summary (Isomonodromic deformations and quantum field theory)

National Research University Higher School of EconomicsFaculty of MathematicsPavlo GavrylenkoIsomonodromic deformations and quantum field theorySummary of the PhD thesisSupervisorAndrei MarshakovDr.Sc., professorMoscow – 2018IntroductionIn this thesis I consider the correspondence between isomonodromic deformations andconformal field theory with W-symmetry. First example of it was found by Gamayun,Iorgov and Lisovyy in 2012: they have found that the general tau-function of the PainlevéVI equation can be given as a series expansion over 4-point conformal blocks in c = 1theory.

From mathematical point of view this formula is a series over a pair of Youngdiagrams and one integer number, with coefficients given by some explicit factorizedexpressions (which come from the AGT formula for conformal block). The generalizationof this correspondence to the case of multi-point conformal blocks and tau-functions ofmore Garnier systems with more that four points, which generalize Painlevé VI equation,was found later.Cases which were known before are related to isomonodromic deformations of thelinear 2 × 2 Fuchsian system of the formndΦ(z) X Ai=Φ(z)dzz−zii=1Isomonodromic deformations of this system are such simultaneous changes of Ai and zithat preserve monodromies of solutions around singular points. These transformationsare governed by the system of non-linear Schlesinger equations:[Ai , Aj ]∂Ai=,∂zjzj − zjX [Ai , Aj ]∂Ai=−∂zizi − zji6=jThe tau-function of this system, in some sense, is the simplest object.

It is correctlydefined by its first derivatives:X tr Ai Aj∂log τ =∂zizi − zjj6=iThe natural question is the question about the generalization of isomonodromy-CFTcorrespondence to rank N higher than 2. This thesis deals exactly with such generalization: I show that the tau-functions of general N × N isomonodromic systems are relatedto correlation functions of the primary fields of WN algebras. Some parts of the thesisare devoted solely to isomonodromic deformations in order to make investigation morerigorous and self-contained.

Some parts are devoted to the study of the W-algebras only,but are inspired by their relation to isomonodromic deformations: namely, I present aconstruction of the primary fields of the W-algebra that generalize Zamolodchikov’s field1of dimension 16. Correlation functions of such fields can be computed explicitly with thehelp of some constructions on the branch covers of the Riemann sphere.Study of the isomonodromy-CFT correspondence for the higher rank case is interesting and important due to the several reasons.

One reason is that as for rank two case,it gives explicit formulas for the isomonodromic tau-functions which were not knownbefore. Another reason is that W-algebras are much more complicated than Virasoro algebra: for example, spaces of their conformal blocks start to become infinite-dimensionalfaster than in the Virasoro case. In the W-algebra case even 3-point conformal blocks1form an infinite-dimensional space, and there is no algebraic way to pick some particular element from this space in order to use it in construction of multi-point conformalblocks. Isomonodormy-CFT correspondence gives a way to resolve this ambiguity byfixing monodromy properties of the vertex operators.This thesis consists of six chapters and the bibliography.

Chapter 1 is an introductionto the subject that hopefully should be clear for the non-experts. It gives the definition ofbasic objects of conformal field theory, such as infinitesimal conformal transformations,operator product expansions, Virasoro algebra. Then there are two simplest examplesof conformal theories with W-symmetry, theory of N free massless bosonic fields andtheory of N massless charged fermions. I explain the definition of W-algebras and thedefinition of their vertex operators together with the simplest examples. I also explainfrom utilitarian point of view what is the AGT relation in conformal field theory. Thenthere follows the explanation of what are Fuchsian systems and what are isomonodromicdeformations. After this I present the dictionary of isomonoromy-CFT correspondence,then the definition of Zamolodchikov’s twist fields and their generalization to WN case.Last part of this chapter contains the outline of the thesis, the list of key results, the briefcontents of each chapter, and the list of publications.Chapter 2In this chapter I formulate the main conjecture that tau-function of the general isomonodromic (Schlesinger) system can be given in terms of conformal blocks of the W-algebra.Then I check this conjecture for 3 × 3 case with 4 singular points.

In order to do this firstI study the structure of solution of the Schlesinger system in the limit when two singularpoints collide. It turns out that solution of the system is given by series in fractionalpowers of t (the distance between colliding points). They contain monomials of the formtk+(w,σ) , where w is an integer vector, and σ is some arbitrary complex vector.I check numerically that the structure of expansion of the tau-function matches exactlythe CFT prediction for the growth of powers of t and has the formX(0t)(1∞)e(β,w) Cw(θ 0 , θ t , σ 0t , µ0t , ν0t )Cw(θ 1 , θ ∞ , σ 0t , µ1t , ν1t )×τ (t) =w∈Q×t1(σ 0t +w,σ 0t +w)− 21 (θ 0 ,θ 0 )− 21 (θ t ,θ t )2Bw ({θ i }, σ 0t , µ0t , ν0t , µ1∞ , ν1∞ ; t)I also check that in the cases when we have the definition and explicit formula for Wconformal block, it coincides with function B.

In this case I conjecture and check that3-point functions are given by explicit formula that generalizes Gamayun-Iorgov-Lisovyyresult:Q=ijG[1 −atN(0t)(1∞)Cw(θ 0 , at , σ)Cw(σ, a1 , θ ∞ ) =+ (ei , θ 0 ) − (ej , σ + w)]G[1 − aN1 + (ei , σ + w) + (ej , θ ∞ )]QG[1 + (αi , σ + w)]iwhere G is a Barnes function: G(z + 1) = Γ(z)G(z).In the next chapters I present the proofs of these statements by different methods.2Chapter 3In this chapter we develop the free-fermionic formalism for the W-algebras. We representW-currents in terms of fermions ψ, ψ̃ by the following formula:∞XttN X tk−1+ )ψασ (z − ) =+Ukσ (z)22t(k−1)!k=1ψ̃ασ (zαThen we construct vertex operators for the W-algebra in axiomatic way. Namely, wepostulate that:1) Vertex operator is a fermionic group-like element.2) Its two-particle matrix elements are expressed through the solution of 3-point Fuchsian system.Then we prove that such operators are W-primaries – it relates them to conformal side ofthe correspondence.

We also prove that solution of the Fuchsian system with n singularpoints is given by (z − w)Kαβ (z, w), where Kαβ (z, w) is a two-fermionic correlator inpresence of the vertex operators:Kαβ (z, w) =hθ ∞ |Vθn−2 (tn−2 ) . . . Vθ1 (t1 )ψ̃αθ0 (z)ψβθ0 (w)|θ 0 ihθ ∞ |Vθn−2 (tn−2 ) . . . Vθ1 (t1 )|θ 0 iAt the same time isomonodromic tau-function is given by denominator:τ (t1 , . . . , tn−2 ) = hθ ∞ |Vθn−2 (tn−2 ) . . . Vθ1 (t1 )|θ 0 iIn this way we relate constructed free-fermionic operators to the both side of the correspondence, W-algebras and isomonodromic deformations. This gives the free-fermionicproof of the conjectures from the Chapter 2.We also show that 4-point tau-function can be written as a Fredholm determinant withsome explicit kernel given in terms of hypergeometric functions: τ (t) = det (1 + Rt ).

Thisformula will be the main object of the study in the next chapter.Chapter 4This chapter is written in a pure mathematical language and absolutely rigorously, so itdoes not require any field-theory background. Here we develop the framework in whichthe Fredholm determinant formula, that was obtained in the previous chapter from thefield-theoretic considerations, and even more general n-point version of it, can be proved.To do this first we cut the sphere with n punctures into n − 2 three-punctured sphere,like on the picture below, and then introduce the spaces of functions on the obtainedboundaries.gkgk+1gn-2g1T [1][1]CoutA1Ak-1[k]CinT[k][k]Cout[k+1]AkCing0T[k+1][k+1]CoutAk+1An-3[n-2]CinT [n-2]gn-13Then we construct two projectors onto the space of functions that can be continuedbetween the different boundaries, PΣ and P⊕ .

These projectors are given explicitly bythe formulas of the following kind:1PΣ f (z) =2πiICΣ−1Ψ̂+ (z) Ψ̂+ (z 0 ) f (z 0 ) dz 0,z − z0CΣ :=n−3[[k][k+1]Cout ∪ Cink=1This formula involves the solution of n-point problem given by Ψ̂+ (z), and the formula forP⊕ involves solutions of auxiliary 3-point problems related to different pants. We restrictconstructed projectors to another space H+ (some combination of spaces of positive andnegative Laurent series): PΣ,+ = PΣ |H+ , P⊕,+ = P⊕ |H+ .

Then we define an infinitedimensional determinant−1τ = det PΣ,+P⊕,+which is then proved to coincide with isomonodromic tau-function. After that we showthat this determinant can be rewritten as a Fredholm determinant with the kernel givenby 3-point solutions. It makes it absolutely explicit in the cases when these solutions areknown.We also expand found Fredholm determinant as a series of principal minors and compute each minor explicitly in the 2 × 2 case. This combinatorial expansion has the formof a series over the collection of Young diagrams and integer AN −1 lattices:τ=XXn−2Y~Y~ k−1,QZ Y~k−1~,QkT [k] ,k~ 1 ,...Q~ n−3 ∈QN Y~1 ,...Y~n−3 ∈YN k=1Q~~ k−1Y,Q[k]Furthermore, expressions Z Y~k−1Tin the 2 × 2 case can be written in terms of~k ,QkNekrasov functions, which identifies expansion of the tau-function with the series overconformal blocks.

We also find explicit reduction from the general Fredholm determinantto simpler scalar case considered by Borodin and Deift.Chapter 5This chapter is devoted to the study of twist fields of W-algebra. We work in the bosonicrealization of the W-algebra in terms of N free fields Jα , so its generators are elementarysymmetric polynomials of bosonic fields:XWk (z) ≡: Jα1 (z) . . . Jαk (z) :α1 <...<αkTwist fields are labelled by the elements of the permutation group SN . Rotation aroundthe twist field permutes N bosonic currents, but leaves the W-generators untouched:Jk (qα + · e2πi )Os (0) = Js(k) (qα + )Os (qα )(1)Such construction leads naturally to consideration of the branch cover above the Riemannsphere, whose branching structure is defined by the twist fields:4CFT considerations in this chapter lead us to explicit formula for the conformal blockof such fields, which generalizes Zamolodchikov’s exact conformal block:(q2 )...OsL (q2L−1 )Os−1 (q2L )i = τSW (q) · τB (q)hOs1 (q1 )Os−11LIn this formula τB (q) is so called Bergmann tau-function, and it does not depend onW-charges.

More interesting part islog τSW =12XI,J1 X i jrα rβ log Θ∗ (A(qαi ) − A(qβj ))−2Ii 6=q jqαβi 1/ld(z(q) − qα ) α 1X i 2i(rα ) lα log−2 ih2∗ (q)aI TIJ aJ +XaI UI (r) +qαiq=qαThis expression contains the period matrix of the branch cover TIJ , W-charges in theintermediate channels aI , some extra U (1) charges rαi , some combinations of Abel mapsUI , odd Riemann theta-function Θ∗ , and corresponding holomorphic 1-form h2∗ .