Диссертация (Isomonodromic deformations and quantum field theory), страница 38

However, the formula (6.144) is valid for any affine Kac-Moody algebra.Below we consider the case G = SO(2N + 1), where the level k = 1 representationscan be constructed using free fermions.(1)SO(2N + 1), N > 2 case. Root system is BN (affine BN ), the dual root system is(1,∨)(2)BN = A2N −1 (affine twisted A2N −1 )Simple roots: α0 = δ−e1 −e2 , αi = ei −ei+1 , 1 ≤ i ≤ N −1, αN = eN .Simple coroots: α0∨ = K − e∨1 −e∨2 , αi∨ = e∨i −e∨i+1 , 1 ≤ i ≤ N −1, αN = 2e∨N .Real coroots: 2mK±2ei , mK±ei ∓ej , mK±ei ±ej , 1 ≤ i < j ≤ N, m ∈ Z.Imaginary coroots: (2m − 1)K of multiplicity N − 1, m ∈ Z2mK of multiplicity N , m ∈ Z \ {0}.XNeik = 1 weights: Λ0 , Λ1 = Λ0 + e1 , ΛN = Λ0 + 21i=1XNXNh = 2N, ρ =(N − j + 12 )ej + (2N − 1)Λ0 , ρ =(N − j + 1)ej .j=1j=1(6.155)Compute again the denominatorY 1−q(ρ,α∨ )2Nmult(α∨ )α∨ ∈∆∨+=∞Y(1 − qk2Nk=1N) ·∞Y(1 − q2k−12N)(6.156)k=1and the numerator in the formula (6.144). Now the numerator for Λ = Λ0 and Λ = Λ1is the sameY Y (ρ+Λ0 ,α∨ )(ρ+Λ1 ,α∨ )1 − q 2N=1 − q 2N=α∨ ∈∆∨+α∨ ∈∆∨+=∞Y(1 − qk2NN) ·∞Y(6.157)(1 + q )k=1k=1180k6.6.

Characters from lattice algebras constructionsbut for Λ = ΛN it is different Y∞∞Y Y(ρ+ΛN ,α∨ )k1N1 − q 2N=(1 − q 2N ) ·(1 + q k− 2 ),α∨ ∈∆∨+k=1(6.158)k=1QQ2k−1Here we used the identities (6.235) and ∞)(1 − q k−1/2 )−1 = ∞k=1 (1 − qk=1 (1 +q k−1/2 ). It is convenient to consider the direct sums of two representations LΛ0 ⊕ LΛ1and LΛN ⊕LΛN since these sums have construction in terms of fermions.

Using (6.144)one getsq−Λ0 (ρ∨ )/hTr(qqρ∨ /h)LΛ0−ΛN (ρ∨ )/h+q−Λ1 (ρ∨ )/hTr(qρ∨ /h)Tr(qLΛNρ∨ /h=2)LΛ1=2∞Y(1 + q k )k=11∞Y(1 + q k− 2 )k=1 (1 − q2k−12N)(1 − q2k−12N),(6.159).The r.h.s. of these equations suggest the existence of the construction of these representation in terms of N -component twisted (principal) Heisenberg algebra and additional fermion (in NS and R sectors correspondingly), exactly this construction hasbeen considered in sect. 6.4.4.On the other hand these characters can be rewritten in terms of the simplestB-lattice theta-functions just using the Jacobi triple product identity2∞Y(1 + q k )n1=q−i(1 + q k− 2N ) =(1 − q) k=1 i=01∞XYj1 PN(1 + q k− 2 )nj )(n2j + Nj=12q=(1 − q k )N,...,n ∈Zk=1k=1==2k−12N∞ Y2NY(N +1)(2N +1)48NXq11(α+ 2N21ρ,α+ 2Nρ)α∈QBNand21∞Y(1 + q k− 2 )2k−1k=11∞Y(1 + q k− 2 )k=1(1 − q k )N,∞ 2N−1∞YYYi1k− 2N=2(1 + q) (1 + q k− 2 ) =(1 − q 2N )k=1 i=0k=1∞XY1 PN(1 + q k−1 )2 (j−1)=q 2 j=1 (nj + N nj ) ·=k )N(1−qn ,...,n ∈Zk=11= q−(6.160)N(6.161)N(N −1)(2N −1)48NX111q 2 (α+ 2N ρ,α+ 2N ρ)α∈QBN +ΛN −Λ0∞Y(1 + q k−1 )k=1(1 − q k )N.where ΛN − Λ0 is the highest weight of the spinor representation of SO(2N + 1).

Ther.h.s. of these formulas are the characters of sums of nontwisted representations of N component Heisenberg algebra with additional infinite-dimensional Clifford algebra(or real fermion). Another point of view that the r.h.s. are the characters of sums ofrepresentations of W (BN )-algebra [Luk].1816. Twist-field representations of W-algebras, exact conformal blocks and character identities(1)(1)Finally, let us point out, that for the root system B2 = C2 (affine B2 ), the dual(1),∨(2)roots system is C2= D3 (affine twisted D3 ).Simple roots: α0 = δ − 2e1 , α1 = e1 − e2 , α2 = 2e2 .Simple coroots: α0∨ = K − e∨1 , α1∨ = e∨1 − e∨2 , α2∨ = e∨2 .Real coroots: mK±e∨1 , mK±e∨2 , 2mK±e∨1 ±e∨2 , 2mK±e∨1 ∓e∨2 , m ∈ Z.Imaginary coroots: (2m − 1)K of multiplicity 1, m ∈ Z(6.162)2mK of multiplicity 2, m ∈ Z \ {0}.k = 1 weights: Λ0 , Λ1 = 1 + Λ0 , Λ2 = Λ0 + 1 + 213h = 4, ρ = 2e1 + e2 + 3Λ0 , ρ = e1 + e2 .22the computation leads to result, coinciding with formulas (6.156), (6.157), (6.158) forN = 2.

Though the root system here has a bit different combinatorial structure, thefermionic construction is the same, using 5 real fermions.Exact conformal blocks of W (so(2N )) twist fieldsGlobal constructionIt has been shown in [GMtw], that conformal block of the generic W (gl(N )) twistfields is given by explicit formula, analogous to the famous Zamolodchikov’s conformal1blocks of the Virasoro twist fields with dimensions ∆ = 16[ZamAT87, ZamAT86,ApiZam]. To generalize the construction of [GMtw] to all twist fields {Og |g ∈ NG (h)}considered in this chapter, one needs to glue local data in the vicinity of all twist fieldto some global structure. We consider below such construction for G = O(2N ), sinceit can be entirely performed in terms of twisted bosons.First, let us remind the local data in the vicinity of Og (0) already discussed insect. 6.4:• 2l-fold cover z = ξ 2l with holomorphic involution σ : ξ 7→ −ξ without stablepoints except for the twist field position.1• Fermionic field η(ξ) with exotic OPE η(ξ)η(σ(ξ 0 )) ∼ ξ−ξ0 .

On the sheets, con2πirnected to each other by [l, e ]+ , one can identify η(ξ) with ordinary complexfermion ψ(ξ) = η(ξ), η(σ(ξ)) = ψ ∗ (ξ), in this case σ permutes ψ ↔ ψ ∗ .• Bosonic field J(z) = (η(σ(z))η(z)), which is antisymmetric J(σ(z)) = −J(z)under the action of involution σ, and has first-order poles coming from zeromode charges in the branch-points corresponding to cycles of type [l, e2πir ]+ .To compute spherical 2M -point conformal block10G0 (q1 , . . .

, q2M ) = hOh1 ·g1 (q1 )Og1−1 (q2 ) . . . OhM ·gM (q2M −1 )Og−1 (q2M )iM10(6.163)In principle, we may choose any monodromies, though in this way we will get complicated twistedrepresentations in the intermediate channels, but as in [GMtw] we restrict ourselves to simpler, butstill quite general case of pairwise inverse (up to diagonal factors hi monodromies.1826.7. Exact conformal blocks of W (so(2N )) twist fieldswe forget about fermion and consider only the twisted boson with current J(z). Nowlet us list the field-theoretic properties which fix this conformal block uniquely.Considering the action of 1-form J(z)dz onto the highest weight vector |0ig of themodule of twist-field Og of order l, due to Jk/l>0 |0ig = 0 one gets, that the mostsingular termdz(6.164)J(z)dz ∼ r + . .

.z→0zin the vicinity of the twist field can be simple pole – in presence of r-charge or a zeromode.Notice, that for two fields with opposite (up to diagonal factor h = diag (e2πia1 , . . . , e2πian ))monodromiesOh·g (z)Og−1 (z 0 ) ∼ 0 (z − z 0 )∆h −2∆g Vh (z 0 ) + descendantsz→zwhere Vh (z 0 ) is a field with fixed charges ~a ∈ h. Hence˛1J(ξ)dξOh·g (z)Og−1 (z 0 ) = aj Og (z)Og−1 (z 0 )2πi(6.165)(6.166)jCz,z0j0where contour Cz,z0 is the j-th preimage of the contour encircling two points z, z onthe base.

We identify such contours with the A-cycles on the cover, and correspondinga’s with A-periods of 1-form J(z)dz.The standard OPE of two currentsdzdz 0+ 4Ť (z 0 ) + . . .J(z)J(z )dzdz = 0z→z (z − z 0 )200(6.167)gives the stress-energy tensorT (z) =XŤ (ξ)π2N (ξ)=z∆g1T (z)Og (0) = 2 Og (0) + ∂Og (0) + . . .zz(6.168)and non-standard coefficient (4 instead of 2) arises due to involution σ. Summarizingthese facts we get:• 2N -sheet branched cover π2N : Σ → P1 with the branch points {q1 , .

. . , q2M }−1}. Inand ramification structure defined by the elements {g1 , g1−1 . . . , gM , gMparticular, Σ is a disjoint union of two curves when all {gi } do not contain [l]−cycles.• Involution of this cover σ : Σ → Σ with the stable points coinciding with [li ]−cyclesπ2NσΣπ2Σ̃πNCP1(6.169)Projections and involution are shown on the diagram: π2N = πN ◦π2 , π2 ◦σ = π2 .1836. Twist-field representations of W-algebras, exact conformal blocks and character identities• Odd meromorphic differential dS(σ(ξ)) = −dS(ξ) with the poles in preimagesof qi and residues given by corresponding r-charges.• Symmetric bidifferential dΩ(ξ, ξ 0 ), satisfying dΩ(σ(ξ), ξ 0 ) = −dΩ(ξ, ξ 0 ), with twopoles:dξdξ 0,(ξ − ξ 0 )2dΩ2 (ξ, ξ 0 ) ∼ 0ξ→ξdΩ2 (ξ, ξ 0 )∼ξ→σ(ξ 0 )−dξdξ 0(ξ − σ(ξ 0 ))2(6.170)and vanishing A-periods.Using this data one can write for two auxiliary correlatorsG1 (ξ|q1 , .

. . , q2M ) = dξhJ(ξ)Oh1 ·g1 (q1 )Og1−1 (q2 ) . . . OhM ·gM (q2M −1 )Og−1 (q2M )iM000G2 (ξ, ξ ) = dξdξ hJ(ξ)J(ξ )Oh1 ·g1 (q1 )Og1−1 (q2 ) . . . OhM ·gM (q2M −1 )Og−1 (q2M )i(6.171)Mtheir explicit expressionsG1 (ξ)G0−1 = dS(ξ),G2 (ξ, ξ 0 )G0−1 = dS(ξ)dS(ξ 0 ) + dΩ2 (ξ, ξ 0 )(6.172)fixed uniquely by their analytic behaviour. Now let us study in detail the structureof the curve Σ in order to construct all these objects.Curve with holomorphic involutionInvolution σ defines thecover π2 : Σ → Σ̃ with the total number of branchPMtwo-fold00points being 2K = 2 i=1 Ki , or exactly the total number of [l]− cycles in all elements{gi , gi−1 }. The Riemann-Hurwitz formula χ(Σ) = 2 · χ(Σ̃) − #BP then gives for thegenus(6.173)g(Σ) = 2g(Σ̃) + K 0 − 1Then a natural way to specify the A-cycles on Σ is the following [Fay]: first to take(1)(1)(2)(2)A1 , .

. . , Ag̃ , A1 , . . . , Ag̃ on each copy of Σ̃, where g̃ = g(Σ̃); and second, all otherA-cycles that correspond to the branch cuts of the cover, connecting the branch points(0)(0)of π2 : A1 , . . . , AK 0 −1 . The action of involution on these cycles is obviously given by(1)(2)σ(Ai ) = Ai ,(0)(2)(1)σ(Ai ) = Ai ,(0)σ(Aj ) = −Aj ,i = 1, . . . , g̃j = 1, . . . , K 0 − 1(6.174)thus we have the decomposition of the real-valued first homology group into the evenand odd partsH1 (Σ, R) = H1 (Σ, R)+ ⊕ H1 (Σ, R)−dim H1 (Σ, R)+ = g(Σ̃) = g̃dim H1 (Σ, R)− = g̃ + K 0 − 1 = g−(6.175)Computeformula for the cover of P1 .

LetPM now g̃ = g(Σ̃), using the Riemann-HurwitzK = i=1 Ki be the total number of [l, e2πir ]+ -type cycles in all elements {gi }, as well1846.7. Exact conformal blocks of W (so(2N )) twist fieldsas K 0 serves for the type [l0 ]− . Then χ(Σ̃) = N · χ(P1 ) − #BP gives (cf. with theformula (2.17) of [GMtw])0KKXXg̃ = 1 − N +(li − 1) +(li0 − 1)i=1i=1so that0g− = g̃ + K − 1 =KX0(li − 1) +i=1KXKXli0 − N(6.177)(li0 − 21 )(6.178)i=1andg = 1 − 2N + 2(6.176)0(li − 1) + 2i=1KXi=1For our purposes the most essential is the odd part H1 (Σ, R)− of the homology. Onecan see these g− A-cycles explicitly as follows: two mutually inverse permutations ofP (1,2)(1,2)with constraints i Ai= 0.