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We propose a nonperturbative approach to nonabelian two-form gauge theory. We formulate the theory on a lattice in terms of plaquette as fundamental dynamical variable, and assign U(N) Chan-Paton colors at each boundary link. We show that, on hypercubic lattices, such colored plaquette variables constitute Yang-Baxter maps, where holonomy is charac...
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... With this purpose, we first introduce the model for electromagnetism, and then we combine both models to obtain the coupling of the Kalb-Ramond field potential to the string charge and to the electric field. Earlier lattice models incorporating Kalb-Ramond fields were proposed in [23,24]. In [23] a Higgs mechanism for the Kalb-Ramond fields is proposed by coupling them to a string that eventually condensates. ...
... In [23] a Higgs mechanism for the Kalb-Ramond fields is proposed by coupling them to a string that eventually condensates. Moreover in Ref. [24] a non-abelian tensor gauge theory is implemented in the cubic lattice through the consideration of Chan-Paton colors in each boundary link. ...
... It would be interesting to apply these methods within this context and compute some quantum observables. We would like for the near future to search for a relation to the results involving Kalb-Ramond fields [23,24]. Moreover, the model of the antisymmetric field studied in the present article could be coupled to the symmetric model (as in Refs. ...
The emergence of a Kalb-Ramond field and string charge in the lattice is discussed. The local bosonic model with rotor variables placed on the faces of a cubic lattice is considered. The coupling model consisting of the Maxwell fields and the Kalb-Ramond field is given. This construction naturally incorporates the emerging coupling between both gauge and string fields. In the process, an object that resembles to a D-brane on the lattice is introduced.
... However until recently there was no way to address statistical properties of large tensors of rank higher than 2 through a 1/N expansion, hence such models have been mostly studied through computer simulations. See also [17, 18, 19] for other related approaches. Group field theory is a special kind of random tensor model in which one adds a Lie group G and a gauge invariance [20, 21, 22, 23, 24] 1 . ...
We prove that an integrated version of the Gurau colored tensor model supplemented with the usual Bosonic propagator on U(1)4 is renormalizable to all orders in perturbation theory. The model is of the type expected for quantization of space-time in 4D Euclidean gravity and is the first example of a renormalizable model of this kind. Its vertex and propagator are four-stranded like in 4D group field theories, but without gauge averaging on the strands. Surprisingly perhaps, the model is of the \({\phi^6}\) rather than of the \({\phi^4}\) type, since two different \({\phi^6}\)-type interactions are log-divergent, i.e. marginal in the renormalization group sense. The renormalization proof relies on a multiscale analysis. It identifies all divergent graphs through a power counting theorem. These divergent graphs have internal and external structure of a particular kind called melonic. Melonic graphs dominate the 1/N expansion of colored tensor models and generalize the planar ribbon graphs of matrix models. A new locality principle is established for this category of graphs which allows to renormalize their divergences through counterterms of the form of the bare Lagrangian interactions. The model also has an unexpected anomalous log-divergent \({(\int \phi^2)^2}\) term, which can be interpreted as the generation of a scalar matter field out of pure gravity.
... The corresponding entropy exponent is γ melons = 1/2 in any dimensions [41]. It is the analog of the string susceptibility exponent γ string = −1/2 of the invariant matrix models for the universality class of pure 2d quantum gravity, Colored random tensors [43] therefore gave the first theory of random geometries in three and more dimensions with analytically tractable geometrogenesis and the subject is rapidly expanding44454647484950. Coupling of statistical mechanical systems to these random geometries in arbitrary dimension has been done in [48, 51, 52], and results at all orders in 1/N have been established for some restricted models [53] (see also [54] for some related developments). Obvious questions then arise. ...
Tensor models generalize random matrix models in yielding a theory of
dynamical triangulations in arbitrary dimensions. Colored tensor models have
been shown to admit a 1/N expansion and a continuum limit accessible
analytically. In this paper we prove that these results extend to the most
general tensor model for a single generic, i.e. non-symmetric, complex tensor.
Colors appear in this setting as a canonical book-keeping device and not as a
fundamental feature. In the large N limit, we exhibit a set of Virasoro
constraints satisfied by the free energy and an infinite family of
multicritical behaviors with entropy exponents \gamma_m=1-1/m.
... Although here we will only study the equations of motion, the additional non-propagating vector field that we introduce is a nonabelian analogue of the auxiliary field in [8] [9]. For recent work that also touch upon some of these issues see [10] [11] [12] [13] [14]. ...
... Although here we will only study the equations of motion, the additional non-propagating vector field that we introduce is a nonabelian analogue of the auxiliary field in [8, 9]. For recent work that also touch upon some of these issues see1011121314. We proceed by studying the closure of the supersymmetry algebra. ...
Using 3-algebras we obtain a nonabelian system of equations that furnish a representation of the (2,0)-supersymmetric tensor multiplet. The on-shell conditions are quite restrictive so that the system can be reduced to five-dimensional gauge theory along with six-dimensional abelian (2,0) tensor multiplets. We briefly discuss possible applications to D4-branes using a spacelike reduction and M5-branes using a null reduction. Comment: 17 pages, Latex; v2: Typos corrected and references added
We discuss the emergence of a Kalb-Ramond field and string charge in the lattice and consider the local bosonic model with rotor variables placed on the faces of a cubic lattice. We give the coupling model consisting of the Maxwell fields and the Kalb-Ramond field. This construction naturally incorporates the emerging coupling between both gauge and string fields. In the process, an object that resembles a D-brane on the lattice is introduced.
We provide a visual and intuitive introduction to effectively calculating in 2-groups along with explicit examples coming from non-abelian 1- and 2-form gauge theory. In particular, we utilize string diagrams, tools similar to tensor networks, to compute the parallel transport along a surface using approximations on a lattice. Although this work is mainly intended as expository, we prove a convergence theorem for the surface transport in the continuum limit. Locality is used to define infinitesimal parallel transport and two- dimensional algebra is used to derive finite versions along arbitrary surfaces with sufficient orientation data. The correct surface ordering is dictated by two-dimensional algebra and leads to an interesting diagrammatic picture for gauge fields interacting with particles and strings on a lattice. The surface ordering is inherently complicated, but we prove a simplification theorem confirming earlier results of Schreiber and Waldorf. Assuming little background, we present a simple way to understand some abstract concepts of higher category theory. In doing so, we review all the necessary categorical concepts from the tensor network point of view as well as many aspects of higher gauge theory.
We study counting invariant operators in general tensor models. We show that representation theory provides an efficient framework to count and classify invariants in tensor models of (gauge) symmetry Gd =U(N1)⊗···⊗U(Nd). In continuation of our earlier work, we present two natural ways of counting invariants, one for arbitrary Gd and another valid for large rank of Gd. We construct basis of invariant operators based on the counting, and compute correlators of their elements. The basis associated with finite rank of Gd diagonalizes two-point function. It is analogous to the restricted Schur basis used in matrix models. We show that the constructions get almost identical as we swap the Littlewood-Richardson numbers in multi-matrix models with Kronecker coefficients in general tensor models. We explore this striking parallel between matrix model and tensor model in depth from the perspective of representation theory and comment on several ideas for future investigation.
We formulate the theory of a 2-form gauge field on a Euclidean spacetime
lattice. In this approach, the fundamental degrees of freedom live on the faces
of the lattice, and the action can be constructed from the sum over Wilson
surfaces associated with each fundamental cube of the lattice. If we take the
gauge group to be $U(1)$, the theory reduces to the well-known abelian gerbe
theory in the continuum limit. We also propose a very simple and natural
non-abelian generalization with gauge group $U(N) \times U(N)$, which gives
rise to $U(N)$ Yang-Mills theory upon dimensional reduction. Formulating the
theory on a lattice has several other advantages. In particular, it is possible
to compute many observables, such as the expectation value of Wilson surfaces,
analytically at strong coupling and numerically for any value of the coupling.
We study the low-energy limit of a compactification of N = 4U(n) super Yang-Mills theory on S
1 with boundary conditions modified by an S-duality and R-symmetry twist. This theory has N = 6 supersymmetry in 2+1D. We analyze the T
2 compactification of this 2+1D theory by identifying a dual weakly coupled type-IIA background. The Hilbert space of normalizable
ground states is finite-dimensional and appears to exhibit a rich structure of sectors. We identify most of them with Hilbert
spaces of Chern-Simons theory (with appropriate gauge groups and levels). We also discuss a realization of a related twisted
compactification in terms of the (2, 0)-theory, where the recent solution by Gaiotto and Witten of the boundary conditions describing D3-branes ending on a (p, q) 5-brane plays a crucial role.
KeywordsDuality in Gauge Field Theories–String Duality–Brane Dynamics in Gauge Theories–Chern-Simons Theories
Random tensor models for a generic complex tensor generalize matrix models in
arbitrary dimensions and yield a theory of random geometries. They support a
1/N expansion dominated by graphs of spherical topology. Their Schwinger Dyson
equations, generalizing the loop equations of matrix models, translate into
constraints satisfied by the partition function. The constraints have been
shown, in the large N limit, to close a Lie algebra indexed by colored rooted
D-ary trees yielding a first generalization of the Virasoro algebra in
arbitrary dimensions. In this paper we complete the Schwinger Dyson equations
and the associated algebra at all orders in 1/N. The full algebra of
constraints is indexed by D-colored graphs, and the leading order D-ary tree
algebra is a Lie subalgebra of the full constraints algebra.
