Here I would like to outline my research, regarded as a modest activity in the field outlined above, in four items as follows:
Going further along these lines in recent joint work Sz. Szabó and myself determined the moduli space of SU(2) anti-instantons of unit energy converging in a certain sense to the trivial flat connection at infinity over the multi-Taub--NUT space [ete-sza]. Several open problems arise as direct continuation of this work as follows.
So far we described the unframed moduli space as a (singular) 5-dimensional differentiable manifold [ete-sza]: it is a singular B^2-fibration over R^3 (where B^2 is the open 2-disk), with a finite number of fibers degenerating to semi-open intervals. It can also be seen as a singular filling of the multi-Taub--NUT space itself. It would be interesting to describe the natural L^2-theoretical Riemannian metric on it. We conjecture that it is asymptotic to the hyperbolic metric on the two-dimensional discs, and behaves like the metric of the multi-Taub--NUT space in the transverse direction. It would be natural to determine the framed moduli space. It is known to be a singular SU(2)-fibration over the unframed moduli space, with a finite number of special fibers equal to the circle S^1. On the other hand, it is known to be a complete smooth hyper-Kähler 8-manifold. Using these facts, it should be possible to describe it explicitly.
A natural extension of our work would be to study moduli spaces of instantons of higher energy and rank. A study on this topic has already been undertaken by Cherkis, using bow diagrams and a Nahm transform-type argument. In this direction, it would be interesting to understand how our explicit construction fits into this general picture. This would perhaps allow for finding some further explicit examples. Unfortunately, the method of Cherkis so far only applies to non-trivial holonomy condition at infinity; therefore, if we are to explore the relation between our example and his methods, the first task is to extend the bow diagram-construction to the case of trivial holonomy. It would be also interesting to understand the middle L^2-cohomologies of the hyper-Kähler moduli spaces and the natural Kac--Moody action on them, which is also of interest in mathematical physics.
We plan to go further in this direction in the future. For example, L^2 cohomology theory indicates that moduli spaces of fractional energy SU(2) instantons over the ALF Gibbons--Hawking spaces are smooth, i.e., do not contain Abelian instantons. This would be in sharp contrast to the previously known examples (like the compact and the ALE cases), hence we would like to obtain an explicit description of these new moduli spaces; this result would be certainly very interesting.
We also hope to be able to use our instanton theoretic methods to obtain some results about the topological or even geometric classification of the asymptotically locally flat spaces. An encouraging step in this direction is [ete]. The energy spectrum of instantons over these spaces was studied in [ete2].
References to this item
[che-hit] Cherkis, S.A., Hitchin, N.J.: Gravitational instantons of type D_k, Commun. Math. Phys. 260, 299-317 (2005);
[che-kap1] Cherkis, S.A., Kapustin, A.: Hyper-Kähler metrics from periodic monopoles, Phys. Rev. D65, 084015 (2002);
[che-kap2] Cherkis, S.A., Kapustin, A.: D_k gravitational instantons and Nahm equations, Adv. Theor. Math. Phys. 2, 1287-1306 (1999);
[ete] Etesi, G.: The topology of asymptotically locally flat gravitational instantons, Phys. Lett. B641, 461-465 (2006);
[ete-hau1] Etesi, G., Hausel, T.: Geometric interpretation of Schwarzschild instantons, Journ. Geom. Phys. 37, 126-136 (2001);
[ete-hau2] Etesi, G., Hausel, T.: Geometric construction of new Yang--Mills instantons over Taub--NUT space, Phys. Lett. B514, 189-199 (2001);
[ete-hau3] Etesi, G., Hausel, T.: On Yang--Mills instantons over multi-centered gravitational instantons, Commun. Math. Phys. 235, 275-288 (2003);
[ete-jar] Etesi, G., Jardim, M.: Moduli spaces of self-dual connections over asymptotically locally flat gravitational instantons, Commun. Math. Phys. 280, 285-313 (2008), Erratum: Commun. Math. Phys. 288, 799-800 (2009);
[ete2] G. Etesi: On the energy spectrum of Yang--Mills instantons over asymptotically locally flat spaces, Comm. Math. Phys. 322, 1-17 (2013), arXiv: 1103.0241 [math.DG] ;
[ete-sza] G. Etesi, Sz. Szabó: Harmonic functions and instanton moduli spaces on the multi-Taub--NUT space, Comm. Math. Phys. 301, 175-214 (2011), arXiv: 0809.0480 [math.DG];
[gib-haw] Gibbons, G.W., Hawking, S.W.: Gravitational multi-instantons, Phys. Lett. B78, 430-432 (1976);
[kro] Kronheimer, P.B.: The construction of ALE spaces as hyper-Kähler quotients, Journ. Diff. Geom. 29, 665-683;
[kro-nak] Kronheimer, P.B., Nakajima, N.: Yang--Mills instantons on ALE gravitational instantons, Math. Ann. 288, 263-307 (1990);
[nak] Nakajima, H.: Moduli spaces of anti-self-dual connections on ALE gravitational instantons, Invent. Math. 102, 267-303 (1990);
[sen] Sen, A.: Dyon-monopole bound states, self-dual harmonic forms on the multi-monopole moduli space, and SL(2, Z) invariance in string theory, Phys. Lett. B329, 217-221 (1994).
Regarding the first problem, which deals with the Hamiltonian formulation of general relativity, I wrote two papers: the first is a global uniqueness theorem for solutions [ete1] the other one is about a rigidity phenomenon for non-vacuum solutions caused by the scalar curvature of a spacelike section of the space-time [ete2].
Recently there has been some interest in gravitational-gauge theoretic dualities, also called AdS/CFT or Maldacena conjecture [mal]. In this framework a natural gravitational interpreatation of the Hitchin equations [hit] (the 2 dimensional reduction of the self-duality equations in gauge theory) appears [ete3].
A related classical long-standing question is the physical interpretation of gauge theoretic instantons over curved space-times: this requires a detailed understanding of the vacuum structure of general gauge theories (i.e., which are defined on generic curved space-times). Regarding this important question (related with the Theta-problem) I could demonstrate that the complexity of the vacuum sector in general is the same as in the Minkowskian case [ete4].
References to this item
[ete1] Etesi, G.: A global uniqueness theorem for stationary black holes, Comm. Math. Phys. 195, 691-697 (1998);
[ete2] Etesi, G.: A rigidity theorem for non-vacuum initial data, Journ. Math. Phys. 43, 554-562 (2002);
[ete3] Etesi, G.: Gravitational interpretation of the Hitchin equations, Journ. Geom. Phys. 57, 1778-1788 (2007);
[ete4] Etesi, G.: Homotopic classification of Yang--Mills vacua taking into account causality, Int. Journ. Theor. Phys. 46, 832-847 (2007);
[hit] Hitchin, N.J.: The self-duality equations on Riemann surface, Proc. London Math. Soc. 55, 59-126 (1987);
[mal] Maldacena, H.: The large N limit of superconformal theories and supergravities, Adv. Theor. Math. Phys. 2, 231-252 (1998);
[wal] Wald, R.M.: General relativity, Univ. of Chicago Press (1984).
There is a particular current interest in constructing seven dimensional manifolds with G_2 and eight dimensional ones with Spin(7) structure. In this setup I constructed new Spin(7) manifolds by making use of Yang--Mills instantons possessing certain symmetry [ete2].
However based on a natural parallelism between the concept of spontaneous symmetry breaking in particle physics and the existence of special geometric structures on manifolds we hope to find new approaches to very old problems in differential geometry. Spontaneous symmetry breakings to discrete subgroups in the SO(3) gauge were classified in [ete1].
References to this item
[ete1] Etesi, G.: Spontaneous symmetry breaking in the SO(3) gauge theory to discrete subgroups, Journ. Math. Phys. 37, 1596-1602 (1996);
[ete2] Etesi, G.: Spin(7) manifolds and symmetric Yang--Mills instantons, Phys. Lett. B521, 391-399 (2001);
[ete3] G. Etesi, Á. Nagy: S-duality in Abelian gauge theory revisited, Journ. Geom. Phys. 61, 693-707 (2011), arXiv: 1005.5639 [math.DG];
[joy] Joyce, D.D.: Compact manifolds with special holonomy, Oxford Univ. Press (2000).
I. Németi and myself wrote a quite influensive paper which demonstrates that in principle such powerful gravitational computers can exist in Kerr spece-times [ete-nem]. We also could present some statements in exactly what extent our machine can go beyond ordinary Turing computability in the computational hierarchy of integer-valued functions. I also suggested the detailed study of a natural link between the Strong Cosmic Censor Hypothesis of R. Penrose in general relativity and the Church--Turing thesis in computability theory [ete] however this idea still requires a detailed presentastion and explanation [ete2].
References to this item
[ear] Earman, J.: Bangs, crunches, whimpers and shrieks, Chapter 4: Supertasks, Oxford Univ. Press, Oxford (1995);
[ete] Etesi, G.: Note on a reformulation of the strong cosmic censor conjecture based on computability, Phys. Lett. B550, 1-7 (2002);
[ete2] G. Etesi: A proof of the Geroch--Horowitz--Penrose formulation of the strong cosmic censor conjecture motivated by computability theory, Int. Journ. Theor. Phys. 52, 946-960 (2013), arXiv: 1205.4550v3 [gr-qc];
[ete-nem] Etesi, G., Németi, I.: Non-Turing computations via Malament--Hogarth space-times, Int. Journ. Theor. Phys. 41, 341-370 (2002);
[mal] Hogarth, M.L.: Non-Turing computers and non-Turing computability, in: Earst Lansing: Philosphy of Science Associtation 1, ed.: Hull, D., Frobens, M., Burian, R.M., 126-138 PSA (1994).