# Research Area D2

# Mathematical challenges in field theory

## PIs: Disertori, Gubinelli, Klemm, Teichner

## Contributions by Borot, Lesch, Zagier, Zirnbauer

## Topic and goals

Quantum field theory (QFT) is one of the most fundamental and successful frameworks of theoretical physics. However, its mathematical foundations are far from being understood and remain a great challenge. Bringing together leading experts from various areas of mathematics and physics, we plan to explore and apply rigorous approaches to QFT. We will pursue three complementary threads and their interactions: a geometrical approach based on duality relations between QFT and gravity, the anti-de Sitter/conformal field theory correspondence (AdS-CFT), a more algebraic approach based on cobordism (functorial QFT), and a more analytic approach based on the connection between functional integrals and stochastic partial differential equations (stochastic PDEs).

## State of the art, our expertise

**AdS-CFT correspondence**. The classical techniques to solve the A-model and the B-model, which amounts to actually calculating all correlation functions, are localization in the moduli space of holomorphic maps and variations of Hodge structures. Dabholkar–Murthy–Zagier advance the theory of meromorphic Jacobi forms to show that BPS-state counting for type II string theories 68 on K3 _ T2 can be expressed by mock modular forms, see also [HKK15]. An earlier success of BPS-state counting via Hodge theory and modular forms is [KMPS10], proving the genus zero Yau–Zaslow conjecture for K3 surfaces. Another approach pioneered by Klemm is the ‘large N technique’. It is based on string/gauge theory duality and, in particular, on the fact that amplitudes calculated for large N gauge groups encode string amplitudes at high genus. For the A-model this leads to the topological vertex formalism and for the B-model to the formalism of topological recursion [GJKS15]. The latter has close links to work of Borot–Eynard–Orantin [BEO15] on geometric and topological recursion.

**Functorial QFT**. Stolz–Teichner [ST12] developed a mathematical model for supersymmetric field theories in terms of functors on geometric bordism categories. The focus is on connections to algebraic topology, in particular elliptic cohomology [ST11]. This approach was pioneered for a topological field theory (TFT) by Atiyah. Lurie’s approach to the cobordism hypothesis is considered a great breakthrough in this area. It enables the complete computation of deformation classes of such TFTs. For example, Freed–Hopkins approached symmetry-protected topological phases in terms of certain invertible TFTs and gave 10 tables of deformation classes for various symmetry groups. Surprisingly, these 10 tables are consistent with previous results from the physics literature, for example with Zirnbauer’s work on another mathematical model describing the ground states of topological insulators and superconductors with symmetries. In the case of translation-invariant systems, the model reduces to vector bundles with Clifford symmetries, which were classified by Kennedy–Zirnbauer [KZ16], now known as the 10-fold way. In the bulk-edge correspondence of such topological insulators, the intersection product in Kasparov’s KK-theory plays a prominent role. Kaad–Lesch [KL13] extended the unbounded picture for this theory by constructing explicit representatives for KK-theoretic intersection products.

**PDEs in QFT**. QFT can be formulated in terms of functional integrals whose rigorous formulation requires the construction of probability measures (for Euclidean QFT) or complex infinitedimensional integrals (for Minkowski QFT) via non-perturbative methods. One strategy is the so-called constructive renormalization group approach. In the context of fermionic theories, this has been pursued by several groups, including Disertori–Rivasseau [DR00]. In a different direction, Disertori explored an approach based on internal symmetries of the model that applies to situations where standard renormalization tools do not. In particular, Disertori–Spencer–Zirnbauer [DSZ10] developed a multiscale method based on an infinite family of identities generated by supersymmetries. In the Euclidean setting, another promising alternative to standard constructive tools is the stochastic quantization approach based on seminal work of Parisi and Wu. Stochastic dynamics can be used to define the functional integrals as equilibrium distributions and brings into play many tools from analysis of PDEs which are unavailable in the standard functional integral formu- 69 lation. In this context, Gubinelli et al. [GIP15, GP15] developed analytical and probabilistic tools to study singular stochastic PDEs and in general nonlinear operations on random distributions.

## Research program

**AdS-CFT correspondence**. Modularity of partition functions in string theory will be one of our central topics. Klemm and Zagier will explore the relation between, on the one hand, the transition matrices for the differential equation of the mirror quintic family and, on the other hand, the periods and quasiperiods (a notion developed specifically for this application) of special Hecke eigenforms. They will also consider the application of the theory of modular and quasi-modular forms to the direct integration method, where it is expected to play a pivotal role. To construct the elliptic genus of new 6d super conformal QFTs in the geometric engineering approach, Klemm wants to explore automorphic symmetries acting on moduli spaces of elliptically fibred Calabi– Yau spaces. Together with Borot, he also plans to use the topological large N technique to study non-perturbative effects with resurgence techniques.

**Functorial QFT**. The classification of symmetry-protected topological phases and its miraculous relation to deformation classes of invertible TFTs will be jointly approached by Zirnbauer, Teichner, Disertori, and Lesch. The goal is to explain the coincidence of the two models and, in particular, to relate Zirnbauer’s 10-fold way to the tangential structures proposed by Freed–Hopkins. Such an explanation will in turn trigger new calculations in equivariant homotopy theory that are motivated by solid state physics. Teichner will also relate functorial QFTs to the approach via factorization algebras that was pioneered by Costello–Gwilliam. The long-term goal is a rigorous mathematical formalism for higher-dimensional supersymmetric sigma models, obtained by varying perturbative quantizations over the stack of classical solutions. Borot and Teichner will promote the functorial construction of amplitudes of two-dimensional QFTs by implementing a gluing procedure, modeled both on Mirzakhani–McShane identities in hyperbolic geometry and on the topological recursion of Eynard–Orantin; see the figure. This construction will take values in the space of fields of the QFT under consideration, which itself can be obtained via various means, e.g., differential geometry of Teichmüller space, vertex operator algebras, or factorization algebras.

**PDEs in QFT**. The non-perturbative formulation of interacting QFT in two and three dimensions in terms of functors on geometric bordism categories will be investigated by Gubinelli and Teichner. The goal is to apply constructive techniques and recent advances in the analysis of stochastic quantization to get candidates for sigma models and relate them to the ones arising in Costello– Gwilliam’s factorization algebras. Disertori and Gubinelli plan to investigate rigorous derivations of the dimensional reduction phenomenon where a (D + 2)-dimensional elliptic stochastic PDE gives rise to a D-dimensional QFT by taking a trace of the realizations of the random field on a codimension-two subspace. This construction requires casting the stochastic PDE in the language of supersymmetric (D+2)- dimensional QFT and provides yet another alternative construction of QFTs via stochastic PDEs. Understanding stochastic quantization dynamics of complex functional integrals arising from the QFT in the Minkowski setting poses new mathematical challenges. To improve the integration process, one can (at least in the finite-dimensional approximations) find integration domains in the complex plane for which the phase factor does not oscillate, a domain usually refereed to as a Lefschetz thimble. Constructive tools to attack this problem will be explored by Gubinelli and Disertori; in a different context Disertori–Lager [DL17] recently used a complex deformation to study interacting Gaussian fields arising in the theory of random band matrices. Initial efforts will be devoted to constructing equilibrium dynamics on the Lefschetz thimble and proving that they can be used to compute expectation values for the QFT, at least in finite-dimensional approximations. The next step is to pass to the continuum limit. Borot and Gubinelli plan to study stochastic processes on Riemann surfaces, e.g., stochastic dynamics, Liouville theory, and Schramm–Löwner equations. The basic idea is to combine constructive and probabilistic ideas to define the local behavior of certain observables and the geometric gluing approach to reach for global objects. One of the first steps would be to localize the recent construction of the Liouville CFT by David, Kupiainen, Rhodes, and Vargas.

## Summary

In various teams of researchers, the long-standing open problem of rigorous construction for non-perturbative QFT will be attacked, using methods from number theory, homotopy theory, non-commutative geometry, and stochastic PDEs. The IRU benefits from close interactions with other research areas, especially with A3 and B1. Two of the main goals are to construct and relate quantized sigma models in higher dimensions, and to understand the role of invertible TFTs in solid state physics. Structural remarks. This IRU will be set up in collaboration with the Bethe Center for Theoretical Physics in the Physics Department of UBonn. A further impulse will come with the hiring of a new director of the Bethe Center, planned for 2018. Moreover, in case of approval, the two clusters of excellence at UBonn, HCM and CASCADE, will join forces to support and extend the research activity of this IRU by jointly establishing a junior research group with one group leader (W2 professor for 5 years), one postdoc, and one PhD student. They will contribute their own research topics within the broad scope of mathematical challenges in field theory and its applications.

## Bibliography

[BEO15] G. Borot, B. Eynard, and N. Orantin. Abstract loop equations, topological recursion and new applications. *Commun. Number Theory Phys., 9(1):51–187, 2015*.

[DL17] M. Disertori and M. Lager. Density of states for random band matrices in two dimensions. *Ann. Henri Poincaré, 18(7):2367–2413, 2017*.

[DR00] M. Disertori and V. Rivasseau. Continuous constructive fermionic renormalization.* Ann. Henri Poincaré, 1(1):1–57, 2000*.

[DSZ10] M. Disertori, T. Spencer, and M. R. Zirnbauer. Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model. *Comm. Math. Phys., 300(2):435–486, 2010*.

[GIP15] M. Gubinelli, P. Imkeller, and N. Perkowski. Paracontrolled distributions and singular PDEs. *Forum Math. Pi, 3:e6, 75, 2015*.

[GJKS15] J. Gu, H. Jockers, A. Klemm, and M. Soroush. Knot invariants from topological recursion on augmentation varieties. *Commun. Math. Phys., 336(2):987–1051, 2015*.

[GP15] M. Gubinelli and N. Perkowski. Lectures on singular stochastic PDEs, volume 29 of Ensaios Matemáticos. *Sociedade Brasileira de Matemática, Rio de Janeiro, 2015*.

[HKK15] M.-x. Huang, S. Katz, and A. Klemm. Topological string on elliptic CY 3-folds and the ring of Jacobi forms. J. High Energ. Phys., 10:125, 2015. [KL13] J. Kaad and M. Lesch. *Spectral flow and the unbounded Kasparov product. Adv. Math., 248:495–530, 2013*.

[KMPS10] A. Klemm, D. Maulik, R. Pandharipande, and E. Scheidegger. Noether-Lefschetz theory and the Yau-Zaslow conjecture. *J. Amer. Math. Soc., 23(4):1013–1040, 2010*.

[KZ16] R. Kennedy and M. R. Zirnbauer. Bott periodicity for Z2 symmetric ground states of gapped free-fermion systems. *Comm. Math. Phys., 342(3):909–963, 2016*.

[ST11] S. Stolz and P. Teichner. Supersymmetric field theories and generalized cohomology.* In Mathematical foundations of quantum field theory and perturbative string theory, volume 83 of Proc. Sympos. Pure Math., pages 279–340. Amer. Math. Soc., Providence, 2011*.

[ST12] S. Stolz and P. Teichner. Traces in monoidal categories. *Trans. Amer. Math. Soc., 364(8):4425–4464, 2012*.