Non-perturbative equivalences among large N gauge theories with adjoint and bifundamental matter fields

We prove an equivalence, in the large N limit, between certain U(N) gauge theories containing adjoint representation matter fields and their orbifold projections. Lattice regularization is used to provide a non-perturbative definition of these theories; our proof applies in the strong coupling, large mass phase of the theories. Equivalence is demonstrated by constructing and comparing the loop equations for a parent theory and its orbifold projections. Loop equations for both expectation values of single-trace observables, and for connected correlators of such observables, are considered; hence the demonstrated non-perturbative equivalence applies to the large N limits of both string tensions and particle spectra.Comment: 40 pages, JHEP styl

Nonassociative deformations of non-geometric flux backgrounds and field theory

In this thesis we describe the nonassociative geometry probed by closed strings in at non-geometric R- ux backgrounds, and develop suitable quantization techniques. For this, we propose a Courant sigma-model on an open membrane with target space M, which we regard as a topological sector of closed string dynamics on Rspace. We then reduce it to a twisted Poisson sigma-model on the boundary of the membrane with target space the cotangent bundle T M. The pertinent twisted Poisson structure is provided by a U(1) gerbe in momentum space, which geometrizes R-space. From the membrane perspective, the path integral over multivalued closed string elds in Q-space (i.e. the T-fold endowed with a non-geometric Q- ux which is T-dual to the R- ux), is equivalent to integrating over open strings in R-space. The corresponding boundary correlation functions reproduce Kontsevich's global deformation quantization formula for the twisted Poisson manifolds, which we take as our proposal for quantization. We calculate the corresponding nonassociative star product and its associator, and derive closed formulas for the case of a constant R- ux. We then develop various versions of the Seiberg{Witten map, which relate our nonassociative star products to associative ones and add uctuations to the R- ux background. We also propose a second quantization method based on quantizing the dual of a Lie 2-algebra via convolution in an integrating Lie 2-group. This formalism provides a categori cation of Weyl's quantization map, and leads to a consistent quantization of Nambu{Poisson 3-brackets. We show that the convolution product coincides with the star product obtained by Kontsevich's formula, and clarify its relation with the twisted convolution products for topological nonassociative torus bundles. As a rst step towards formulating quantum gravity on non-geometric spaces, we develop a third quantization method to study nonassociative deformations of geometry in R-space, which is analogous to noncommutative deformations of geometry (i.e. noncommutative gravity). We nd that the symmetries underlying these nonassociative deformations generate the non-abelian Lie algebra of translations and Bopp shifts in phase space. Using a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions, and the exterior di erential calculus in R-space. We de ne a suitable integration on these nonassociative spaces, and nd that the usual cyclicity of associative noncommutative deformations is replaced by weaker notions of 2-cyclicity and 3-cyclicity. In this setting, we consider extensions to non-constant R- ux backgrounds as well as more generic twisted Poisson structures emerging from non-parabolic monodromies of closed strings. As a rst application of our nonassociative star product quantization, we develop nonassociative quantum mechanics based on phase space state functions, wherein 3-cyclicity is instrumental for proving consistency of the formalism. We calculate the expectation values of area and volume operators, and nd coarse-graining of the string background due to the R- ux. For a second application, we construct nonassociative deformations of elds, and study perturbative nonassociative scalar eld theories on R-space. We nd that nonassociativity induces modi cations to the usual classi cation of Feynman diagrams into planar and non-planar graphs, which are controlled by 3-cyclicity. The example of '4 theory is studied in detail and the one-loop contributions to the two-point function are calculated

A minimal approach to the scattering of physical massless bosons

Tree and loop level scattering amplitudes which involve physical massless bosons are derived directly from physical constraints such as locality, symmetry and unitarity, bypassing path integral constructions. Amplitudes can be projected onto a minimal basis of kinematic factors through linear algebra, by employing four dimensional spinor helicity methods or at its most general using projection techniques. The linear algebra analysis is closely related to amplitude relations, especially the Bern-Carrasco-Johansson relations for gluon amplitudes and the Kawai-Lewellen-Tye relations between gluons and graviton amplitudes. Projection techniques are known to reduce the computation of loop amplitudes with spinning particles to scalar integrals. Unitarity, locality and integration-by-parts identities can then be used to fix complete tree and loop amplitudes efficiently. The loop amplitudes follow algorithmically from the trees. A range of proof-of-concept examples is presented. These include the planar four point two-loop amplitude in pure Yang-Mills theory as well as a range of one loop amplitudes with internal and external scalars, gluons and gravitons. Several interesting features of the results are highlighted, such as the vanishing of certain basis coefficients for gluon and graviton amplitudes. Effective field theories are naturally and efficiently included into the framework. The presented methods appear most powerful in non-supersymmetric theories in cases with relatively few legs, but with potentially many loops. For instance, iterated unitarity cuts of four point amplitudes for non-supersymmetric gauge and gravity theories can be computed by matrix multiplication, generalising the so-called rung-rule of maximally supersymmetric theories. The philosophy of the approach to kinematics also leads to a technique to control color quantum numbers of scattering amplitudes with matter.Comment: 65 pages, exposition improved, typos correcte

Games on partial orders and other relational structures

This thesis makes a contribution to the classification of certain specific relational structures under the relation of n-equivalence, where this means that Player II has a winning strategy in the n-move Ehrenfeucht-Fraı̈ssé game played on the two structures. This provides a finer classification of structures than elementary equivalence, since two structures A and B are elementarily equivalent if and only if they are n-equivalent for all n. On each move of such a game, Player I picks a member of either A or B, and Player II responds with a member of the other structure. Player II wins the game if the map thereby produced from a substructure of A to a substructure of B is an isomorphism of induced substructures. Certain ordered structures have been studied from this point of view in papers by Mostowski and Tarski, for ordinals [22], and Mwesigye and Truss, for ordinals [25], some scattered orders, and finite coloured linear orders [24]. Here we extend the known results on linear orders by classifying them all up to 3-equivalence (which had previously been done for 2-equivalence), of which there are 281, using the method of characters. We also classify all partial orders up to 2-equivalence (there are 39), and discuss the difficulties of extending this to 3-equivalence, since the method of characters is not as effective as in the linear case. We classify (total) circular orders up to 3-equivalence, and relate the classification of partial circular orders to both these and to partial orders. A variety of related structures are discussed: trees, directed and undirected graphs, and unars (sets with a single unary function), which we categorise up to 2-equivalence. In a pebble game, the players of an otherwise standard Ehrenfeucht-Fraı̈ssé game are in addition provided with two identical sets of k distinguishable pebbles, and on each move they place a pebble on their chosen point. On each move, Player I may choose either to move a pebble to another point, or else use a new pebble, if any remain, and Player II must place the corresponding partner pebble. Such games correspond to logics in which there are only k variables, and moving a pebble corresponds to reusing the variable. Here we extend some work of Immerman and Kozen [14] on pebble games played on linear orders

The Plane-Wave/Super Yang-Mills Duality

We present a self-contained review of the Plane-wave/super-Yang-Mills duality, which states that strings on a plane-wave background are dual to a particular large R-charge sector of N=4, D=4 superconformal U(N) gauge theory. This duality is a specification of the usual AdS/CFT correspondence in the "Penrose limit''. The Penrose limit of AdS_5 S^5 leads to the maximally supersymmetric ten dimensional plane-wave (henceforth "the'' plane-wave) and corresponds to restricting to the large R-charge sector, the BMN sector, of the dual superconformal field theory. After assembling the necessary background knowledge, we state the duality and review some of its supporting evidence. We review the suggestion by 't Hooft that Yang-Mills theories with gauge groups of large rank might be dual to string theories and the realization of this conjecture in the form of the AdS/CFT duality. We discuss plane-waves as exact solutions of supergravity and their appearance as Penrose limits of other backgrounds, then present an overview of string theory on the plane-wave background, discussing the symmetries and spectrum. We then make precise the statement of the proposed duality, classify the BMN operators, and mention some extensions of the proposal. We move on to study the gauge theory side of the duality, studying both quantum and non-planar corrections to correlation functions of BMN operators, and their operator product expansion. The important issue of operator mixing and the resultant need for re-diagonalization is stressed. Finally, we study strings on the plane-wave via light-cone string field theory, and demonstrate agreement on the one-loop correction to the string mass spectrum and the corresponding quantity in the gauge theory. A new presentation of the relevant superalgebra is given.Comment: RevTeX 4 format; 91 pages; 7 figures. Prepared for Reviews of Modern Physics. Please send comments to darius, jabbari @ itp.stanford.edu. v3: Minor typos fixe