ) M ··· ··· exp β 2 χ4(h) E h PM ∈h = − M . (122) E exp β χ (h) 4 4 4 M ∈h − 2
4 = + + h | ( 1){zp =2} 2h, (115b) 0 4 − p − P
6     pX=0,4   V. RAMIFICATIONS where h Z 0 is the number of tori in the connected ∈ ≥ sum. Since, we can always reproduce eq. (115a) by set- A. Fermion/boson picture ting h =0, we have, Typically, in the 1/ expansion of a random matrix E N χ4( )=2 2h, (116) theory such as in eq. (96), terms corresponding to the M − 4 4 4 4 topology, χ4(S #S # #S ) = χ4(S ) = 2 (planar for all connected sums. Consequently, we note that, vacuum Feynman diagrams)··· would be expected to dom- χ ( E)= χ ( E) implies, 2 A 4 M inate the expansion.(Gurau and Rivasseau, 2011) How- ever, here, the sign arising from opposite-direction h = g + b/2. (117a) Wick rotations slightly± complicates the situation. To understand how, recall that we are considering R = 0, Thus, when E is compact, the contribution to the A where R is the Ricci scalar. We can proceed by the Gauss-Bonnet theorem from the geodesic curvature van- 4 E assumption above that the S topology dominates with ishes, b = 0 and h corresponds to the genus of . E P E 1, where the manifold energy h = 0 is A M ≃ MBH ∈ the central mass, E = M and, HM ∈h=0 E. Average energy of manifolds χ ( E )= χ (S4)=2, (123) 4 MBH 4 The energies associated with a given manifold can be E E with BH the manifold corresponding to a space-time obtained from eq. (95) by, E (β) = ∂ E (β)/∂β, M HM − IM with a single Schwarzschild black hole of mass, M while which yields, the sign of the energy function chosen to correspond to the Wick rotation, t = it . Even though black hole M − E ME = χ4(h) solutions are vacuum solutions of EFE, this cannot be the H ∓ 2 ground state of our theory since we could always lower M 4 E E the energy of the manifold by removing the central mass, 2 d x det(gµν )R, (118) ∓ 2β ME∈h by setting, M = 0. Z q 24

¯ In fact, the ground state corresponding to ME = as expected. Here, R = β ψψ and β = 8πGM have 0 is characterized by three quantum states:H (M = been used from eq. (49),− andh u0i(~x) = dt/dτ = dtE/ds E E E 0,χ4( ) = 0), (M = 0,χ4( ) = 0) and (M = with s = iτ the proper distance and t = it Eu- ME 6 6 M − 0,χ4( ) = 0) states; hence it is three-fold degenerate. clidean time, where the normalization condition given in Nonetheless,M since the 1/ expansion requires we take eq. (76a) applies. Thus, we conclude that the Euler char- the ’t Hooft limit (correspondingN to , λ =1/16G acteristic is proportional to the fermion number operator fixed), and eq. (84b) requires M 2 NπN → ∞= ln , acting on the quantum states (vacuum state, h = 1 and this lifts the degeneracy by selecting∝ the uniqueN ground → ∞ the single fermion state, h =0), state to be (M = 0,χ( E) = 0). Thus, for our ap- 6 M † † † proach to be physical, (M =0,χ ( E)) ought to include χˆ4 =2c c = c c cc +1, (126a) 4 M − Minkowski space-time/Euclidean space, albeit alongside ν E h 1 , (126b) other Riemannian manifolds with non-trivial topologies, | i ≡ |M ∈ ≤ i χ ( E) = 2 2h. This is encouraging since it classi- where h = cc† and ν =1 h = c†c is the winding number 4 − fiesM relevant6 manifolds− consistent with the assumption, given in eq. (28) in order for, E h in the partition function, ZE . M ∈ QG M Proceeding, we already have, χ ( E h)=2 2h, ˆ = (ˆχ 1), (126c) 4 H 2 4 − with h =0, 1. To avoid negative massM states,∈ M <−0 we ˆ ν = E (ν) ν . (126d) observe that, the manifold states with h 1 correspond H| i HM ∈h | i ≥ i.e. to a Wick rotation in the opposite direction, t = The expressions are consistent with the plus sign option +itE. In fact, the Wick rotations, t = itE correspond ± of the Hamiltonian given in eq. (118). to the following 1/ expansions, Similarly, the manifolds with h 1 correspond to N ≥ −2 0 bosonic states. In particular, performing a Wick rotation ΨQG(t,h) = + , (124a) in the opposite direction, t = +itE selects the appropri- E N N t=−it ,h≤1 ate sign corresponding to the bosonic average energy, 0 −2 2−2h ΨQG(t,h) = + + , (124b) E N N ···N βM t=+it ,h≥1 h=∞ χ (h)exp χ (h) M 4 2 4 0 = ′ where both select the unique ground state, = 1 for hHi − 2 h =∞  βM ′ N h=1 ′ exp χ4(h ) large . Moreover, there is a sense in which the h = 1 X h =1 2 N h=∞   state is fermionic whereas the h > 1 states are bosonic ∂ P βM in nature. This follows from their average energy expres- = ln exp χ4(h) −∂β 2 ! sions, hX=1   ∂ = ln (1 exp( βM)) h=1 χ (h)exp βM χ (h) ∂β − − M 4 − 2 4 = ′ M hHi 2 h =1  βM ′ † ′ exp χ (h ) = = M a a , (127a) h=0 h =0 2 4 exp(βM) 1 h i X − − P exp( βM)  = M − where a†,a are the bosonic creation and annihilation op- 1+exp( βM) − erators respectively satisfying the commutation relations, M † = = M c†c , (125a) [a,a ] = 1. However, caution should be exercised to re- exp(βM)+1 h i place the Dirac field in the trace of EFE with R = βρ, − where c†,c are the fermionic creation and annihilation arising from the trace with a dust energy-momentum ten- sor, T µν T µν = Mρuµuν , where ρu0 is interpreted operators respectively, satisfying the anti-commutation Dirac → Dust relations, c,c† = 1. Thus, when the Ricci scalar in eq. as the bosonic probability density normalized as, (118) is non-vanishing,{ } R = 0, the presence of a massive 6 3 E 0 Dirac field restores the fermionic zero-point energy term, d x det(gµν )ρ(~x)u (~x)=1, (127b) V M/2 missing in eq. (125). Consequently, a substitution Z q of− the trace of EFE into eq. (118) yields, to unity. Thus, we obtain,

M 4 E E M 4 E E 2 d x det(gµν )R 2 d x det(gµν )R 2β ME − 2β ME Z q Z q M β M β 3 E ¯ 0 3 E 0 = ds d x det(gµν ) ψ(~x)ψ(~x) u (~x) = ds d x det(gµν )ρ(~x)u (~x) −2β 0 V h i 2β 0 V Z Z q Z Z q M β M M β M = ds = , (125b) = ds = , (127c) −2β − 2 2β 2 Z0 Z0 25 as expected. Consequently, parallel to the fermionic case, probability density. A second adjustment is to relax the the Euler characteristic operator is proportional to the expression for temperature, β = 8πGM, since this was boson number operator, and the manifold states span motivated by black hole thermodynamics6 , and hence is the states of the quantum harmonic oscillator, not necessarily guaranteed to apply to this scenario, as we shall see. Nonetheless, the condition, eq. (85a) can † † † χˆ4 = 2a a = a a aa +1, (128a) be expected to be robust across varied (quantum) grav- − − − ν E h 1 , (128b) itational systems, since it simply corresponds to the old | i ≡ |M ∈ ≥ i quantization.(Ishiwara, 2017; Pauling and Wilson, 2012) as expected, where h = aa† and ν = h 1 = a†a is the To simplify the problem, we assume that the periphery winding number given in eq. (28), in order− for, of the galaxy contains a negligible amount of baryonic matter compared to the interior, where ρ(x) ρD(x) M largely corresponds to the number density contribution≃ ˆ = (ˆχ4 1), (128c) H − 2 − from dark matter.(Persic et al., 1996) In addition, we im- ˆ µ ν = ME∈h ν . (128d) pose the time-like Killing vector, ξ on the space-time H| i H | i manifold, in order to guarantee the equilibrium condi- The expressions are consistent with the minus sign option tions, of the Hamiltonian given in eq. (118). µ Finally, eq. (126) and eq. (128) satisfy the free energy ξ ∂µρD =0, (130a) equation, ρ (x)= ρ exp( βM¯ Φ(x)), (130b) D c − ˆ = ˆ β−1 , (129a) hFi hHi ± S where ρc is a critical number density to be defined. Re- spectively, eq. (130) follows from the Killing equations where ˆ = M χˆ is the minimized free energy opera- F ± 2 4 in eq. (54) while assuming the equilibrium distribu- tor. By inspection, we see that eq. (129) corresponds to tion takes the form of a Boltzmann factor with Φ(x) the quantum version of the classical Gauss-Bonnet the- the Newtonian potential appropriately defined. We take orem given in eq. (112), where the Gaussian curvature the dark matter particles to be equivalent to a Bose- term is responsible for the black hole entropy, whereas Einstein condensate(Kanyolo and Masese, 2021) of the the random acceleration term is responsible forS the 1/ N colors, k = N/2 where k is the number of bosons/Cooper- expansion of the free energy since, pairs(Tinkham, 2004) forming the condensate normaliza- − tion condition, ˆ = β 1 ln( E ), (129b) hFi − ZQG 3 2 0 Consequently, geodesics (kg(s) = 0) are allowed if and d x det(gµν ) Ψ(~x) u (~x)= k, (131a) V − | | only if M = 0. However, since M = 0 is imposed by Z q the ’t Hooft limit, geodesics must be6 precluded even for with Ψ(~x) 2 = ρ (~x) the number density, u0(~x) the time the vacuum state (h = 1) unless R = 0. Conversely, D component| | of the n =4-velocity vector and Ψ(~x) satisfies R = 0 gives rise to a finite zero-point term, M/2 that eq. (69). For brevity, we consider the Newtonian limit cannot6 otherwise vanish. Consequently, E must± be non- corresponding to, compact when M = 0 unless R = 0 andA h = 1, since the geodesic curvature6 term would otherwise identically 2Φ(~x)=4πGM Ψ(~x) 2 vanish, contradicting our previous statements. ∇ | | =4πGMρ exp( 2kΦ(~x)), (131b) c − B. Asymptotic behavior in galaxy rotation curves where u0(x) 1 and k =2βM¯ from eq. (85a). Here, we have used the≃ time-like Killing vector, ξµ = ( 1,~0) and µ − A direct way of deriving the empirical baryonic Tully- ξ ∂µR =8πGM∂ρD/∂t =0, from eq. (130), in order to Fisher relation(McGaugh, 2012; McGaugh et al., 2000), guarantee, ∂Φ(x)/∂t = 0. relevant for explaining the asymptotic behavior of galaxy Assuming the conditions of spherical symmetry rotation curves is to employ eq. (85a) with additional mi- of the dark matter halo(Kramer and Randall, 2016; nor considerations.(Kanyolo and Masese, 2021) First, we Persic et al., 1996), we can re-write eq. (131b) in spher- shall consider the EFE coupled to a pressure-less dust, ical coordinates as, where R = 8πGMρ is the trace. This corresponds to − the bosonic case given in eq. (127b), where M is the 1 ∂ ∂Φ(r) r2 =4πGMρ exp( 2kΦ(r)), (132) mass of the baryonic matter in the galaxy. However, since r2 ∂r ∂r c − EFE are analogous to the Fokker-Planck equations, we   can modify the normalization condition and by treating and proceed to solve it. This yields the potential and ρ as the number density of bosonic particles instead of a number density corresponding to the singular isothermal 26 profile(Keeton, 2001), function ought to define the size of the manifold, and not necessarily the halo. Thus, it is reasonable Vto set 1 Φ(r)= ln Kr2 , (133a) the cut-off scale to be 1/a0 3/Λ, which is compara- 2k ble to the size of the de Sitter≃ universe, where Λ is the p ρc 1
cosmological constant. Consequently, from eq. (135), we ρD(r)= 2 = 2 , (133b) Kr 4πGMkr have, K ρc = , (133c) 1 1 4πGMk k = . (136) √GMa ≃ 1/2 where K is a constant. However, the isothermal den- 0 GM(Λ/3) sity profile is singular at the origin. This may not be a However, the number of bosonsp , k in eq. (136) appears priori unphysical since it is not peculiar for vortices in not to take positive integer values. Of course, this is not a nature to be described by singularities at the origin. In problem since a finite cosmological constant requires the µ fact, Φ(r)= pµdx takes the form of a logarithmic spi- kinetic term of Ψ to spontaneously break SU(N) gauge ral with arctan(k) = θ the pitch angle, where eq. (66) R p symmetry in eq. (83), and hence does not require eq. and eq. (107) imply Φ(r) is the quantum phase. How- (84b) nor eq. (85a) to be satisfied. This corresponds ever, since we have only considered conditions where dark to breaking conformal invariance of the gauge theory, matter dominates over baryonic matter at the periphery as earlier discussed. Equivalently, whether k is a good of the spiral galaxy, it is entirely feasible that predictions quantum number or not depends on the number/phase with eq. (133) better approximate the asymptotic be- regime(Kanyolo, 2020; Kanyolo and Shimada, 2020) of havior of spiral galaxies where dark matter dominates, the condensate governed by the commutation relation, rather than in the interior where there would be a sig- nificant energy density contribution from the neglected [k, Φ] = i. (137) baryonic matter. Consequently, we shall introduce the − condensate wave function as, For a more rigorous treatment, the commutation re- lation should be replaced by the Susskind–Glogower Ψ= √ρD exp(ikΦ(r)) = √ρc exp(k(1+2i)Φ) , (134) operators.(Susskind and Glogower, 1964) Thus, we are considering here the phase regime of the condensate, implying the approach admits the complex charge, Q = where the quantum phase Φ(r) obeys classical equa- Q + i2Q corresponding to the subsequent breaking of tions of motion analogous to the Josephson relations in SU(N) to U(1), Fµν µν , where in eq. (69), µν large tunnel junctions(Josephson, 1974), and the Cooper- is the U(1) field strength→ F and K = R i isF the µν µν − Fµν pair number, k is not a good quantum number. Mean- complex-valued object. In a previous paper, we argued while, the galactic size is determined by the binding that the ensuing U(1) gauge field strength corresponds condition(Kanyolo and Masese, 2021), Φ(r) 0, which to ξ , where ξ is the time-like Killing vector, ≤ µν ν µ µ corresponds to rc 1/√K, where ρ(rc)= ρc. The total andF thus∝ ensuring∇ dark matter remains effectively non- ≤ mass, Mtot.(r) within radius, r rc becomes, interacting and charge-less.(Kanyolo and Masese, 2021) ≤ Moreover, the normalization condition in eq. (131a) M (r)= M dr4πr2ρ(r)= M + M, (138a) can be exploited to find an expression for the number of tot. D Z r bosons, k, ′ ′2 ′ r MD = M dr 4πr ρ(r )= , (138b) 0 Gk 1/a0 Z 3 2 k = d xρD(r)= 4πr ρD(r) where MD is the mass of dark matter and is defined as the ZV Z0 mass, M of baryonic matter, arising as the integration 1/a0 dr 1 = = , (135) constant. GMk GMka Z0 0 Consequently, a star of mass, m⋆ in orbit with speed, v at the periphery of the galaxy will experience a gravi- where we have introduced a cut-off scale, 1/a for the 0 tational attractive force, n = 3 manifold, since the integral would otherwise V diverge. In addition, since the logarithmic spiral is Gm M (r) m GMm F (r)= ⋆ tot. = ⋆ ⋆ , (139a) scale invariant but the cut-off introduces a length scale, g − r2 − kr − r2 this suggests that conformal invariance must be broken, while preserving scale invariance.(Milgrom, 2017) Conse- as well as the centrifugal repulsion, quently, one would suspect that the cut-off scale would 2 m⋆v be comparable to the size of the dark matter halo. How- Fc = . (139b) † r ever, since ρD =Ψ Ψ in eq. (131a) is the number density of the bosons (which are quantum mechanical objects), The first term in eq. (139a) is the gravitational con- the cut-off in the normalization of the condensate wave tribution from dark matter, arising from the particular 27 solution of the non-homogeneous equation (eq. (132)) 0

1 1/4 KC8, RbC8 and CsC8, including their in- vc = (GMa0) , (140) ≃ √k termediate compositions, for instance, KC12n (n > 1), LiC6n (n > 1) and LiC9n where a0 takes the form predicted by MOdified New- (n 2);(Dresselhaus and Dresselhaus, 2002, 1981; tonian dynamics (MOND) for the acceleration param- Guerard≥ and Herold, 1975; Hosaka et al., 2020; eter.(McGaugh, 2012; McGaugh et al., 2000) However, Jian et al., 2015) our path to this result vastly differs from standard MOND, since we explicitly rely on a pressure-less source (iv) Layered polyanion-based compounds compris- at the right-hand side of EFE as well as the con- ing pyrophosphates such as Na2CoP2O7 and straints introduced herein which ultimately reproduce K2MP2O7 (M = Co, Ni, Cu), pyrovanadates eq. (140), consistent with the empirical baryonic Tully- such as K2MnV2O7 and Rb2MnV2O7, oxyphos- Fisher relation.(McGaugh, 2012; McGaugh et al., 2000) phates such as NaVOPO4 and LiVOPO4, layered KVOPO4, diphosphates such as Na3V(PO4)2, fluorophosphates such as Na2FePO4F, hydrox- C. Layered materials with cationic vacancies ysulphates such as LiFeSO4OH and oxysilicates such as Li2VOSiO4.(Barpanda et al., 2018, 2012; The derivation of eq. (140) largely makes use of the old Jin et al., 2020; Liao et al., 2019; Liu et al., 2018; quantization condition, βM¯ = N = 2k from eq. (85a) Masquelier and Croguennec, 2013; Niu et al., 2019; and a single (time-like) Killing vector, ξµ = ( 1,~0). For Prakash et al., 2006; Yahia et al., 2007) a gravitational system with two Killing vectors− such as the Schwarzschild black hole (time-like and azimuthal- In a majority of the aforementioned exemplars of lay- like), eq. (130) requires that the gravitational potential, ered materials, the mobility of the cations can be Φ cannot depend on more that two coordinates out of the traced to extremely weak chemical bonds whose strength n = 4 coordinates of . However, due to the spherical is correlated with the strength of emergent forces symmetry, the radialM coordinate, r(x,y,z) depends on such as Van der Waals interactions and the inter- three Cartesian coordinates, x,y,z instead of two, which layer distance between the slabs.(Delmas et al., 2021; means the Newtonian limit of the theory is effectively Dresselhaus and Dresselhaus, 1981; Kanyolo et al., 2021; three dimensional. Sun et al., 2019; Whittingham, 2004) Here, we are interested in a two dimensional emergent Layered transition metal oxides display a wide swath of quantum gravitational system whose dynamics exploit crystal structural versatility and composition tuneability. the Gauss-Bonnet theorem in n 2 dimensions, partic- They have thus been the subject of passionate research ularly in the Newtonian limit, where− n = 4 is the di- in various realms of solid-state (electro)chemistry, ma- mensions of the manifold, and 2 is the number of terials science and condensed matter physics.(He et al., translation Killing vectors.M Following our approach, we 2012; Kalantar-zadeh et al., 2016; Kanyolo et al., 2021; shall consider a condensed matter system with desirable Kubota, 2020; Liu et al., 2019; McClelland et al., 2020; properties which favor the topological aspects discussed Schnelle et al., 2021) A specific class of layered transition herein to emerge. In particular, we proceed to highlight metal oxide materials has recently emerged adopting, a wide class of layered materials where positively charged inter alia, chemical compositions embodied mainly mobile ions (cations) are sandwiched between the layers by A4MDO6, A3M2DO6 or A2M2DO6 wherein A of immobile ions forming adjacent series of slabs within represents an alkali-ion (Li, Na, K, etc.) or coinage a stable crystalline structure, viz., metal ions suchlike Ag, whereas M is mainly a transition metal species such as Co, Ni, Cu, Zn, etc. and D depicts (i) Layered transition metal oxides such as AxMO2 a pnictogen or chalcogen metal species such as Sb, Bi, (where A = Li, Na, K, Ag, etc., M is a transi- Te and so forth. (Berthelot et al., 2012; Brown et al., tion metal or a combination of multiple transi- 2019; Derakhshan et al., 2007; Evstigneeva et al., tion metals and 0 < x < 1), AyV2O5 (where 2011; Grundish et al., 2019; Kumar et al., 2012; 28

Nagarajan et al., 2002; Nalbandyan et al., 2013; meager amounts via doping techniques in order to im- Politaev et al., 2010; Roudebush et al., 2013; prove resolution.(Lee et al., 2000; Pan et al., 2002) Con- Seibel et al., 2013; Skakle et al., 1997; Smirnova et al., sequently, their overall effects on the diffusion properties 2005; Stratan et al., 2019; Uma and Gupta, 2016; are expected to be negligible. Nonetheless, if the vacan- Viciu et al., 2007; Yadav et al., 2019; Zvereva et al., cies are treated as holes it is expected that this particle- 2013, 2016, 2012) In these materials, mobile A cations hole symmetry is rather befitting to cationic Majorana are sandwiched between slabs entailing M atoms coor- modes e.g. with twist defects(Beenakker, 2013; Bomb´ın, dinated with oxygen around D atoms in a hexagonal 2010; Zheng et al., 2015) which could be exploited to in- (honeycomb) arrangement. We shall thus refer to these corporate fermionic behavior in the formalism.(Kanyolo, materials as honeycomb layered oxides.(Kanyolo et al., 2019) 2021) Considering emergent gravity to describe defects in Of particular interest are the dynam- crystals is not entirely a novel idea.(Holz, 1988; Kleinert, ics of the cations within the aforementioned 1988, 1987, 2005; Ver¸cin, 1990; Yajima and Nagahama, materials,(Kanyolo and Masese, 2020) since their 2016) For instance, it has long been proposed that con- diffusion contributes a net current when a sufficient sidering finite torsion (non-symmetric Christoffel sym- ρ ρ external electric field arises in an electrode-electrolyte bols/affine connection, Γ µν = Γ νµ) within the con- setup forming a cell or battery.(Goodenough and Park, text of Einstein-Cartan theory6 ought to capture various 2013) The polarity of the electric field defines the intriguing aspects related to disclinations and disloca- charging and discharging processes corresponding to tions within crystals.(Holz, 1988; Kleinert, 1988, 1987; de-intercalation (cation extraction) and intercalation Ver¸cin, 1990; Yajima and Nagahama, 2016) Moreover, it (cation insertion/reinsertion) processes respectively. has been further argued that Einstein gravity can still Theoretical computations show that the diffusion paths emerge in a crystal whose kinetic energy order terms are largely restricted to honeycomb pathways in hon- are restricted to second-order in derivatives(Kleinert, eycomb layered tellurates (for instance,K2Ni2TeO6 2005) in accordance with Lovelock’s theorem.(Lovelock, and Na2Ni2TeO6)(Bera and Yusuf, 2020; Masese et al., 1971) Thus, since the connection considered herein is ρ ρ 2018), where locations of the cations are correlated torsion-free (Γ µν = Γ νµ) and topological defects can with specific sites defined by the honeycomb octahe- still be non-vanishing even for torsion-free manifolds as dral structures within the slabs. Thus, the Van der long as higher order derivatives of the Gauss-Bonnet Waals forces initially localize the cations, forming a type(Lovelock, 1971) are present in the crystal, herein loosely-bound two dimensional hexagonal lattice where we shall consider a description of bosonic cationic vacan- mobility of the cations is only possible when sufficient cies as topological defects. activation energy can offset this localization leaving In particular, the radial distribution function (pair cor- cationic vacancies.(Matsubara et al., 2020; Wang et al., relation function), g(~xa) is the conditional probability 2018) Since the number of cationic vacancies should cor- density that a cation will be found at ~xa at each inter- respond to the number of mobile cations if the material layer, relative to another within the same inter-layer. had no initial vacancy defects, it is reasonable to expect Equivalently, it is the average density of the cations at that the diffusion in the material can be completely ~xa relative to a tagged particle.(Chandler, 1987) This a captured either by the dynamics of the cations within means the number density, ρbg(~xa)= ρ2D(~xa) is normal- the lattice or by the dynamics of the cationic vacancies. ized as(Tuckerman, 2010), In particular, when the cations are bosons, a Fermi level does not exist, implying that a particle-hole pic- 2 a a d xa det(gij (~xa)) ρ2D(~xa)= Na 1, (141) ture, where the particle and the vacancy carry separate AE − Z a pieces of information is precluded. Thus, the vacancies q cannot be treated as holes, but an equivalent descrip- where Na is the number of cations within the inter-layer, tion for the dynamics of the cations carrying the same a, ρb is the bulk number density and the integration is E (thermodynamic) information. Consequently, a Bose- performed over the Euclidean n = 2 manifolds, a at a A Einstein condensate of the cations(Kanyolo and Masese, each inter-layer with a metric tensor, gij (~xa). Note that 2020) avails a prime avenue for an emergent geomet- Einstein summation convention should not be applied for ric description of such vacancies as topological defects the index, a. The 1 in the normalization can be thought − within a theory of diffusion in the context of emergent to arise from excluding the contribution of the reference quantum gravity. Conversely, describing the diffusion cation, as per the above definition. The center of mass co- in layered materials comprising fermionic cations such ordinates describe average diffusion properties and hence as 6Li with this approach would pose some significant must obey the uncoupled Langevin equations, challenge. Nonetheless, since their magnetic moment is 2 readily trace-able in Nuclear Magnetic Resonance exper- d ~xa = ~p (s)+ ~η (s), (142) iments, the fermionic cations are typically introduced in ds2 − a a 29 where s = it is Euclidean time, ~η (s) is the acceleration where 2 = ∂2/∂x2 + ∂2/∂y2 is the two dimensional ± a ∇a a a and ~pa =md~x ¯ a(s)/ds are the center of mass momenta Laplacian operator. Thus, to be consistent with eq. withm ¯ the average mass of the cations and 1/m¯ playing (141), the number density normalization is given by, the role of a mean time between collisions, assumed to be equivalent in all slabs due to translation invariance, N Σ= d 3x exp(2Φ(x))ρ(x) za+1 = za + ∆za along the vector, ~n = (0, 0, 1) normal − E ZV to the slabs (the z direction). This guarantees that in Σ 2 the continuum limit and when the energy of cations is = lim ∆za d xa exp(2Φa(~xa))ρa(~xa) ∆za→0 E conserved, the crystal must admit not only a time-like a=1 Aa X Z but also a z-like Killing vector. Σ Observe that, we can follow eq. (112) and define the = (N 1), (147a) a − Gauss-Bonnet theorem at each inter-layer by making the a=1 X identification, Σ where N = a=1 Na is the total number of cations, Σ d2~x is the total number of two dimensional manifolds corre- a = ka(s), (143) P ds2 g sponding to the space between the layers (inter-layers) of a the material and, a = M kg (s)ds, (144) H E Ca a Z ∆zaρa(~xa)= ρ2D(~xa) (147b) where a is the Hamiltonian at each inter-layer, a and M a exp(2Φ(~xa)) = det(g (~xa)), (147c) is theH total mass of the mobile cations in the structure. ij Since we shall be concerned with momentum conserva- q 2 2 a with ∆za the inter-layer spacing. tion, d ~xa/ds =0, we can set kg (s) = 0, which implies Thus, we recognize eq. (146c) as Liouville’s equation, the Gauss-Bonnet theorem expression is devoid bound- with the Gaussian curvature given by, ary terms and hence simplifies to, 1 Ka(~xa)= ρa(~xa). (148a) pa = ⋆Ka = dηa =2π(2 2ha), (145) −m¯ ∂AE AE AE − Z a Z a Z a This means that the n = 2 manifold, E is described by Aa E the conformal metric, where ha is the genus of a . If we interpret the cationic E A vacancies in as the genus, then eq. (145) reveals that 2 2 2 2 the energy neededA to create vacancies in the vacuum must dσa = f (dxa + dya), (148b) always balance the energy due to motion, since pa = 0 6 2 a when h = 1. where f = exp(2Φ(~xa)) = det(gij (~xa)) is the confor- a 6 On the other hand, the Fokker-Planck equation corre- mal factor. To find the orderq of magnitude for ∆za, we sponds to the Newtonian limit, should use eq. (145), eq. (148a) and eq. (147b) to find,

1 h = N , (149a) 2Φ(x)= ρ(x) exp(2Φ(x)), (146a) a a m¯ ∇ ∆za =1/4πm.¯ (149b) 0 where we have used u (x) = exp(Φ(x)) =1 and 1/m¯ A potential problem with the model is that the cations 4πGM = β/2, requiring m¯ to play the6 role of temper-↔ are charged while typically the gravitational field, Φa(~xa) ature. In order to guarantee ρ(~x) can be related to the is not. However, this poses no problem since we only need two dimensional number density, we impose time-like, µ ~ µ ~ to require that the SU(N) symmetry in our approach to ξ¯ = ( 1, 0) and z like, ξz¯ = (0, 1) Killing vectors which µ t − − break to F F = dA, where A = Aµdx is the elec- guarantee that ∂ρ(x)/∂t = ∂ρ(x)/∂z = 0, where, tromagnetic→(U(1)) gauge potential and hence requiring the wave function, Ψ to be charged. To consistently in- ρ(x)= ρ exp( βM¯ Φ(x)), (146b) b − troduce the electric field, we shall require the (random) accelerations and the friction terms respectively, in the is the Boltzmann factor with β¯ = 1/m¯ . Moreover, since Langevin equation to take the forms, m¯ is also defined as the average mass of the cations, we have βM¯ = N as required by eq. (85a). ~ηa = qm(~n E~a), (150a) Thus, the Newtonian potential ought to split into z- × ~ Φ(~x )= ~p , (150b) like inter-layer slices labeled by index a, each satisfying, ∇a a a 1 ~ E 2 where Ea = (Exa , Eya , 0) is the electric field on a re- aΦa(~xa)= ρa(~xa) exp(2Φa(~xa)), (146c) A ∇ m¯ sponsible for the (de-)intercalation process in the cell, qm 30 is the magnetic charge and ~n = (0, 0, 1) is the unit vector certain thermodynamic and topological features in gen- normal to the slabs. The Gauss-Bonnet theorem then re- eral relativity and specifically quantum gravity. In the quires that the Dirac quantization condition(Dirac, 1931) present work, we have shown that the foundations of our is satisfied, approach reveal a rich complex-Hermitian structure anal- ogous to Cayley-Dickson algebras(Schafer, 1954), which can be exploited to formulate the appropriate action Φa = pa = ηa CE CE principle to yield general relativity in n = 4 dimen- Z Z sions as the effective theory, albeit with a Gauss-Bonnet ~ 2 ~ ~ = qm d~xa (~n Ea)= qm d x a Ea (topological) term. Moreover, we treat the ill-defined CE · × AE ∇ · Z Z Euclidean path integral measure of quantum gravity, = qmqe =2πνa = πχ2(ha), (150c) E [gµν ] by taking the sum over topologically distinct D E 2 ~ ~ manifolds, [gµν ] ME∈h, which yields a parti- where qe = AE d x a Ea is the electric charge, D → a ∇ · tion function reminiscent of the 1/ expansion of an ~ ~ 2 R P N a Ea = Rf Ka/2qm, νa is the monopole number unknown large theory.(Freidel, 2005; Gielen et al., and∇ · χ (h )− = 2 2h . Notably, Φ is analogous N 2 a − a a 2013; Gurau, 2011, 2012; Gurau and Rivasseau, 2011; to the Aharonov-Casher phase(Aharonov and Casher, Maldacena, 1999; Oriti, 2006; Thorn, 1994) 1984), where the magnetic moment corresponds to, ~µ = However, the (pseudo-)Riemannian manifold in our ap- qm~n. Thus, it is intuitive to view the cations diffusing proach is not emergent, rather it is introduced ab initio along curves around neutral vacancies with a magnetic in the treatment. Nonetheless, large theories have the moment. feature that the topologies of the vacuumN Feynman dia- Consequently, the pair correlation function can be grams pave an emergent manifold, with dimensions equal written as a Boltzmann factor, to the rank of the tensor group theory. Moreover, it has been conjectured that specific large theories are equiv- M N g(h ) = exp πχ (h ) = χ2(ha), (151) alent to some string theories.(Freidel, 2005; Gielen et al., a m¯ 2 a N   2013; Gurau, 2011, 2012; Gurau and Rivasseau, 2011; Maldacena, 1999; Oriti, 2006; Thorn, 1994) Thus, it where = exp(πN) and M/m¯ = N. Thus, we rec- would be interesting to investigate whether these ideas ognizeN the 1/ expansion factor appearing in the pair are compatible with the work herein. Nonetheless, the correlation function.N We expect the pair correlations to expectation is that rank n = 4 tensor group theo- be calculated for varied cationic vacancies as cations are ries are prime candidates for the large theory we created/annihilated on the manifold. Thus, a weighted seek(Gurau, 2011, 2012; Gurau and RivasseauN , 2011), sum over the distinct topologies yields, where the group field, ϕ need not be space-time depen- dent. A particularly curious fact is that, unlike random g = P (λ) χ2 (ha), (152) h ai ha N matrix large N theories in n = 4 dimensions, /λ is not h Xa dimensionless, but has dimensions of 1/(mass)N2. where P (λ) is the probability for the topology h to For illustration purposes, we can substitute, in eq. ha a N occur in a given (de-)intercalation process and λ(¯m) is (96), the bosonic action, S(ϕ)= λ Tr(γ(ϕ)) with γ(ϕ)= ϕ2/2 ϕ4/4g2, where the free propagator after appropri- a parameter that ought to depend only on m¯ and hence − ¯−1 ate re-scaling, ϕ ϕ/g corresponds to ϕ2 λ/g2 the temperature, β =m ¯ . Consequently, eq. (152) → h i ≃ N corresponds to the sum of vacuum Feynman diagrams of and g is a coupling constant with mass dimensions. Thus, a large theory, where the ’t Hooft limit ( , the field, ϕ becomes dimensionless after re-scaling, is as- keeping Nλ fixed) makes physical sense, sinceN it can → ∞ be sumed Hermitian and need not be space-time dependent. interpreted as a consequence of considering a fixed equi- Assuming ϕ transforms as a vector under the group, U( ) constrained by λ/π = g2N = ln( ) = , where librium temperature environment with a large number of N N S particles in the material. the ’t Hooft limit corresponds to N, at fixed λ/π. The free energy contribution fromN any → ∞ vacuum Feyn- man diagram, (N, ) at a given loop level, F is pro- FhF N VI. DISCUSSION portional to the product of three terms(’t Hooft, 1993),

V −E Gravity, as formulated by Einstein, is not simply a g2 g2 (N, )= f N N F force like electro-magnetism or the weak and strong FhF N F λ λ N     nuclear forces but rather manifests itself as space- = (πN)F −χ2(h)f χ2(h), (153) time curvature in a pseudo-Riemannian n = 4 mani- F N fold.(Thorne et al., 2000) Thus, the emergence of such where F is the number of loops, E is the number of propa- manifolds from an underlying theory with desirable clas- gators, V is the number of vertices, corresponding to two- sical or quantum properties would be sufficient to test particle interactions, fF is the proportionality constant 31 which depends only on F , and F E+V = χ2(h)=2 2h identifications in eq. (36) is possible since the number of is the Euler characteristic of some− emergent n = 2− di- anti-symmetric SU(N) generatorsis given by (N 2 N)/2, mensional Riemannian manifold, E, with the Feynman which is equivalent to the number of SO(n) generators,− diagrams acting as the simplex triangulationA of the mani- (n2 n)/2, which allows the decomposition of ω in eq. fold with F faces, E edges and V vertices. Consequently, (35)− when N = n is the number of dimensions. Conse- the free energy contribution at a given topology level, quently, due to eq. (106), eq. (23c) can be interpreted h = g +b/2 (where g is the genus of E and b the bound- as a relation between the central charge of a black hole, ary contribution) corresponds to theA sum over loop dia- k and the number of dimensions, n. Such a relationship grams, has been conjectured in the context of the sphere pack- ing optimization problem in n dimensions and quantum +∞ gravity.(Hartman et al., 2019) (N, )= (N, )= f (N) χ2(h), (154a) Fh N FhF N h N In summary, we have employed the thermodynamic ex- FX=0 pressions for entropy and temperature of Schwarzschild where, black holes to constrain an SU(N) gauge theory lead- ing to the emergence of a quantum framework for grav- +∞ ity, where the gravitational degrees of freedom are the −χ2 (h) F fh(N) = (πN) (πN) fF . (154b) N Z 0 colors and a condensate comprising color ∈ ≥ FX=0 pairs, k = N/2, appropriately coupled to the Yang-Mills In fact, eq. (154a) is robust and is obtained by most rank gauge field. The foundations of our approach reveal a complex-Hermitian structure, constructed as [Ricci ten- n = 2 large group theories, albeit with differing fh(N) N sor √ 1 Yang-Mills field strength], whose structure values which completely characterize the specific theory ± − under consideration. is analogous to Cayley-Dickson algebras(Schafer, 1954), E E which aids in formulating the appropriate action princi- In our approach, we argued that χ2( )= χ4( ) is possible, where E is the n = 4 dimensionalA compactM ple. The SU(N) gauge group is broken into an effective M SU(4) SO(4) SO(1, 3) field theory with two terms: Riemannian manifold corresponding to the connected → ↔ sum of h number of n = 4-tori (T 4) and an arbitrary the Einstein-Hilbert action and a Gauss-Bonnet topolog- number of n = 4-spheres (S4). Thus, the 1/ expan- ical term. Moreover, the Euclidean path integral is con- sion in eq. (95) corresponds to a large theoryN with sidered as the sum over manifolds with distinct topolo- 2 N gies, h Z 0 homeomorphic to connected sums of fh(N) exp( g λ) = exp( πN)=1/ , where the ∈ ≥ coupling≃ constant− satisfies, − N an arbitrary number of n = 4-spheres and h number of n = 4-tori. β Consequently, the partition function takes a rem- −2 1 E M πN g = det(gµν )R = ds = . (155) iniscent form of the sum of the vacuum Feynman −π ME 2λ 0 λ Z q Z diagrams for a large = exp(βM/2) theory, provided N Here, we have used eq. (98) in Euclidean signature and = βM/2 = πN is the Schwarzschild black hole S eq. (100b) respectively. Consequently, due to the form entropy, β = 8πGM is the inverse temperature, G is of eq. (154b), it is prudent to re-scale the path integral gravitational constant, M is the black hole mass and χ4(h) measure as, E E (πN) in order for horizon area, = 2GβM = 4πGN is pixelated in units M ∈h → M ∈h A fF =1/F ! to appropriately yield eq. (95) with the quan- of 4πG. This leads us to conclude that the partition tum gravity partitionP functionP corresponding to the 1/ function for quantum gravity is equivalent to the vac- expansion, E = (N, ). The fermionic caseN uum Feynman diagrams of a yet unidentified large QG h h N should followZ similar treatmentF N and considerations with theory in n = 4 dimensions. Our approach also sheds a suitable group fieldP theory. new light on the asymptotic behavior of dark matter- Finally, using the temperature and entropy of the dominated galaxy rotation curves (the empirical bary- Kerr-Newmann black hole(Bekenstein, 2008) instead of onic Tully-Fisher relation)(Eisenstein and Loeb, 1997; Schwarzschild’s should alter the expressions considered, Famaey and McGaugh, 2012; Kanyolo and Masese, but not the conclusions herein. Nonetheless, we are 2021; Keeton, 2001; Martel and Shapiro, 2003; left to puzzle how Unruh-Hawking radiation(Hawking, McGaugh, 2012; McGaugh et al., 2000; Persic et al., 1974, 1976a; Unruh, 1977), and more importantly, 1996) and emergent gravity in condensed matter the black hole information paradox(Hawking, 1976b; systems with defects(Holz, 1988; Kleinert, 1987; Mathur, 2009) fits into this picture. Since a space-time Kleinert and Zaanen, 2004; Ver¸cin, 1990; Zaanen et al., with an evaporating black hole will be subject to fluctu- 2004) such as layered materials(Kanyolo et al., 2021; ation-dissipation effects(Banerjee and Majhi, 2020), one Masese et al., 2021a,b,c, 2018) which admit cationic expects that our approach lacks key features which ought vacancies as topological defects.(Kanyolo and Masese, to be incorporated in future works. Another interest- 2020) ing observation is that breaking SU(N) to SO(N) by the While we acknowledge quantum gravity is a broad sub- 32 ject whose rapid progress largely remains uncovered by Becker, Katrin, Melanie Becker, and John H Schwarz (2006), the present work, we consider the demonstration of the String theory and M-theory: A modern introduction (Cam- intriguing ramifications, not only in quantum gravity re- bridge university press). search but also in condensed matter and materials sci- Beenakker, CWJ (2013), “Search for majorana fermions in su- perconductors,” Annu. Rev. Condens. Matter Phys. 4 (1), ence, as warranting considerable merit. 113–136. Bekenstein, Jacob D (2005), “How does the en- tropy/information bound work?” Foundations of Physics 35 ACKNOWLEDGEMENTS (11), 1805–1823. Bekenstein, Jacob D (2008), “Bekenstein-Hawking entropy,” Scholarpedia 3 (10), 7375. The authors would like to acknowledge the financial Bekenstein, Jacob D, and VF Mukhanov (1998), “Quantum support of TEPCO Memorial Foundation, Japan So- gravity and Hawking radiation,” Quantum Gravity , 141. ciety for the Promotion of Science (JSPS KAKENHI Bekenstein, Jacob D, and Viatcheslav F Mukhanov (1995), Grant Numbers 19K15685 and 21K14730) and Japan “Spectroscopy of the quantum black hole,” Physics Letters 360 Prize Foundation. The authors also acknowledge fruit- B (1-2), 7–12. ful discussions with D. Ntara during the cradle of the Bekenstein, JD (1973), “Black holes and entropy,” Phys. Rev. D 7, 2333. ideas herein, and especially the rigorous proofreading Bera, Anup Kumar, and Seikh M Yusuf (2020), work on the manuscript done by Edfluent. Both authors “Temperature-dependent Na-ion conduction and its are grateful for the unwavering support from their fam- pathways in the crystal structure of the layered battery ily members (T. 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