Enhanced homotopy theory of the period integrals of hypersurfaces

Transcription

1 Enhanced homotopy theory of the period integrals of hypersurfaces Jeehoon Park (Joint Work with Jae-suk Park at IBS) Department of Mathematics Postech Glenn Stevens 60th birthday, June 6, 2014

2 I would like to thank the organizers and my thesis advisor Glenn Stevens and it is a real honor to be his student and talk at his 60th birthday conference. He suggested, as a thesis problem, a problem of finding p-adic theory of Weil (or oscillator) representation which would govern the p-adic θ-correspondence. But there was no genuine progress and I thought that one of the reasons is that I did not understand the physical nature or the role of the Weil representation in quantum field theory (no p-adic Haar measure no measure for the Feynmann path integral) By chance, I met a theoretical physicist Jae-Suk Park(the current collaborator) who had worked in string theory; he is an expert on QFT(quantum field theory). In particular, He made an algebraic formulation of QFT may help to develop a p-adic theory of Schrödingier representation and Weil representation. The key language is the cochain complex enhanced with a binary product (correlations of Feynman path integrals) and induced infinite homotopy theory (notably, L -homotopy theory and A -homotopy theory). (Today s talk) We apply this algebraic formulation to period integrals of projective (or toric) hypersurfaces and complete intersections; we develop a (0+0)-dimensional quantum field theory QF T X associated to such an algebraic variety X such that the period integral of X =the Feymann path integral in QF T X. This leads us to a notion of a p-adic (0+0)-dimensional QFT which gives rise to the p-adic Dwork complex enhanced with a binary product (an ongoing project); this would shed some light (??) on p-adic properties of Weil representations? (maybe his 70th birthday) My recent mathematical journey - find higher homotopy structures appearing in number theory and algebraic geometry and try to find applications.

3 The goal of my talk is to reveal hidden structures on the Betti cohomology and period integrals of differential forms on smooth projective hypersurfaces over a field in terms of BV(Batalin-Vilkovisky) algebras and L -homotopy deformation theory. I will concentrate on BV algebras aspects in this talk (I will NOT touch L -homotopy related issues here). A BV algebra over (which arises from a quantization scheme in physics) is a cochain complex ( = m m,k) over a field equipped with a (super)commutative binary product such that (,K,l K 2 ) is a differential graded Lie algebra and (,,lk 2 ) is a Possion graded algebra, where l K 2 (a, b) := K(a b) K(a) b ( 1) a xk(b); Kl K 2 (a, b)+lk 2 (Ka, b)+( 1) a l K 2 (a,k b) = 0, lk 2 (a b, c) = ( 1) a a l K 2 (b, c)+( 1) b c l K 2 (a, c) b.

4 Let n and d be positive integers. Let =. Let X = X G be a smooth projective hypersurface in the complex projective n-space P n defined by a homogeneous polynomial G(x) = G(x 0,, x n ) of degree d in [x 0,, x n ]. Let [X ] be the homogeneous coordinate ring. Let = H n 1 (X, ) be the middle-dimensional primitive cohomology of X. prim The decreasing Hodge filtration F i l on. A cup product polarization on ;, ω η ω η. X For each γ H n 1 (X, ), we define a period integral of X ; C [γ] :, ω ω, γ We will enhance all the above invariants at the level of BV algebras. As one application, we show how such an enhancement (especially the binary product structure) can be useful; we provide an explicit algorithm to compute the Gauss-Manin connections of families of smooth projective hypersurfaces of degree d.

5 Let S(y, x) := y G(x), which we call the Dwork potential. := [y][η] = [y 1, y 0,, y n ][η 1,η 0,,η n ], y = y 1, x 0 = y 0,, x n = y n n S(y, x) K := + n S(y, x), Q :=, := K Q. y i= 1 i y i η i y i= 1 i η i We define three additive gradings on = [y][y] with respect to the multiplication, called ghost number gh, charge ch and physical dimension pd, by the following rules; gh(y 1 ) = 0, gh(x j ) = 0, gh(η 1 ) = 1, gh(η j ) = 1, ch(y 1 ) = d,ch(x j ) = 1, ch(η 1 ) = d, ch(η j ) = 1, pd(y 1 ) = 1, pd(x j ) = 0, pd(η 1 ) = 0, pd(η j ) = 1, where j = 0,, n. Then the ghost number is same as the cohomology degree. 0 (n+2) K (n+1) K K 1 K 0 0 = g h (pd),ch = j (w),λ gh,pd,ch (n+2) j 0 w 0 λ 0 F i l m := (0) (1) (m), F i l m := F i l n 1 m Then (F i l,k) becomes a filtered cochain complex.

6 A BV (Batalin-Vilkovisky) algebra associated to Theorem (J. Park 2 ) Assume that d = n + 1 (for simplicity of presentation). Let r := d yη 1 + n i=0 x i η i and let θ be a formal element with gh(θ) = 2,ch(θ) = 0,pd(θ) = 1. The quadruple ( X := 0 [θ],,k X = Q + r θ +,lk 2 ) is a BV -algebra which satisfies The triple ( X,,Q X := Q + r ) is a commutative differential graded algebra θ (CDGA) and its cohomology is concentrated on degree (ghost number) 0 and H QX ( X ) [X ], where [X ] is a homogeneous coordinate ring of X. The cohomology ( X,K X ) is also concentrated on degree 0 and is quasi-isomorphic to (,0) = (H n 1 (X, ),0); prim J : ( X,K X ) (,0). The quasi-isomorphism J sends the pd filtration F i l m X to the Hodge filtration F i l m. In fact, the filtered complex (F i l X,K X ) gives rise to a spectral sequence whose E 1 -term is H QX ( X ) and E 2 -term is H KX ( X ) which degenerates at E 2. The binary product in the cochain complex ( X,K X ) has consequences on ; the product will induce a formal Frobenius manifold structure on, which gives the Gauss-Manin connection in a special case.

7 BV realization of the period integral and the polarization on We define a -linear map : X such that is a zero map on j if j 0, otherwise; X X X, u v u v, 1 f := (2πi) n+2 X (ɛ) C f ydy S S x 0 x n where C is a closed path on with the standard orientation around y = 0 and X (ɛ) = x n+1 G(x) x i = ɛ > 0, i = 0,1,, n. d x 0 d x n, For each γ H n 1 (X G, ), we define a -linear map C γ : X such that C γ is a zero map on j if j 0, otherwise; X C γ (u) := 2πi Re s u e yg(x) d yω n, u 0 X (2) γ 0 where Ω n = n i=0 ( 1)i x i (d x 0 d ˆ x i d x n ). Let n (X ) be the n-th rational de Rham cohomology group of P n regular outside X G. (P. Griffiths) The residue isomorphism Re s : n (X ). Theorem (J. Park 2 ) Under the isomorphism J : H KX ( X ), (1) induces the cup product polarization on and (2) induces the period integral C [γ]. (1)

8 Applications Assume that d = n + 1. Let {e α } α I be a -basis of = p+q=n 1 p,q and divide I = I 0 I 1 I n 1 according to pd. Let {t α } α I be its dual -basis. Let y k F [k]a (x), k I a,a = 0,, n 1 be representatives of the cohomology classes e α,α I. Γ (t) = t0 a F [0]a(x) + y t1 a F [1]a (x) + + y n 1 tn 1 a F [n 1]a (x) 0 X [[t]]. (3) a I 0 a I 1 a I n 1 There exists a unique 3-tensor A γ (t) [[t]] (explicitly computable based on the Gröbner αβ basis algorithm and depends only on L -homotopy types of Γ (t)) such that α Γ (t) β Γ (t) = γ A αβ γ (t) γ Γ (t) + K X (Λ αβ (t)) + l K 2 (Γ (t),λ αβ (t)), α = t α (4) for some homotopy Λ αβ (t) 1 [[t]]. This PDE is a generalization of the Picard-Fuchs X ODE. Then the following matrix of 1-forms with coefficients in power series in t 1 A γ β (t 1 ) := α I 1 dt α 1 A αβ γ (t) Γ (t) t a j =0,a I j,j 1, β,γ I, becomes the connection matrix of formal Gauss-Manin connection along the geometric deformation G t1 (x) = G(x) + a I 1 t a 1 F [1]a (x) given by the n 2,1 -component of the L -homotopy type of f. We have γ ω α β (X G t1 ) ρ I A γβ ρ (t 1 ) ω α ρ (X G t1 ) = 0, γ I 1,α J,β I, where ω α β (X G t1 ) is the period matrix of a deformed hypersurface X Gt1.

9 Happy Birthday! Glenn!