This page is about an AMS volume in preparation:
Branislav Jurco, Hisham Sati, Urs Schreiber (eds.)
Mathematical Foundations of Quantum Field and Perturbative String Theory
Proceedings of Symposia in Pure Mathematics
AMS
This page contains
a preliminary version of a call for papers
a preliminary version of a preface
a preliminary list of confirmed contributors
Conceptual progress in fundamental theoretical physics is linked with the search for the suitable mathematical structures that model the physics in question.
There are many indications that today we are in a period where the fundamental mathematical nature of quantum field theory and of the worldvolume aspects of string theory is being identified. It is not unlikely that future generations will think of the turn of the millennium and the beginning of the 21st century as the time when it was fully established that and how QFT in general – and in particular that of worldvolume theories that are summed over in perturbative second quantizations such as notably string theory – is precisely the representation theory of higher categories of cobordisms with structure. And dually the theory of factorization algebras and their siblings.
While on the one hand significant insights on these matters have been gained in the last years, the full impact of these insights has possibly not come yet to the wide public perception that it deserves, notably not among most of the theoretical but pure physicists for whom it should be of utmost relevance. At the same time those who do appreciate the mathematical structures involved may wonder how it all fits into the big physical picture of quantum field and string theory.
We therefore imagine creating a volume that collects original presentations as well as reviews/surveys of recent progress in the unravelling of mathematical structures underlying the very nature of quantum field and worldvolume string theory. As a rough guide, we imagine articles fitting into the following outline of what we feel are the cornerstones of the emerging development:
Here is a rough keyword list of topics that would be imagined contributions might contribute to. More details below in the Preface .
The Schrödinger picture of quantum field theory: functors on cobordism categories
the cobordism hypothesis and the classification theorem of topological QFTs
analogous structure theorems for categories of cobordisms with extra structure,
notably in 2-dimensional conformal QFT
and in super-Euclidean QFT
the generalization to QFT with defects and boundaries
The Heisenberg picture of quantum field theory: assignments of algebras of observables to local patches
the local nets in Haag-Kastler algebraic QFT
the various proposed modern formalizations
their relation to other concepts:
to the cobordisms reps mentioned above
construction of worldvolume sigma-model theories from background field – the basic building blocks of second-quantized perturbation series as in string theory
formalization of the background structures in terms of differential cohomology
formalization of the quantization prescription by pull-push operations
examples of extended QFT refinements of theories such as Dijkgraaf-Witten,Chern-Simons theory, Gromov-Witten theory, etc.
Here are some sketches of what a preface to such a book might look like, here serving for the moment as an annotated outline to show how contributions might fit together to a coherent whole.
The history of theoretical fundamental physics is the story of a search process for the suitable mathematical notions and structural concepts that naturally model the physical structures in question.
It may be worthwhile to recall some examples, for instance
the identification of symplectic geometry as the underlying structure of classical Hamiltonian mechanics;
or the identification of Riemannian differential geometry as the underlying structure of gravity.
Or, based on more recent developments, for instance
which has a close connection to the issues to be discussed here.
All these examples exhibit the identification of mathematical languages that naturally capture the physics in question. While each of these languages was upon its introduction into theoretical physics originally met with some scepticism or even hostility – just compare the Gruppenpest complaint by Wolfgang Pauli – we do know in retrospect that the modern insights and results in the respective areas of theoretical physics would have been literally unthinkable without usage of these languages.
Much time has passed since the last major such formalization success in theoretical physics. The rise of quantum field theory in the middle of the last century and its stunning successes despite its notorious lack of formal structural underpinning made theoretical physicist confident enough to attempt an attack on the next open structural question – that for the quantum theory of gauge forces including gravity – without much more of a structural guidance than some folklore called the path integral, however useful that has proven to be.
While everyone involved readily admitted that nobody knew the full answer to
“What is string theory?”
it was maybe gradually forgotten that nobody even knew the full answer to
“What is quantum field theory?” .
While a huge discussion ensued on the “landscape” moduli space of backgrounds for string theory, it was maybe forgotten that nobody even had anything close to a full answer to
“What is a string theory background?”
or even to what should be a simpler question:
“What is a classical string theory background?”
which in turn is essentially the question:
“What is a full 2d conformal field theory?” .
Most of the literature on 2d conformal field theory describes just what is called chiral conformal field theory formalized in terms of vertex operator algebras or local conformal nets. But this only describes the holomorphic and low-genus aspect of conformal field theory and is just one half of the data required for a full CFT, the remaining piece being the full solution of the sewing constraints that makes the theory well defining on all genera.
With these questions – fundamental as they are for perturbative string theory – seemingly too hard to answer, a plethora of related model and toy model quantum field theoretic systems found attention instead. A range of topological quantum field theories either approximates the physically relevant conformal field theories as in the topological A-model and B-model, or encodes these in their boundary theory as for 3d Chern-Simons theory and its toy model, Dijkgraaf-Witten theory.
At the same time these interrelationships were seen to repeat in some way in a -dimensional pattern, leading to the consideration of worldvolume theories for NS-5-branes, superconformal field theories in 6-dimensions and “little string theories”.
This way a wealth of worldvolume quantum field theories appears that in some way or other is thought to encode information about string theory. And in each case what matters really is the full worldvolume QFT: the rule that assigns correlators to all possible worldvolume cobordisms, because this is what is needed to even write down the corresponding second quantized perturbation series. But despite this urgent necessity for understanding QFT on arbitrary cobordisms, the tools to study or even formulate precisely this were for a long time mainly unavailable.
Of course proposals for how to make these questions accessible to the development of suitable mathematical machinery have been known for a long time.
Early on it was suggested, based on the topological examples, that the path integral and the state-propagation operators that it is supposed to yield is nothing but a representation of a category of cobordisms. And it was noticed that this prescription is not restricted to TQFTs and conformal field theories were proposed to be axiomatized as representations of categories of conformal cobordisms.
In parallel to that another school developed a dual picture, now known as AQFT, where not the state-propagation – the Schrödinger picture – of QFT is axiomatized and made accessible to high-powered machinery, but the assignment of algebras of observables – the Heisenberg picture of QFT.
While these axiomatizations were known and thought of highly by a few select who thought about them, they were mostly happily ignored by the quantum field theory and string theory community at large. And to a good degree rightly so: nobody should trust an axiom system that hasn’t proven its worth yet by providing some useful theorems and describing some nontrivial examples of interest. But neither the study of cobordism representations nor that of systems of algebras of observables could for a long time – apart from a few isolated exceptions – claim to add much to the world-view of those who enjoy formal structures in physics, but not a priori formal structures in mathematics.
And it is precisely this that is changing now.
Major structural results have been proven about the axioms of functorial quantum field theory in the form of cobordism representations and dually those of local nets of algebras and factorization algebras, filling these axiom systems with life and providing a wealth of examples for systems realizing them.
On the FQFT side this involves:
in the topological case the proof of the cobordism hypothesis which says that an extended -dimensional TQFT is entirely encoded by a “fully dualizability”-structure on the space of states that it assigns to the point. This hugely facilitates constructing interesting examples of extended -dimensional TQFTs;
as a special case the understanding that the topological string A-model and B-model 2dTQFTs are specical cases of -TQFTs called topological conformal field theories whose classification in terms of their Calabi-Yau A-∞ categories of branes realizes Kontsevich’s old reformulation of mirror symmetry in string theory as homological mirror symmetry ;
in this context crucial aspects of Edward Witten’s observation in Chern-Simons Gauge Theory as a String Theory could be made precise and thus a rigorous handle on the effective background theory (here Chern-Simons theory) induced by a string perturbation series over all genera (here the A-model or B-model topological strings) is obtained.
in the conformal case the full classification of rational 2d CFTs. While the rational case is still “too simple” for the most interesting applications in string theory, its full solution shows that already here vastly more interesting structure is to be found than suggested by the naive considerations in much of the physics literature.
in the super-Euclidean case the full proof from the axioms that the partition function of a supersymmetric 2d-QFT indeed is a modular form, as suggested by the physics literature on the heterotic string.
On the AQFT/factorization algebra side this involves:
the explicit connection of the old Haag-Kastler axioms for local nets in AQFT – which produced such fundamental QFT results, such as the PCT theorem or the spin-statistics theorem, but wasn’t well supplied with examples for a long time – to standard constructions in perturbative quantum field theory as practiced in most theoretical physics departments.
the refinement of the traditionally strong point of AQFT applied to 2d chiral super CFT from local nets to factorization algebras and the organization of these into a monoidal tricategory that appears to be the right generalization of the bicategory of Clifford algebras and hence of spin geometry to stringy spinors and string geometry.
the general development of factorization algebras into a powerful formal framework for Wilsonian quantum field theory.
a rigorous construction of the Witten genus as the partition function of a QFT encoded by a suitable factorization algebra;
finally, in the incarnation of factorization algebras in terms of blob homology and topological chiral homology the understanding of the passage between these Heisenberg-picture formalizations and the Schrödinger-picture cobordisms reps.
This indicates the closure of a grand circle of ideas and makes the outline of a comprehensive fundamental formalization of full higher-genus quantum field theory visible.
Maybe even more importantly in the long run for physics is that with the supposed outcome of the path integral quantization process thus identified precisely, it becomes possible to think about what this process itself is, precisely. This is in particular relevant in applications of QFT as worldvolume theories in string theory, where one wishes to explicitly consider QFTs that arise as the quantization of sigma-models with specified background fields. A good understanding of this quantization is what connects the world-volume to the target space theory and hence the abstract algebraic description of the worldvolume QFT to the phenomenological interpretation of its correlators in its target space.
Progress in understanding the nature of sigma-model backgrounds includes:
the observation that higher background gauge fields – such as the Yang-Mills field, the Kalb-Ramond field, the RR field
are cocycles in generalized differential cohomology
modeled geometrically by bundles with connection, gerbes with connection and in general ∞-bundles with connection.
whose anomaly cancellation such as in the Green-Schwarz mechanism is reflected in cohomological twistings.
the explicit refinement of this statement to the crucial higher equivariant cases of background fields on orbifolds and notably orientifolds.
the beginning of an understanding of how the quantization of the corresponding sigma models arises from some kind of general abstract push-forward-operation on these differential cocycles.
Taken together, these developments should go a long way towards understanding the fundamental nature of quantum field theory on arbitrary cobordisms. But even in the light of all these develoments, the reader accustomed to the prevailing phyiscs literature may still complain that none of this progress in quantum field theory on cobordisms of all genera yields a definition of what string theory really is. And of course this is true if by “string theory” one understands its non-perturbative definition. But this supposed non-perturbative definition of string theory is little more than a dream of a dream for the time being. Marvelling – with a certain pride about their daringness – at how ill-understood this is has made the community forget that something much more mundane, the perturbation series over CFT correlators that defines perturbative string theory , has been ill defined all along: only the machinery of full CFT in terms of cobordism representations gives a precise meaning to what exactly it is that the string pertubation series is a series over. Maybe it causes feelings of disappointment to be thrown back from the realm of speculations about non-perturbative string theory to just the perturbation series. But at least this time one lands on solid ground. Which is the only ground that serves as a good jump-off point for further speculation.
In string theory it has been the tradition to speak of major conceptual insights into the theory as revolutions of the theory. The community speaks of a first and a second superstring revolution and a certain longing for the third one to arrive can be sensed. With a large part of the community busy attacking grand structures with arguably insufficient tools, it doesn’t seem out of the question that when the third one does arrive, it will have come out of the math departments.