Geometric phases in quantum control disturbed by classical stochastic processes

We describe the geometric (Berry) phases arising when some quantum systems are driven by control classical parameters but also by outer classical stochastic processes (as for example classical noises). The total geometric phase is then divided into an usual geometric phase associated with the control parameters and a second geometric phase associated with the stochastic processes. The geometric structure in which these geometric phases take place is a composite bundle (and not an usual principal bundle), which is explicitely built in this paper. We explain why the composite bundle structure is the more natural framework to study this problem. Finally we treat a very simple example of a two level atom driven by a phase modulated laser field with a phase instability described by a gaussian white noise. In particular we compute the average geometric phase issued from the noise

Cyclic Foam Topological Field Theories

This paper proposes an axiomatic for Cyclic Foam Topological Field theories. That is Topological Field theories, corresponding to String theories, where particles are arbitrary graphs. World surfaces in this case are two-manifolds with one-dimensional singularities. We proved that Cyclic Foam Topological Field theories one-to-one correspond to graph-Cardy-Frobenius algebras, that are families $(A,B_\star,\phi)$, where $A=\{A^s|s\in S\}$ are families of commutative associative Frobenius algebras, $B_\star = \bigoplus_{\sigma\in\Sigma} B_\sigma$ is an graduated by graphes, associative algebras of Frobenius type and $\phi=\{\phi_\sigma^s: A^s\to (B_\sigma)|s\in S,\sigma\in \Sigma\}$ is a family of special representations. There are constructed examples of Cyclic Foam Topological Field theories and its graph-Cardy-Frobenius algebrasComment: 14 page

The sl_3 web algebra

In this paper we use Kuperberg’s sl3-webs and Khovanov’s sl3-foams to define a new algebra KS, which we call the sl3-web algebra. It is the sl3 analogue of Khovanov’s arc algebra. We prove that KS is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of q-skew Howe duality, which allows us to prove that KS is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group K0 (WS )Q(q) , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that KS is a graded cellular algebra.info:eu-repo/semantics/publishedVersio

Fivebranes and 4-manifolds

We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d N=(0,2) theories, we obtain a number of results, which include new 3d N=2 theories T[M_3] associated with rational homology spheres and new results for Vafa-Witten partition functions on 4-manifolds. In particular, we point out that the gluing measure for the latter is precisely the superconformal index of 2d (0,2) vector multiplet and relate the basic building blocks with coset branching functions. We also offer a new look at the fusion of defect lines / walls, and a physical interpretation of the 4d and 3d Kirby calculus as dualities of 2d N=(0,2) theories and 3d N=2 theories, respectivelyComment: 81 pages, 18 figures. v2: misprints corrected, clarifications and references added. v3: additions and corrections about lens space theory, 4-manifold gluing, smooth structure

Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams

We show how Feynman amplitudes of standard QFT on flat and homogeneous space can naturally be recast as the evaluation of observables for a specific spin foam model, which provides dynamics for the background geometry. We identify the symmetries of this Feynman graph spin foam model and give the gauge-fixing prescriptions. We also show that the gauge-fixed partition function is invariant under Pachner moves of the triangulation, and thus defines an invariant of four-dimensional manifolds. Finally, we investigate the algebraic structure of the model, and discuss its relation with a quantization of 4d gravity in the limit where the Newton constant goes to zero.Comment: 28 pages (RevTeX4), 7 figures, references adde

A Class of Topological Actions

We review definitions of generalized parallel transports in terms of Cheeger-Simons differential characters. Integration formulae are given in terms of Deligne-Beilinson cohomology classes. These representations of parallel transport can be extended to situations involving distributions as is appropriate in the context of quantized fields.Comment: 41 pages, no figure

Super-A-polynomials for Twist Knots

We conjecture formulae of the colored superpolynomials for a class of twist knots $K_p$ where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the formulae, we compute both the classical and quantum super-A-polynomial for the twist knots with small values of p. The results support the categorified versions of the generalized volume conjecture and the quantum volume conjecture. Furthermore, we obtain the evidence that the Q-deformed A-polynomials can be identified with the augmentation polynomials of knot contact homology in the case of the twist knots.Comment: 22+16 pages, 16 tables and 5 figures; with a Maple program by Xinyu Sun and a Mathematica notebook in the ancillary files linked on the right; v2 change in appendix B, typos corrected and references added; v3 change in section 3.3; v4 corrections in Ooguri-Vafa polynomials and quantum super-A-polynomials for 7_2 and 8_1 are adde

An Invitation to Higher Gauge Theory

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincar\'e 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a 'tangent 2-group', which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an 'inner automorphism 2-group', which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an 'automorphism 2-group', which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a 'string 2-group'. We also touch upon higher structures such as the 'gravity 3-group' and the Lie 3-superalgebra that governs 11-dimensional supergravity.Comment: 60 pages, based on lectures at the 2nd School and Workshop on Quantum Gravity and Quantum Geometry at the 2009 Corfu Summer Institut

Geometric phases in adiabatic Floquet theory, abelian gerbes and Cheon's anholonomy

We study the geometric phase phenomenon in the context of the adiabatic Floquet theory (the so-called the $(t,t')$ Floquet theory). A double integration appears in the geometric phase formula because of the presence of two time variables within the theory. We show that the geometric phases are then identified with horizontal lifts of surfaces in an abelian gerbe with connection, rather than with horizontal lifts of curves in an abelian principal bundle. This higher degree in the geometric phase gauge theory is related to the appearance of changes in the Floquet blocks at the transitions between two local charts of the parameter manifold. We present the physical example of a kicked two-level system where these changes are involved via a Cheon's anholonomy. In this context, the analogy between the usual geometric phase theory and the classical field theory also provides an analogy with the classical string theory.Comment: This new version presents a more complete geometric structure which is topologically non trivia