In nitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Lie Algebras
Tom Copeland, Japan, [email protected] Nov. 24, 2012
Consider bases comprised of sequences of polynomials pn(x) characterized by ladder operators (up / down, creation / annihilation, or raising / lowering) de ned by
R pn(x) = pn+1(x) and L pn(x) = n pn−1(x)
with p0(x) = 1, giving the number operator # pn(x) = RL pn(x) = n pn(x). ======n For the power basis pn(x) = x , R = x and L = D = d/dx, and n −1 for the divided power basis pn(x) = x /n!, R = D and L = D x D. ======
Then the lower triangular Pascal matrix PMat (A007318) is a rep of I: P = exp(RLR) = exp(#R) 1) P = exp(xDx) for the power basis, and 2) P = exp(x) for the divided power basis. OEIS-A132440 presents relations between the in nitesimal generators,
(in nigens) as in nite dimensional matrices and operators, of PMat and P . ======The upper triangular transpose of the Pascal matrix T is a rep of PMat II: P T = exp(L) 3) P T = exp(D) for the basis set xn, and 4) P T = exp(DxD) for the set xn/n! (see A218272). ======
The lower triangular padded Pascal matrix P dMat (A097805) is a rep of III: P d = exp(R2L) = exp(R#) 5) P d = exp(x2 D) for the set xn, and 6) P d = D−1 exp(t · x) D for the set xn/n! (see A218234). ======
1 Given the invariantly diagonal matrix exp(RL) = exp(#) = E, the three operators/matrices P T , E, and P d can be used to form an in nite dimensional rep of the special linear group SL2 associated to Möbius / linear fractional transformations. This is most easily seen in the basis set zn where 7) (P T )af(z) = exp(a · D)f(z) = f(a + z), a linear translation, 8) Ebf(z) = exp(b · zD)f(z) = f(eb · z), a scaling or dilation, and 9) (P d)cf(z) = exp(c · z2D)f(z) = f(z/(1 − c · z)), a special conformal or Moebius / linear fractional / projective transformation. (The in nigens D, zD, and z2D are a subgroup of the in nite-dimensional Witt Lie algebra.) ======The special conformal transformation can be related back to P by noting the conjugate z (zDz) z−1 = z2D, so that 10) z P c z−1f(z) = z exp(c · zDz) z−1f(z) = exp(c · z2D) f(z) = (P d)cf(z) = f(z/(1 − c · z)). (See A038207 to relate this formalism to a relation between powers of the Pascal matrix, face vectors of hypercubes, and the evolution of an exponential distribution governed by the Fokker-Planck equation.) ======The transformations above are easily derived and are related to ow elds: For or2 , let x1−m , and (−1) 1/(1−m), m = 0, 1, y = fm(x) = 1−m x = fm (y) = [(1 − m)y]
d m d d d then gm(x) = x = = , so dx dx d(fm(x) dy
d d exp[t · g (x) ]W (x) = exp[t · ]W [f (−1)(y)] = W [f (−1)(y + t)] = W {f (−1)[f (x) + t]}. m dx dy m m m m (See A145271 for more on this relation.) 11) : , (−1) , and d d , so m = 0 y = f0(x) = x x = f0 (y) = y g0(x) dx = dx d . exp[t · dx ]W (x) = W (x + t) 12) : two-sided limits give , (−1) y, and m = 1 y = f1(x) = ln(x) x = f1 (y) = e d d , so d t . g1(x) dx = x dx exp[t · x dx ]W (x) = W [exp(ln(x) + t)] = W (e x) 13) : , (−1) , d 2 d , so m = 2 y = f2(x) = −1/x x = f2 (y) = −1/y g2(x) = x h i h i dx dx exp[t · x2 d ]W (x) = W −1 = W x . dx −1 +t 1−t·x There are eigenfunctions and Fourierx transforms associated with each operator. ======(−1) Translating the curves so that fm(0) = fm (0) = 0, there are binomial Sheer sequences with raising and lowering operators (see Mathemagical Forests) and formal group laws (FGLs) associated with each operator.
2 1−m Choose (1+x) −1 , then (−1) 1/(1−m) , y = fm(x) = 1−m x = fm (y) = [1 + (1 − m)y] − 1 d m d d d and gm(x) = (1 + x) = = . dx dx d(fm(x) dy Following the analysis above with t replaced by fm(y), m d (−1) F GLm(x, y) = exp[fm(y) · (1 + x) dx ] x = fm [fm(x) + fm(y)] 1 = [1 + (1 + x)1−m − 1 + (1 + y)1−m − 1] 1−m − 1. 14) : , (−1) , d d , m = 0 y = f0(x) = x x = f0 (y) = y g0(x) dx = dx and F GLo(x, y) = x + y. 15) : , (−1) y , m = 1 y = f1(x) = ln(1 + x) x = f1 (y) = e − 1 d d , and g1(x) dx = (1 + x) dx F GL1(x, y) = x + y + x y. 16) : x , (−1) y , d 2 d , m = 2 y = f2(x) = 1+x x = f2 (y) = 1−y g2(x) dx = (1 + x) dx and x+y+2 x·y F GL2(x, y) = 1−x·y . ======An operator analogous to 16) can be constructed from the full group of Möbius a b transformations by letting y = h(x) = a·x+b with δ = det = ad − bc 6= 0. c·x+d c d
h i−1 2 Then (−1) d·y−b , dh(x) (cx+d) , and x = h (y) = −c·y+a g(x) = dx = δ 2 h 2 i h 2 i −1 (δ+tcd) x+d t , so (cx+d) (δ+tcd)·x+d t h [t+h(x)] = −c2t x+(δ−tcd) exp t · δ D f(x) = f −c2t·x+(δ−tcd) . For x in the upper half of the complex plane, the trajectory of the point induced (δ + tcd) d2t by the op will remain in the same half plane since det = δ2 > 0. −c2t (δ − tcd) h 2 i The operator can be transformed to obtain (cx+d) ¯ 2 d exp t · δ Dx f(x) = exp t · x¯ Dx¯ f(¯x − c ) h i 2 x¯ d , just as for the op in 9), 10), and 13), with ¯ t·c = f −t¯·x¯+1 − c t = δ and d . x¯ = x + c Alternatively, with , 2 2 2, ¯ 2, c δ = 1 r = c + d t = t · r cos(θ) = r , and sin(θ) = d , r h 2 i 1 0 exp t · (cx+d) D f(x) = exp t¯· [cos(θ) · x + sin(θ)]2D f(x) = f R(θ) RT (θ) {x} δ x x −t¯ 1 cos(θ) −sin(θ) 1 −tan( θ ) 1 0 1 −tan( θ ) where R(θ) = = 2 2 sin(θ) cos(θ) 0 1 sin(θ) 1 0 1 a b and {z} = a·z+b . θ = 0 gives the pure vertical shear or translation c d c·z+d 1 0 1 t¯ , and θ = π gives the pure horizontal shear or translation . −t¯ 1 2 0 1 ====== The special linear fractional transformation of 9), 10), and 13) 1 0 −t 1 is a vertical shear mapping, and, as William Cliord noted,
3 "A succession of shears will enable us to reduce any gure bounded by straight lines to a triangle of equal area." a b In general, any element A = of SL2(R) (det(A) = 1) with c d |tr(A)| = |a + d| = 2 is classi ed as parabolic and a shear mapping; with |tr(A)| > 2, as hyperbolic and a squeeze mapping; and with |tr(A)| < 2, as elliptic and (δ + tcd) d2t a rotation. For our general mapping B = , −c2t (δ − tcd) |tr(B)| = 2|det(A)| = 2|δ|, det(B) = δ2, and the characteristic polynomial det(B − λI) = (δ − λ)2, so that (δI − B)2 = 0. If |δ| = 1, B is parabolic, independent of the value of , with the unique xed point c t zF = − d . (See Visual Complex Analysis by T. Needham for an excellent presentation of Möbius motions, in particular pages 168-9 and 308-10 for parabolic matrices.) ======In Gromov-Witten Invariants and Quantization of Quadratic Hamiltonians, − 1 m − 1 Alexander Givental introduces the operators Lm−1 = x 2 (xDx) x 2 for m = 0, 1, 2, ... to construct reps of the in nite-dimensional Witt Lie algebra and the Virasoro algebra. − 1 m − 1 − 1 −1 2 m − 1 −1 − 1 2 m 1 17) Lm−1 = x 2 (xDx) x 2 = x 2 x (x D) x x 2 = x x 2 (x D) x 2 , 2 m so L¯m = xLm−1 and (x D) are a conjugate pair of operators, and, with ¯n ¯ , L. = Ln 1 1 f(x/(1+t·x)) 18) ¯ − 2 t 2 . G f(x) = exp(−t · L.) f(x) = x (P d) x f(x) = (1+t·x)1/2 (For an overview of the importance of the Virasoro algebra in mathematics and physics, see "Panorama Around the Virasoro Algebra" by S. Palcoux.) ======In addition, 1 1 1 1 1 1 (− 1 ) ¯ n − n n n − n n n− n 2 Ln = x 2 (xDx) x 2 = x 2 x D x x 2 = x x 2 D x 2 = x n!Lagn (−xDc ) n 1 n n − 2 where D = d/dx, xDc = x D , and Lagn (x) are generalized orthogonal Laguerre polynomials, which are related to the even 1 √ (− 2 ) n 2n subgroup of the Hermite polynomials by n!Lagn (−x) = (−1) H2n(i x)/2 . Also, see A132681, and the operator (x2D)n can be decomposed into a sum of the "state number" ops (xD)k involving the Stirling numbers of the rst kind A094638 or a sum of the xkDk ops involving Lah numbers (ref. in A130561). ======(− 1 ) n n− 1 j 2 P j 2 x . And, as a consistency check, observe Lagn (x) = j=0 (−1) 1 j! j− 2 (− 1 ) k n n− 1 k k n+k− 1 k −k− 1 k 2 x P 2 x 2 x n 2 x . Lagn (−xDc ) k! = j=0 1 j k! = n k! = (−1) n k! j− 2 This result can be used to re-derive the eigenvalue eqn. for G below by −k− 1 1 interchanging sums and products and noting P 2 n −k− 2 . n≥0 n (t · x) = (1 + t · x) ======
4 (− 1 ) The rst few polynoms of n 2 are (2y) n!Lagn (−x/2y) = Lagg n(x, y)
Lagg 0(x, y) = 1
Lagg 1(x, y) = y + x
2 2 Lagg 2(x, y) = 3y + 6xy + x 3 2 2 3 Lagg 3(x, y) = 15y + 45y x + 15yx + x 4 3 2 2 3 4 Lagg 4(x, y) = 105y + 420y x + 210y x + 28yx + x
h 1 i2 with 2 2 x (1 + 2D) Lagg n(x, 1) = (1 + x) + 2(1 + 2x)D + 4xD Lagg n(x, 1) = Lagg n+1(x, 1) and D 1+2D Lagg n(x, 1) = n Lagg n−1(x, 1). ======(Many statements in this paragraph remain to be veri ed.) The integer coe cients of this pyramid (OEIS-A176230) are related to many combinatoric structures: (A119743, A100861, A122848, A144299, A181386)
1. k-matchings of the complete graph K2n, 2. involutions of 2n with k pairs,
3. self-inverse permutations of (1, 2, ..., 2n) having exactly k cycles, 4. partitions of an n-set into k nonempty subsets, each of size at most 2, 5. ways of choosing m disjoint subsets of 2 members from an original set of 2n members.
Row sums A066223 and A000085:
1. self-inverse permutations on 2n letters or involutions, 2. Young tableaux on 2n elements, 3. arc connections of 2n points on a line,
4. Hosoya index of 2n-complete graph K2n, 5. number of 2nx2n symmetric permutation matrices,
6. sum of the degrees of the irreducible reps of the symmetric group S2n, 7. partitions of a set of n distinguishable elements into sets of size 1 and 2,
8. tableaux on the edges of the star graph of order 2n ( S2n),
5 9. lattice paths
10. ballot sequences (or lattice permutations) of length 2n. Note that simple complete graphs are projections of hypertetrahedra onto a 2-D plane. (The coe cient of 11 of is x Lagg 27(x, y) 150,738,274,937,250 which is the number of possible plugboard settings for a WWII German Enigma Enciphering Machine. See A181386.) ======The Hermite polynomials have rich associations to the heat equation, brownian motion, gaussian probabliltiy distribution, the quantum harmonic oscillator, and enumerative combinatorics. The associated Laguerre polynomials of various orders have rich associations with the con uent hypergeometric functions, the radial component of probability wave functions for the electron orbital of a hydrogen atom, and combinatorics (see F. Bergeron, G. Labelle and P. Leroux, Introduction to the Theory of Species of Structures). Also see the generating function of the polynomials on page 20 of Virasoro Correlation Functions for Vertex Operator Algebras by D. Hurley and P. Tuite. ======(Dec. 11, 2013: Changed T to S below, introducing missing factors.) The in nigen for the in nite matrix rep for the 's is, Lagg n(1, y) with q = −1/2 and z = 2y, 0 0 0 0 0 0 0 . .. (q + 1) z 0 0 0 0 0 . .. 0 2(q + 2) z 0 0 0 0 .. S(q; z) = 0 0 3(q + 3) z 0 0 0 . .. 0 0 0 4(q + 4) z 0 0 . .. 0 0 0 0 5(q + 5) z 0 . ...... 0 ......
0 0 0 0 0 0 0 . 1 .. 2(0 + 2 ) y 0 0 0 0 0 . 1 .. 0 4(1 + 2 ) y 0 0 0 0 . 1 1 . S(− ; 2y) = 0 0 6(2 + ) y 0 0 0 . 2 2 .. 0 0 0 8(3 + 1 ) y 0 0 . 2 .. 0 0 0 0 10(4 + 1 ) y 0 . 2 ...... 0 ......
6 0 0 0 0 0 0 0 . .. 1 · 1 y 0 0 0 0 0 . .. 0 2 · 3 y 0 0 0 0 .. = 0 0 3 · 5 y 0 0 0 . .. 0 0 0 4 · 7 y 0 0 . .. 0 0 0 0 5 · 9 y 0 . ...... 0 ......
0 0 0 0 0 0 0 . .. 1 y 0 0 0 0 0 . .. 0 6 y 0 0 0 0 .. = 0 0 15 y 0 0 0 . .. 0 0 0 28 y 0 0 . .. 0 0 0 0 45 y 0 . ...... 0 ......
These types of in nigens, their associated exponentiations (fundamental or generalized factorial matrices), their relation to multiplication along diagonals and reciprocals of Taylor series (e.g.f.s), weighted surjections, recursions, convolutions, permutahedra, and partition polynomials are sketched in A133314. The matrix associated to the 's is then Mf Lagg n(1, y)
Mf = exp(S(−1/2; 2y)) = exp(2y · S(−1/2; 1))
1 0 0 0 0 0 0 . .. y 1 0 0 0 0 . 2 .. 3y 6y 1 0 0 0 .. = 15y3 45y2 15y 1 0 0 . .. 105y4 420y3 210y2 28y 1 0 . .. 945y5 4725y4 3150y3 630y2 45y 1 . ...... 10395y6 ......
7 with inverse
Mf(−1) = exp(S(−1/2; −2y)) = exp(−2y · S(−1/2; 1)). ======If this matrix is further divided along its diagonals by the double factorial numbers (A001147) in the rst column, the matrix A086645 of h-vectors of polytopes formed from the classical root lattice Cn, or coordinator polynomials, is obtained:
1 0 0 0 0 0 0 . .. y 1 0 0 0 0 . 2 .. 1y 6y 1 0 0 0 .. 1y3 15y2 15y 1 0 0 . .. 1y4 28y3 70y2 28y 1 0 . .. 1y5 45y4 210y3 210y2 45y 1 . ...... 1y6 ...... ======(Direct, correct argument introduced in Sept. 2014.) Returning to the basic operators, but introducing an extra sign, let −1 − 1 n+1 − 1 −1 − 1 2 n+1 1 ln = −Ln = −x L¯n+1 = −x 2 (xDx) x 2 = −x x 2 (x D) x 2 −1 n+1 1 since the action = −x x xD + n + 1 − 2 n+1 n s n n n s n s n s So, (xDx) x = x D x x = x (s + n)n x = x (xD + n)n x . s −1 m+1 1 −1 n+1 1 s lmlnx = x x xD + m + 1 − 2 m+1 x x xD + n + 1 − 2 n+1 x −1 m+n+1 1 1 s = x x s + m + n + 1 − 2 m+1 s + n + 1 − 2 n+1 x 1 −1 m+n+1 1 s = (s + n + 1 − 2 ) x x s + m + n + 1 − 2 m+n+1 x 1 s; therefore, the commutator for the ops is = −(s + n + 1 − 2 ) lm+n x [lm, ln] = (m − n) lm+n, describing a Witt algebra. ======(Dec. 11, 2013: Expanded on T relations.) Givental goes on to quantize the Witt algebra, obtaining a Virasoro algebra.
8 Consider the in nigen for associated Laguerre matrices (cf. OEIS A132681)
0 0 0 0 0 0 0 . .. (q + 1) z 0 0 0 0 0 . .. 0 (q + 2) z 0 0 0 0 .. T (q; z) = 0 0 (q + 3) z 0 0 0 . .. 0 0 0 (q + 4) z 0 0 . .. 0 0 0 0 (q + 5) z 0 . ...... 0 ......
With q = 0 and z = 1, this is the in nigen for the Pascal matrix. With q = −1/2
and z = 1, this is the in nigen for an associated Laguerre matrix of order -1/2
0 0 0 0 0 0 0 . 1 .. (0 + 2 ) 0 0 0 0 0 . 1 .. 0 (1 + 2 ) 0 0 0 0 . 1 1 . T (− ; 1) = 0 0 (2 + ) 0 0 0 . 2 2 .. 0 0 0 (3 + 1 ) 0 0 . 2 .. 0 0 0 0 (4 + 1 ) 0 . 2 ...... 0 ......
The matrix operation 1 can be characterized in several ways b = T (− 2 ; 1) · a in terms of the coe cients a(n) and b(n) or their o.g.f.s A(x) and B(x): 1 19) b(0) = 0, b(n) = (n − 2 ) · a(n − 1) (for n > 0) 1 − 1 20) B(x) = x 2 (xD x) x 2 A(x) = L¯1 A(x) 1 − 2 1 = x · Lag1 (−xDc ) A(x) = x · [ 2 + xD] A(x), so the exponentiated operation in matrix form has vector components n n− 1 21) {exp[t · T (− 1 ; 1)] a} = P 2 tn−ja(j), 2 n j=0 j− 1 or in operator form acting on the o.g.f. 2 1 − 1 22) exp(t · Top) A(x) = x 2 exp(t · xDx) x 2 A(x) (− 1 ) ¯ P n 2 = exp(t · L.) A(x) = n≥0 (t · x) Lagn (−xDc ) A(x) t·u exp( 1−t·u ·xDc ) A(x/(1−t·x)) = (1−t·u)1/2 A(x)(eval at u = x) = (1−t·x)1/2 , a generalized Euler transformation for an o.g.f., as in 18). ======
9 T (−1/2; z) is present in the terms independent of ~ of Givental's Virasoro operators as
T (−1/2; t · ∂) Q = Mat I − T (−1/2; t · ∂) = T (−1/2; t · ∂) + [T (−1/2; t · ∂)]2 + ... =
0 0 0 0 0 0 0 .. 1 . (0 + 2 )t∂ 0 0 0 0 0 .. 1 1 2 1 . (1 + )(0 + )(t∂) (1 + )t∂ 0 0 0 0 2 2 2 . . 1 1 1 3 1 1 2 1 . (2 + )(1 + )(0 + )(t∂) (2 + )(1 + )(t∂) (2 + )t∂ 0 0 0 2 2 2 2 2 2 . . . 3 2 . . (3 + 1 )(2 + 1 )(1 + 1 )(t∂) (3 + 1 )(2 + 1 )(t∂) (3 + 1 )t∂ 0 0 . 2 2 2 2 2 2 . . . . . . . . (4 + 1 )(3 + 1 )(2 + 1 )(t∂)3 (4 + 1 )(3 + 1 )(t∂)2 (4 + 1 )t∂ 0 . 2 2 2 2 2 2 ...... ......
||
0 0 0 0 0 0 0 .. 1 . (0 + 2 )t∂ 0 0 0 0 0 .. 3 1 2 1 . (0 + )(0 + )(t∂) (1 + )t∂ 0 0 0 0 2 2 2 . . 5 3 1 3 3 1 2 1 . (0 + )(0 + )(0 + )(t∂) (1 + )(1 + )(t∂) (2 + )t∂ 0 0 0 . 2 2 2 2 2 2 . . . 3 2 . . (1 + 5 )(1 + 3 )(1 + 1 )(t∂) (2 + 3 )(2 + 1 )(t∂) (3 + 1 )t∂ 0 0 . 2 2 2 2 2 2 . . . . . . . . (2 + 5 )(2 + 3 )(2 + 1 )(t∂)3 (3 + 3 )(3 + 1 )(t∂)2 (4 + 1 )t∂ 0 . 2 2 2 2 2 2 ...... ...... ======Summing over the Givental quantized Virasoro operators gives
2 3 Lˆ−1 + t · Lˆ0 + t Lˆ1 + t Lˆ2 + ... =
0 0 2 (q∂) (q∂) q0 /(2~) 1 1 (q∂) (q∂) 0 2 2 2 (q∂) 1 (q∂) ~ ∂0 /8 1 1 1 1 ... q · 3 + · Q · 3 + 1 t t2 t3 ... (q∂) ∂ Mat (q∂) 3~ ∂0∂1/4 (q∂)4 (q∂)4 . . . . . .
10 where q and ∂ commute and, after all operations are completed, the powers of ∂ are placed to the right of the the powers of q and then both quantities n k ∂ 0 ∂ are umbrally evaluated as q ∂ = qn∂k = qn with ∂ = ∂0 = ∂qk ∂q0 0 and q = q0. (This could be just as easily expressed as an e.g.f.) For easy reference, the operators in Givental are ˆ 2 P L−1 = q0/(2~) + m≥0 qm+1∂m ˆ P 1 L0 = m≥0 (m + 2 ) qm∂m ˆ 2 P 1 3 L1 = ~ ∂0 /8 + m≥0 (m + 2 )(m + 2 ) qm∂m+1 ˆ P 1 3 5 . L2 = 3 ~∂0∂1/4 + m≥0 (m + 2 )(m + 2 )(m + 2 ) qm∂m+2 A detailed derivation of Givental's V ops are in B. Dubrovin and Y. Zhang's Normal forms of hierarchies of integrable PDEs, Frobenius manifolds, and Gromov-Witten invariants. ======Returning to 18), let's look at possible eigenfunctions for the operator G. iπ·x One eigenfunction is the Dedekind eta function 12 Q 2iπn·x η(x) = e n≥1(1−e ) − iπ·t with eigenvalue e 12 for integer t and for x in the upper half of the complex plane. η x 1 1 tx+1 iπ · t ¯ − 2 2 2 G η(x) = exp(−t · L.) η(x) = x exp −t · x D x η(x)= 1 $ exp − η(x), (tx + 1) 2 12
The rst three equalities hold for real t. Only the last equality requires t to be an integer, and I ag this constraint by using the symbol $. Re-expressed,
1 2 2 1 x x iπ · t 1 exp −t · x D x 2 η(x) = η exp − x 2 η(x). tx + 1 tx + 1 $ 12 ======The inverse of G(t, x) is G(−1) = G(−t, x), implying that the eigenvalues (t) of an eigenfunction fe(x) satisfy (t)(−t) = 1; i.e.,
− 1 2 1 − 1 2 1 − 1 2 1 fe(x) = x 2 exp t · x D x 2 x 2 exp −t · x D x 2 fe(x) = x 2 exp t · x D x 2 (t) fe(x) = (t)(−t) fe(x).
======Again
1 2 2 1 x x iπ · t 1 exp −t · x D x 2 η(x) = η exp − x 2 η(x). tx + 1 tx + 1 $ 12
Recall that the last equality holds only for integer values of t, but the rst equality holds for all real t, at least as long as x is in the upper half of the
11 complex plane. Also, the operator side is annihilated by ∂ 2 ∂ . ∂t + x ∂x Changing variables to 1 , we get p = − x
1 2 ∂ 1 1 1 1 1 iπ · t 1 1 1 exp −t · (− ) 2 η(− ) = η exp − (− ) 2 η(− ) ∂p p p t − p t − p $ 12 p p
with the annihilator ∂ ∂ (for the rst equality only), and, using the well- ∂t + ∂p 1 known formula 1 2 , this becomes (with and ) η(− z ) = (−iz) η(z) z = p −t 7→ t ∂ iπ · t exp t · η(z) = η(z + t) exp η(z) ∂z $ 12
with the annihilator ∂ ∂ (again for the rst equality only). ∂t − ∂z iπt −πy −2πny Along the imaginary axis, we have 12 12 Q 2iπn·t 12 , η(i · y + t) = e e n≥1(1 − e e ) iπt which rapidly decays as y increases, giving η(i · y + t) ≈ e 12 η(i · y) for even for real , and ∂ iπ y = 1.5 t Dz η(z) = −i ∂y η(iy) ≈ ( 12 ) η(iy) or ∂ −π along the axis. This suggests ¯ iπ ∂y η(iy) ≈ ( 12 ) η(iy) Lη(z) ≈ ( 12 )η(z), 1 1 1 2 1 or 2 2 iπ 2 , and x x iπ·t 2 , z D z η(z) ≈ 12 z η(z) tx+1 η tx+1 ≈ exp − 12 x η(x) even for real t , to a good approximation along the imaginary axis for |z| 5 .6. For
η z 1 1 tz+1 iπ · t ¯ − 2 2 2 G η(z) = exp(−t · L.)η(z) = z exp −t · z D z η(z)= 1 $ exp − η(z), (tz + 1) 2 12
1 1 the annihilator is ∂ − 2 2 ∂ 2 . ∂t + z z ∂x z ======
Emphasizing ow elds, with 1 (−1) , (−1) x , f(x) = − x = f (x) H(t, x) = f [t + f(x)] = −tx+1 H(t, H(s, x)) = H(t + s, x) and
1 2 1 x x i2πt 1 [H(t, x)] 2 η[H(t, x)] = η exp( ) x 2 η(x). −tx + 1 −tx + 1 $ 24
Changing coordinates and introducing the ow eld (−1) (1+t)x+t with x and (−1) x , F (t, x) = f [t + f(x)] = −tx+(1−t) f(x) = 1+x f (x) = 1−x then F (t, F (s, x)) = F (t + s, x), and
2 1 1 i2πt 1 exp t · (1 + x) D (1 + x) 2 η(1 + x) = [1 + F (t, x)] 2 η[1 + F (t, x)] exp( ) (1 + x) 2 η(1 + x). $ 24
12 ======In Superconformal Algebras and Mock Theta Functions 2. Rademacher Expansion for K3 Surface, T. Eguchi and K. Hikami present the Poincare-Maass series Ps(τ), apparently another eigenfunction candidate of G with eigenvalue − iπ·t e 4 for integer t and x in the upper half of the complex plane. ======These eigenvalue equations are reminiscent of wave equations with periodic boundary conditions, and E. Ghys in "Knots and Dynamics" on pg. 20 (see also A New Twist in Knot Theory, Terrence Tao's blog, and here) presents the equation
aτ + b a b 24 log η = 24 log(η(τ)) + 6 log(−(cτ + d)2) + 2πi R cτ + d c d where R is the Rademacher function, which he relates to the linking number between two knots related to modular/Lorenz ow: For every hyperbolic element A in PSL(2,Z), the linking number between the [modular/Lorenz] knot kA and the trefoil knot l is equal to R(A) .... Taking the log of G η(x), we can identify a b in our case, suggesting corresponds to a R c d = t t linking number (however, our A is parabolic), but that remains to be explored another time. ======
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