Research:

For any speculation which does not at first glance look crazy,
there is no hope. – Freeman Dyson

[ Thought-Bite ][So far] [Now] [Plans] [Publications] [Research Students Wanted]

Special thanks to the High Energy Program of the US Department of Energy, for their generous support.

If this is all gibberish to you but you don't want to give up, please visit (listed in no particular order):
Fundamental Particles & Interactions, The Particle Adventure, Particle Data Group, String Theory, Superstrings!, Advance of Physics: Strings, String Theory On-Line, 2nd Revolution, ThinkQuest sites first, use your internet search engine, or see if you like my own attempt at introducing the concepts of the contemporary search for The Theory of (More than) Everything: Through Strings to Cosmic Strings and Why.

On a related note, you may want to read some other attempts, in book form, to explain string theory and the related quest for a theory of everything:

John D. Barrow	Jeremy Bernstein	Stephen Hawking	Brian Greene		John Gribbin	Robert Oerter	Steven Weinberg
Theories of Everything:…	A Theory for Everything	Quest for a Theory of Everything	The Elegant Universe	The Fabric of the Cosmos	The Search for Superstrings…	The Theory of Almost Everything	Dreams of a Final Theory

The (by now classical) technical reference on (super)string theory is the 2-volume set by Michael Green, John Schwarz, and Edward Witten: “Introduction” and “Loop Amplitudes, Anomalies and Phenomenology”

If you prefer a more recent and serious (and technical) reading on string theory, I recommend Joe Polchinski‘s 2-volume text on the subject. It will provide both a historical setting and a good introduction to the contemporary ideas in this speedily developing research field.

Subsequently appeared Clifford Johnson's book on D-branes, which can be read either together with Polchinski's 2-volume set or thereafter. This triptich reveals why string theory is not just a theory of strings, but also of other things: D-branes are higher-dimensional objects that play a vital role in (the increasingly misnamed) string theory.

Further sources—for those not faint of heart (or brain)—are mostly found on the arXiv, but also in the collections:

<= A herculean collaboratorial effort by Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandaripande, Richard Thomas, Cumrun Vafa and Eric Zaslow (as editors) contains almost 1000 pages of material contributed from "both sides of the fence", (mathematical) physicists as well as (physical) mathematicians.

Structurally similar, but in many ways complementary is the 2-volume set (totaling 40% mora pages!) by Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, and Edward Witten (as editors). =>

Thought-Bites: Two 1-page efforts… [back]

• A Power-point slide [ PDF ]
• A 3-paragraph summary [ 3¶Σ ]

My research: The story so far… [back]

“The most exciting phrase to hear in science, the one that heralds new discoveries,
is not ‘Eureka!’ but ‘That's funny...’” – Isaac Asimov

I have been studying the vacuum; since 1983 in publication, but actually since well before that.

In so-called grand-unification models, this relates to the pattern(s) of symmetry breaking, in superstrings--to ‘compactification’. Besides the ‘purely’ mathematical physics issues in these problems, I was also interested in implications to phenomenology. So, the techniques developed in articles [E.1-6]* to study the pattern of symmetry breaking in grand-unified models could be applied to investigate the possibility that the symmetry of strong interactions, SU(3), is itself broken [E.9]. In general though, these group theory boosted field theory methods are applicable rather more widely and still await for new applications. Some have been successfully applied to the study of nuclear structures [N.1-10].

Since 1986, most of my research has focused on the study of superstring ground states, the corresponding Calabi-Yau manifolds and related field theories, notably the so-called Landau-Ginzburg orbifolds. This has brought about an unsuspected wealth different techniques and methods, borrowed, modified and developed from various branches of physics and mathematics. It transpires that the broad perspective provided by superstring theory enables relations between rather different approaches and so the use of standard results and methods of one field to obtain novel information in another.

For example, Ref. [E.42] answers certain questions about a class of highly nonlinear constrained sigma-model, both by an easy algebro-geometric calculation and also a simple particle physics argument. Also, Refs. [E.43, E.49] prove the perfect correspondence between the semiclassical results for Landau-Ginzburg orbifolds, developed by C.Vafa and others, and the so-called Koszul computations (classical geometry) for the corresponding complete intersection manifolds, developed in Refs. [E.17, E.21, E.33, E.36, E.38]. Moreover, the latter technique is also closely related to the so-called BRST treatment of constraints [E.49,52]. As an anecdotal remark: whereas the technique of Koszul computations is 'in principle' widely known to mathematicians, it took a concatenation of felicitous events and two years of inquiry to bring about a one-year transcontinental e-collaboration [E.33] which resolved the issue in all technical detail and firmly establish the method.

It has become gradually clearer [E.43, E.49, E.50,51], that the semiclassical Landau-Ginzburg orbifold methods and the Koszul computations overlap to a great extent, but by far not entirely. Many of these two (and possibly all) types of models have since been shown by E.Witten [“Phases”] to be suitable limits of a more general linear sigma-model. From a different approach, the article [E.52] presents a residue-theoretic formulation of the Koszul computation, links it to the (BRST) gauge-theoretic treatment of constraints, so-called polynomial deformations, homological algebra, special geometry etc. This also relates these to the technique for calculating the Yukawa couplings and the Zamolodchikov metric as functions over the whole moduli space [E.32,49-51].

My book Calabi-Yau Manifolds-A Bestiary for Physicists was meant as a comprehensive and hopefully comprehensible introduction and reference to the rapidly developing study of superstring ground states. It covers the portion of this field which is likely to have settled in its definite form, notably constructions of such manifolds, their (co)homological analysis and the general facts and ideas about the relation to field theories and superstring ground states.

*All alpha-numerical references cite items from my publication list.

Present... [back]

Some of the topics of my current research are:

• The general assumptions about the physical spacetime in superstring models have undergone a series of generalizations in the past decade or so. Some rather general facts about the admissible spacetimes in superstring theory [E.54] offer an amusing and cosmologically novel type of candidate solutions. Some concrete, if toy, models have been studied in the series of articles [E.63-68], and demonstrate that it is possible to construct cosmological models which:
   (1)  "our" spacetime is immersed in a bigger one,
   (2)  there is an exponential hierarchy between "our" Plank energy and that in higher dimensions,
   (3)  gravity is localized to "our" spacetime, with experimentally acceptable corrections,
   (4)  "our" spacetime metric is de Sitter, and
   (5)  the features of the model are induced from stringy properties in higher dimensions.
• Conifolds [E.30] are at finite distance [E.25] from smooth manifolds in the web of superstring ground states [E.22]. The passage within this web through a conifold implies a phase transition of sorts [E.23], the physics of which remains a mystery with the desirable N<2 supersymmetry in the effective 4 dimensional spacetime. Circumstantial evidence (and the simplicity of models with N>1 supersymmetry) however provides an operational definition of a Stringy Singular Cohomology [E.55] for (at least the mild) conifolds, and which turns out to be rather novel and unexplored. A detailed study of Witten's gauged σ-model [E.69] naturally provides a partial solution to this problem.
• Supersymmetry
• Multivariate complex analysis and (complex) algebraic geometry are becoming increasingly closer related. For example, the supersymmetric analogue of the Hodge star has a concrete operator representation and introduces several new bilinear forms into field theory [E.56]. The standard Lefschetz SL(2) action on forms has a complexified double acting on the Hilbert space of (2,2)-supersymmetric models [E.59]. Also, the target spaces of N=2 supersymmetric 4-dimensional models turn out to be algebraic varieties [E.58], and algebro-geometric properties prove the inequivalence of Chiral and Linear superfields [E.60].
• Representation theory of supersymmetry is surprisingly incomplete: while supersymmetries have been studied for over four decades, linear, unitary, finite-dimensional, off-shell representations turn out to be far from well understood when supersymmetry has more than 8 generators (N=2 in 4-dimensional spacetime). The collaboration of [E.71, E.78, E.81, E.83] presents Adinkras, faithfully depicting of supermultiplets of worldline N-extended supersymmetry. This provides: (1) a recursive generation of all supermultiplets with the same chromotopology from any one of them, (2) all Adinkra chromotopologies as quotients of a cube by a doubly even binary linear block code, (3) a superfield rendition of every Adinkra, (4) a semi-infinite sequence of indecomposable representations of supersymmetry constructed from tensor products of one. [E.72] relates supermultiplets to filtered Clifford spermodules, thereby providing a rigorous foundation to this program. [E.73, E.79] demonstrates the use of Adinkras in practical applications. [E.88, E.89] extend this analysis to worldsheet supermultiplets.
• The measurement conundrum seems to have plagued quantum mechanics for so long that impressions of an inconsistency amongst its axioms have spawned. A demonstration that this is fictitious [Q.01] turns out to unveil an exclusion principle of sorts: quantum mechanics cannot be simultaneously linear and introspective (self-observing). The nonlinearity of the latter approach permits a quantum mechanical description of the entire measuring process, and also an application to the entire Universe, for which ‘external observer’ (inherent of the linear QM) is an oxymoron. Further study [Q.02] shows that the quantum measurement may well be understood in terms of spontaneous superposition (and symmetry) breaking.

Future... [back]

Certain to flow from the efforts of the past, my future research plans include the study of:

1. the category of superstring spacetimes, their geometry and the induced dynamics for matter;
2. supersymmetry and supersymmetric systems, superspace and related techniques;
3. mathematical techniques for the analysis of the above;
4. other applications of these techniques.

© Tristan Hübsch, 2011