Kontsevich-Penner model and open intersection numbers Geometry of - - PowerPoint PPT Presentation

Kontsevich-Penner model and open intersection numbers

Geometry of Integrable Systems SISSA, Trieste, 7-9 June 2017 Giulio Ruzza, SISSA Joint work with Marco Bertola, SISSA/Concordia University

Moduli spaces of Riemann surfaces

Mg,n = {stable compact Riemann surfaces of genus g with n marked points}/isomorphism (Deligne and Mumford, 1969). Mg,n is a compact smooth complex orbifold.

• dimC Mg,n = 3g − 3 + n.

Stability condition: 2g − 2 + n > 0.

Moduli spaces of Riemann surfaces

Mg,n = {stable compact Riemann surfaces of genus g with n marked points}/isomorphism (Deligne and Mumford, 1969). Mg,n is a compact smooth complex orbifold.

• dimC Mg,n = 3g − 3 + n.

Stability condition: 2g − 2 + n > 0.

Free energy and Witten-Kontsevich theorem

Psi classes ψi := c1(Li) ∈ H2 Mg,n, Q

• , Li tautological line bdle
• f cotangent spaces at the i−th marked point.

Intersection numbers τd1···τdn=τ r0

0 τ r1 1 ··· :=

• Mg,n ψd1

1 ∧···∧ψdn n

(rj=♯{i: di=j})

Generating function

F(t0,t1,...):=

• exp

j≥0

tjτj

• =

r∗

τ r0

0 τ r1 1 ··· tr0 0 tr1 1 ··· r0!r1!··· = t3 6 + t1 24+ t0t2 24 + t2 1 24 + t2 0 t3 48 +···

Theorem (E. Witten - M. Kontsevich, 1991)

exp F is a KdV tau function. In particular U := ∂t0F satisfies ∂U ∂t1 = U ∂U ∂t0 + 1 12 ∂3U ∂t3

Free energy and Witten-Kontsevich theorem

Psi classes ψi := c1(Li) ∈ H2 Mg,n, Q

• , Li tautological line bdle
• f cotangent spaces at the i−th marked point.

Intersection numbers τd1···τdn=τ r0

0 τ r1 1 ··· :=

• Mg,n ψd1

1 ∧···∧ψdn n

(rj=♯{i: di=j})

Generating function

F(t0,t1,...):=

• exp

j≥0

tjτj

• =

r∗

τ r0

0 τ r1 1 ··· tr0 0 tr1 1 ··· r0!r1!··· = t3 6 + t1 24+ t0t2 24 + t2 1 24 + t2 0 t3 48 +···

Theorem (E. Witten - M. Kontsevich, 1991)

exp F is a KdV tau function. In particular U := ∂t0F satisfies ∂U ∂t1 = U ∂U ∂t0 + 1 12 ∂3U ∂t3

Free energy and Witten-Kontsevich theorem

Psi classes ψi := c1(Li) ∈ H2 Mg,n, Q

• , Li tautological line bdle
• f cotangent spaces at the i−th marked point.

Intersection numbers τd1···τdn=τ r0

0 τ r1 1 ··· :=

• Mg,n ψd1

1 ∧···∧ψdn n

(rj=♯{i: di=j})

Generating function

F(t0,t1,...):=

• exp

j≥0

tjτj

• =

r∗

τ r0

0 τ r1 1 ··· tr0 0 tr1 1 ··· r0!r1!··· = t3 6 + t1 24+ t0t2 24 + t2 1 24 + t2 0 t3 48 +···

Theorem (E. Witten - M. Kontsevich, 1991)

exp F is a KdV tau function. In particular U := ∂t0F satisfies ∂U ∂t1 = U ∂U ∂t0 + 1 12 ∂3U ∂t3

Free energy and Witten-Kontsevich theorem

Psi classes ψi := c1(Li) ∈ H2 Mg,n, Q

• , Li tautological line bdle
• f cotangent spaces at the i−th marked point.

Intersection numbers τd1···τdn=τ r0

0 τ r1 1 ··· :=

• Mg,n ψd1

1 ∧···∧ψdn n

(rj=♯{i: di=j})

Generating function

F(t0,t1,...):=

• exp

j≥0

tjτj

• =

r∗

τ r0

0 τ r1 1 ··· tr0 0 tr1 1 ··· r0!r1!··· = t3 6 + t1 24+ t0t2 24 + t2 1 24 + t2 0 t3 48 +···

Theorem (E. Witten - M. Kontsevich, 1991)

exp F is a KdV tau function. In particular U := ∂t0F satisfies ∂U ∂t1 = U ∂U ∂t0 + 1 12 ∂3U ∂t3

Kontsevich matrix integral

Zn(Y ) :=

• Hn dM exp Tr
• i M3

3 − M2Y

• Hn dM exp Tr (−M2Y )

Hn = Rn2 = n × n hermitian matrices, Y = diag(y1, ..., yn).

• Zn(Y ) is a KdV tau function in Miwa variables

Tk(Y ) := − 2− 2k+1

3

(2k + 1)!! Tr Y −(2k+1)

• Feynman diagramatic expansion as n → ∞ for large Y of

log Zn(Y ) is F(t0(Y ), t1(Y ), ...) where tk(Y ) := −2− 2k+1

3 (2k − 1)!! Tr Y −(2k+1)

Kontsevich matrix integral

Zn(Y ) :=

• Hn dM exp Tr
• i M3

3 − M2Y

• Hn dM exp Tr (−M2Y )

Hn = Rn2 = n × n hermitian matrices, Y = diag(y1, ..., yn).

• Zn(Y ) is a KdV tau function in Miwa variables

Tk(Y ) := − 2− 2k+1

3

(2k + 1)!! Tr Y −(2k+1)

• Feynman diagramatic expansion as n → ∞ for large Y of

log Zn(Y ) is F(t0(Y ), t1(Y ), ...) where tk(Y ) := −2− 2k+1

3 (2k − 1)!! Tr Y −(2k+1)

Kontsevich matrix integral

Zn(Y ) :=

• Hn dM exp Tr
• i M3

3 − M2Y

• Hn dM exp Tr (−M2Y )

Hn = Rn2 = n × n hermitian matrices, Y = diag(y1, ..., yn).

• Zn(Y ) is a KdV tau function in Miwa variables

Tk(Y ) := − 2− 2k+1

3

(2k + 1)!! Tr Y −(2k+1)

• Feynman diagramatic expansion as n → ∞ for large Y of

log Zn(Y ) is F(t0(Y ), t1(Y ), ...) where tk(Y ) := −2− 2k+1

3 (2k − 1)!! Tr Y −(2k+1)

The Riemann-Hilbert problem

Question

Zn is genuinely analytic for Re yk > 0. Does F represent an asymptotic expansion?

M0 M1 M2 M3

Answer: consider RHP in the λ-plane

• Γ(n)

+ = Γ(n) − Mj

Γ(n)(λ) ∼ λ− σ3

4 1+iσ1

√ 2 (1 + O(λ− 1

2 ))

λ → ∞ Mj := D−1

− e−θ−Sjeθ+D+

S0 := [ 1 1

0 1 ] S1 := [ 1 0 1 1 ] S2 :=

0 1

−1 0

• S3 := [ 1 0

1 1 ]

θ := 2 3λ

3 2 σ3

D :=

n

• j=1

√

λj+ √ λ

√

λj− √ λ

The Riemann-Hilbert problem

Question

Zn is genuinely analytic for Re yk > 0. Does F represent an asymptotic expansion?

M0 M1 M2 M3

Answer: consider RHP in the λ-plane

• Γ(n)

+ = Γ(n) − Mj

Γ(n)(λ) ∼ λ− σ3

4 1+iσ1

√ 2 (1 + O(λ− 1

2 ))

λ → ∞ Mj := D−1

− e−θ−Sjeθ+D+

S0 := [ 1 1

0 1 ] S1 := [ 1 0 1 1 ] S2 :=

0 1

−1 0

• S3 := [ 1 0

1 1 ]

θ := 2 3λ

3 2 σ3

D :=

n

• j=1

√

λj+ √ λ

√

λj− √ λ

Kontsevich matrix integral as isomonodromic tau function

The jumps of Ψn := Γne−θD−1 do not depend on λ, λ1, ..., λn ⇒ isomonodromy equations ∂

∂λΨn(λ; λ1, ..., λn) = An(λ; λ1, ..., λn)Ψn(λ; λ1, ..., λn) ∂ ∂λj Ψn(λ; λ1, ..., λn) = Un,j(λ; λ1, ..., λn)Ψn(λ; λ1, ..., λn)

⇒ isomonodromic tau function τn(λ1, ..., λn) (M. Jimbo, T. Miwa and K. Ueno, 1981) ∂ ∂λj log τn(λ1, ..., λn) = resλ=λj dλ Tr A2

n(λ; λ1, ..., λn)

Theorem (M. Bertola - M. Cafasso, 2016)

τn(λ1, ..., λn) = Zn(Y ), λj = y2

j .

Kontsevich matrix integral as isomonodromic tau function

The jumps of Ψn := Γne−θD−1 do not depend on λ, λ1, ..., λn ⇒ isomonodromy equations ∂

∂λΨn(λ; λ1, ..., λn) = An(λ; λ1, ..., λn)Ψn(λ; λ1, ..., λn) ∂ ∂λj Ψn(λ; λ1, ..., λn) = Un,j(λ; λ1, ..., λn)Ψn(λ; λ1, ..., λn)

⇒ isomonodromic tau function τn(λ1, ..., λn) (M. Jimbo, T. Miwa and K. Ueno, 1981) ∂ ∂λj log τn(λ1, ..., λn) = resλ=λj dλ Tr A2

n(λ; λ1, ..., λn)

Theorem (M. Bertola - M. Cafasso, 2016)

τn(λ1, ..., λn) = Zn(Y ), λj = y2

j .

Formulae for closed intersection numbers

Theorem (M. Bertola - B. Dubrovin - D. Yang, 2015)

Let

Θ(λ) :=

       

− 1

2

• g≥1

(6g−5)!! 24g−1(g−1)!λ−6g+4

−

g≥0 (6g−1)!! 24gg! λ−6g

• g≥0

6g+1 6g−1 (6g−1)!! 24gg! λ−6g+2 1 2

• g≥1

(6g−5)!! 24g−1(g−1)!λ−6g+4

       

Fn(λ1,...,λn):=

∞

• k1,...,kn=0
• n

j=1 (2kj +1)!! λ 2kj +1 j

τkj

• Then

F1(λ)=

∞

• g=1

(6g−3)!! 24g g! λ−6g+2

Fn(λ1,...,λn)=− 1

n

• σ∈Sn

Tr(Θ(λσ(1))···Θ(λσ(n)))

• j∈Z/nZ
• λ2

σ(j)−λ2 σ(j+1) −δn,2 λ2 1+λ2 2

(λ2

1−λ2 2) 2

n≥2

Moduli spaces of open Riemann surfaces

Mg,k,l=moduli spaces of open (i.e.with boundary) Riemann surfaces (g =doubled genus, k = ♯ bdry markings, l = ♯ int. markings). Rigorous study initiated by Pandharipande, Solomon and Tessler, 2015.

• •
• Main challenges: Mg,k,l is a real orbifold with real boundary,

possibly nonorientable ⇒ difficulties in the definition of intersection numbers. dimR Mg,k,l = 3g − 3 + k + 2l. Stability condition: 2g − 2 + k + 2l > 0.

Moduli spaces of open Riemann surfaces

Mg,k,l=moduli spaces of open (i.e.with boundary) Riemann surfaces (g =doubled genus, k = ♯ bdry markings, l = ♯ int. markings). Rigorous study initiated by Pandharipande, Solomon and Tessler, 2015.

• •
• Main challenges: Mg,k,l is a real orbifold with real boundary,

possibly nonorientable ⇒ difficulties in the definition of intersection numbers. dimR Mg,k,l = 3g − 3 + k + 2l. Stability condition: 2g − 2 + k + 2l > 0.

Moduli spaces of open Riemann surfaces

Mg,k,l=moduli spaces of open (i.e.with boundary) Riemann surfaces (g =doubled genus, k = ♯ bdry markings, l = ♯ int. markings). Rigorous study initiated by Pandharipande, Solomon and Tessler, 2015.

• •
• Main challenges: Mg,k,l is a real orbifold with real boundary,

possibly nonorientable ⇒ difficulties in the definition of intersection numbers. dimR Mg,k,l = 3g − 3 + k + 2l. Stability condition: 2g − 2 + k + 2l > 0.

Open intersection numbers

Write g = 2h + b − 1, h = ♯handles, b = ♯bdry components, Mh,b,k,l =submoduli of Mg,k,l with fixed h, b. (Refined) open intersection numbers: τα1 · · · ταlσβ1 · · · σβkb := “

• Mh,b,k,l

ψα1

1 · · · ψαl l φβ1 1 · · · φβk k ”

(= 0 only when 2 αi+2 βi=3g−3+k+2l=6h−6+3b+k+2l). (Rigorous definitions of the above are yet to come). (Refined) generating function of open intersection numbers: F o(t0, t1, ...; s0, s1, ...; N) :=

• b≥0

Nb

• exp
• j≥0

(tjτj + sjσj)

• b

Open intersection numbers

Write g = 2h + b − 1, h = ♯handles, b = ♯bdry components, Mh,b,k,l =submoduli of Mg,k,l with fixed h, b. (Refined) open intersection numbers: τα1 · · · ταlσβ1 · · · σβkb := “

• Mh,b,k,l

ψα1

1 · · · ψαl l φβ1 1 · · · φβk k ”

(= 0 only when 2 αi+2 βi=3g−3+k+2l=6h−6+3b+k+2l). (Rigorous definitions of the above are yet to come). (Refined) generating function of open intersection numbers: F o(t0, t1, ...; s0, s1, ...; N) :=

• b≥0

Nb

• exp
• j≥0

(tjτj + sjσj)

• b

Kontsevich-Penner integral and open intersection numbers

• Z(Λ; N) :=
• Hn d

M det

• 1 +
• M

Λ

−N exp Tr

• −
• M3

6 − Λ M 2

• Hn d

M exp Tr

• − Λ

M 2

• Conjecture (A. Alexandrov - B. Safnuk - ...)

Identify tk = (2k − 1)!! Tr Λ−2k−1, sk = 2kk! Tr Λ−2k−2. Then F o(t0, t1, ...; s0, s1, ...; N) gives the expansion of Z(Λ; N) in Miwa variables, for large Λ and n → ∞.

• N = 1 ⇒ conjecture is true (A. Alexandrov, 2015), i.e.
• exp

j≥0

(tjτj + sjσj)

• gives the expansion of Zn(Λ; 1).
• N = 0 ⇒ closed case.

Kontsevich-Penner integral and open intersection numbers

• Z(Λ; N) :=
• Hn d

M det

• 1 +
• M

Λ

−N exp Tr

• −
• M3

6 − Λ M 2

• Hn d

M exp Tr

• − Λ

M 2

• Conjecture (A. Alexandrov - B. Safnuk - ...)

Identify tk = (2k − 1)!! Tr Λ−2k−1, sk = 2kk! Tr Λ−2k−2. Then F o(t0, t1, ...; s0, s1, ...; N) gives the expansion of Z(Λ; N) in Miwa variables, for large Λ and n → ∞.

• N = 1 ⇒ conjecture is true (A. Alexandrov, 2015), i.e.
• exp

j≥0

(tjτj + sjσj)

• gives the expansion of Zn(Λ; 1).
• N = 0 ⇒ closed case.

Kontsevich-Penner integral and open intersection numbers

• Z(Λ; N) :=
• Hn d

M det

• 1 +
• M

Λ

−N exp Tr

• −
• M3

6 − Λ M 2

• Hn d

M exp Tr

• − Λ

M 2

• Conjecture (A. Alexandrov - B. Safnuk - ...)

Identify tk = (2k − 1)!! Tr Λ−2k−1, sk = 2kk! Tr Λ−2k−2. Then F o(t0, t1, ...; s0, s1, ...; N) gives the expansion of Z(Λ; N) in Miwa variables, for large Λ and n → ∞.

• N = 1 ⇒ conjecture is true (A. Alexandrov, 2015), i.e.
• exp

j≥0

(tjτj + sjσj)

• gives the expansion of Zn(Λ; 1).
• N = 0 ⇒ closed case.

Kontsevich-Penner integral and open intersection numbers

• Z(Λ; N) :=
• Hn d

M det

• 1 +
• M

Λ

−N exp Tr

• −
• M3

6 − Λ M 2

• Hn d

M exp Tr

• − Λ

M 2

• Conjecture (A. Alexandrov - B. Safnuk - ...)

Identify tk = (2k − 1)!! Tr Λ−2k−1, sk = 2kk! Tr Λ−2k−2. Then F o(t0, t1, ...; s0, s1, ...; N) gives the expansion of Z(Λ; N) in Miwa variables, for large Λ and n → ∞.

• N = 1 ⇒ conjecture is true (A. Alexandrov, 2015), i.e.
• exp

j≥0

(tjτj + sjσj)

• gives the expansion of Zn(Λ; 1).
• N = 0 ⇒ closed case.

Kontsevich-Penner integral as isomonodromic tau function

Consider the Kontsevich-Penner integral normalized as Zn(Y ; N) := det(iY )N

Hn dM exp Tr

i

3M3 − YM2 − N log(M + iY )

• Hn dM exp Tr(−YM2)

Theorem (M. Bertola - G. R., 2017)

Zn(Y ; N) is the isomonodromic tau function τn(Y ; N) of a suitable (N + 2) × (N + 2) Riemann-Hilbert problem depending on parameters y1, ..., yn and N. This Riemann-Hilbert problem is obtained by a sequence of Schlesinger transformations of the ODE ∂N(∂2 − λ)Ψ = 0

Limiting isomonodromic problem

We can single out a 3 × 3 limiting isomonodromic problem (= n → ∞), depending on N (not necessarily nonnegative integer) as well as on isomonodromic times t1, t2, t3, .... tk = tk(λ1, λ2, ...) = 1 k

• j≥1

λ

− k

2

j

(Miwa variables) The Ψ−function has the expansion Ψ(λ; t1, t2, ...; N) = S · G · YN(λ) · λT0 · exp

• j≥1

λ

j 2 (tj − δj,3)θj

S := diag

• λ− 1

2 , λ− 1 2 , 1

• , G :=

0 0 1

−1 1 0 1 1 0

• , YN(λ) = 1 + O
• λ− 1

2

• T0 = diag

1 4 + N 2 , 1 4 + N 2 , 1 2 − N

• , θj := diag((−1)j, 1, 0)

Limiting isomonodromic problem

We can single out a 3 × 3 limiting isomonodromic problem (= n → ∞), depending on N (not necessarily nonnegative integer) as well as on isomonodromic times t1, t2, t3, .... tk = tk(λ1, λ2, ...) = 1 k

• j≥1

λ

− k

2

j

(Miwa variables) The Ψ−function has the expansion Ψ(λ; t1, t2, ...; N) = S · G · YN(λ) · λT0 · exp

• j≥1

λ

j 2 (tj − δj,3)θj

S := diag

• λ− 1

2 , λ− 1 2 , 1

• , G :=

0 0 1

−1 1 0 1 1 0

• , YN(λ) = 1 + O
• λ− 1

2

• T0 = diag

1 4 + N 2 , 1 4 + N 2 , 1 2 − N

• , θj := diag((−1)j, 1, 0)

Formulae for open intersection numbers

Applying Bertola-Dubrovin-Yang scheme we can derive explicit and computationally efficient formulae. E.g. 1-point functions are given by the following polynomials in N 1 (3k − 1)!!

• k1+k2=k

res

ξ=0 res η=0

1 + ξ 1 + η N (−1)k1(2k1 − 1)!!(2k2 − 1)!!

• ξ2 + ξ3

3

k1+ 1

2

η2 + η3

3

k2+ 1

2

k even/odd ⇐ ⇒ interior/bdry marked point:

τ1= 1+12N2

24

, τ4= 1+56N2+16N4

1152

, τ7= 25+5508N2+3120N4+192N6

2073600

, τ10= 1225+1030896N2+848736N4+102144N6+2304N8

9754214400

,··· τ 5

2 = N+N3 12

, τ 11

2 = 12N+25N3+3N5 2880

, τ 17

2 = 116N+357N3+84N5+3N7 725760

, τ 23

2 = 704N+2764N3+945N5+66N7+N9 139345920

,···

THANKS FOR YOUR ATTENTION!!