Topological M-theory Research Articles

Mathematical structures of cohomological field theories

A mathematical framework of cohomological field theories (CohFTs) is formulated in the language of bigraded manifolds. Algebraic properties of operators in CohFTs are studied. Methods of constructing CohFTs, with or without gauge symmetries, are discussed. In particular, a generalization of the Mathai-Quillen formalism is given. Examples such as topological quantum mechanics, topological sigma model, topological M-theory, and topological Yang-Mills theory can be obtained uniformly using this new formalism.

Evidence for an algebra of G2 instantons

In this short note, we present some evidence towards the existence of an algebra of BPS G2 instantons. These are instantonic configurations that govern the partition functions of 7d SYM theories on local G2 holonomy manifolds mathcal{X} . To shed light on such structure, we begin investigating the relation with parent 4d mathcal{N} = 1 theories obtained by geometric engineering M-theory on mathcal{X} . The main point of this paper is to substantiate the following dream: the holomorphic sector of such theories on multi-centered Taub-NUT spaces gives rise to an algebra whose characters organise the G2 instanton partition function. As a first step towards this program we argue by string duality that a multitude of geometries mathcal{X} exist that are dual to well-known 4d SCFTs arising from D3 branes probes of CY cones: all these models are amenable to an analysis along the lines suggested by Dijkgraaf, Gukov, Neitzke and Vafa in the context of topological M-theory. Moreover, we discuss an interesting relation to Costello’s twisted M-theory, which arises at local patches, and is a key ingredient in identifying the relevant algebras.

General relativity and topological string duality through Penrose\u2013Ward transform

This paper discusses the relation between topological M-theory, self-dual Yang–Mills and general relativity. We construct a topological membrane field action from Witten’s cubic string field theory, which reduces to topological Yang–Mills on a one-parameter family of conifolds. It turns out that this can be interpreted as the twistor space of the four-dimensional Lagrangian submanifold M for large momenta. From the viewpoint of the target, we find that A-model and B-model on M unify in the topological membrane theory through the Penrose–Ward transform. The partition function is constructed and it is shown that, in the weak-coupling regime, it is equal to the partition function of Donaldson-Witten theory. Additionally, homological mirror symmetry, background independence as well as role of knot cobordisms as topological two-branes is discussed. It is outlined that all types of Floer homology are part of the topological membrane theory. Additionally, we find evidence that in the non-perturbative regime, the partition function of the membrane field action and that of the partially twisted (2,0) SU(N) superconformal field theory on the worldvolume of N topological fivebranes must coincide.

BTZ entropy from topological M-theory

By determining the relation between topological M-theory and the Chern-Simons actions for a gauge field constructed from the Lie algebra of either SL(2,R)x SL(2,R) or SL(2,C)x SL(2,C), depending on the sign of the space-time curvature, we show that the standard and exotic actions of 3-dimensional gravity can be recovered from topological M-theory. With this result, we provide a concrete realisation of a conjecture by Dijkgraaf et al. stating that the partition function of topological M-theory is equivalent to the partition function of a black hole in a related theory. We do this for the standard and exotic BTZ black holes in 3-dimensional gravity.

Self-dual formulation of gravity in topological M-theory

Inspired by the low wave-length limit of topological M-theory, which re-constructs the theory of 3 + 1D gravity in the self-dual variables’ formulation, and by the realization that in Loop Quantum Gravity (LQG) the holonomy of a flat connection can be non-trivial if and only if a non-trivial (space-like) line defect is localized inside the loop, we argue that non-trivial gravitational holonomies can be put in correspondence with space-like M-branes. This suggests the existence of a new duality, which we call [Formula: see text] duality, interconnecting topological M-theory with LQG. We spell some arguments to show that fundamental S-strings are serious candidates to be considered in order to instantiate this correspondence to classes of LQG states. In particular, we consider the case of the holonomy flowers in LQG, and show that for this type of states the action of the Hamiltonian constraint, from the M-theory side, corresponds to a linear combination of appearance and disappearance of a SNS1-strings. Consequently, these processes can be reinterpreted, respectively, as enucleations or decays into open or closed strings.

A new duality between topological M-theory and loop quantum gravity

Inspired by the long wave-length limit of topological M-theory, which re-constructs the theory of $3+1$D gravity in the self-dual variables' formulation, we conjecture the existence of a duality between Hilbert spaces, the ${\bf H}$-duality, to unify topological M-theory and loop quantum gravity (LQG). By ${\bf H}$-duality non-trivial gravitational holonomies of the kinematical Hilbert space of LQG correspond to space-like M-branes. The spinfoam approach captures the non-perturbative dynamics of space-like M-branes, and can be claimed to be dual to the S-branes foam. The Hamiltonian constraint dealt with in LQG is reinterpreted as a quantum superposition of SM-brane nucleations and decays.

Topological M-theory, self-dual gravity and the Immirzi parameter

Working in the first order formalism of gravity, we propose an action that combines the self and (anti)self-dual parts of the curvature and comprises all the diffeomorphism invariant Lagrangians that one can consider in this formalism. We use this action to derive the (2+1)-dimensional version of the Immirzi parameter. Our derivation relates explicitly the Immirzi parameter to the existence of two classically equivalent actions for the description of gravity in (2+1) dimensions, namely the standard and exotic actions introduced by Witten in the description of (2+1) gravity as a gauge theory. Furthermore, we identify the two decompositions of the seven dimensional manifold of topological M-theory that lead to three dimensional Chern–Simons actions which are the building blocks of several gravitational theories, including the standard and exotic actions, as well as the action for topologically massive AdS gravity. Finally, we motivate the three dimensional Immirzi ambiguity from the structure of topological M-theory. We conclude that this ambiguity originates in the S-duality of the theory, and can be geometrically interpreted as a choice of the volume of the 7 dimensional G2 manifold.

Relating field theories via stochastic quantization

This note aims to subsume several apparently unrelated models under a common framework. Several examples of well-known quantum field theories are listed which are connected via stochastic quantization. We highlight the fact that the quantization method used to obtain the quantum crystal is a discrete analog of stochastic quantization. This model is of interest for string theory, since the (classical) melting crystal corner is related to the topological A-model. We outline several ideas for interpreting the quantum crystal on the string theory side of the correspondence, exploring interpretations in the Wheeler–De Witt framework and in terms of a non-Lorentz invariant limit of topological M-theory.

A note on topological M5-branes and string-fivebrane duality

We derive the stability conditions for the M5-brane in topological M-theory using k-symmetry. The non-linearly self-dual 3-form on the world-volume is necessarily non-vanishing, as is the case also for the 2-form field strengths on coisotropic branes in topological string theory. It is demonstrated that the self-duality is consistent with the stability conditions, which are solved locally in terms of a tensor in the representation 6 of SU(3) subset of G(2). The double dimensional reduction of the M5-brane is the D4-brane, and its direct reduction is an NS5-brane. We show that the equation of motion for the 3-form on the NS5-brane wrapping a Calabi-Yau space is exactly the Kodaira-Spencer equation, providing support for a string-fivebrane duality in topological string theory.

G2 Hitchin functionals at one loop

We consider the quantization of the effective target space description of topological M-theory in terms of the Hitchin functional whose critical points describe seven manifolds with a G2 structure. The one-loop partition function for this theory is calculated and an extended version of it, that is related to the generalized G2 geometry, is compared with the topological G2 string. We relate the reduction of the effective action for the extended G2 theory to the Hitchin functional description of the topological string in six dimensions. The dependence of the partition functions on the choice of background G2 metric is also determined.

Topological Membranes

Introducing membranes as fundamental object in topological M-theory, we explore the global supersymmetry as well as the local κ-symmetry of the theory. Promoting the supersymmetry generators to be BRST operators, it is shown that the action can be chosen to be BRST exact. The presentation is given along the lines of [L. Bao, V. Bengtsson, M. Cederwall and B. E. W. Nilsson, JHEP 0601, 150 (2006) [arXiv:hep-th/0507077]].

Computing amplitudes in topological M-theory

We define a topological quantum membrane theory on a seven dimensional manifold of $G_2$ holonomy. We describe in detail the path integral evaluation for membrane geometries given by circle bundles over Riemann surfaces. We show that when the target space is $CY_3\times S^1$ quantum amplitudes of non-local observables of membranes wrapping the circle reduce to the A-model amplitudes. In particular for genus zero we show that our model computes the Gopakumar-Vafa invariants. Moreover, for membranes wrapping calibrated homology spheres in the $CY_3$, we find that the amplitudes of our model are related to Joyce invariants.

Membranes for topological M-theory

We formulate a theory of topological membranes on manifolds with G_2 holonomy. The BRST charges of the theories are the superspace Killing vectors (the generators of global supersymmetry) on the background with reduced holonomy G_2. In the absence of spinning formulations of supermembranes, the starting point is an N=2 target space supersymmetric membrane in seven euclidean dimensions. The reduction of the holonomy group implies a twisting of the rotations in the tangent bundle of the branes with ``R-symmetry'' rotations in the normal bundle, in contrast to the ordinary spinning formulation of topological strings, where twisting is performed with internal U(1) currents of the N=(2,2) superconformal algebra. The double dimensional reduction on a circle of the topological membrane gives the strings of the topological A-model (a by-product of this reduction is a Green-Schwarz formulation of topological strings). We conclude that the action is BRST-exact modulo topological terms and fermionic equations of motion. We discuss the role of topological membranes in topological M-theory and the relation of our work to recent work by Hitchin and by Dijkgraaf et al.

Topological membrane theory from Mathai-Quillen formalism

It is suggested that topological membranes play a fundamental role in the recently proposed topological M-theory. We formulate a topological theory of membranes wrapping associative three-cycles in a seven-dimensional target space with G_2 holonomy. The topological BRST rules and BRST invariant action are constructed via the Mathai-Quillen formalism. In a certain gauge we show this theory to be equivalent to a membrane theory with two BRST charges found by Beasley and Witten. We argue that at the quantum level an additional topological term should be included in the action, which measures the contributions of membrane instantons. We construct a set of local and non-local observables for the topological membrane theory. As the BRST cohomology of local operators turns out to be isomorphic to the de Rham cohomology of the G_2 manifold, our observables agree with the spectrum of d=4, N=1 G_2 compactifications of M-theory.

From current algebras for p-branes to topological M-theory

In this note we generalize a result by Alekseev and Strobl for the case of $p$-branes. We show that there is a relation between anomalous free current algebras and "isotropic" involutive subbundles of $T\oplus \wedge^p T^*$ with the Vinogradov bracket, that is a generalization of the Courant bracket. As an application of this construction we go through some interesting examples: topological strings on symplectic manifolds, topological membrane on $G_2$-manifolds and topological 3-brane on $Spin(7)$ manifolds. We show that these peculiar topological theories are related to the physical (i.e., Nambu-Goto) brane theories in a specific way. These topological brane theories are proposed as microscopic description of topological M/F-theories.

Topological M-theory from pure spinor formalism

We construct multiloop superparticle amplitudes in 11d using the pure spinor formalism. We explain how this construction emerges in the superparticle limit of the multiloop pure spinor superstring amplitudes prescription. We then argue that this construction points to some evidence for the existence of a topological M theory based on a relation between the ghost number of the full-fledged supersymmetric critical models and the dimension of the spacetime for topological models. In particular, we show that the extensions at higher orders of the previous results for the tree and one-loop level expansion for the superparticle in 11 dimensions is related to a topological model in 7 dimensions.

Topological M-theory as unification of form theories of gravity

We introduce a notion of topological M-theory and argue that it provides a unification of form theories of gravity in various dimensions. Its classical solutions involve G_2 holonomy metrics on 7-manifolds, obtained from a topological action for a 3-form gauge field introduced by Hitchin. We show that by reductions of this 7-dimensional theory, one can classically obtain 6-dimensional topological A and B models, the self-dual sector of loop quantum gravity in four dimensions, and Chern–Simons gravity in 3 dimensions. We also find that the 7-dimensional M-theory perspective sheds some light on the fact that the topological string partition function is a wavefunction, as well as on S-duality between the A and B models. The degrees of freedom of the A and B models appear as conjugate variables in the 7-dimensional theory. Finally, from the topological M-theory perspective, we find hints of an intriguing holographic link between non-supersymmetric Yang–Mills in four dimensions and A model topological strings on twistor space.