Jacobi Sums and String Theory

By Lucian Ionescu

Algebra Seminar 9/28/2016

To “know” a space, analyze its functions ...

- Taylor series: log(1-x)=x+x²/2+x³/3 + ...

- Fourier series: f(t)=cos(t)+½ cos(2t) + ⅓ cos(3t)+ …

- Haar analysis: wavelet basis, instead of cos(nt) & sin(nt)

- General case: Hilbert Spaces and Linear Operators

(Infinite dimensional “Euclidean geometry” - with Topology & Analysis toppings … )

… to “understand” Z/nZ, we have to see what is:

Fourier Analysis for Finite Abelian Groups

It has lots of applications in Sciences, including to String Theory!

A Basis of Functions on Z/nZ

- Additive characters on Z/nZ are an analog of linear functions

e_k(x)=exp(2πi kx/n) [cos(...) + i sin(...)]

- The Fourier coefficients of a function f:Z/nZ->C are:

<f,e_k>=Sum f(x) e_k(x)* [“dot product”]

- A very important class of functions on Z/nZ are the multiplicative characters: “linear functions”, but defined on the multiplicative group of invertible elements Zn*:=U(Z/nZ)

- Note: the invertible elements are the conformal symmetries of the discrete vector space Z/nZ (Z-module).

Analyzing the symmetries of the space

[Possible conserved quantities of a dynamics of that space]

- Def. A Dirichlet character modulo n, is the 0-extension to Zn of a multiplicative character c:Zn*->C [Zn*⇔symmetries of Zn].

- Ex. n=p=7, c:Z₇*->C, c(x)=e_p-1(3x) is an order 2 Dirichlet character (when extended c(0)=0); it therefore coincide with the Legendre symbol (k/p)=k^(p-1)/2 mod p.

- (Traditional) Def. the Gauss sums of the Dirichlet character c(x) of modulus n are:

g_k(c)=Sum _{x in Zn} c(x) e_x(k)

- Ex. Quadratic Gauss sums g_a(L_p(t))=𝚺_{t in Zp} L_p(t) z_p^at, g²=(-1)^p*p

Properties of Gauss sums

- Since c is multiplicative, g_k(c)=c(k^-1) g₁(c), with g(c)=g₁(c) denoting the principal Gauss sum:

g_k(c)=Sum _x c(x) e₁(kx)=Sum _x c(xk^-1) e(x)=c(k^-1) g₁(c).

- Prop.1 The Gauss sums are opposite to the Fourier coefficients of the Dirichlet characters:

g_k(c)= Sum _x c(x) e_k(x)=<c, e_-k>=FT(c)(-k)=FT(c)^v (k),

where we introduce the notation f^v(k)=f(-k).

Note: In general FT²=Id^v is a reflection “shy” of being a complex structure.

… and many more properties (e.g. |g|²=p) ...

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Gauss Sums and Fourier Transform

- Th. (Folk) Dirichlet characters c(t) are skew eigenfunctions of Discrete FT, with Gauss sums g(c) as eigenvalues:

FT(c)^v=g(c) c* [FT(c)=c(-1) g(c) c*]

Proof: FT(c)(-k)=g_k(c)=g(c) c(k^-1)=g(c) c*(k).

- Def. The Gauss transform GT(c)(k)=g_k(c)=c(-1) FT^-1(c)

- So, FT diagonalizes on Dirichlet characters c1,...,cn:

FT[c₁,...,c_n]=(c₁(-1),...,c_n(-1))

[Q: Even/odd chars => super-graded? or symplectic str?]

- SAGE Plotting roots of unity and Gauss sums

Jacobi Sums and Intersection Numbers

- Def. Jacobi sum J(c,c’)=Sum _a+b=1 c(a)c(b)

- Prop.

1) JS are convolution values: J(c,c’)=(c*c’)(1);

2) J=dg are 2-cocycles: J(c,c’)=g(c)g(c’)/g(cc’);

- Prop. (4, p.93) Given a Fermat curve FC_n: xⁿ+yⁿ=1 mod p

|FC_n|=Sum _{order c,c’ =n} Sum_a+b=1 cⁱ(a)c^j(b)=Sum _c,c’ J(c,c^’),

- Examples:

Circles x²+y²=1, |C|=p+J(c,c), ord(c)=2;

Elliptic Curve x³+y²=1, |C|=p+Sum _j=1,2 J(c₂, c₃^j)

What is going on … !?

- Characters have a kernel: project onto some roots of unity; when considering “Weil curves” (linear combinations of two power functions), solutions correspond to partitions of 1 into powers, e.g. x³+y²=1; or whether 1-x³ has a square-root, i.e. how the image of f(x)=1-x³ intersects the squares [subgroup].

- How far is this intersection numbers “game” from a “static” intersection of cycles in Fp (or of two cylinders in FpxFp), from being a discrete analog of an interaction of two strings?

Virasoro-Shapiro / Veneziano Amplitudes

- Strings interaction is modeled as Riemann surfaces with boundary, embedded in a (e.g.) 1+3+6 dim. Space-Time (World-sheet swept by incoming strings, joined & then outgoing. One takes the strings to infinity: punctured Riemann Surface, with operators attached to punctures describing the QM transition of states.)

- Virasoro-Shapiro amplitude (open strings), and Veneziano amplitude (closed strings) were written in terms of Gamma functions (as an Euler Beta function B(a,b)=∫_[0,1]|z|^a-1|1-z|^b-1dz), since it gave the right poles (a=-2(½ + s), … with s,t,u M-vars):

C(a,b)=B(2a,2b)=G(a)G(b)G(c)/G(a+b+c) [3]

[4-point Veneziano Amplitude, with parameters a+b+c=1]

Importing CFT “stuff” for Finite ST ...

- Witt algebra generators l_n of diffeo. of the circle S¹:

[l_n,l_m]=(n-m)l_n+m ,

and ITS Virasoro central extension (classical -> quantum / proj. rep. -> lin. rep. L_n=L[l_n]; unique: dim H²(Diff.S¹)=1):

[L_n,L_m]=(n-m)L_n+m+ ½ (n-1)n(n+1)/3! c d_n+m,0 [p(n,m)=2-cocycle]

- What are the discrete analogues for the finite string Z_n~S¹ and finite torus? (free bosonic string / Riemann cylinder ~ Z_n x Z_m)

Aut_Ab(Z/nZ) and multiplicative / Dirichlet characters, D_k (finite difference & 2-cohomology), l_n=z^k D_k (on the group ring), Galois reps & Frobenius ...

… and Jacobi sum (2-cocycle) as a Veneziano amplitude?

Jacobi sum & Veneziano Amplitude

- Cycle intersection in ambient discrete torus

- Veneziano amplitude (4-point function), i.e. the amplitude of interaction of two strings can also be written as [2]:

A(s,t)=G(x)G(y)/G(x+y), x=-a(s), y=-a(t).

[Compare with Jacobi sum in terms of Gauss sum!]

- Gamma function G(x)=Int_R+ t^x-1 e^-t dt

Is it a Mellin transform MT(e^-x) (an exp. of FT), or a

Fourier coefficient of the character c_x(t)=t^x?

[Connection with Elliptic Curves or Gauss sums …]

Examples of counting points of curves

(Computing integrals!)

- |C| = “length” of a curve (zero locus …)

- Discrete Riemann Surfaces y²=f(x) over Fp; ex. f(x)=1-x³

- Relation with number fields, splitting primes and Weil zeros which satisfy the Riemann Hypothesis:

|EC|=p-a_p; a_p=J(c,d)+J(c,d²), c²=1=d³ [see Wang]

a) If 3 does not divide p-1 (inert in Eisenstein integers, as p-cyclotomic extension), then a_p=0, N=p and w=+/-i p^½

b) If 3|p-1, ww*=p [see Wang D=1 & SAGE file]

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Conclusions & Further developments

- Briefly, Jacobi sums measure intersections of subgroups, analogous to interactions of cycles [like Bezout Th. in algebraic geometry]; a comparison with String Theory may be helpful: JS <-> V-Amp. [and mutually beneficial!].

- There is a rich algebraic structure to be studied from a modern point of view of algebraic quantum groups

[Convolution algebra of functions (characters), the 2-cocycle J=dg (central extension/projective rep), with Gauss sums the eigenvalues of Fourier Transform, the group ring/cyclo ext. -> conv. Alg. duality; relation with Weil zeros & Riemann spectrum …].

Bibliography

[1] Ireland and Rosen, A classical introduction to modern number theory.

[2] Introduction to String Theory, Gerard ‘t Hooft.

[3] David Tong, Lectures on String Theory.

[4] L. M. Ionescu, The Finite Strings Theory Project.

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… and much more! (just ask :)

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