From de Jonquières Counts to Cohomological Field Theories

Transcription

1 From de Jonquières Counts to Cohomological Field Theories Mara Ungureanu Women at the Intersection of Mathematics and High Energy Physics 9 March 2017

2 What is Enumerative Geometry? How many geometric structures of a given type satisfy a given collection of geometric conditions?

3 What is Enumerative Geometry? Appolonius problem (approx. 200 BC) 8 circles tangent to 3 other circles

4

5 What is Enumerative Geometry? 3264 conics tangent to 5 given conics (1864 Chasles)

6 What is Enumerative Geometry? 3264 conics tangent to 5 given conics (1864 Chasles) 27 lines on cubic surface (1849 Cayley-Salmon)

7 What is Enumerative Geometry? 3264 conics tangent to 5 given conics (1864 Chasles) 27 lines on cubic surface (1849 Cayley-Salmon) 2875 lines on a quintic threefold (1886 Schubert)

8 What is Enumerative Geometry? 3264 conics tangent to 5 given conics (1864 Chasles) 27 lines on cubic surface (1849 Cayley-Salmon) 2875 lines on a quintic threefold (1886 Schubert) conics on a quintic threefold (1985 Katz)

9 What is Enumerative Geometry? 3264 conics tangent to 5 given conics (1864 Chasles) 27 lines on cubic surface (1849 Cayley-Salmon) 2875 lines on a quintic threefold (1886 Schubert) conics on a quintic threefold (1985 Katz) cubics on a quintic threefold (1991 Ellingsrud-Strømme)

10 What is Enumerative Geometry? Question Number of rational curves of any degree on quintic threefold?

11 What is Enumerative Geometry? Question Number of rational curves of any degree on quintic threefold? Answer Mirror symmetry! 1991 Candelas, de la Ossa, Green, Parkes

12 What is Enumerative Geometry? Question Number of rational curves of any degree on quintic threefold? Answer Mirror symmetry! 1991 Candelas, de la Ossa, Green, Parkes 1995 Kontsevich 1996 Givental Clemens conjecture?

13 What is Enumerative Geometry? Question How many points in the plane lie at the intersection of two given lines?

14 What is Enumerative Geometry? Question How many points in the plane lie at the intersection of two given lines? Answer It depends! Lines in general position exactly one Parallel lines none Lines coincide an infinite number of points

15 What is Enumerative Geometry? Parameter (moduli) space Compactify! Do excess intersection theory

16 Ernest de Jonquières

17 Disclaimer

18 de Jonquières multitangency conditions C smooth, genus g f : C P r non-degenerate Degree of f = #{f(c) H} =: d d = 3

19 de Jonquières multitangency conditions C smooth, genus g f : C P r non-degenerate Degree of f = #{f(c) H} =: d d = 3

20 de Jonquières multitangency conditions C smooth, genus g f : C P r non-degenerate Degree of f = #{f(c) H} =: d d = 3

21 de Jonquières multitangency conditions C smooth, genus g f : C P r non-degenerate Degree of f = #{f(c) H} =: d f 1 {f(c) H} = p 1 + 2p 2

22 de Jonquières multitangency conditions de Jonquières counts the number of pairs (p 1, p 2 ) such that there exists a hyperplane H P r with f 1 {f(c) H} = p 1 + 2p 2

23 de Jonquières multitangency conditions de Jonquières (and Mattuck, Macdonald) count the n-tuples (p 1,..., p n ) such that there exists a hyperplane H P r with f 1 {f(c) H} = a 1 p a n p n where a a n = d

24 de Jonquières multitangency conditions The (virtual) de Jonquières numbers are the coefficients of t 1... t n in (1 + a 2 1t a 2 nt n ) g (1 + a 1 t a n t n ) d r g

25 Constructing the moduli space An embedding f : C P r of degree d is given by A pair (L, V ) a line bundle L of degree d on C an (r + 1)-dimensional vector space V of sections of L

26 Constructing the moduli space An embedding f : C P r of degree d is given by A pair (L, V ) a line bundle L of degree d on C an (r + 1)-dimensional vector space V of sections of L Choose (σ 0,..., σ r ) basis of V f : C P r p [σ 0 (p) :... : σ r (p)]

27 Constructing the moduli space Space of all divisors of degree d on C For example C d = C... C /S }{{} d d times p 1 + 2p 2 C 3

28 Constructing the moduli space Space of all divisors of degree d on C For example We define de Jonquières divisors C d = C... C /S }{{} d d times p 1 + 2p 2 C 3 p p n C n such that f 1 {f(c) H} = a 1 p a n p n

29 Constructing the moduli space D = p p n is de Jonquières divisor there exists a section σ whose zeros are a 1 p a n p n

30 Constructing the moduli space D = p p n is de Jonquières divisor there exists a section σ whose zeros are a 1 p a n p n the map has nonzero kernel β D : V V a1 p a np n σ σ a1 p a np n

31 Constructing the moduli space the map β D : V V a1 p a np n σ σ a1 p a np n has nonzero kernel rank(β D ) dim V 1 = r

32 Constructing the moduli space V β D V a1 p a np n D = p p n C n

33 Constructing the moduli space V O Cn β F C n De Jonquières divisors: DJ n = {D C n rank(β D ) r} determinantal variety

34 To summarise... Fix curve C of genus g Fix embedding given by (L, V ) C n := space of divisors of degree n Defined de Jonquières divisors via multitangency conditions Described space DJ n of de Jonquières divisors as determinantal variety over C n

35 Analysing the moduli space dim DJ n n d + r

36 Analysing the moduli space dim DJ n n d + r Relevant questions n d + r < 0 non-existence of de Jonquières divisors n d + r 0 existence of de Jonquières divisors n d + r = 0 finite number of de Jonquières divisors dim DJ n = n d + r

37 Taking a variational perspective Allow C to vary in M g,n Vary the de Jonquières structure with it

38 Taking a variational perspective M g,n = moduli space of smooth curves of genus g with n marked points (C; p 1,..., p n ) M g,n dim M g,n = 3g 3 + n compactification M g,n

39 Taking a variational perspective M g,n = moduli space of smooth curves of genus g with n marked points (C; p 1,..., p n ) M g,n dim M g,n = 3g 3 + n compactification M g,n Question What is the cohomology of M g,n?

40 Taking a variational perspective L = K C = bundle of differential forms on C (L, V ) = (K C, Γ(C, K C )) Now d = 2g 2 and r = g 1

41 Taking a variational perspective L = K C = bundle of differential forms on C (L, V ) = (K C, Γ(C, K C )) Now d = 2g 2 and r = g 1 Fix partition µ = (a 1,..., a n ) of 2g 2 H g (µ) = {(C; p 1,..., p n ) such that K C admits the de Jonquières divisor a 1 p a n p n }

42 Taking a variational perspective H g (µ) M g,n determinantal subvariety flat surfaces, dynamical systems, Teichmüller theory: Masur, Eskin, Zorich, Kontsevich,... Bainbridge-Chen-Gendron-Grushevsky-Möller ( 16),... algebraic geometry: Diaz ( 84), Polishchuk ( 03), Farkas-Pandharipande ( 15)

43 Taking a variational perspective H g (µ) M g,n Take closure: H g (µ) M g,n Question What is the fundamental class [H g (µ)]? Answer (potentially) Cohomological field theory!

44 M g,n and 2-dimensional CFT CFT 2-dimensional QFT invariant under conformal transformations defined over compact Riemann surfaces Stick to holomorphic side CFT = 2-dimensional QFT covariant w.r.t. holomorphic coordinate changes

45 M g,n and 2-dimensional CFT Infinitesimal change of holomorphic coordinate Local holomorphic vector field z z + ɛf(z) f(z) d dz

46 M g,n and 2-dimensional CFT Infinitesimal change of holomorphic coordinate Local meromorphic vector field z z + ɛf(z) f(z) d dz

47 M g,n and 2-dimensional CFT Infinitesimal change of holomorphic coordinate Local meromorphic vector field z z + ɛf(z) f(z) d dz Virasoro algebra: L n = z n+1 d dz [L n, L m ] = (m n)l m+n, n Z etc...

48 M g,n and 2-dimensional CFT Local meromorphic vector field f(z) d dz Infinitesimal deformation of complex structure Infinitesimal deformation of an algebraic curve

49 M g,n and 2-dimensional CFT (C; p 1,..., p n ) M g,n λ = (λ1,..., λ n ) representation labels V λ (C; p 1,..., p n ) space of conformal blocks

50 M g,n and 2-dimensional CFT (C; p 1,..., p n ) M g,n λ = (λ1,..., λ n ) representation labels V λ (C; p 1,..., p n ) space of conformal blocks

51 M g,n and 2-dimensional CFT ( C; p 1,..., p n, q +, q ) M g,n+2 λ = (λ1,..., λ n, λ, λ ) representation labels V λ ( C; p 1,..., p n, q +, q ) space of conformal blocks

52 M g,n and 2-dimensional CFT Verlinde bundle V λ M g,n Each fibre is given by space of conformal blocks V λ (C; p 1,..., p n ) (C; p 1,..., p n )

53 M g,n and CohFT The characters ch(v λ ) define a CohFT on M g,n!

54 M g,n and CohFT The characters ch(v λ ) define a CohFT on M g,n! A CohFT a vector space of fields U a non-degenerate pairing η a distinguished vector 1 U a family of correlators Ω g,n H (M g,n, Q) (U ) n satisfying gluing...

55 M g,n and CohFT Quantum multiplication on U η(v 1 v 2, v 3 ) = Ω 0,3 (v 1 v 2 v 3 ) Q (U, ) Frobenius algebra of the CohFT Teleman: classification of all CohFT with semisimple Frobenius algebra

56 M g,n and CohFT H g (µ) = {(C; p 1,..., p n ) such that K C admits the de Jonquières divisor a 1 p a n p n } [H g (µ)] =? Maybe [H g (µ)] is one of the Ω g,n

57 H g (µ) and CohFT [H g (µ)] is not a CohFT class! Conjecture (Pandharipande, Pixton, Zvonkine): it is related to one Witten R-spin class Wg,µ R H 2g 2 (M g,n, Q)

58 To summarise... Tour of enumerative geometry

59 To summarise... Tour of enumerative geometry Described de Jonquières divisors on fixed curve C with fixed embedding (L, V )

60 To summarise... Tour of enumerative geometry Described de Jonquières divisors on fixed curve C with fixed embedding (L, V ) Allowed C to vary in moduli

61 To summarise... Tour of enumerative geometry Described de Jonquières divisors on fixed curve C with fixed embedding (L, V ) Allowed C to vary in moduli Obtained subspace of M g,n for particular case L = K C

62 To summarise... Tour of enumerative geometry Described de Jonquières divisors on fixed curve C with fixed embedding (L, V ) Allowed C to vary in moduli Obtained subspace of M g,n for particular case L = K C What if L K C?