The problem of quantizing a symplectic manifold (M, ω) can be formulated in terms of the A-model of a complexification of M .This leads to an interesting new perspective on quantization.From this point of view, the Hilbert space obtained by quantization of (M, ω) is the space of (B cc , B ′ ) strings, where B cc and B ′ are two A-branes; B ′ is an ordinary Lagrangian A-brane, and B cc is a space-filling coisotropic A-brane.B ′ is supported on M , and the choice of ω is encoded in the choice of B cc .As an example, we describe from this point of view the representations of the group SL(2, R).Another application is to Chern-Simons gauge theory.
Inorder tostudy vertex operatorsfortheType IIBsuperstring onAdS space, we derive supersymmetric constraint equations for the vertex operators in AdS3S 3 backgrounds with Ramond-Ramond flux, using Berkovits-Vafa-Witten variables. These constraints are solved to compute the vertex operators and show that they satisfy the linearizedD=6,N=(2;0) equations of motion for a super- gravity and tensor multiplet expanded around theAdS3S 3 spacetime.
These are notes on the theory of supermanifolds and integration on them, aiming to collect results that are useful for a better understanding of superstring perturbation theory in the RNS formalism.
In these notes, I will sketch a new approach to Khovanov homology of knots and links based on counting the solutions of certain elliptic partial differential equations in four and five dimensions. The equations are formulated on four and five-dimensional manifolds with boundary, with a rather subtle boundary condition that encodes the knots and links. The construction is formally analogous to Floer and Donaldson theory in three and four dimensions. It was discovered using quantum field theory arguments but can be described and understood purely in terms of classical gauge theory. (Based on a lecture at the conference Low-Dimensional Manifolds and High-Dimensional Categories, University of California at Berkeley, June 2011).
In a modern understanding of particle physics, global symmetries are approximate and gauge symmetries may be emergent. This view, which has echoes in condensed-matter physics, is supported by a variety of arguments from experiment and theory.
This paper is devoted to a systematic discussion of the supersymmetric index Tr (-1)^F for the minimal supersymmetric Yang-Mills theory -- with any simple gauge group G -- primarily in four spacetime dimensions. The index has refinements that probe confinement and oblique confinement and the possible spontaneous breaking of chiral symmetry and of global symmetries, such as charge conjugation, that are derived from outer automorphisms of the gauge group. Predictions for the index and its refinements are obtained on the basis of standard hypotheses about the infrared behavior of gauge theories. The predictions are confirmed via microscopic calculations which involve a Born-Oppenheimer computation of the spectrum as well as mathematical formulas involving triples of commuting elements of G and the Chern-Simons invariants of flat bundles on the three-torus.
This article consists of a very short introduction to classical and quantum information theory. Basic properties of the classical Shannon entropy and the quantum von Neumann entropy are described, along with related concepts such as classical and quantum relative entropy, conditional entropy, and mutual information. A few more detailed topics are considered in the quantum case.
Caron-Huot has recently given an interesting formula that determines OPE data in a conformal field theory in terms of a weighted integral of the four-point function over a Lorentzian region of cross-ratio space. We give a new derivation of this formula based on Wick rotation in spacetime rather than cross-ratio space. The derivation is simple in two dimensions but more involved in higher dimensions. We also derive a Lorentzian inversion formula in one dimension that sheds light on previous observations about the chaos regime in the SYK model.
We discuss some aspects of symmetry breakings in string theories. We mainly focus on the algebra of bosonic strings, but suspect similar ideas may be used for superstring theories. We point out a curious relation between the partition function of certain bosonic string theories and superstring theories.
Thearticleisdevoted toaquantumeld theoryexplanation oftherelationship between theVerlindealgebra ofthegroup U(k)atlevel N k and the\quantum cohom ology oftheGrassm annian ofcom plex k planesin N space.In x2,Iexplain therelation between theVerlindealgebra and thegauged W ZW m odelof G=G;in x3,Idescribe the quantum cohom ology and itsorigin in a quantumeld theory; and in x4,Ipresenta path integralargum entform apping between them .
The vanishing of the cosmological constant and absence of a massless dilaton might be explained by a duality between a supersymmetric string vacuum in three dimensions and a non-supersymmetric string vacuum in four dimensions.
A bstract We propose an algebra of operators along an observer’s worldline as a background-independent algebra in quantum gravity. In that context, it is natural to think of the Hartle-Hawking no boundary state as a universal state of maximum entropy, and to define entropy in terms of the relative entropy with this state. In the case that the only spacetimes considered correspond to de Sitter vacua with different values of the cosmological constant, this definition leads to sensible results.
Yang-Mills theory in four dimensions formally admits an exact Chern-Simons wavefunction. It is an eigenfunction of the quantum Hamiltonian with zero energy. It is known to be unphysical for a variety of reasons, but it is still interesting to understand what it describes. We show that in expanding around this state, positive helicity gauge bosons have positive energy and negative helicity ones have negative energy. Some of the negative energy states would have negative norm. We also show that the Chern-Simons state is the supersymmetric partner of the naive fermion vacuum in which one does not fill the fermi sea. Finally, we give a sort of explanation of ``why'' this state exists. Similar properties can be expected for the analogous Kodama wavefunction of gravity.
