IDEAS AND SURPRISES OF MATHEMATICAL PHYSICS IN SIMPLEST EXAMPLES
Course by Andrey Losev, 

Mathematical Department of HSE Research University and System Research Institute, Moscow. 

e-mail：aslosev2@yandex.ru

Mathematical physics is a theoretical science that studies worlds that are somehow similar to the world we live in. 

As a piece of physics it is a laboratory developing theoretical models that we would like to compare to observations in physics. Even if the model is definitely different from the real world it still is useful since it could contain phenomena similar to what we observe in the real world. In the model world such phenomena could look simpler and allow exact theoretical study.

As a piece of mathematics it may be considered as geometry 2.0. Actually geometry started from studying the shapes that we meet in our world but later it turned on that its object of study are all possible shapes (spaces). We call mathematical physics as geometry 2.0 because it generalizes from shapes of the world to laws of the world (note that shape is one but not the only one law of the world).

Just like geometry is unified with algebra Mathematical physics shares many ideas with modern mathematics, not only using them but providing different view on these ideas. Sometimes it even goes ahead of pure mathematical development.

This course is aimed to give an introduction to some ideas of mathematical physics that surprisingly intertwine with ideas of algebraic geometry, supergeometry and homological algebra. I will also give a brief overview of developments of ideas of modern physics and explain how mathematical physics may be applied in real physics and in development of theoretical physics. I will also outline how ideas and models of mathematical physics could lead to new phenomena in mathematics and show it on simplest examples.

We would start from the basics so the 3-4 year students can attend this course. Prerequisites include calculus and linear algebra including classical Lie groups such as SO(d) and U(d). No knowledge of physics is required. In considering examples I would often say – in such and such conditions the construction describes such and such notion or phenomena in physics. 

In this way we gradually will get up to the Standard Model coupled to gravity. Standard Model is  not a model but a tested theory of fundamental laws of the Universe. The history of it’s discovery explains the name. It was invented as a model among other models. Later this model turned out to fit experiments better than other models and it was called standard model to distinguish it from competitors. Finally, it was tested and other models disappear.

Students from both sides, from theoretical physics and mathematics are very welcome. The aim of this course is not only to teach some stuff (that student can apply in his research) but to get students the feeling of the subject and possibly to get them interested in studying it in more detail.  

To get students interested I will not only illustrate everything that I say in simplest examples, but I will do my best to surprise students by unexpected analogies between pure mathematics and theoretical physics.

The course will be given by zoom (zoom: 798-399-4886), twice a week. 

On Tuesdays we will start at 16.00 Beijing time and work until 18.00 (11.00 – 13.00 Moscow time)

On Thursdays we will start at 15.00 Beijing time and work until 18.00 (10.00-13.00 Moscow time).

Extra topics for interested students would be discussed on Fridays, time would be decided.

The first lecture would be on 11 of November (Thursday) and the main program will end around 30 of May (there would be no lectures during Chinese New year vacations).

In summer I am planning to cover extra topics for interested listeners.

To enter the course one has to register at Euler Institute site.

Videos of lecture will be available, and I hope to make notes such that it would be possible to catch up if you have to miss the course.

Highlights of the course.

To get you a feeling about the course I picked up 27 statements from the list of what I call surprises. 

Please take a look. If you know the concepts I am talking about you may decide if it surprises you.

If you do not I will explain these concepts in the course and I hope then you will be able to evaluate surprise. 

1.Newton laws (as published) are incomplete and missing the axiom of material point.

The notion of Force mistakenly excludes theories with nonbinary potentials.

2.The first consideration of theoretical physics (Galileo argument on the acceleration of bodies

Falling on Earth that he considered universal) is actually false in General relativity because it

ignored (due to notion of Force) nonbinary effective potentials. We actually see these potentials in the motion of Mercury on the sky.

3.Newton’s mechanic is not a classical physics, it is a preclassical physics. Classical physics is an extremal action principle.

4.The classical theory of point particles and fields does contain ultraviolet divergences that have to be renormalized.

5.Figures are not collections of points.

6.In Einstein-Cartan formulation of gravity fundamental degrees of freedom (coframes) behave as quarks (and could be called graviquark) while metric is like a meson (white object constructed from two graviquarks)

7.In Einstein-Cartan description one can finally answer a question what spins in electron spin.

The graviquark gets condensed and the only combination of truly rotational symmetry and global part of “gauge” symmetry is preserved. Invariance under this symmetry is related to

conservation of rotational momentum.

8.Berezin odd integral can be understood by looking at integrals of functions on a circle with invariant measure.

9.Take a nondegenerate top form on X. It identifies polyvectors with differential forms by contraction with the top form. Then map from forms to polyvectors may be considered as

an odd Fourier transform, De Rham vector field (first order operator) goes to BV operator (second order operator). It was missed by mathematicians since they considered canonical operations while BV operator depends on the choice of the top form. Action of BV operator

on polyvectors generalizes the notion of divergence of a vector field (previously understood

as a Lie derivative of the Log of the top form).

If we use metric to identify polyvectors and differential forms, odd Fourier transform becomes

 Hodge * operation, while BV operator would become d*.

10. Maxwell equations in all dimensions can be formulated in 3 letters d*dA=0 where A-is a 1-form.

Equivalently, as a system of two equations dF=0 and d*F=0. Electromagnetic duality for Maxwell

Theory in 2 dimensions can be written as a change F_D=*F, where F_D can be written as dA_D,

So A_D is 1-form. Generalization of this duality to arbitrary dimension and arbitrary degree of the form A shows that k-forms in dimension D are electromagnetic dual to D-k-2 forms in dimension D. In particular, Maxwell theory for 3 dimensional spacetime has a dual description

In terms of scalars (0-forms), that is an ether description!

11.Quantum field theory (in Dirac-Segal formulation)  is a monoidal functor from cobordisms to monoidal category of vector spaces.

12.Actually quantum theory contains no Plank constant, Plank constant is a description of how close quantum theory is to a classical one. So quantization is not so natural, the natural process is “classization”.

13.In most cases Plank constant is an inverse of an integer, it gets dimension due to dimensional transmutation. Like natural coordinates on a sphere do not have dimension, coordinates we use got dimensions only because of existence of the radius of the sphere. 

14.Higher topological theories are those where category of vector spaces is replaced by category of

complexes, functions of geometrical data are promoted to differential forms and the functor is closed.

15.All quantum mechanics can be completed to topological ones, this is a Polyakov completion.

16.Canonical higher differential in spectral sequence for bicomplex and canonical Massey  operations (as described in the textbooks) are partially defined on cohomology of the first differential. Higher differentials are defined on cohomology of preceding higher differential and Massey operation is defined only if some products of cohomology vanish. However, they can be refined in a way, depending on contracting homotopy. Then they are correctly defined and form infinity structure, while canonical operations correspond to moduli of such operations under the action of natural symmetries acting on higher operations.

17.Higher operations in homological algebra are close analogues of scattering amplitudes in quantum field theory. Namely, scattering amplitudes are higher operations in Polyakov higher topological Quantum Mechanics.

18.Gauge theories contain maps to algebra Lie with inversed parity, the inverse image of linear coordinates are Faddeev-Popov c-ghosts

19.In the process of gauge fixing we integrate over conormal bundle to gauge fixing constraints,

Linear coordinates along the fibers are Faddeev-Popov b-ghosts.

20.Mathai-Quillen integral representatives of delta-forms on solutions to Bogomolny equations in different dimensions are topological theories obtained from supersymmetric theories by A-twisting, the smoothening of the delta-form is a coupling constant in these theories.

21.Scalar fields in d=4 N=2 Super Yand Mills theory are Faddeev -Popov ghosts for gauge fixing of odd counterpart of the gauge symmetry. That is why they are even.

22.The BV formalism implies the BV generalization of the concept of symmetry, called L-infinity symmetry, that maps the external algebra of the Lie algebra to polyvectors on the space X,

where constant is mapped to invariant function. The L-infinity conditions mean that vector fields represent Lie algebra up to derivatives of S. This phenomena was first discovered in supersymmetric theories and was called closeness of the symmetry algebra on-shell.

23.The notion of complex structure determined by Beltrami differential is a particular case by a more general notion determined by polyvector with values in (0,*) differential forms, so Beltrami corresponds to vector field with values in (0,1) forms. The much simpler case corresponding to functions was originally discovered by physicists and was called superpotential. The complex Hodge theory can be generalized to a general case with superpotential is called N= 4 Supersymmetric Quantum Mechanics.

24.All known Lagrangian topological field theories in different dimensions can be considered as particular cases of AKSZ model that is a Q-manifold of maps between two Q-manifolds.

25.Tropicalization of algebraic geometry being applied to functional integral leads to classical theory.

26.In Polyakov string theory the notion of the space-time equipped with the geometry satisfying Einstein-like equation is replaced by the concept of conformal field theory. The space-time becomes the emergent, i.e. asymptotic concept. Families of conformal field theories can have

different geometric limits that is revealed, in particular, in the phenomena of mirror symmetry. 

27.From the point of view of topological quantum field theory the number of degree 1 maps from the genus 3 surfaces to CP^1 equals to 8.

Plan of the course (version 1)

As you may expect, the logical structure of the course is Y-shaped. One horn comes from mathematics and another from physics. At the beginning they look separated at the pretty far distance and have nothing to do with each other. Still I have to explain both horns and at the beginning some lectures would be called “mathematics” and some  “physics”. Later horns would come closer and it will turn out that physics actually heavily uses the language developed in the “mathematical” horn. First it allows to write down simply looking formulas, later it becomes the necessary language to formulate concepts. At some moment horns join and I will use concepts from both horns together. However, time is linear, so I would start by giving one lecture from the “mathematical” horn (on Tuesdays) and one from “physics” horn (on Thursdays). When they join I will use both languages. Still, in describing phenomena I will try to explain it as physicist would do and also as a mathematician would do. In this way I will try to teach you how to “translate” from one language to the other.

MATHEMATICAL HORN.

1.Basics of categories and functors.

1.1.Motivation and definition of category

1.2.Examples: sets, linear algebra, smooth spaces, paths

1.3.Semigroups and groups as categories with one object

1.4.Definitions of covariant and contravariant functor

1.5.Monodromy as a covariant functor

1.6.Contravariant functor between vector spaces and dual vector spaces

1.7.Contravariant functor between spaces and algebras of functions on these spaces

1.8.Categories and functors in mathematical physics are used: in definition of affine schemes and superschemes, in Dirac-Segal’s definition of QFT, in D-branes in string theory (open string amplitudes are (higher) compositions in A-infinity category).

2.Algebra-geometric correspondence

2.1.Ideals, why they have such a funny name and how do they correspond to figures

2.2.Operations on ideals and corresponding operations on figures

2.3.Maximal ideals as points and how to understand the double root of equations

Surprise: figures are not collections of points

2.4.New spaces, corresponding to algebras with nilpotents, double point as an example

2.5.Differentiations of the algebra as a replacement of the notion of vector fields, differentiations as module over the algebra (very bad name while good notion), Lie algebra of differentiations, examples for simplest schemes

2.6.What kind of spaces do appear in physics, why smooth manifolds are not enough

3.Superalgebra and supergeometry, I, general topics.

3.1.External algebra of the finite dimensional vector space and duality between components of complement degree

3.2.Applications: basis free definition of determinant and trace, meaning of “vector product” in dimension 3, magnetic field in 3 dimensions as element of the external square and geometry of Lorentz force.

3.3.Concept of Z_2 grading and supercommutativity of the external algebra,

Z_2 graded supercommutative algebras

3.4.Affine superspace as a geometric object, corresponding to supercommutative algebra

3.5.Even and odd superdifferentiations, simplest examples of superdifferentiation of external algebra, analogue of Jacobi identity for superdifferetiations and notion of superalgebra Lie. Simplest examples.

3.6.Application: Dirac problem of taking square root out of Dalambertian, gamma matrices and spinors in term of external algebra. Why they were called spinors – story of a severe puzzle in the physics solved by mathematicians.

3.7.Superalgebra of supersymmetries as most well-known superalgebra Lie, representation of superalgebra as differentiations of the superspace.

3.8.Homological vector fields, example on the external algebra – surprise – quadratic vector fields are in one-to-one correspondence with ordinary Lie algebras 

(Chevalley differential). 

3.9.Quadratic homological vector fields on superlinear space (corresponding to polynomials times external algebra) correspond to superalgebra Lie.

3.10.Differential  and cohomology in general : Z_2 graded vector space and odd operator that squares to zero. Explanation of the origin of strange names (“differential” and “cohomology”) would appear later.

3.11.Linear homological vector field on superlinear space as the simplest example,

Linear and quadratic homological vector field as differential graded superalgebra Lie.

3.12.If we make polynomial change of coordinates we will get polynomial homological vector field. In general, we may consider germs of the homological vector fields, such objects are called L-infinity algebras. In physics, Yang-Mills equation (we will meet it later) together with the gauge transformation may be considered as a homological polynomial homological vector field with the third degree of polynomial. Much more general homological vector fields are represented by tree level scattering amplitudes (that we will consider later).

3.13.Special homological vector fields a superlinear space multiplied by NxN matrices.

Associativity condition, differential graded associative algebra and A-infinity algebra.

Graphical representation in terms of ribbon trees (and Gershtenhaber bracket).

3.14.Berezin integral over odd linear supermanifold and comparison with the integral of functions over a circle in U(1) invariant measure. Odd delta-function and odd Fourier transform. Integral of exponent of quadratic expression in odd variables. 

4.Superalgebra and supergeometry, II.  Differential forms.

4.1.  Oriented manifolds, boundary operation on submanifolds, square of boundary operator equals zero.

4.2.    Integration of differential forms over an oriented submanifold and

 De Rham operator as operator conjugated to the boundary operator, 7 authors theorem

4.3.   Supermanifold T[1]X as dual to algera of differential forms, De Rham operator as an odd vector field,

Lie derivatives and contractions of vector fields as even and odd differentiations of T[1]X, Cartan formula

4.4.   Homology and cohomology of De Rham operator, relation to topology

4.5.    Nondegenerate top differential form, geometrical meaning of divergence,

algebra of polyvector fields and its identification with algebra of differential forms on the spaces with nondegenerate top form,

BV operator as the image of De Rham operator under this correspondence

4.6.    Schouten bracket on polyvectors as a generalization Lie bracket on vectors

4.7.   Metric and another identification of differential forms and polyvector fields, Hodge * operation,  BV operator becomes d* operator

4.8.    Maxwells equations in 3 letters and electro-magnetic duality,  their generalizations to different dimensions, quasi-ether theory in 2+1 dimensions  

4.9.    Harmonic forms on compact manifolds and their equivalence to De Rham cohomology (Real Hodge theory or geometric example of N=2 SQM)

4.10.  Overview of complex generalization - Kahler and Calabi-Yau manifolds,  Dolbeault operator and complex Hodge theory (geometric N=4 SQM)

4.11. Supersymmetric Quantum mechanics and Extended Supersymmetric Quantum Mechanics, superpotential, examples and generalizations

4.12. Integration of differential forms along fibers (direct image), closeness of the integral

4.13.  Mathai-Quillen representation of the smoothed delta-form on cycles, interpretation in terms of integral against the supermanifold

4.14.  BV action, classical and quantum BV equations

4.15.  BV integral over Lagrangian submanifolds

4.16. BV integral as a Fourier image of  direct image

4.17. Application: integral proof that inverse of the invertible sнmplectic form is Poisson bivector

4.18. Symmetric systems in BV language, polyvectors and on-shell closeness

4.19.  BV integral and construction of polyvector on-shell actions

4.20.  Applications to supersymmetry in dimension 0

4.21.  Quantization of symmetries, anomalies, examples

HORN OF PHYSICS

0.Phenomenology SU(3)xSU(2)xU(1) Standard model: a bit of handwaving

1.Development of ideas in physics up to classical physics

1.1.Preclassical physics: Newton’s laws, its incompleteness and restrictions

1.2.Classical physics – fields coupled to extended objects, extremal action principle, Euler-Lagrange equations of motion

1.3.Classical mechanics of a particle, Hamiltonian an symmetries, symplectic Potential and Lorentz force

1.4.Relativity and nonrelativistic limit, Lorentz force

1.5. Symplectic Potential and Poisson mechanics 

1.6. Hamiltonian and symmetries

1.7.Gauge theories and Yang-Mills equations

1.8.Gravity in Hilbert-Einstein formulation

1.9.Gravity in Einstein-Cartan formulation, coframe as graviquarks and metric components as gravimesons

1.10.Spinors in physics, coupling to gravity, torsion exchange 

2.Quantum mechanics

2.1.Space of states and Schrodinger’s equation

2.2.Observations and probability, Heisenberg’s equation

2.3.Representation of SU(2) as simplest example

2.4.Classical limit and appearance of  the phase space as a spectrum

2.5.Classical limit of Heisenberg equations

3.Segal’s treatment of QFT as a functor 

3.1.  Symmetric monoidal functor from cobordisms with geometrical data

to linear algebra.

3.2.  Figure observables and deformation of QFT by local observables

3.2.  Dirac solution to Segal’s axioms in one dimensional case, Hamiltonian

And deformations

3.3. Theories with quadratic actions and solutions to Segal axioms, examples

In different dimensions.

3.4. Vortexes in different dimensions

4.Functional integral (Feynman) approach.

4.1.Definition and Feynman diagrams

4.2.Divergencies

4.3.Feynman integral in gaussian theories,

Determinants (as D-modules), example of harmonic oscillator

4.4.Partition function of particle on a circle, Poisson resummation formula predicted and proved.

4.5.Two-dimensional gaussian theory with circle as a target space,

4.6.T-duality

4.7.Sigma-models and geometric limits of OFT, T-duality as an example 

4.8.Free chiral theories in dimension 2

4.9.Chiral algebras and energy-momentum tensor

4.10. Conserved charges equations as Segal-like approach to 2 dimensional QFT.

4.11.Modular invariance of partition function in gaussian models

4.12.Standard model and Higgs phenomena

5.Higher topological theories

5.1.Segal’s definition of higher topological theories

5.2.Topological quantum mechanics, general expression

5.3.Laplacian and  Cartan topological quantum mechanics

5.4.Polyakov topological quantum mechanics

5.5.Deformations of topological quantum mechanics and obstructions

6.AKSZ formulation of QFT

6.1.AKSZ solution to Master Equation, examples

6.2.AKSZ containing Lie algebra with inversed parity as a gauge theory

6.3.Gauge fixing as a choice of Lagrangian submanifold, b-ghosts as coordinates in the fiber of conormal bundle

6.4.One-dimensional AKSZ theories, gauged Quantum Mechanics and

Integral representation for the number of states

6.5.BF , Chern-Simons and Landau-Ginzburg theories as examples of AKSZ

6.6.Type A and Type B theories in different dimensions

6.7.Correlator of closed zero-observables in type A in 2 dimensions,

Quantum cohomology

6.8.Correlator of closed zero-observables in Landau-Ginzburg model 

6.9.Handle gluing operator, explicite computations, surprise with 8 maps

from genus 3 surface to CP^1 with degree 1.

6.10.Nontopological theories as AKSZ theories in external backgrounds

7.Instantonic theories beyond cohomology

7.1.Instantonic-like equations in dimensions 1,2 and 4

7.2.Mathai-Quillen representatives for QFT of type A, coupling constants as smoothening of the delta-form

7.3.Deformed instantons and definition of all correlators in terms of finite-dimensional space over slightly deformed instantons

7.4.Cartan topological quantum mechanics as instantonic quantum mechanics

7.5.Two-dimensional Gromov-Witten instantonic theory

7.6.Four dimensional gauge instantonic theory as a version of Yang-Mills theory, appearance of scalar fields as ghosts

8.Lagrangians of supersymmetric theories in different dimensions

8.1.Superfields – chiral, twisted chiral and vector.

8.2.Supersymmetric Lagrangians – D-terms and F-terms

8.3.Elimination of axillary field and generation the on-shell supersymmetry

with byvectors

8.4.General form of Lagrangian  of (2,2) supersymmetric sigma-model,

free field examples

8.5.Gauged sigma-models and supersymmetric gauge theories,

R-symmetry, free field examples

8.6.A and B twisting of supersymmetric theories, Gromov-Witten instantonic

theory as an A-twisted theory, Donaldson-Witten A-twisted theory as a deformed instantonic theory.

9.A bit of strings

9.1.String theory as a higher topological conformal theory in two dimensions

9.2.Observables, descendants and amplitudes

9.3.Gromov-Witten string theory, geometrical meaning of amplitudes

9.4.Polyakov string theory