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ZMP Seminar

The ZMP Seminar each term is about a specific topic. Fold out the informations below for details about each semesters topic and see further below for the list of talks in each semester.

Summer Semester 20226 “Resurgence, BPS states, and holomorphic Floer theory”

The problem to give meaning to divergent series occurs in many problems of mathematics and theoretical physics. Intricate algebraic structures can govern the Stokes jumps relating the summations associated to different regimes of the parameters. Such structures, sometimes referred to as wall-crossing structures, can encode non-perturbative effects such as the BPS states in supersymmetric quantum field theories. They may furthermore encode subtle topological information, or enumerative invariants determining geometric structures.

Introduction to resurgence (I)

How can we define analytic functions from divergent series? The goal of this part would be to introduce basic notions such as Borel summation, and transseries.

Literature: There are several good reviews from different perspectives, e.g. [Sa, ABS, MS, Do, Ma]

Stokes phenomena, wall-crossing and BPS-states (I)

How do the results of summations depend on the regimes of parameters being considered? The answer to this question turns out to be related to geometric structures such as cluster varieties, and to the intricate algebraic structures governing wall-crossing phenomena. A relatively simple, but very instructive family of examples is provided by the theory of Sturm-Liouville, or Schrödinger type differential operators on Riemann surfaces [IN].

Option 1: Applications to integrable models (S)

In order to understand the scope and the potential of resurgence, it is useful to study the cases, for which exact results are known by other means, as provided by integrable models. This may offer insight into the question how canonical the non-perturbative completions of quantum theories are. Recent work in this direction offers some insights in examples related to integrable many-particle models [DL], or integrable quantum field theories [BBV].

Option 2: Holomorphic Floer theory (S)

Resurgence is the basis for an ambitious program of Kontsevich and Soibelman [KS]. This program aims to relate categories of representations of the non-commutative algebras arising from the deformation quantisation of complex symplectic manifolds to the Fukaya categories of the same manifold. The Fukaya categories naturally define sheaves on a skeleton of the manifold. The Stokes (or wall-crossing) phenomena of resurgence describe the gluing of the local data characterising these sheaves into global objects. Deformation quantisation, on the other hand, is expected to define algebras of differential or finite difference operators. The functor assigning objects in the Fukaya category to modules of the algebras from deformation quantisation is interpreted by Kontsevich and Soibelman as an ingredient of a generalisation of the Riemann-Hilbert correspondence.

Literature: We could extract basic ideas from [KS]; Reference [GKNPS] offers a physics perspective.

 


References

[Sa] D. Sauzin, Introduction to 1-summability and resurgence, arXiv:1405.0356 [math.DS].
[ABS] I. Aniceto, G. Basar and R. Schiappa, A Primer on Resurgent Transseries and Their Asymptotics, Phys. Rept. 809 (2019) 1–135, [1802.10441].
[MS] C. Mitschi and D. Sauzin, Divergent Series, Summability and Resurgence I, vol. 2153 of Lecture Notes in Mathematics. Springer, 2016.
[Do] D. Dorigoni, An Introduction to Resurgence, Trans-Series and Alien Calculus, Annals Phys. 409 (2019) 167914, arXiv:1411.3585.
[Ma] Marino, Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory. Cambridge University Press, 2015.
[IN] K. Iwaki, T. Nakanishi, Exact WKB analysis and cluster algebras, J. Phys. A: Math. Theor. 47 (2014) 474009.
[DL] F. Del Monte, P. Longhi, Monodromies of Second Order q-difference Equations from the WKB Approximation, arXiv:2406.00175.
[BBV] Z. Bajnok, J. Balog, I. Vona, The complete trans-series for conserved charges in integrable field theories, arXiv:2501.16435.
[KS] M. Kontsevich, Y. Soibelman, Holomorphic Floer theory I: exponential integrals in finite and infinite dimensions, arXiv:2402.07343.
[KaS] M. Kapranov, Y. Soibelman, Resurgence and perverse sheaves, arXiv:2512.22718.
[GKNPS] S. Gukov, P. Koroteev, S. Nawata, D. Pei, I. Saberi, Branes and DAHA representations, Springer Briefs in Mathematical Physics 48.

 
Contact

Jörg Teschner and Roberta Angius.

Winter Semester 2025/26 “Semiclassical conformal field theory, QFT, and gravitational physics”

Results from the semiclassical (large central charge) limits of conformal field theory have recently found some striking applications to quantum field theory and gravitational physics. The mathematical basis for these applications is the observation that several ordinary differential equations on Riemann surfaces such as the Heun equation, in the following called "oper equations", are obtained as limits of the Belavin-Polyakov-Zamolodchikov (BPZ) equations satisfied by the Virasoro conformal blocks. Methods and ideas from conformal field theory can therefore shed new light on classical problems related to the solution theory of the oper equations such as the Riemann-Hilbert problem. It has turned out, on the other hand, that there are several important problems of mathematical physics which require a detailed understanding of the solutions to the oper equations. Two types of problems have recently attracted particular interest. The first type of applications is motivated or inspired by the relations between supersymmetric quantum field theories, conformal field theory, and opers equations discovered by Alday, Gaiotto and Tachikawa, as well as Nekrasov, Rosly and Shatashvili. A second type of applications concerns black hole physics. It is based on the fact that the Heun-equation, an ordinary differential equation on a punctured Riemann sphere, arises in gravitational physics in the study of perturbations of the Kerr black hole solution. The fact that the Heun equation can not be solved exactly represents an obstacle for many directions of research in black hole physics, and many potential applications related to the quasi normal modes, as well as tidal response of black holes and Love numbers. The relations between the Heun equation and conformal field theory have inspired new computational methods for these problems.

The goal of the ZMP seminar would be to introduce both into the mathematical background from conformal field theory and the Riemann-Hilbert problem, and into some of the above-mentioned applications.

For questions about the seminar, or to get involved in any way, please contact Jörg Teschner.

Plan

  1. Overview; What's the mathematical problem? (⁓ 30 min, J.T.?)
    Background: ODE on Riemann surfaces like Heun equation, confluence limits, monodromy, RH-problems, connection problems

  2. CFT-background (⁓ Seminars 1&2, Jonah & Giovanni?)

    • gluing construction of conformal blocks
    • BPZ-equations
    • Fusion and braiding
    • semiclassical limit of conformal blocks

    Literature: Section 3 of [ILT], lecture notes of J.T.'s CFT-course 

  3. Relations to N=2, d=4 SUSY QFT: Seiberg-Wittten theory, AGT-conjecture and Nekrasov-Shatashvili Program
    Literature: [AGT], [HK],

    1. Physics application I: Kerr black holes
      Literature: [BIPT21], [LN22], [CC21], [BBITZ]
      Relation between Kerr black holes and Heun equation: See Section 2 of [BITP21], and references cited therein
      "Trieste formula" for connection coefficients: [BIPT22]
      Simplified derivation of "Trieste formula": Section 2 of [LN22]
    2. Physics application II: Holographic thermal correlators
      Literature: [DGILZ]

Literature

[AGH] G. Aminov, A. Grassi, Y. Hatsuda, Black hole quasi normal modes and Seiberg-Witten theory, Ann. Henri Poincaré 23, (2022), 1951–1977; arXiv:2006.06111 [hep-th].

[AGT] L.F. Alday, D.Gaiotto, Y.Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys. 91, (2010), 167–197; arXiv:0906.3219 [hep-th].

[BBITZ] Y.F. Bautista, G. Bonelli, C. Iossa, A. Tanzini, Z. Zhou, Black Hole Perturbation Theory Meets CFT2: Kerr Compton Amplitudes from Nekrasov-Shatashvili Functions, arXiv:2312.05965

[BIPT21] G. Bonelli, C. Iossa, D. Panea Lichtig, A. Tanzini, Exact solution of Kerr black hole perturbations via CFT2 and instanton counting. Greybody factor, quasinormal modes and Love numbers, Phys. Rev. D105, (2022), 044047; arXiv:2105.04483 [hep-th]. (Key reference on Kerr-Heun-CFT-connection)

[BIPT22] G. Bonelli, C. Iossa, D. Panea Lichtig, A.Tanzini, Irregular Liouville correlators and connection formulae for Heun functions, arXiv:2201.04491 [hep-th]. (Original reference for "Trieste formula")

[CC21] B. Carneiro da Cunha, J.P. Cavalcante, Confluent conformal blocks and the Teukolsky master equation, Phys. Rev. D102, (2020), 105013; arXiv:1906.10638 [hep-th].

[DGILZ] M. Dodelson, A.Grassi, C. Iossa, D.Panea Lichtig, A. Zhiboedov, Holographic thermal correlators from supersymmetric instantons, arXiv:2206.07720 [hep-th].

[HK] L. Hollands, O. Kidwai Higher length-twist coordinates, generalized Heun's opers, and twisted superpotentials, arXiv:1710.04438

[ILT] N. Iorgov, O. Lisovyy, J. Teschner, Isomonodromic tau-functions from Liouville conformal blocks, Comm. Math. Phys. 336, (2015), 671–694; arXiv:1401.6104 [hep-th].

[LN22] O. Lisovyy, A. Naidiuk Perturbative connection formulas for Heun equations arXiv:2208.01604 (Simplified approach to Heun-CFT-connection and "Trieste formula" + rigorous checks)

Contact

Jörg Teschner

Summer Semster 2025 “Global symmetries, cobordisms and quantum gravity”

Among the general properties which every consistent quantum gravity theory in more than two dimensions is believed to satisfy is the absence of exact global symmetries: According to a by now well known conjecture, all symmetries in quantum gravity (in d > 2) are either gauged or ultimately broken. The absence of exact global symmetries has been reformulated more recently as the postulate that the cobordism group of quantum gravity should be trivial. Applied to compactifications of string theory, this conjecture predicts new types of (non-BPS) branes, whose role is to trivialize the cobordism group in the effective field theory.

In this ZMP seminar we will discuss the physics and mathematics around this circle of ideas. On the physics side we begin from a heuristic motivation of the no-global-symmetry conjecture from the point of view of black hole physics, we will review holographic arguments for it in the special case of quantum gravity in Anti-de Sitter spacetime.

On the mathematics side we describe global symmetries and their gauging from a categorical perspective, give a gentle introduction to cobordism groups and how to compute them, as well as their relation to invertible topological field theories. We connect cobordism groups and quantum gravity theories and exemplify how the triviality of the cobordism group of quantum gravity requires the introduction of certain defects, leading to the prediction of new types of branes.

For questions about the seminar, or to get involved in any way, please contact Timo Weigand, Ingo Runkel or Julian Holstein.

Literature

Talk notes

Note Junior ZMP

Note also the Junior ZMP for graduate students and postdocs which meets before the ZMP seminars for free Pizza and informal talks.

Contact

Timo Weigand, Ingo Runkel or Julian Holstein

Winter Semester 2024/25 “Separation of Variables”

Winter Term 2024/25

DESY/UHH

See also the Junior ZMP Seminar

Abstract, including references

More literature

Literature

Babelon, Bernard, Talon: Introduction to Classical Integrable Systems

Sklyanin: The Quantum Toda Chain

Sklyanin: Separation of variables: New trends (solv-int/9504001)

Kuznetsov, Nijhoff, Sklyanin: Separation of variables for the Ruijsenaars system (solv-int/9701004)

Derkachov, Korchemsky, Manashov: Separation of variables for the quantum SL(2,R) spin chain (hep-th/0210216)

Seminars

Seminar 1: Classical SoV 1 (Lukas Johannsen)
Lecture notes 1

Seminar 2: Classical SoV 2: Spectral Curve (Jonah Baerman)
Lecture notes 2

Seminar 3: Classical SoV 3: Action-Angle Variables (Albert Bekov)
Lecture notes 3

Seminar 4: Quantum SoV 1: Sklyanin Measure and Baxter TQ equation (Torben Skrzypek)
Lecture notes 4

Seminar 5: Quantum SoV 2: Modern approach and higher rank (Paul Ryan)
Lecture notes 5

Seminar 6: Q-operators from quantum group representations (Christopher Raymond)
Lecture notes 6
Literature:

Seminar 7: Functional Separation of Variables (Paul Ryan)
Lecture notes 7

Seminar 8: Separation of Variable and the Analytic Langlands Correspondence (Federico Ambrosino)
In this talk I will discuss a crucial role played by the Separation of Variable method in the context of the Analytic Langlands program. Indeed, the SoV transform between sl(2) WZW model and Liovuille CFT provides a “quantum” generalization of the Analytic Langlands correspondence. I will explain how this relation makes Langlands computable via CFT methods and allows us to test and verify many conjectures and results proposed in the mathematical literature. I will finish by commenting on how the relation to the Langlands program may provide crucial in constructing systematically the SoV transform for theories based on different Lie Algebras.
Lecture notes 8

Summer Semster 2024 “Generalised Geometry and Homological Algebra”

The notion of a generalized complex manifold was introduced by Hitchin as a common framework in which complex manifolds and symplectic manifolds can be described from a unified perspective.  Deformations of a projective variety X, considered as a generalised complex manifold, provide an extended deformation theory of X that agrees with the first order deformations of the category of coherent sheaves on X.
In this seminar we will first study fundamentals of generalised geometry and derived categories in order to be able to look at this coincidence. We will then consider further notions of generalised geometry, like generalized Kähler or generalized Calabi-Yau structures, and the generalized Ricci flow, and look at applications to physics, in particular cosmic inflation.

We plan to talk about generalized complex structures and their deformations (2 talks) , derived categories and their deformations (2 talks) and then the generalised Ricci flow (2 talks). Some references are below.

For questions about the seminar, or to get involved in any way, please contact Vicente Cortés or Julian Holstein.

Some References

Talks 1 & 2

  • Marco Gualtieri, Generalized complex geometry, PhD thesis Oxford, 2003.
  • Marco Gualtieri, Generalized complex geometry, Annals of Mathematics 174 (2011), 75–123.
  • Vicente Cortés, Generalized Geometry, MSc lecture course summer semester 2023.

Talks 3 & 4

  • D. Huybrechts: Fourier-Mukai Transforms in Algebraic Geometry, OUP (2006).
  • J J. Block: Duality and equivalence of module categories in noncommutative geometry I, arXiv:0509284
  • M. Gerstenhaber, S. Schack: Algebraic Cohomology and Deformation Theory, Springer (1988).
  • Y. Toda: Deformations and Fourier-Mukai transforms, J.Differential Geometry (2009).

Talks 5 & 6

  • Mario Garcia-Fernandez and Jeffrey Streets. Generalized Ricci Flow. University Lecture Series. AMS, 2020.
  • Jeffrey Streets, Charles Strickland-Constable, Fridrich Valach: Ricci flow on Courant algebroids, arXiv:2402.11069

Further reading on connections between generalized geometry and string theory as well as T-duality:

  • Anthony Ashmore, Charles Strickland-Constable, David Tennyson and Daniel Waldram, Heterotic backgrounds via generalised geometry: moment maps and moduli, arXiv:1912.09981.
  • Gil Cavalcanti and Marco Gualtieri, Generalized complex geometry and T-duality, A Celebration of the Mathematical Legacy of Raoul Bott, CRM Proceedings and Lecture Notes, American Mathematical Society (2010), p. 341--366.
  • David Baraglia and Pedram Hekmati, Transitive Courant algebroids, String structures and T-duality, Adv. Theor. Math. Phys. 19 (3) (2015), 613--672.
  • Vicente Cortés and Liana David, T-duality for transitive Courant algebroids, Journal of symplectic geometry, Volume 21, Number 4, 775--856, 2023.

Note Baby ZMP

Note also the Baby ZMP for graduate students and postdocs which meets before the ZMP seminars for free Pizza and informal talks.

Contact

Vicente Cortés or Julian Holstein

Winter Semster 2023/24 “Knot Homology”

The topic of the ZMP Seminar this term will be Knot Homology. For questions about the seminar, please contact Paul Wedrich.

Seminar Notes

Some References

  • Khovanov, Mikhail. A categorification of the Jones polynomial. Duke Math. J. 101 (2000), no. 3, 359--426 (doi:10.1215/S0012-7094-00-10131-7)
  • Khovanov, Mikhail. Triply-graded link homology and Hochschild homology of Soergel bimodules. Internat. J. Math. 18 (2007), no. 8, 869--885 (doi:10.1142/S0129167X07004400)
  • Aganagic, Mina. Homological Knot Invariants from Mirror Symmetry. (arXiv:2207.14104)
  • Morrison, Scott and Walker, Kevin and Wedrich, Paul. Invariants of 4-manifolds from Khovanov-Rozansky link homology. (arXiv:1907.12194)
  • Stroppel, Catharina. Categorification: tangle invariants and TQFTs. Proceedings of the ICM 2022. (arXiv:2207.05139)
  • Liu, Yu Leon and Mazel-Gee, Aaron, and Reutter, David and Stroppel, Catharina and Wedrich, Paul. A braided (infinity,2)-category of Soergel bimodules. (arXiv:2401.02956)

Contact

Paul Wedrich

List of talks per semester