Narrow Results

We review the AKSZ construction as applied to the topological open membranes and Poisson sigma models. We describe a generalization to open topological p-branes. Also, we propose a related (not necessarily BV) Nambu–Poisson sigma model.

We discuss the asymptotic properties of quantum states density for fundamental p-branes which can yield a microscopic interpretation of the thermodynamic quantities in M-theory. The matching of the BPS part of spectrum for superstring and supermembrane gives the possibility of getting membrane's results via string calculations. In the weak coupling limit of M-theory, the critical behavior coincides with the first-order phase transition in the standard string theory at temperature less than the Hagedorn's temperature T_H. The critical temperature at large coupling constant is computed by considering M-theory on manifold with topology . Alternatively we argue that any finite temperature can be introduced in the framework of membrane thermodynamics.

We show that the Witten covariant phase space for p-branes with thickness in an arbitrary background is endowed of a symplectic potential, which although is not important to the dynamics of the system, plays a relevant role on the phase space, allowing us to generate a symplectic structure for the theory and therefore give a covariant description of canonical formalism for quantization.

The cosmological constant on a D-brane is analyzed. This D-brane is in the background produced by the p-brane solutions. The energy–momentum tensor in this model has been found and the form of the cosmological constant has been derived. This energy–momentum tensor is interpreted as an energy–momentum tensor for a perfect fluid on the D-brane. The energy density and the pressure for this fluid have been derived. As it turned out the pressure is negative but the speed of sound is real.

We discuss different aspects of the (2+2)-signature from the point of view of the quatl theory. In particular, we compare two alternative approaches to such a spacetime signature, namely the (1+1)-matrix-brane and the (2+2)-target spacetime of a string. This analysis also reveals hidden discrete symmetries of the (2+2)-brane action associated with the (2+2)-dimensional sector of a (2+10)-dimensional target background.

In this paper we show the equivalence of various (nonthreshold) bound state solutions of branes, or equivalently branes in background potentials, in ten- and eleven-dimensional supergravity. We compare solutions obtained in two very different ways. One method uses a zero mode analysis to make an ansatz which makes it possible to solve the full nonlinear supergravity equations. The other method utilizes T duality techniques to turn on the fields on the brane. To be specific, in eleven dimensions we show the equivalence for the (M2,M5) bound state, or equivalently an M5-brane in a C₍₃₎ field, where we also consider the (MW,M2,M2′,M5) solution, which can be obtained from the (M2,M5) bound state by a boost. In ten dimensions we show the equivalence for the ((F,D1),D3) bound states as well as the bound states of (p,q) five-branes with lower dimensional branes in type IIB, corresponding to D3-branes in B₍₂₎ and C₍₂₎ fields and (p,q) five-branes in B₍₂₎, C₍₂₎ and C₍₄₎ fields. We also comment on the recently proposed V duality related to infinitesimally boosted solutions.

It is shown how actions corresponding to antisymmetric non-Abelian tensorial gauge field theories of (p+1)-dimensional diffeomorphisms yield p-brane actions associated with their (p+1)-dimensional worldvolume evolution. We conclude with a discussion of how to obtain p-brane actions from the large N limit of covariant matrix models based on generalized hypermatrices. A deformation quantization of Nambu classical mechanics furnishing Nambu quantum mechanics by constructing the n-ary noncommutative product of n functions f₁ •f₂ • ⋯ •f_n, the n-ary version of the Moyal bracket, and the analog of the Weyl–Wigner–Groenowold–Moyal map among operators and c-functions remains an open problem. A solution to this problem will reveal important relations between the physics of p-branes and matrix models based on generalized hypermatrices in the large N limit.

In this paper, we examine the emergence of conserved charges on the horizon of a particular class of extremal nondilatonic black $p$-branes (which reduce to extremal dilatonic black holes in $D=4$ dimensions upon toroidal compactification) in the presence of a probe massless scalar field in the bulk. This result is achieved by writing the black $p$-brane geometry in a Gaussian null coordinate system which allows us to get a nonsingular near-horizon geometry description. We find that the near-horizon geometry is $Ad{S}_{p+2}\times S^2$ and that the $Ad{S}_{p+2}$ section has an internal structure which can be seen as a warped product of $Ad{S}_2\times S^p$ in Gaussian null coordinates. We show that the bulk scalar field satisfying the field equations is expanded in terms of nonnormalizable and normalizable modes, which for certain suitable quantization conditions are well-behaved at the boundary of $Ad{S}_{p+2}$ space. Furthermore, we show that picking the normalizable modes results in the existence of conserved quantities on the horizon. We discuss the impact of these conserved quantities in the late-time regime.

Recently Nieto has proposed a link between oriented matroid theory and the Schild type action of p-branes. This particular matroid theory satisfies the local condition, i.e., the degenerate form must be closed. This allows us to explain the dynamics of p-branes in terms of Nambu–Poisson structure. In this paper using an infinitesimal canonical transformation of Nambu brackets we show that the helicity is conserved in the dynamics of p-branes. Applying Filippov algebra (or quantum Nambu bracket) we define a generalized Yang–Mills action in 4k space. We show that this action is equivalent to Dolan–Tchrakian type action.

We formalize higher-dimensional and higher gauge WZW-type sigma-model local prequantum field theory, and discuss its rationalized/perturbative description in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language used in the supergravity literature). We show generally how the intersection laws for such higher WZW-type σ-model branes (open brane ending on background brane) are encoded precisely in (super-)L_∞-extension theory and how the resulting "extended (super-)space-times" formalize spacetimes containing σ-model brane condensates. As an application we prove in Lie n-algebra homotopy theory that the complete super-p-brane spectrum of superstring/M-theory is realized this way, including the pure σ-model branes (the "old brane scan") but also the branes with tensor multiplet worldvolume fields, notably the D-branes and the M5-brane. For instance the degree-0 piece of the higher symmetry algebra of 11-dimensional (11D) spacetime with an M2-brane condensate turns out to be the "M-theory super-Lie algebra". We also observe that in this formulation there is a simple formal proof of the fact that type IIA spacetime with a D0-brane condensate is the 11D sugra/M-theory spacetime, and of (prequantum) S-duality for type IIB string theory. Finally we give the non-perturbative description of all this by higher WZW-type σ-models on higher super-orbispaces with higher WZW terms in stacky differential cohomology.

The AGT-W relation is a conjecture of the nontrivial duality between 4-dim quiver gauge theory and 2-dim conformal field theory. We verify a part of this conjecture for all the cases of quiver gauge groups by studying on the property of 3-point correlation function of conformal theory. We also mention the relation to algebra as one of the promising direction towards the proof of the remaining part.

The following sections are included:

• Introduction

• The Leblanc-Harms WKB Theory

• Inconsistent String and p-Brane Quantization

• Acknowledgments

• References