Harald Grosse - Academia.edu (original) (raw)

Papers by Harald Grosse

Journal of Geometry and Physics, Jun 1, 2001

We study the q-deformed fuzzy sphere, which is related to D-branes on SU(2) WZW models, for both ... more We study the q-deformed fuzzy sphere, which is related to D-branes on SU(2) WZW models, for both real q and q a root of unity. We construct for both cases a differential calculus which is compatible with the star structure, study the integral, and find a canonical frame of one-forms. We then consider actions for scalar field theory, as well as for Yang-Mills and Chern-Simons-type gauge theories. The zero curvature condition is solved.

Nuclear Physics B, Feb 1, 2005

Physical review, Apr 23, 2010

We show how the fields and particles of the standard model can be naturally realized in noncommut... more We show how the fields and particles of the standard model can be naturally realized in noncommutative gauge theory. Starting with a Yang-Mills matrix model in more than 4 dimensions, a SU (n) gauge theory on a Moyal-Weyl space arises with all matter and fields in the adjoint of the gauge group. We show how this gauge symmetry can be broken spontaneously down to which couples appropriately to all fields in the standard model. An additional U (1) B gauge group arises which is anomalous at low energies, while the trace-U (1) sector is understood in terms of emergent gravity. A number of additional fields arise which we assume to be massive, in a pattern that is reminiscent of supersymmetry. The symmetry breaking might arise via spontaneously generated fuzzy spheres, in which case the mechanism is similar to brane constructions in string theory. * The case D = 10 is of particular interest. In this case it is possible to impose a Majorana-Weyl condition on Ψ, and the model admits an extended supersymmetry . On a 4-dimensional Moyal-Weyl background as discussed below, the model then reduces to the N = 4 SYM on R 4 θ , which is expected to be well-behaved upon quantization.

Journal of Geometry and Physics, Sep 1, 2002

Nucleation and Atmospheric Aerosols, 2009

We discuss how a matrix model recently shown to describe emergent gravity may contain extra degre... more We discuss how a matrix model recently shown to describe emergent gravity may contain extra degrees of freedom which reproduce some characteristics of the standard model, in particular the breaking of symmetries and the correct quantum numbers of fermions.

Journal of Physics A, Oct 11, 2022

We review the construction of the λφ 4 -model on noncommutative geometries via exact solutions of... more We review the construction of the λφ 4 -model on noncommutative geometries via exact solutions of Dyson-Schwinger equations and explain how this construction relates via (blobbed) topological recursion to problems in algebraic and enumerative geometry.

Journal of High Energy Physics, Jul 1, 2010

It was shown recently that the lagrangian of the Grosse-Wulkenhaar model can be written as lagran... more It was shown recently that the lagrangian of the Grosse-Wulkenhaar model can be written as lagrangian of the scalar field propagating in a curved noncommutative space. In this interpretation, renormalizability of the model is related to the interaction with the background curvature which introduces explicit coordinate dependence in the action. In this paper we construct the U 1 gauge field on the same noncommutative space: since covariant derivatives contain coordinates, the Yang-Mills action is again coordinate dependent. To obtain a two-dimensional model we reduce to a subspace, which results in splitting of the degrees of freedom into a gauge and a scalar. We define the gauge fixing and show the BRST invariance of the quantum action.

Advances in Theoretical and Mathematical Physics, 2008

The noncommutative self-dual φ 3 model in six dimensions is quantized and essentially solved, by ... more The noncommutative self-dual φ 3 model in six dimensions is quantized and essentially solved, by mapping it to the Kontsevich model. The model is shown to be renormalizable and asymptotically free, and solvable genus by genus. It requires both wavefunction and coupling constant renormalization. The exact ("all-order") renormalization of the bare parameters is determined explicitly, which turns out to depend on the genus 0 sector only. The running coupling constant is also computed exactly, which decreases more rapidly than predicted by the 1-loop beta-function. A phase transition to an unstable phase is found.

Vietnam journal of mathematics, Sep 6, 2018

Over many years, we developed the construction of the φ 4 -model on four-dimensional Moyal space.... more Over many years, we developed the construction of the φ 4 -model on four-dimensional Moyal space. The solution of the related matrix model Z[E, J ] = d exp(tr(J -E 2λ 4 4 )) is given in terms of the solution of a non-linear equation for the 2-point function and the eigenvalues of E. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. Locality is fulfilled. The Schwinger 2-point function is reflection positive in special cases.

Journal of Geometry and Physics, May 1, 2002

Using the theory of quantized equivariant vector bundles over compact coadjoint orbits we determi... more Using the theory of quantized equivariant vector bundles over compact coadjoint orbits we determine the Chern characters of all noncommutative line bundles over the fuzzy sphere with regard to its derivation-based differential calculus. The associated Chern numbers (topological charges) arise to be noninteger, in the commutative limit the well-known integer Chern numbers of the complex line bundles over the two-sphere are recovered.

$Research paper thumbnail of The <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="normal">Φ</mi><mn>4</mn><mn>3</mn></msubsup></mrow><annotation encoding="application/x-tex">\Phi^3_4</annotation></semantics></math>Φ43 and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msubsup><mi mathvariant="normal">Φ</mi><mn>6</mn><mn>3</mn></msubsup></mrow><annotation encoding="application/x-tex">\Phi^3_6</annotation></semantics></math>Φ63 matricial QFT models have reflection positive two-point function$

arXiv (Cornell University), Dec 22, 2016

We extend our previous work (on D = 2) to give an exact solution of the Φ 3 D large-N matrix mode... more We extend our previous work (on D = 2) to give an exact solution of the Φ 3 D large-N matrix model (or renormalised Kontsevich model) in D = 4 and D = 6 dimensions. Induction proofs and the difficult combinatorics are unchanged compared with D = 2, but the renormalisation -performed according to Zimmermann -is much more involved. As main result we prove that the Schwinger 2-point function resulting from the Φ 3 D -QFT model on Moyal space satisfies, for real coupling constant, reflection positivity in D = 4 and D = 6 dimensions. The Källén-Lehmann mass spectrum of the associated Wightman 2-point function describes a scattering part |p| 2 ≥ 2µ 2 and an isolated fuzzy mass shell around |p| 2 = µ 2 . declare the equations as exact and construct exact solutions. Whereas the Φ 3 6 -Kontsevich model with imaginary coupling constant is asymptotically free [12], our real Φ 3 6 -model has positive β-function. But this is not a problem; there is no Landau ghost, and the theory remains well-defined at any scale! In other words, the real Φ 3 6 -Kontsevich model could avoid triviality. It is instructive to compare our exact results with a perturbative BPHZ renormalisation of the model. In D = 6 dimensions the full machinery of Zimmermann's forest formula is required. We provide in sec. 5 the BPHZ-renormalisation of the 1-point function up to two-loop order. One of the contributing graphs has an overlapping divergence with already 6 different forests. Individual graphs show the full number-theoretic richness of quantum field theory: up to two loops we encounter logarithms, polylogarithms Li 2 and ζ(2) = π 2 6 . The amplitudes of the graphs perfectly sum up to the Taylor expansion of the exact result. The original BPHZ scheme with normalisation conditions at a single scale leads in justrenormalisable models to the renormalon problem which prevents Borel resummation of the perturbation series. We also provide in sec. 5 an example of a graph which shows the renormalon problem. But all these problems cancel as our exact correlation functions are analytic(!) in the coupling constant. Exact BPHZ renormalisation is fully consistent (for the model under consideration)! The most significant result of this paper is derived in sec. 6. Matrix models such as the Kontsevich model Φ 3 D arise from QFT-models on noncommutative geometry. The prominent Moyal space gives rise to an external matrix E having linearly spaced eigenvalues with multiplicity reflecting the dimension D. In [21] two of us (H.G.+R.W.) have shown that translating the type of scaling limit considered for the matrix model correlation functions back to the position space formulation of the Moyal algebra leads to Schwinger functions of an ordinary quantum field theory on R D . Euclidean symmetry and invariance under permutations are automatic. The most decisive Osterwalder-Schrader axiom , reflection positivity, amounts for the Schwinger 2-point function to the verification that the diagonal matrix model 2-point function is a Stieltjes function. We proved in [14] that for the D = 2-dimensional Kontsevich model this is not the case. To our big surprise and exaltation, we are able to prove: Theorem 1.1. The Schwinger 2-point function resulting from the scaling limit of the Φ 3 D -QFT model on Moyal space with real coupling constant satisfies reflection positivity in D = 4 and D = 6 dimensions. As such it is the Laplace-Fourier transform of the Wightman 2-point function of a true relativistic quantum field theory [24] (θ, δ are the Heaviside and Dirac distributions). Its Källén-Lehmann mass spectrum ̺( M 2 µ 2 ) [25, 26] is explicitly known and has support on a scattering part with M 2 ≥ 2µ 2 and an isolated fuzzy mass shell around M 2 = µ 2 of non-zero width.

arXiv (Cornell University), Jun 30, 2014

We provide further analytical and first numerical results on the solvable λφ 4 4 -NCQFT model. We... more We provide further analytical and first numerical results on the solvable λφ 4 4 -NCQFT model. We prove that for λ < 0 the singular integral equation has a unique solution, whereas for λ > 0 there is considerable freedom. Furthermore we provide integral formulae for partial derivatives of the matrix 2-point function, which are the key to investigate reflection positivity. The numerical implementation of these equations gives evidence for phase transitions. The derivative of the finite wavefunction renormalisation with respect to λ is discontinuous at λ c ≈ -0.39. This leads to singularities in higher correlation functions for λ < λ c . The phase λ > 0 is not yet under control because of the freedom in the singular integral equation. Reflection positivity requires that the two-point function is Stieltjes. Implementing Widder's criteria for Stieltjes functions we exclude reflection positivity outside the phase [λ c , 0]. For the phase λ c < λ ≤ 0 we show that refining the discrete approximation we satisfy Widder to higher and higher order. This is clear evidence, albeit no proof, of reflection positivity in that phase. The much simpler case λ > 0 was already treated in under the (as we prove: false) assumption that the non-trivial solution of the homogeneous Carleman equation can be neglected. A first hint about reflection positivity can be obtained from a computer simulation of the equations. Widder's criteria for Stieltjes functions [13] need derivatives of arbitrarily high order, which is impossible for a discrete approximation of the equation. We therefore derive in sec. 3 an integral formula for arbitrary partial derivatives of the 2-point function. In sec. 4 we present first results of a numerical simulation of this model using Mathematica T M . The source code is given in the appendix. Starting point is the fixed point equation ( ) for the boundary 2-point function. We view G 0b as a piecewise-linear function and (2) as recursive definition of a sequence {G i 0b } i . We convince ourselves that this sequence converges in Lipschitz norm. For given λ, a sufficiently precise G i 0b is then used to compute characterising data of the model. In this way we find clear evidence for a phase transition at λ c ≈ -0.39 where the function ∂ 2 G 0b (λ) ∂b∂λ b=0 of λ is discontinuous. Within numerical error bounds we have 1 G 0b ≡ 1 for 0 ≤ b 0 we confirm an inconsistency due to neglecting the freedom with the homogeneous Carleman equation. This leaves the region [λ c , 0] as the only interesting phase, and precisely here we seem to have reflection positivity for the 2-point function. Of course, a discrete approximation by piecewise-linear functions cannot be Stieltjes. We show that the order where the Stieltjes property fails increases significantly when the approximation is refined; and this refinement slows down exactly at the same value λ c ≈ -0.39. We view this as overwhelming support for the conjecture that the boundary and diagonal 2-point functions G 0b and G aa , respectively, are Stieltjes functions. Together with this would imply reflection positivity of the Schwinger 2-point function. In we have studied the λφ 4 4 -model on noncommutative Moyal space in matrix representation. We showed that the two-point function G |ab| satisfies a closed non-linear equation in a scaling limit which simultaneously sends the volume V = ( θ 4 ) 2 and the size N of the matrices to infinity with the ratio this limit, the 2-point function G ab depends on 'continuous matrix indices' a, b ∈ [0, Λ 2 ] and satisfies a non-linear integral equation I a [G •b ] = 0. It was convenient to replace this equation by the coupled system I a [G •b ] -I a [G •0 ] = 0 and I a [G •0 ] = 0. The difference equation admitted a wavefunction renormalisation Z → (1 + Y) which reduced the problem to a linear singular integral equation [7] for the difference D ab := a G ab -G a0 b . We treat this equation an its solution G ab [G •0 ] in sec. 2.1. In sec. 2.2 we show that the boundary equation I a [G •0 ] = 0 gives no other information than the solution G ab [G •0 ] plus symmetry G ab = G ba . 1 We prove in the appendix of that G 0b = 1 is an exact solution of (2) for any λ < 0 and Λ 2 → ∞. This solution seems numerically unstable under small perturbations.

Physical review, Nov 8, 2001

We study the feasibility of detecting noncommutative (NC) QED through neutral Higgs boson (H) pai... more We study the feasibility of detecting noncommutative (NC) QED through neutral Higgs boson (H) pair production at linear colliders (LC). This is based on the assumption that H interacts directly with photon in NCQED as suggested by symmetry considerations and strongly hinted by our previous study on π 0 -photon interactions. We find the following striking features as compared to the standard model (SM) result: (1) generally larger cross sections for an NC scale of order 1 TeV; (2) completely different dependence on initial beam polarizations; (3) distinct distributions in the polar and azimuthal angles; and (4) day-night asymmetry due to the Earth's rotation. These will help to separate NC signals from those in the SM or other new physics at LC. We emphasize the importance of treating properly the Lorentz noninvariance problem and show how the impact of the Earth's rotation can be used as an advantage for our purpose of searching for NC signals.

$Research paper thumbnail of The <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math>β -function in duality-covariant non-commutative <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>ϕ</mi><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\phi^4</annotation></semantics></math>ϕ4 -theory$

European Physical Journal C, Jun 1, 2004

Communications in Mathematical Physics, May 1, 1996

In the framework of noncommutative geometry we describe spinor fields with nonvanishing winding n... more In the framework of noncommutative geometry we describe spinor fields with nonvanishing winding number on a truncated (fuzzy) sphere. The corresponding field theory actions conserve all basic symmetries of the standard commutative version (space isometries and global chiral symmetry), but due to the noncommutativity of the space the fields are regularized and they contain only finite number of modes.

Communications in Mathematical Physics, Dec 2, 2004

General Relativity and Gravitation, May 20, 2010

We argue that some features of the standard model, in particular the fermion assignment and symme... more We argue that some features of the standard model, in particular the fermion assignment and symmetry breaking, can be obtained in matrix model which describes noncommutative gauge theory as well as gravity in an emergent way. The mechanism is based on the presence of some extra (matrix) dimensions. These extra dimensions are different from the usual ones which give to a noncommutative geometry of the Grönewold-Moyal type, and are reminiscent of the Connes-Lott model, although the action is very different.

Birkhäuser Basel eBooks, Feb 13, 2007

Field theories on deformed spaces suffer from the IR/UV mixing and renormalization is generically... more Field theories on deformed spaces suffer from the IR/UV mixing and renormalization is generically spoiled.

EPL, Aug 7, 2007

Inspired by the renormalizability of the non-commutative φ 4 model with added oscillator term, we... more Inspired by the renormalizability of the non-commutative φ 4 model with added oscillator term, we formulate a non-commutative gauge theory, where the oscillator enters as a gauge fixing term in a BRST invariant manner. All propagators turn out to be essentially given by the Mehler kernel and the bilinear part of the action is invariant under the Langmann-Szabo duality. The model is a promising candidate for a renormalizable non-commutative U (1) gauge theory.

Nuclear Physics B, Aug 1, 2006

We examine the UV/IR mixing property on a κ-deformed Euclidean space for a real scalar φ 4 theory... more We examine the UV/IR mixing property on a κ-deformed Euclidean space for a real scalar φ 4 theory. All contributions to the tadpole diagram are explicitly calculated. UV/IR mixing is present, though in a different dressing than in the case of the canonical deformation.