Ordered exponential and its features in Yang-Mills effective action

A V Ivanovand N V Kharuk

1 St.Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, 27 Fontanka, St.Petersburg 191023, Russia

2 Leonhard Euler International Mathematical Institute, 10 Pesochnaya nab., St.Petersburg 197022, Russia

Abstract In this paper,we discuss some non-trivial relations for ordered exponentials on smooth Riemannian manifolds.As an example of application,we study the dependence of the four-dimensional quantum Yang-Mills effective action on the special gauge transformation with respect to the background field.Also, we formulate some open questions about a structure of divergences for a special type of regularization in the presence of the background field formalism.

Keywords: Yang-Mills effective action, ordered exponential, heat kernel, Green’s function,

1.Introduction

Ordered exponentials play a crucial role in mathematical and theoretical physics, see [1, 2], because they have explicit geometrical and physical meanings.Indeed, on the one hand, they appear naturally in the differential geometry, see formulae(10.13a) and (10.17) in [3], as a solution for the ‘parallel transport’equation on principle bundles.On the other hand side,they allow us to make a transition to the Fock-Schwinger gauge condition in the non-Abelian gauge theories[4],and,hence,they are elements of the gauge transformation group.Also, formally,from the mathematical point of view,the ordered exponential is a product integral, which appears in different applications quite frequently [5].Of course, there are a lot of ways to apply the exponentials and an ordering itself, including the theory of integrable models[6],the theory of the heat kernel method[7-9],and many others [10].

In this work, we present some non-trivial properties of the ordered exponential on a smooth Riemannian manifold in the case of a compact semisimple Lie group, see [4, 8, 11].We apply these properties to the pure four-dimensional Yang-Mills theory [12-14] to make some useful conclusions about the properties of the effective action regarding its dependence on the special gauge transformation with respect to the background field.We show that actually, the action depends on the field stress tensor and its covariant derivatives.Also, we formulate several crucial questions about the structure of divergences for a special type of regularization in the presence of the background field formalism,described in section 4.In particular,this would be useful for the renormalization scenario[15-17]in the case of a cutoff regularization.

The paper has the following structure.In section 2,we give the problem statement and formulate the basic properties of the ordered exponentials.Then, in section 3, we present some additional non-trivial properties and prove them.In section 4, we discuss the application of the obtained equalities to the fourdimensional Yang-Mills effective action and formulate actual problems for further work.In the conclusion,we give some useful remarks and discuss the text of the work and alternative proofs.

2.Ordered exponentials

LetGbe a compact semisimple Lie group, and g is its Lie algebra, see [11].Lettabe the generators of the algebra g,where (a= 1,…,dim g),such that the relations hold

wherefabcare antisymmetric structure constants for g, and‘tr’ is the Killing form.We work with an adjoint representation.It is easy to verify that the structure constants have the following crucial property

Let (M,g) be a smooth Riemannian manifold of dimension dimM=d∈N.Then,we introduce two elements equation

Such type of operators have some useful properties,which can be formulated in the form

x,y∈U,whereU⊂Mis a smooth convex open domain,and Greek letters denote the coordinate components.Of course, we assume that the metric components form the symmetric matrix-valued operator, so we havegμν(x)=gνμ(x).The corresponding local formula for the Christoffel symbols of the second kind can be written as

Further,by symbolBμ(x)=Bμa(x)ta,whereBμ(·)∈C∞(U,g)for all values ofμ,we define the components of a Yang-Mills connection 1-form, see [3].The operatorBμ(x)as an element of the Lie algebra acts as a commutator according to the adjoint representation.Hence,from now we considerBμ(x)as the matrix-valued operator with the componentsfadbBμd(x).

For the purposes of the section, we need to introduce several additional objects.First of all, we describe a number of derivatives.Letbe a tensor-valued operator, andbe its matrix components at the pointx, then we define

whereαstands instead of the correspondingith index, and

The main object of the paper, as it was noted in the introduction, is a path-ordered exponential.Let us give an appropriate definition by the following formula

whereγ(·) :[u0,u1]→Uis the parameterized geodesic, see[2],such thatγ(u0)=yandγ(u1)=x,and the dotdenotes the derivative dγ(s)/ dsby the parameter of parametrization.We note that the geodesic satisfies the following differential

where the pointsx,y,z∈Ulie on the same geodesic.In other words, it means there is such a points∈R that the equalityγ(s)=zholds.The proofs of the properties described above can be found in [4, 7, 8].

Then, we need to introduce Synge’s world functionσ(x,y),see [18], as half the square of the geodesic length fromxtoy.It is a symmetric two-point scalar.Let us define some convenient abbreviations, suggested in Synge’s monograph [19], as follows

where the symbol ‘＇’ means that we work with the use of the second argument of a two-point function.Let us note that ifUis a domain of Rdwithgμν(x) =δμν,then Synge’s world functionσ(x,y)is equal to ∣x-y∣2/2.

In the rest of the paper,in all derivatives(4)-(6)we writeρandρ＇ instead ofxρandyρ＇, respectively, because it does not cause any confusion.Therefore, we can formulate differential equations for exponential (7) in the form

The proof can be achieved by the direct differentiation of(7)and the integration by parts,see the papers mentioned above.

3.Additional properties

In this section, we present a number of formulae for representing the path-ordered exponential and some additional properties, which are useful in gauge theories.We start with two series representations, which actually generalize the covariant Taylor representation from[20]on the case of nonzero Yang-Mills connection components.

Lemma 1.Letx,y∈U,and1denotetheunitfunction.Then,undertheconditionsdescribedabove,wehave

Proof.It is obvious that the initial condition φab(y,y) =δabholds, becauseσρ(x,y)andσρ(x,y)tend to zero whenx→y.Let us check that the right hand side of(13)satisfies both equations from (12).We start with the second one.Let us apply the operatorσρ(x,y).Due to the fact that the derivative acts at the pointy,we get the answer instantly after using the following additional property of Synge’s world function

which actually follows from differentiation of2σ(x,y)=σρ＇(x,y)σρ＇(x,y),see formula (2.31) in [19].

Let us move on to the first equation from (12).In this case,we have the derivative at the pointx.So we are going to use the covariant Taylor expansion in the form

It does not follow from ordinary covariant expansion,because the last formula contains both functions,σρ(x,y)andσρ＇(x,y),and we need to decompose the first function in terms of another one.The simplest way to do this is to use the formalism of geodesic lines,successfully applied to prove the covariant expansions,see appendix in [20].First of all,let us rewrite the derivatives of Synge’s world function as

Hence,using the standard Taylor expansion,we get the chain of equalities

The expression in the large parentheses can be transformed in the manner (an analog of (4.112) from [20])

which can be proved by mathematical induction with the usage of(8).Thereby,we obtain statement(16)after applying the second relation from (17).

Further, after performing the differentiation of the right hand side of (13), using the last equality and one additional property of Synge’s world function

we get a number of relations for each degree ofσμ＇(x,y)σμ1＇(x,y)…σμ＇k(x,y)leads to the symmetrization of the tensor in

The last sum is equal to zero.Indeed, the factor large parentheses.This means, in particular, that we can change the order of derivatives.Hence, the first term cancels the others after factorizing the first degree of the Yang-Mills connection.

Therefore, we have obtained the first statement of the lemma.The second equality follows from the first one, the second property from formula(9),the following permutationsa,x↔b,y,and the relationfabc= -fcba.

Notice one more representation for the path-ordered exponential.

Lemma 2.Letx,y∈U,and1denotetheunitfunction.Also,Nr(x,y)andNl(x,y)denotematrix-valuedoperatorsrespectively.Then,undertheconditionsdescribedabove,wehave

Proof.For simplicity,we work with matrix-valued operators.Then,using property(15),we can write the following relation

It is obtained with the use of mathematical induction.Applying the changex↔yand transposition of the matrices,we can get the relation for the left operators.Let us note,that the right hand side of (24) contains the product of commutative (with each other) operators.

Hence, we can rewrite the following chain of equalities for representation (14)

from which we obtain the first relation of the lemma.The second one follows from the changex,r↔y,land the matrix transposition.

From statements (13)and (14)it follows that for smooth connection components we have a covariant Taylor expansion for the path-ordered exponential(7).Moreover,it allows us to write out the answer for the ordered exponential in terms of the fieldBdμand its covariant derivatives.At the same time,we need to emphasize that the first ordinary derivatives (left and right) of Synge’s world function are uniquely related to each other with the usage of covariant Taylor series, see formula (16) forBρd(x) =δρiwith fixedi∈{1,… ,d}.Hence,the degrees ofσρ(x,y)and degrees ofσρ＇(x,y)lead to equivalent expansions for two-point functions, like the degrees of (x-y)ρin Rd.

Now we are ready to formulate the next relation.

Lemma 3.Letx,y∈U.Then,undertheconditionsdescribed above,wehave

Proof.Let us apply the operatorσρ(x,y)∂ρto the left hand side of formula(27).In this case,the derivative acts on every φ-factor.So we get three terms.

Then we need to combine formulae (5) and (12) and substitute-σρ(x,y)fadbBρd(x)φbc(x,y)instead ofσρ(x,y)∂xρφac(x,y).After that we obtain three new terms

Therefore,if we apply property(2)to the first term,it can be converted into the second and the third terms, but with opposite signs.This means we have proven that

Now we need to use the representation for the pathordered exponential derived above, see (14).From this, it follows that we can expand the ordered exponential in the covariant Taylor series in powers ofσρ(x,y),because the fieldBμ(x)is smooth by definition.

Hence,formula(29)means thatfabcaeφ(x,y)φbd(x,y)φcg(x,y)does not depend on the variablex.So we can take any convenient value ofx∈U.If we choosex=yand use the property from (9), then we get the first statement of the lemma.The second statement is the consequence of the first one.

Here we have presented the proof in terms of mathematical analysis.An alternative algebraic proof with an appropriate interpretation can be found in the conclusion.

The last lemma is devoted to a product of three pathordered exponentials.For convenience,we need to define the field strength tensor, components of which are equal to

wherex∈U.

Lemma 4.Letx,y,z∈U.Then,undertheconditions describedabove,wehavethefollowingrelation

where

Moreover,letgμν(·) =δμνfor all points fromU.Also,1,Bμ,andFμνdenote the matrix-valued operators with the elementsδab,fadbBμd,andfadbFμdν,respectively.Also, we define an additional derivative of a special type by the formula Dxμ·=∂xμ· + [Bμ(x) , ·].So, we obtain

where (x-z)μ1…μk= (x-z)μ1· …· (x-z)μk.The first terms in decomposition (30) have the form

where‘three dots’denotes terms,which have the total degree of a monomial more than four.Parentheses denote index symmetrization without division by the corresponding factorial.

Proof.Let us introduce two gauge transformed derivatives according to the formulae

and rewrite the equations from (12) in the following form

Moreover, the combination Ψac(x,y;z)= φab(z,x)φbd(x,y)φdc(y,z)satisfies the following initial condition Ψac(y,y;z) =δac,which is the consequence of (9).This means that Ψac(x,y;z)is the ordered exponential for the connection components of the form(·-z,z).Hence, we have obtained statement (30).Formula (32) is the particular case and has been derived in the paper [8].Then, expansion(33) follows from the results of (30) and the definition from(7) with the use of the explicit form for geodesicγ(s)=y+s(x-y).

Let us write a small summary for this section.We have obtained four results, and it would be useful to emphasize their positions in the general theory.Lemmas 1 and 2 provide new representation formulae for the ordered exponential,where the first one generalizes the very well-known result,mentioned above.Then, lemma 3 is known in the context of the algebraic approach, see Conclusion, but we have presented its new derivation in terms of mathematical analysis with the use of the differential equation.Further, lemma 4 gives a description of the transition to the Fock-Schwinger gauge condition.Actually, this is intuitively and partially known result[8],which we have formulated in such form for clarity and simplicity of various applications.

4.Application to the Yang-Mills theory

Let us introduce an effective action for the four-dimensional quantum Yang-Mills theory [12, 14].For this we use the main conditions of section 2, but with some additional restrictions:M= R4,gμν(·) =δμνfor all points from R4,and the fieldBμ(x)satisfies the quantum equation of motion,which follows from the use of the path integral formulation[14, 21] and the background field method [22-26].This field is called the background field.

For further work we need to define some additional constructions, such as the classical action of the Yang-Mills theory

the following two Laplace-type operators

and the corresponding Green’s functionsG0andG,1which follow from the equalities

with appropriate boundary conditions.These conditions have a physical nature and, actually, are not studied well enough.They should be such that the problem for the quantum equation of motion would be well posed.

Now we are ready to introduce the effective action of the Yang-Mills theory for small values of a coupling constantg.This is a function of the background fieldBμ,which has the following asymptotic expansion forg→+0

where the symbol Λ denotes some type of regularization,such that the Green’s functions are deformed near the diagonal(x～y).Removing the regularization corresponds to the limit Λ →+∞.Using the language of quantum field theory, we can declare thatWn[B,Λ] is the (n+ 1)th quantum correction,corresponding to the (n+1) -loop contribution, see [15, 16].Precise formulae for these corrections can be found in papers[21,27].Fortunately,explicit expressions do not matter in our calculations, that is why we have introduced the effective action in such a general form.

Also, we need to note that the parameter Λ symbolizes some type of regularization, not necessarily a cutoff one,despite the fact that it looks like the standard momentum of cut in the momentum representation.The assumption that the regularization is removed at the limit transition Λ →+∞actually does not matter,because this condition always can be chosen after an appropriate linear fractional transformation.

Let us draw attention that the regularization is necessary because all corrections to the classical action in (39) contain divergent integrals.Some first terms have been studied earlier.For example,the divergent part of‘ln det’has been calculated explicitly in[28-30].The two-loop contribution also has been computed with the use of different regularizations, see [27,30-32].

Then, we should note that the corrections from the last sum can be constructed with the use of the integration operator over4R and the following elementary blocks (with the regularization applied)

which are connected to each other only by the following combinations of the structure constants:

It is quite important to note two conditions.Firstly, all the variables should be under the integration.Secondly, if three(four)basic blocks are connected to each other with the usage offabc(fabefecd), then the corresponding ends of the blocks have the same variables.Other combinations do not exist.Also, we have noted above that the blocks from (40) should contain regularized Green’s functions instead ofG0andG.1Now we want to introduce one restriction on a type of regularization.We assume that regularized Green’s functions respect covariance.This means that after applying the gauge transformation, we obtain

and the regularized Green’s functions get the following changes

whereω(·)∈C∞(R4,G).Of course, non-regularized functions have this property by construction.Using the fact that in further calculations we want to make a special type of gauge transformation, we conclude that for simplicity we can produce all manipulations with (40) instead of the regularized ones because the deformation does not break the covariance.Also, we need to draw attention that such regularizations exist.As an example, we can suggest the explicit cutoff regularization in coordinate representation, recently studied in [33].

For further study, we need to give some basic concepts and results on the heat kernel expansion, [20, 34-36].Let us introduce a Laplace-type operatorA, which has a more general view than one from (37).Locally, it has the following form

whereIis ann×nunit matrix withn∈N, andvab(x) is ann×nmatrix-valued smooth potential, such that the operatorAis symmetric.If we taken=4,(I)μν=δμν,and(vab)μν(x)=2facbFμcν(x),then we obtain the operator(x).Also,for convenience,we will not write the matrixIin the rest of the text, because this does not make any confusion.

Then we move on to a definition of the local heat kernel.As we know, the standard heat kernel is the solution of the problem

paired with boundary conditions for (45).At the same time,the local heat kernel [37] is defined as a solution of (46),which for small values of the proper timeτ→+0 has the following asymptotic expansion

We note that an asymptotic expansion for the ordinary heat kernel ofAcan be presented as the sum of (47) and some additional corrections.The coefficient aab(x,y)of expansion(47) is the Seeley-DeWitt coefficient (also, it is named after Hadamard, MinakshiSundaram [38], and Gilkey [35]), see[39,40],and can be calculated recurrently[7,8,35],because they satisfy the following system of equations

Then,we are going to make a substitution.Letz∈R4be an auxiliary point, that will play a role of an additional parameter.Then, we define

Further,we substitute formula(49)into relation(48)and introduce gauge transformed operators, according to the papers [8, 9], in the form

where we have used the Fock-Schwinger connection components (32) and formula (37).Hence, the system of the equations from (48) can be rewritten in the following form

Now we are ready to note a piece of very useful information.It follows from formulae (32), (34), (35), (50), and (51),that if the potentialvis a function only of the field strength and its D-derivatives,then the decomposition of the Seeley-DeWitt coefficients ackb(x,y;z)can be constructed by using only the field stress tensor and its D-derivatives.As we can see, the operatorsM0abandfor the Yang-M ills theory are suitable for this statement.Additionally, we emphasize that formulae(48)-(52) are valid for all dimensions.

Further,using the formulae introduced above,we can write out the following asymptotic behavior near the diagonal for the Green’s function in the four-dimensional space [39, 41]

where

Nazbis a non-local part [41], depending on the boundary conditions of a spectral problem, and Z Mazbis a number of local zero modes to satisfy the problem.Let us separately note,it was shown in the paper[42],that an infrared part in the second loop does not depend on Z Mazb.Moreover, in the calculation process, we can choose Z Mazbin such a way [27], that the nonlocal partzabN would have the following behavior near the diagonalx～y

Now we move on to the property mentioned in the introduction.For this we should show the possibility of the following replacementfadbBμd(·)→fμab(·-z,z)in the effective action,wherez∈R4is an auxiliary point andfμabis from (32).The procedure can be carried out with the application of the gauge transformation using the ordered exponential φab(· ,z).In the case we have formulae (34), (35), and

Using the last substitutions and formulae(38),(50),and(51),we can conclude, that the new covariant derivativesandand the Green’s functionsG0ab(xi,xj;z) andare constructed only with the use of the following connection components,see (32), which are functions of the field stress tensorFμνand its D-derivatives.

At the same time after the gauge transformation, every building block from (40) is multiplied by the ordered exponential φea(xi,z)on the left hand side and by φbg(z,xj)on the right hand side.Further,let us remember,that the top indices of the blocks should be connected to each other by thefabcorfabefecd.Hence,using the results of lemma 3,we can exclude all additional factors φea(xi,z)and φbg(z,xj)from the blocks.Therefore, we obtain the following equalities

where the functionals actually do not depend onz.In other words,the procedure described above can be understood as a re-expansion of the blocks near some auxiliary point.It should be noted that in the calculation we have used only covariance of the regularized Green’s functions(43)and(44).We did not assume any additional gauge conditions.The socalled Fock-Schwinger gauge(32)was generated inside eachWnby substituting the unitI=ωω-1between all adjacent blocks.Additionally, we want to emphasize that we have proven a more general result (58).It is correct for each Feynman graph separately.

In the same manner, we replace the operators in ‘ln det’by the gauge transformed ones.Then, in particular, using the invariance of the classical action with respect to the gauge transformations, we get

In the last part of the paper we want to formulate some actual open questions and make several useful remarks about divergencies searching.It is known that infrared divergences in the coordinate representation (or ultraviolet in the momentum one) follow from the appearance of non-integrable densities in (39), because the behavior (53) of the Green’s function near the diagonal (x～y) includes such functions as (54).

At the same time, decomposition (53) is applicable only in the vicinity of the diagonal,so we need to split the Green’s function into two parts, in some neighborhood of thex=yand out of it.Formally, we can write this as

whereμ1> 0is an auxiliary fixed finite number of momentum dimension, andχ(inequality)is the Heaviside step function of a domain from R4.While for the first part,we can write the standard decomposition (53) near the diagonal, the second part, actually, is unknown to us, because it and its properties depend on the boundary conditions and the form of the background field.Anyway, we assume that the ‘out’ part has a good enough behavior at infinity, so the effective action does not include other types of divergences.Otherwise, we need to introduce one more regularization and study these non-infrared singularities in the coordinate representation.

Now we are ready to formulate some questions.

· Does the effective action contain the infrared singularity in coordinate representation,depending on the‘out’part?For the positive answer, it is sufficient to provide an explicit expression for someWn(B, Λ) .We draw attention that the two-loop contributionW1(B,Λ) includes only singularities, related to the ‘in’ part because in this case the quantum correction is split into a sum of ‘in’ and‘out’ parts, without mixing.Hence, without any fundamental justifications and assumptions, we are interested in the three-loop calculation.Moreover, it would be meaningful to describe the structure of such contributions for a cutoff regularization.

· Does the effective action contain the infrared singularity in coordinate representation, depending on the non-local part Nab?This is one more non-obvious issue.The fact is that in the two-loop contribution,a special combination of nonlocal terms occurs,but it is converted to the local part with the use of the Seeley-DeWitt coefficients.The appearance or absence of the non-local part in the multi-loop terms is quite an interesting challenge because it leads to the possibility of using simplifications in the calculations.

The last two questions are related to the form of the coefficients,their locality and dependence on the boundary conditions.However,we have a problem,corresponding to a structure of singularities itself.Indeed, in the quantum corrections after the regularization introduced, we have some dimensional parameters: Λ is the parameter of regularization,μis the parameter from (54), controlling the presence of local zero modes in the Green’s functions,andμ1is the auxiliary parameter from(60),which actually does not lead to any new dependence.Hence, the answer contains some dimensionless combinations of them.The question is the following.

· Leti,j∈N ∪{0 }andn>0.What combinations of(Λ/m)ilnj(Λ/m)does the correctionWn(B,Λ) contain,wherem∈{μ,μ1}is a dimension parameter? For example, it was shown [27, 32] that the two-loop correction includes only ln(Λ/μ).And it is expected that for other orders we havei= 0andj∈{1,…,n}.

If the last hypothesis is correct,then for the local singular contribution, depending on the ‘in’ part, we have only one type of the coefficients with non-zero trace

5.Conclusion

In this paper,we have studied some non-trivial properties of the path-ordered exponentials.Additionally, we have shown that such equalities give the ability to make some useful conclusions about the structure of the four-dimensional Yang-Mills effective action.We believe that the result can be useful both in the general theory of renormalization of the Yang-Mills theory and in the investigation of some other models with non-Abelian structures.For example, in the sigma-models.

Furthermore, we would like to emphasize the relevance of our study and comment on some ways of further development.It should be noted that lemmas in section 3 are new and they provide the possibility to analyze multi-loop calculations in gauge theories, including Yang-Mills theory on Riemannian manifolds.The results of section 4 are devoted to one example of applying the obtained formulae for the ordered exponential to the effective action.Shortly, it can be written using two relations (58) and (59).From a physical point of view,our result guarantees covariance of the singular parts and, in particular, shows the absence of the termBμBμ,the so-called mass term, after the renormalization.

Also, we need to repeat one more time a very important observation.The covariance under the gauge transformations of the Feynman diagram blocks actually does not exclude nonlogarithmic divergencies.To formulate the proposition in a stronger form,we need some additional investigations which are not from the scope of this work.They are formulated in the previous section as open questions.Also,we emphasize that the standard gauge invariance of the Yang-Mills action (with the ghost fields) is not connected with the transition to the special gauge, generated in the effective action by hand - they are different.Indeed,the first one is the symmetry of the action,which can be violated by the regularization, while the second one follows from the covariance of the deformed Green’s functions with respect to the background field(not the Yang-Mills field).

In addition, we need to comment on another way to prove lemma 3.Let us introduce three auxiliary arbitrary elementshi=hiata,wherei=1, 2, 3,of the Lie algebra g.Then, an elementgof the Lie groupGacts on this algebra according to the adjoint representation asAdg(hi) =ghig-1.Letgabdenote the corresponding matrix elements, such that(Adg(hi))ata=gabhibta.After that we can write the following chain of equalities

from which we obtain an alternative proof.Despite the fact that this derivation looks shorter, we note that it includes the usage of some additional properties, presented in(9).Indeed,we need to make sure that there is such an elementginGthat the relationgab= φab(x,y)holds.So, we believe that the proof of lemma 3, written above, is more elegant and instructive in the context of covariant Taylor expansions.

Let us give some comments about the Yang-Mills effective action.In the paper, we have used the definition without using the path ordered formulation, because the last construction contains a lot of mathematical questions not related to the main topic.Due to this, we did not face the discussion of the measurement properties and their existence.Also, we study the effective action itself,without any counterterms,because we are interested in the properties of ‘pure’ effective action before applying the renormalization procedure.Of course, the main propositionB→f(·-z,z)can be expanded on the standard counterterms,because,according to the general theory,they also have the covariance.Unfortunately, some regularizations break the gauge invariance,and this leads to the necessity of restoring this symmetry and,as a consequence,adding some extra terms.We cannot guarantee that these terms(additional building blocks of the diagram technique)have covariance because they depend on a concrete view of the deformation.It would be useful to conduct an additional investigation on the issue.

Acknowledgments

Both authors are supported by the Ministry of Science and Higher Education of the Russian Federation, agreement 075-15-2022-289.Also, A V Ivanov is supported in parts by the Foundation for the Advancement of Theoretical Physics and Mathematics ‘BASIS’, grant ‘Young Russian Mathematics’.

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