Permutations and the combinatorics of gauge invariants for general N

Talk given at workshop on non-commutative field theory and gravity, Corfu 2015Talk given at workshop on non-commutative field theory and gravity, Corfu 2015Talk given at workshop on non-commutative field theory and gravity, Corfu 2015Talk given at workshop on non-commutative field theory and gravity, Corfu 2015Talk given at workshop on non-commutative field theory and gravity, Corfu 2015Talk given at workshop on non-commutative field theory and gravity, Corfu 2015Group algebras of permutations have proved highly useful in solving a number of problems in large N gauge theories. I review the use of permutations in classifying gauge invariants in one-matrix and multi-matrix models and computing their correlators. These methods are also applicable to tensor models and have revealed a link between tensor models and the counting of branched covers. The key idea is to parametrize $U(N)$ gauge invariants using permutations, subject to equivalences. Correlators are related to group theoretic properties of these equivalence classes. Fourier transformation on symmetric groups by means of representation theory offers nice bases of functions on these equivalence classes. This has applications in AdS/CFT in identifying CFT duals of giant gravitons and their perturbations. It has also lead to general results on quiver gauge theory correlators, uncovering links to two dimensional topological field theory and the combinatorics of trace monoids

New Modular Hopf Algebras related to rational kk sl(2)^\widehat {sl(2)}

30 pages (minor typos corrected, refs added)30 pages (minor typos corrected, refs added)30 pages (minor typos corrected, refs added)We show that the Hopf link invariants for an appropriate set of finite dimensional representations of $U_q SL(2)$ are identical, up to overall normalisation, to the modular S matrix of Kac and Wakimoto for rational $k$ $\widehat {sl(2)}$ representations. We use this observation to construct new modular Hopf algebras, for any root of unity $q=e^{-i\pi m/r}$, obtained by taking appropriate quotients of $U_q SL(2)$, that give rise to 3-manifold invariants according to the approach of Reshetikin and Turaev. The phase factor correcting for the `framing anomaly' in these invariants is equal to $e^{- {{i \pi} \over 4} ({ {3k} \over {k+2}})}$, an analytic continuation of the anomaly at integer $k$. As expected, the Verlinde formula gives fusion rule multiplicities in agreement with the modular Hopf algebras. This leads to a proposal, for $(k+2)=r/m$ rational with an odd denominator, for a set of $\widehat {sl(2)}$ representations obtained by dropping some of the highest weight representations in the Kac-Wakimoto set and replacing them with lowest weight representations. For this set of representations the Verlinde formula gives non-negative integer fusion rule multiplicities. We discuss the consistency of the truncation to highest and lowest weight representations in conformal field theory

Counting Tensor Model Observables and Branched Covers of the 2-Sphere

56 pages, 14 Figures56 pages, 14 FiguresLattice gauge theories of permutation groups with a simple topological action (henceforth permutation-TFTs) have recently found several applications in the combinatorics of quantum field theories (QFTs). They have been used to solve counting problems of Feynman graphs in QFTs and ribbon graphs of large $N$, often revealing inter-relations between different counting problems. In another recent development, tensor theories generalizing matrix theories have been actively developed as models of random geometry in three or more dimensions. Here, we apply permutation-TFT methods to count gauge invariants for tensor models (colored as well as non-colored), exhibiting a relationship with counting problems of branched covers of the 2-sphere, where the rank $d$ of the tensor gets related to a number of branch points. We give explicit generating functions for the relevant counting and describe algorithms for the enumeration of the invariants. As well as the classic count of Hurwitz equivalence classes of branched covers with fixed branch points, collecting these under an equivalence of permuting the branch points is relevant to the color-symmetrized tensor invariant counting. We also apply the permutation-TFT methods to obtain some formulae for correlators of the tensor model invariants

On the Polarization of Unstable D0-Branes into Non-Commutative Odd Spheres

We consider the polarization of unstable type IIB D0-branes in the presence of a background five-form field strength. This phenomenon is studied from the point of view of the leading terms in the non-abelian Born Infeld action of the unstable D0-branes. The equations have SO(4) invariant solutions describing a non-commutative 3-sphere, which becomes a classical 3-sphere in the large N limit. We discuss the interpretation of these solutions as spherical D3-branes. The tachyon plays a tantalizingly geometrical role in relating the fuzzy S^3 geometry to that of a fuzzy S^4.Comment: 18 pages, Te

From Matrix Models and quantum fields to Hurwitz space and the absolute Galois group

54 pages, 17 figures54 pages, 17 figures54 pages, 17 figuresWe show that correlators of the hermitian one-Matrix model with a general potential can be mapped to the counting of certain triples of permutations and hence to counting of holomorphic maps from world-sheet to sphere target with three branch points on the target. This allows the use of old matrix model results to derive new explicit formulae for a class of Hurwitz numbers. Holomorphic maps with three branch points are related, by Belyi's theorem, to curves and maps defined over algebraic numbers \bmQ. This shows that the string theory dual of the one-matrix model at generic couplings has worldsheets defined over the algebraic numbers and a target space \mP^1 (\bmQ). The absolute Galois group Gal (\bmQ / \mQ) acts on the Feynman diagrams of the 1-matrix model, which are related to Grothendieck's Dessins d'Enfants. Correlators of multi-matrix models are mapped to the counting of triples of permutations subject to equivalences defined by subgroups of the permutation groups. This is related to colorings of the edges of the Grothendieck Dessins. The colored-edge Dessins are useful as a tool for describing some known invariants of the Gal (\bmQ / \mQ) action on Grothendieck Dessins and for defining new invariants

Uniqueness of canonical tensor model with local time

Canonical formalism of the rank-three tensor model has recently been proposed, in which "local" time is consistently incorporated by a set of first class constraints. By brute-force analysis, this paper shows that there exist only two forms of a Hamiltonian constraint which satisfies the following assumptions: (i) A Hamiltonian constraint has one index. (ii) The kinematical symmetry is given by an orthogonal group. (iii) A consistent first class constraint algebra is formed by a Hamiltonian constraint and the generators of the kinematical symmetry. (iv) A Hamiltonian constraint is invariant under time reversal transformation. (v) A Hamiltonian constraint is an at most cubic polynomial function of canonical variables. (vi) There are no disconnected terms in a constraint algebra. The two forms are the same except for a slight difference in index contractions. The Hamiltonian constraint which was obtained in the previous paper and behaved oddly under time reversal symmetry can actually be transformed to one of them by a canonical change of variables. The two-fold uniqueness is shown up to the potential ambiguity of adding terms which vanish in the limit of pure gravitational physics.Comment: 21 pages, 12 figures. The final result unchanged. Section 5 rewritten for clearer discussions. The range of uniqueness commented in the final section. Some other minor correction