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Harald Grosse

Publications and source records attributed to Harald Grosse.

At least 55 records · Page 3Linked to original sources
arXiv2007

Induced Gauge Theory on a Noncommutative Space

We consider a scalar $ϕ^4$ theory on canonically deformed Euclidean space in 4 dimensions with an additional oscillator potential. This model is known to be renormalisable. An exterior gauge field is coupled in a gauge invariant manner to the scalar field. We extract the dynamics for the gauge field from the divergent terms of the 1-loop effective action using a matrix basis and propose an action for the noncommutative gauge theory, which is a candidate for a renormalisable model.

hep-th↗
arXiv2007

Exact renormalization of a noncommutative ϕ^3 model in 6 dimensions

The noncommutative selfdual ϕ^3 model in 6 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 one-loop beta function. A phase transition to an unstable phase is found.

hep-th↗
arXiv2007

Renormalization and Induced Gauge Action on a Noncommutative Space

Field theories on deformed spaces suffer from the IR/UV mxing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this desease by adding one more marginal operator. We review these ideas, show the application to $ϕ^3$ models and use heat kernel expansion methods for a scalar field theory coupled to an external gauge field on a $θ$-deformed space and derive noncommutative gauge actions.

hep-th↗
arXiv2006

Noncommutative QFT and Renormalization

Field theories on deformed spaces suffer from the IR/UV mixing and renormalization is generically spoiled. In work with R. Wulkenhaar, one of us realized a way to cure this disease by adding one more marginal operator. We review these ideas, show the application to $ϕ^3$ models and use the heat kernel expansion methods for a scalar field theory coupled to an external gauge field on a $θ$-deformed space and derive noncommutative gauge field actions.

hep-th↗
arXiv2005

On $κ$-Deformation and UV/IR Mixing

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.

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arXiv2005

On deformation theory of quantum vertex algebras

We study an algebraic deformation problem which captures the data of the general deformation problem for a quantum vertex algebra. We derive a system of coupled equations which is the counterpart of the Maurer-Cartan equation on the usual Hochschild complex of an assocative algebra. We show that this system of equations results from an action principle. This might be the starting point for a perturbative treatment of the deformation problem of quantum vertex algebras. Our action generalizes the action of the Kodaira-Spencer theory of gravity and might therefore also be of relevance for applications in string theory.

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arXiv2004

The universal envelope of the topological closed string BRST-complex

We construct a universal envelope for any Poisson- and Gerstenhaber algebra. While the deformation theory of Poisson algebras seems to be partially trivial, results from string- and M-theory suggest a rich deformation theory of Gerstenhaber algebras. We apply our construction in this case to well known questions on the topological closed string BRST-complex. Finally, we find a similar algebraic structure, as for the universal envelope, in the SU(2)-WZW model.

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arXiv2004

A remark on the motivic Galois group and the quantum coadjoint action

It was suggested by Kontsevich that the Grothendieck-Teichmueller group GT should act on the Duflo isomorphism of su(2) but the corresponding realization of GT turned out to be trivial. We show that a solvable quotient of the motivic Galois group - which is supposed to agree with GT - is closely related to the quantum coadjoint action on U_q(sl_2) for q a root of unity, i.e. in the quantum group case one has a nontrivial realization of a quotient of the motivic Galois group. From a discussion of the algebraic properties of this realization we conclude that in more general cases than U_q(sl_2) it should be related to a quantum version of the motivic Galois group. Finally, we discuss the relation of our construction to quantum field and string theory and explain what we believe to be the "physical reason" behind this relation between the motivic Galois group and the quantum coadjoint action. This might be a starting point for the generalization of our construction to more involved examples.

hep-th↗
arXiv2004

Power-counting theorem for non-local matrix models and renormalisation

Solving the exact renormalisation group equation a la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties--typically arising from orthogonal polynomials--which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative R^D in matrix formulation.

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arXiv2004

Renormalisation of ϕ^4-theory on noncommutative R^4 in the matrix base

We prove that the real four-dimensional Euclidean noncommutative ϕ^4-model is renormalisable to all orders in perturbation theory. Compared with the commutative case, the bare action of relevant and marginal couplings contains necessarily an additional term: an harmonic oscillator potential for the free scalar field action. This entails a modified dispersion relation for the free theory, which becomes important at large distances (UV/IR-entanglement). The renormalisation proof relies on flow equations for the expansion coefficients of the effective action with respect to scalar fields written in the matrix base of the noncommutative R^4. The renormalisation flow depends on the topology of ribbon graphs and on the asymptotic and local behaviour of the propagator governed by orthogonal Meixner polynomials.

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arXiv2004

Finite Gauge Theory on Fuzzy CP^2

We give a non-perturbative definition of U(n) gauge theory on fuzzy CP^2 as a multi-matrix model. The degrees of freedom are 8 hermitian matrices of finite size, 4 of which are tangential gauge fields and 4 are auxiliary variables. The model depends on a noncommutativity parameter 1/N, and reduces to the usual U(n) Yang-Mills action on the 4-dimensional classical CP^2 in the limit N -> \infty. We explicitly find the monopole solutions, and also certain U(2) instanton solutions for finite N. The quantization of the model is defined in terms of a path integral, which is manifestly finite. An alternative formulation with constraints is also given, and a scaling limit as R^4_θis discussed.

hep-th↗
arXiv2004

Exact solution of a 1D many-body system with momentum dependent interactions

We discuss a 1D many-body model of distinguishable particles with local, momentum dependent two-body interactions. We show that the restriction of this model to fermions corresponds to the non-relativistic limit of the massive Thirring model. This fermion model can be solved exactly by a mapping to the 1D boson gas with inverse coupling constant. We provide evidence that this mapping is the non-relativistic limit of the duality between the massive Thirring model and the quantum sine-Gordon model. We also investigate the question if the generalization of this model to distinguishable particles is exactly solvable by the coordinate Bethe ansatz and find that this is not the case.

math-ph↗
arXiv2004

Renormalisation of ϕ^4-theory on noncommutative R^4 to all orders

We present the main ideas and techniques of the proof that the duality-covariant four-dimensional noncommutative ϕ^4-model is renormalisable to all orders. This includes the reformulation as a dynamical matrix model, the solution of the free theory by orthogonal polynomials as well as the renormalisation by flow equations involving power-counting theorems for ribbon graphs drawn on Riemann surfaces.

hep-th↗
arXiv2004

The β-function in duality-covariant noncommutative ϕ^4-theory

We compute the one-loop β-functions describing the renormalisation of the coupling constant λand the frequency parameter Ωfor the real four-dimensional duality-covariant noncommutative ϕ^4-model, which is renormalisable to all orders. The contribution from the one-loop four-point function is reduced by the one-loop wavefunction renormalisation, but the β_λ-function remains non-negative. Both β_λand β_Ωvanish at the one-loop level for the duality-invariant model characterised by Ω=1. Moreover, β_Ωalso vanishes in the limit Ω\to 0, which defines the standard noncommutative ϕ^4-quantum field theory. Thus, the limit Ω\to 0 exists at least at the one-loop level.

hep-th↗
arXiv2003

Renormalisation of ϕ^4-theory on noncommutative R^2 in the matrix base

As a first application of our renormalisation group approach to non-local matrix models [hep-th/0305066], we prove (super-)renormalisability of Euclidean two-dimensional noncommutative ϕ^4-theory. It is widely believed that this model is renormalisable in momentum space arguing that there would be logarithmic UV/IR-divergences only. Although momentum space Feynman graphs can indeed be computed to any loop order, the logarithmic UV/IR-divergence appears in the renormalised two-point function -- a hint that the renormalisation is not completed. In particular, it is impossible to define the squared mass as the value of the two-point function at vanishing momentum. In contrast, in our matrix approach the renormalised N-point functions are bounded everywhere and nevertheless rely on adjusting the mass only. We achieve this by introducing into the cut-off model a translation-invariance breaking regulator which is scaled to zero with the removal of the cut-off. The naive treatment without regulator would not lead to a renormalised theory.

hep-th↗
arXiv2002

Fuzzy Line Bundles, the Chern Character and Topological Charges over the Fuzzy Sphere

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 non-integer, in the commutative limit the well known integer Chern numbers of the complex line bundles over the 2-sphere are recovered.

math-ph↗
arXiv2001

Pair Production of Neutral Higgs Bosons through Noncommutative QED Interactions at Linear Colliders

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.

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