Resurgence and Non-Perturbative Physics

This is a new research program in quantum field theory, based on a relatively new development in the mathematical theory of asymptotics [the study of problems when a parameter, such as a temperature or a chemical potential or a coupling strength becomes relatively large or small], known as “resurgent asymptotics”. In physical terms, resurgent asymptotics provides a framework in which perturbative and non-perturbative information is combined into a consistent object known as a “trans-series”. A trans-series goes beyond a conventional perturbative Taylor or Laurent series in the sense that it encodes the full analytic continuation properties of the quantity being computed. This is important in the physics of strongly-interacting quantum field theory, because semi-classical expansions are a powerful analytic tool for analyzing the strong-coupling regime. This new formalism has been applied with remarkable results in matrix models, string theory, quantum mechanics and now quantum field theory. For an introductory review of some of these methods and their applications in physical systems, see some of my recent summer/winter school lectures, for example Introduction to Resurgence, Trans-series and Non-perturbative Physics.pdf (International Centre for Theoretical Sciences, Bangalore, 2018),  or online lectures at the Kavli Pan-Asian Winter School: Strings, Particles and Cosmology, Sogang University, Seoul, January 2019 [;;;; and also the KITP Program Resurgent Asymptotics in Physics and Mathematics from Fall 2017.

    My current interests lie in several directions:

    1. making practical computational sense of the Minkowski (real-time) path integral, with applications to the “sign problem” of finite density quantum systems and non-equilibrium physics. Resurgent trans-series and their associated Lefschetz thimbles (functional steepest descent contours for path integrals) provide a new approach to such problems, both analytically and computationally.

    2. resurgence and phase transitions: as an external parameter varies, the structure of the trans-series can transmute as the dominant saddle configurations (which may be complex) can change, casting phase transitions in terms of the Stokes phenomenon.

    3. semi-classical interpretation of infrared renormalons and the structure of the operator product expansion.

    4. novel resummation and extrapolation methods, using resurgence to dramatically improve the precision of Borel summation. This has applications to dualities (connecting strong and weak coupling) and the precise extraction of critical exponents.

    5. resurgence, holography and duality.

    Some papers addressing some of these questions are:

    G. V. Dunne and M. Unsal,
    “New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifold Equivalence to Bions and Resurgence,”
    Ann. Rev. Nucl. Part. Sci. 66, 245 (2016), [arXiv:1601.03414]

    G. V. Dunne and M. Unsal,
    “Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model,”
    JHEP 1211, 170 (2012), [arXiv:1210.2423];
    “Continuity and Resurgence: towards a continuum definition of the CP(N-1) model,”
    Phys. Rev. D 87, 025015 (2013), [arXiv:1210.3646].

    G. V. Dunne and M. Unsal,
    “What is QFT? Resurgent trans-series, Lefschetz thimbles, and new exact saddles,”
    PoS LATTICE 2015, 010 (2016), [arXiv:1511.05977].

    A. Behtash, G. V. Dunne, T. Schaefer, T. Sulejmanpasic and M. U†nsal,
    “Complexified path integrals, exact saddles and supersymmetry,”
    Phys. Rev. Lett. 116, no. 1, 011601 (2016), [arXiv:1510.00978];
    “Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence,”
    Annals of Mathematical Sciences and Applications Volume 2, No. 1 (2017), [arXiv:1510.03435];
    “Critical Points at Infinity, Non-Gaussian Saddles, and Bions,”
    JHEP 1806, 068 (2018), arXiv:1803.11533

    G. Basar, G. V. Dunne and M. Unsal,
    “Quantum Geometry of Resurgent Perturbative/Nonperturbative Relations,”
    JHEP 1705, 087 (2017), [arXiv:1701.06572];
    “Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lam’e systems,”
    JHEP 1502, 160 (2015), [arXiv:1501.05671]; WKB and Resurgence in the Mathieu Equation,”
    in Resurgence, Physics and Numbers, F. Fauvet et al (Eds), Edizioni Della Normale (2017), [arXiv:1603.04924].

    G. V. Dunne,
    “Resurgence, Painleve Equations and Conformal Blocks,”

    P. V. Buividovich, G. V. Dunne and S. N. Valgushev,
    “Complex Path Integrals and Saddles in Two-Dimensional Gauge Theory,”
    Phys. Rev. Lett. 116, no. 13, 132001 (2016), [arXiv:1512.09021].

    A. Ahmed, G. V. Dunne,
    “Transmutation of a Trans-series: The Gross-Witten-Wadia Phase Transition,”
    JHEP 1711, 054 (2017), [arXiv:1710.01812].