We present a simple compact formula for a topologically nontrivial map $S^7 \to Spin(7)$ associated with the fiber bundle $Spin(7) \stackrel{G_2}{\to} S^7$. The homotopy group $π_7[Spin(7)] = \mathbb{Z}$ brings about the topologically nontrivial 8-dimensional gauge field configurations that belong to the algebra $spin(7)$. The instantons are special such configurations that minimize the functional $\int {\rm Tr} \{F\wedge F \wedge \star(F \wedge F)\} $ and satisfy non-linear self-duality conditions, $ F \wedge F \ =\ \pm \star (F\wedge F)$.
$Spin(7) \subset SO(8)$, and $Spin(7)$ instantons represent simultaneously $SO(8)$ instantons of a new type. The relevant homotopy is $π_7[SO(8)] = \mathbb{Z} \times \mathbb{Z}$, which implies the existence of two different topological charges. This also holds for all groups $SO(4n)$ with integer $n$. We present explicit expressions for two topological charges and calculate their values for the conventional 4-dimensional and 8-dimensional instantons and also for the 8-dimensional instantons of the new type.
Similar constructions for other algebras in different dimensions are briefly discussed.
We consider an eight-dimensional local octonionic theory with the seven-sphere playing the role of the gauge group. Duality conditions for two- and four-forms in eight dimensions are related. Dual fields--octonionic instantons--solve an 8D generalization of the Yang-Mills equation. Modifying the ADHM construction of 4D instantons, we find general $k$-instanton 8D solutions which depends on $16k-7$ effective parameters.
We consider a gauge theory in which a nonassociative Moufang loop takes the place of a structure group. We construct Belavin-Polyakov-Schwartz-Tyupkin (BPST) and t'Hooft like instanton solutions of the gauge theory in seven and eight dimensions.
In this letter, we study the instanton moduli space of the eight-dimensional solutions of the self-duality equation $F\wedge F= \ast F\wedge F$. Using the known ADHM-construction of such instantons, we compute the dimension of the space of solutions to this construction. We find a general $N$-instanton solution of the 't Hooft type which depends on $8N+1$ free parameters and investigate the cases $N=2$ and $N=3$ in detail.
We construct some classes of instanton solutions of eight dimensional noncommutative ADHM equations generalizing the solutions of eight dimensional commutative ADHM equations found by Papadopoulos and Teschendorff, and interpret them as supersymmetric $D0$-$D8$ bound states in a NS $B$-field. Especially, we consider the $D0$-$D8$ system with anti-self-dual $B$-field preserving 3/16 of supercharges. This system and self-duality conditions are related with the group $Sp(2)$ which is a subgroup of the eight dimensional rotation group SO(8).
In this paper, we study the (anti-)self-duality equations $\ast F\wedge F=\pm F\wedge F$ in the eight-dimensional Euclidean space. Using properties of the Clifford algebra $Cl_{0,8}(\mathbb{R})$, we find a new solution to these equations.
We search for an abelian description of the Yang-Mills instantons on certain eight dimensional manifolds with the special holonomies $Spin(7)$ and SU(4). By mimicing the Seiberg-Witten theory in four dimensions, we propose a set of monopole-like equations governing the 8-dimensional U(1) connections and spinors, which are supposed to be the dual theory of the nonabelian instantons. We also give a naive test of the generalized $S$-duality in the abelian sector of 8-dimensional Yang-Mills theory. Some problems in this approach are pointed out.
In this paper, we study the 't Hooft type instantons in eight dimensions, which satisfy the (anti)self-dual equations $F\wedge F=\pm\ast_8F\wedge F$. Using various designs of such instantons, we find new soliton solutions to the low-energy effective theory of the heterotic fivebrane. We investigate conditions under which these instanton configurations can be identified with the $D$-instantons embedded in the $D7$-brane world volume. Finally, we discuss the relationship between eight-dimensional periodic instantons and monopoles.
We classify supersymmetric D0-Dp bound states with a non-zero B-field by considering T-dualities of intersecting branes at angles. Especially, we find that the D0-D8 system with the B-field preserves 1/16, 1/8 and 3/16 of supercharges if the B-field satisfies the ``(anti-)self-dual'' condition in dimension eight. The D0-branes in this system are described by eight dimensional instantons on non-commutative R^8. We also discuss the extended ADHM construction of the eight-dimensional instantons and its deformation by the B-field. The modified ADHM equations admit a sort of the `fuzzy sphere' (emb…
We study the noncommutative version of the extended ADHM construction in the eight dimensional U(1) Yang-Mills theory. This construction gives rise to the solutions of the BPS equations in the Yang-Mills theory, and these solutions preserve at least 3/16 of supersymmetries. In a wide subspace of the extended ADHM data, we show that the integer $k$ which appears in the extended ADHM construction should be interpreted as the $D4$-brane charge rather than the $D0$-brane charge by explicitly calculating the topological charges in the case that the noncommutativity parameter is anti-self-dual. We a…
We show that the eight-dimensional instanton solution, which satisfies the self-duality equation $F \wedge F = *_8 F \wedge F$, realizes the static Skyrmion configuration in eight dimensions through the Atiyah-Manton construction. The relevant energy functional of the Skyrme field is obtained by the formalism developed by Sutcliffe. By comparing the Skyrmion olution associated with the extreme of the energy, with the Atiyah-Manton solution constructed by the instantons, we find that they agree with high accuracy. This is a higher-dimensional analogue of the Atiyah-Manton construction of Skyrmi…
We construct eight-dimensional gravitational instantons by solving appropriate self-duality equations for the spin-connection. The particular gravitational instanton we present has $Spin(7)$ holonomy and, in a sense, it is the eight-dimensional analog of the Eguchi-Hanson 4D space. It has a removable bolt singularity which is topologically S^4 and its boundary at infinity is the squashed S^7. We also lift our solutions to ten and eleven dimensions and construct fundamental string and membrane configurations that preserve 1/16 of the original supersymmetries.
Multi-instanton solutions in the eight and seven dimensional Yang-Mills fields theory is obtained. Extended-soliton solutions to the low-energy heterotic-field-theory equations of motion is constructed from this higher-dimensional multi-instantons.
We study the ADHM construction of (anti-)self-dual instantons in eight dimensions. We propose the general scheme to construct the (anti-)self-dual gauge field configurations $F \wedge F = \pm *_8 F \wedge F$ whose finite topological charges are given by the fourth Chern number. We show that our construction reproduces the known $\text{SO}(8)$ one-instanton solution. We also construct multi-instanton solutions of 't Hooft and Jackiw-Nohl-Rebbi (JNR) types in the dilute instanton gas approximation. The well-separated configurations of multi-instantons reproduce the correct topological charges wi…
We construct octonionic multi-instantons for the eight and seven dimensional Yang-Mills theory. Extended soliton solutions to the low-energy heterotic field theory equations of motion are constructed from these octonionic multi-instantons. The solitons describe a string in ten-dimensional Minkowski space, and preserve one and two of the sixteen space-time supersymmetries correspondingly.
We obtain the exact non-perturbative thresholds of $R^4$ terms in IIB string theory compactified to eight and seven dimensions. These thresholds are given by the perturbative tree-level and one-loop results together with the contribution of the D-instantons and of the (p,q)-string instantons. The invariance under U-duality is made manifest by rewriting the sum as a non-holomorphic modular function of the corresponding discrete U-duality group. In the eight-dimensional case, the threshold is the sum of a order-1 Eisenstein series for SL(2,Z) and a order-3/2 Eisenstein series for SL(3,Z). The se…
We show that $(1+2)$ nonlinear Klein-Gordon equation with negative coupling admits an exact solution which appears to be the linear superposition of the plane wave and the nonsingular rational soliton. We show that the same approach allows to construct the solution of similar properties for the Euclidean $ϕ^4$ model with broken symmetry. Interestingly, this regular solution will be of instanton type only in the $D\le 5$ Euclidean space.
We discuss higher-dimensional gravitational instantons by studying appropriate self-duality equations for the spin connection. In seven and in eight dimensions, the corresponding spaces admit a covariantly constant spinor and have holonomies in G_2 and Spin(7), respectively. We find a non-compact solution to the self-duality equations in eight dimensions in which the self-dual space has an elliptically-fibered structure.
Using spinorial geometry techniques, we classify the supersymmetric solutions of euclidean ${\cal N}=4$ super Yang-Mills theory. These backgrounds represent generalizations of instantons with nontrivial scalar fields turned on, and satisfy some constraints that bear a similarity with the Hitchin equations, and contain the Donaldson equations as a special subcase. It turns out that these constraints can be obtained by dimensional reduction of the octonionic instanton equations, and may be rephrased in terms of a selfduality-like condition for a complex connection. We also show that the supersym…
We compute the principal contribution to the index in the supersymmetric quantum mechanical systems which are obtained by reduction to 0+1 dimensions of $\mathcal{N}=1$, $D=4,6,10$ super-Yang-Mills theories with gauge group SU(N). The results are: ${1\over{N^{2}}}$ for $D=4,6$, $\sum_{d | N} {1\over{d^{2}}}$ for D=10. We also discuss the D=3 case.
We study ${\cal F}^4$-threshold corrections in an eight dimensional S-dual pair of string theories, as a prototype of dual string vacua with sixteen supercharges. We show that the orbifold CFT description of D-string instantons gives rise to a perturbative expansion similar to the one appearing on the fundamental string side. By an explicit calculation, using the Nambu-Goto action in the static gauge, we show that the first subleading term agrees precisely on the two sides. We then give a general argument to show that the agreement extends to all orders.
M-theoretic construction of N=2 gauge theories implies that the instanton partition function is expressed as the scalar product of coherent states (Whittaker states) in the Verma module of an appropriate two dimensional conformal field theory. We present the characterizing conditions for such states that give the partition function with fundamental hypermultiplets for SU(3) theory and SU(2) theory with a surface operator. We find the states are no longer the coherent states in the strict sense but we can characterize them in terms of a few annihilation operators of lower levels combined with t…
We study a class of generalized self-duality relations in gauge theories on the complex projective space with the Fubini-Study metric. Our theories consist of only gauge fields with gauge group U(n). The pseudo-energies which we consider contain higher orders of field strength and are labeled by an integer p smaller than or equal to [n/2]. For making the Bogomol'nyi completion we need non-single trace terms in the pseudo-energies, unlike the models defined on spheres, which were studied previously. We construct an explicit solution of codimension 2n to generalized self-duality equations as Bog…
We interpret a class of 4k-dimensional instanton solutions found by Ward, Corrigan, Goddard and Kent as four-dimensional instantons at angles. The superposition of each pair of four-dimensional instantons is associated with four angles which depend on some of the ADHM parameters. All these solutions are associated with the group $Sp(k)$ and are examples of Hermitian-Einstein connections on $\bE^{4k}$. We show that the eight-dimensional solutions preserve 3/16 of the ten-dimensional N=1 supersymmetry. We argue that under the correspondence between the BPS states of Yang-Mills theory and those o…
We present a unified eight-dimensional approach to instanton equations on several seven-dimensional manifolds associated to a six-dimensional homogeneous nearly Kähler manifold. The cone over the sine-cone on a nearly Kähler manifold has holonomy group Spin(7) and can be foliated by submanifolds with either holonomy group G_2, a nearly parallel G_2-structure or a cocalibrated G_2-structure. We show that there is a G_2-instanton on each of these seven-dimensional manifolds which gives rise to a Spin(7)-instanton in eight dimensions. The well-known octonionic instantons on R^7 and R^8 are contai…
A class of generalized Taub-NUT gravitational instantons is reported in five - dimensional Einstein gravity coupled to a non-linear sigma model. The geodesic dynamics of a spinless particle of unit mass on these static gravitational instantons is studied. This is accomplished by finding a generalized Runge-Lenz vector. Unlike the Kepler problem, or, the dynamics of a spinless particle on the familiar Taub-NUT gravitational instantons, the orbits are not conic sections.
The total space of the spinor bundle on the four dimensional sphere S^4 is a quaternionic line bundle that admits a metric of Spin(7) holonomy. We consider octonionic Yang-Mills instanton on this eight dimensional gravitational instanton. This is a higher dimensional generalization of (anti-)self-dual instanton on the Eguchi-Hanson space. We propose an ansatz for Spin(7) Yang-Mills field and derive a system of non-linear ordinary differential equations. The solutions are classified according to the asymptotic behavior at infinity. We give a complete solution, when the gauge group is reduced to…
When eight-dimensional instantons, satisfying F \wedge F = \pm \star_8 (F\wedge F), shrink to zero size, we find stringy objects in higher order ten-dimensional Yang-Mills (viewed as a low-energy limit of open string theory). The associated F^4 action is a combination of two independent parts having a single-trace and a double-trace structure. As a result we get a D-string from the single-trace term and a fundamental string from the double-trace. The latter has (8,0) supersymmetry on the worldsheet and couplings to the background gauge fields of a heterotic string. A correlation between the co…
We study the 1/4 BPS equations in the eight dimensional noncommutative Yang-Mills theory found by Bak, Lee and Park. We explicitly construct some solutions of the 1/4 BPS equations using the noncommutative version of the ADHM-like construction in eight dimensions. From the calculation of topological charges, we show that our solutions can be interpreted as the bound states of the $D0$-$D4$-$D8$ with a $B$-field. We also discuss the structure of the moduli space of the 1/4 BPS solutions and determine the metric of the moduli space of the U(2) one-instanton in four and eight dimensions.
We study the superconformal index of five-dimensional SCFTs and the sphere partition function of four-dimensional gauge theories with eight supercharges in the presence of co-dimension two half-BPS defects. We derive a prescription which is valid for defects which can be given a "vortex construction", i.e. can be defined by RG flow from vortex configurations in a larger theory. We test the prescription against known results and expected dualities. We employ our prescription to develop a general computational strategy for defects defined by coupling the bulk degrees of freedom to a Gauged Linea…
We study the non-perturbative corrections generated by exotic instantons in U(N) gauge theories in eight and four dimensions. As it was shown previously, the eight-dimensional prepotential can be resummed using a plethystic formula showing only a dependence from the center of mass and from a U(1) gauge factor. On the contrary, chiral correlators in eight and four dimensions display a non-trivial dependence from the full gauge group. Furthermore the resolvent, the generating function for the eight and four dimensional correlators, can be written in a compact form both in the eight and four dime…
We construct nearly topological Yang-Mills theories on eight dimensional manifolds with a special holonomy group. These manifolds are the Joyce manifold with $Spin(7)$ holonomy and the Calabi-Yau manifold with SU(4) holonomy. An invariant closed four form $T_{μνρσ}$ on the manifold allows us to define an analogue of the instanton equation, which serves as a topological gauge fixing condition in BRST formalism. The model on the Joyce manifold is related to the eight dimensional supersymmetric Yang-Mills theory. Topological dimensional reduction to four dimensions gives non-abelian Seiberg-Witte…
We identify exact gauge-instanton-like solutions to (super)-string theory using the method of dimensional reduction. We find in particular the Polyakov instanton of 3d QED, and a class of generalized Yang-Mills merons. We discuss their marginal deformations, and show that for the $3d$ instanton they correspond to a dissociation of vector- and axial-magnetic charges.
We calculate multi-instanton effects in a three-dimensional gauge theory with N=8 supersymmetry and gauge group SU(2). The k-instanton contribution to an eight-fermion correlator is found to be proportional to the Gauss-Bonnet-Chern integral of the Gaussian curvature over the centered moduli-space of charge-k BPS monopoles, \tilde{M}_{k}. For k=2 the integral can be evaluated using the explicit metric on \tilde{M}_{2} found by Atiyah and Hitchin. In this case the integral is equal to the Euler character of the manifold. More generally the integral is the volume contribution to the index of the…
Generalized Weierstrass formulae for surfaces in four-dimensional space $\Bbb{R}^{4}$ are used to study (anti)self-dual rigid string configurations. It is shown that such configurations are given by superminimal immersions into $\Bbb{R}^{4}$. Explicit formulae for generic (anti)instantons are presented. Particular classes of surfaces are also analyzed.
Using the idea of a generalized Kaehler structure, which is a pair of commuting generalized complex structures, we construct bihermitian metrics on the projective plane and the product of two projective lines, and show that any such structure on a compact 4-manifold M defines one on the moduli space of anti-self-dual connections on a fixed principal bundle over M. We highlight the role of holomorphic Poisson structures in all these constructions.
We describe some recent investigation about the structure of generic D=4,5 theories obtained by generalized dimensional reduction of D=5,6 theories with eight supercharges. We relate the Scherk-Schwarz reduction to a special class of N=2 no-scale gauged supergravities.
We study the problem of string propagation in a general instanton background for the case of the complete heterotic superstring. We define the concept of generalized HyperK\" ahler manifolds and we relate it to (4,4) superconformal theories. We propose a generalized h-map construction that predicts a universal SU(6) symmetry for the modes of the string excitations moving in an instanton background. We also discuss the role of abstract $N$=4 moduli and, applying it to the particular limit case of the solvable SU(2) X R instanton found by Callan et al. we show that it admits deformations and cor…
Generalized Bose operators correspond to reducible representations of the harmonic oscillator algebra. We demonstrate their relevance in the construction of topologically non-trivial solutions in noncommutative gauge theories, focusing our attention to flux tubes, vortices, and instantons. Our method provides a simple new relation between the topological charge and the number of times the basic irreducible representation occurs in the reducible representation underlying the generalized Bose operator. When used in conjunction with the noncommutative ADHM construction, we find that these new ins…
We consider generalized self-duality equations for U(2r) Yang-Mills theory on R^{4k} with quaternionic structure and self-dual Moyal deformation. We employ the extended ADHM method in 4k dimensions to construct new noncommutative generalizations of the 't Hooft as well as of the BPST instantons. It is shown that in the commutative limit the BPST-type configurations coincide with the standard instantons on HP^n written in local coordinates.
We consider Ward's generalized self-duality equations for U(2r) Yang-Mills theory on R^{4k} and their Moyal deformation under self-dual noncommutativity. Employing an extended ADHM construction we find two kinds of explicit solutions, which generalize the 't Hooft and BPST instantons from R^4 to noncommutative R^{4k}. The BPST-type configurations appear to be new even in the commutative case.
We investigate compactifications of type II and M-theory down to $AdS_5$ with generic fluxes that preserve eight supercharges, in the framework of Exceptional Generalized Geometry. The geometric data and gauge fields on the internal manifold are encoded in a pair of generalized structures corresponding to the vector and hyper-multiplets of the reduced five-dimensional supergravity. Supersymmetry translates into integrability conditions for these structures, generalizing, in the case of type IIB, the Sasaki-Einstein conditions. We show that the ten and eleven-dimensional type IIB and M-theory K…
Working within the framework of both the $A_\infty$ and the Berkovits open superstring field theory, we derive a necessary and sufficient condition for a Neveu-Schwarz marginal deformation to be exact up to third order in the deformation parameter. For a specific class of backgrounds, we find that this condition localizes on the boundary of the worldsheet moduli space, thus providing a very simple computational prescription for recovering algebraic constraints (generalized ADHM equations) which need to be satisfied by the moduli. Applying our results to the $\mathrm{D(-1)/D3}$ system, we confi…
We discuss higher derivative interactions in the type IIB superstring in ten dimensions. From the fundamental string point of view, the non-perturbative corrections are due to D-instantons. We argue that they can alternatively be understood as arising from $(p,q)$-strings. We derive a non-renormalization theorem for eight-derivative bosonic interactions, which states that terms involving either NS-NS or R-R fields occur at tree-level and one-loop only. By using the $SL(2, Z)$ symmetry of M-theory on $T^2$, we show that in order for the possible $R^{3m+1} (m=1,2,...)$ interactions in M-theory t…
We consider an eight-dimensional Einstein-Yang-Mills theory to explore whether Yang-Mills instantons formed in extra dimensions can induce the dynamical instability of our four-dimensional spacetime. We show that the Yang-Mills instantons in extra dimensions can trigger the expansion of our universe in four-dimensional spacetime as well as the dynamical compactification of extra dimensions. We also discuss a possibility to realize a reheating mechanism via the quantum back-reaction from the contracting tiny internal space with a smeared instanton.
We compute the partition functions of D(-1)D7 systems describing the multi-instanton dynamics of SO(N) gauge theories in eight dimensions. This is the simplest instance of the so called exotic instantons. In analogy with the Seiberg-Witten theory in four space-time dimensions, the prepotential and correlators in the chiral ring are derived via localization formulas and found to satisfy relations of the Matone type. Exotic prepotentials of SO(N) gauge theories with N=2 supersymmetries in four-dimensions are also discussed.
We consider a complex vector bundle E endowed with a connection A over the eight-dimensional manifold R^2 x G/H, where G/H = SU(3)/U(1)xU(1) is a homogeneous space provided with a never integrable almost complex structure and a family of SU(3)-structures. We establish an equivalence between G-invariant solutions A of the Spin(7)-instanton equations on R^2 x G/H and general solutions of non-Abelian coupled vortex equations on R^2. These vortices are BPS solitons in a d=4 gauge theory obtained from N=1 supersymmetric Yang-Mills theory in ten dimensions compactified on the coset space G/H with an…
A variant of the topological twist, involving SL(2,Z) dualities and hence named topological duality twist, is introduced and explicitly applied to describe a U(1) N=4 super Yang-Mills theory on a Kaehler space with holomorphically space-dependent coupling. Three-dimensional duality walls and two-dimensional chiral theories naturally enter the formulation of the duality twisted theory. Appropriately generalized, this theory is relevant for the study of Euclidean D3-brane instantons in F-theory compactifications. Some of its properties and implications are discussed.
BPS instantons are discussed in Lifshitz-type anisotropic field theories. We consider generalizations of the sigma model/Yang-Mills instantons in renormalizable higher dimensional models with the classical Lifshitz scaling invariance. In each model, BPS instanton equation takes the form of the gradient flow equations for "the superpotential" defining "the detailed balance condition". The anisotropic Weyl rescaling and the coset space dimensional reduction are used to map rotationally symmetric instantons to vortices in two-dimensional anisotropic systems on the hyperbolic plane. As examples, w…
We show that the four derivative terms in the effective action of three-dimensional N=8 Yang-Mills theory are determined by supersymmetry. These terms receive both perturbative and non-perturbative corrections. Using our technique for constraining the effective action, we are able to determine the exact form of the eight fermion terms in the supersymmetric completion of the $F^4$ term, including all instanton corrections. As a consequence, we argue that the integral of the Euler density over $k$ monopole moduli space in SU(2) Yang-Mills is determined by our non-renormalization theorem for all …
We discuss instantons in dimensions higher than four. A generalized self-dual or anti-self-dual instanton equation in n-dimensions can be defined in terms of a closed (n-4) form $Ω$ and it was recently employed as a topological gauge fixing condition in higher dimensional generalizations of cohomological Yang-Mills theory. When $Ω$ is a calibration which is naturally introduced on the manifold of special holomony, we argue that higher dimensional instanton may be locally characterized as a family of four dimensional instantons over a supersymmetric (n-4) cycle $Σ$ with respect to the calibrati…
We propose a new prescription for computing the Nekrasov partition functions of five-dimensional theories with eight supercharges realized by gauging non-perturbative flavor symmetries of three five-dimensional superconformal field theories. The topological vertex formalism gives a way to compute the partition functions of the matter theories with flavor instanton backgrounds, and the gauging is achieved by summing over Young diagrams. We apply the prescription to calculate the Nekrasov partition functions of various five-dimensional gauge theories such as $\mathrm{SO}(2N)$ gauge theories with…
We present a simple alternative to Mackey's account of the (infinite) inequivalent quantizations possible on a coset space G/H. Our reformulation is based on the reduction ${\rm G \rightarrow G/H}$ and employs a generalized form of Dirac's approach to the quantization of constrained systems. When applied to the four-sphere $S^4 \simeq {\rm Spin(5)/Spin(4)}$, the inequivalent quantizations induce relativistic spin and a background BPST instanton; thus they might provide a natural account of both of these physical entities.
We consider heterotic string theories compactified on a K3 surface which lead to an unbroken perturbative gauge group of Spin(32)/Z2. All solutions obtained are combinations of two types of point-like instanton --- one ``simple type'' as discovered by Witten and a new type associated to the ``generalized second Stiefel-Whitney class'' as introduced by Berkooz et al. The new type of instanton is associated to an enhancement of the gauge symmetry by Sp(4) and the addition of a massless tensor supermultiplet. It is shown that if four simple instantons coalesce at an orbifold point in the K3 surfa…
We apply equivariant integration technique, developed in the context of instanton counting, to two dimensional N=2 supersymmetric Yang-Mills models. Twisted superpotential for U(N) model is computed. Connections to the four dimensional case are discussed. Also we make some comments about the eight dimensional model which manifests similar features.
We introduce a particular embedding of seven dimensional self-duality membrane equations in C^3\times R which breaks G_2 invariance down to SU(3). The world-volume membrane instantons define SU(3) special lagrangian submanifolds of C^3. We discuss in detail solutions for spherical and toroidal topologies assuming factorization of time. We show that the extra dimensions manifest themselves in the solutions through the appearance of a non-zero conserved charge which prevents the collapse of the membrane. We find non-collapsing rotating membrane instantons which contract from infinite size to a f…
