Topological recursion from an algebraic perspective Gatan Borot HU - - PowerPoint PPT Presentation

topological recursion from an algebraic perspective
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Topological recursion from an algebraic perspective Gatan Borot HU - - PowerPoint PPT Presentation

Algebraic and combinatorial perspectives in mathematical sciences May Angeli Topological recursion from an algebraic perspective Gatan Borot HU Berlin Oct. 2, 2020 I. Bottom-up : how 2d topology arises from algebra II. Two examples : 2d


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SLIDE 1

Algebraic and combinatorial perspectives in mathematical sciences

Topological recursion from an algebraic perspective

Gaëtan Borot HU Berlin

  • Oct. 2, 2020

May Angeli

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SLIDE 2
  • I. Bottom-up : how 2d topology arises from algebra
  • II. Two examples : 2d TQFT and Virasoro constraints
  • III. Topological expansions in hermitian matrix models
  • IV. Top-down: from geometric to topological recursion
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SLIDE 3

I How 2d topology arises from algebra

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SLIDE 4

Let be a vector space over

  • I. How 2d topology arises from algebra — Airy structures

V

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C

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Choose a basis of linear coordinates (xi)i∈I

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The Weyl algebra is the graded algebra of differential operators on V

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W~

V = C[~]hxi, ~∂xi i 2 Ii

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deg xi = deg ~∂xi = 1

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deg ~ = 2

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An Airy structure is a linear map such that L : V → W~

V

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deg 1 condition : ideal condition : [L(V ), L(V )] ⊂ ~W~

V · L(V )

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Li = ~∂xi + O(2)

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SLIDE 5

Let be a vector space over

  • I. How 2d topology arises from algebra — Airy structures

V

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C

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Choose a basis of linear coordinates (xi)i∈I

<latexit sha1_base64="l3ZIqJTLC/b9WwfHBUogwk5Ch+k=">ADFnicZVLbhMxFHWGVwmPprBkM5BNkUZRBqVqu4hUiQ3sCiJtpcwosj03jRU/RrYnpLmP5DYwm+wQ2zZ8hX8AnYyQkw4Gx+fc+/13DuXlJwZOxz+6kS3bt+5e2/vfvfBw0eP93sHTy6MqjSFCVc6SuCDXAmYWKZ5XBVasCcLgky9fBv1yBNkzJD/amhFzga8nmjGLrpVlv/3A9Yy9njsUZk/HbetbrDwfDeL/SdqQPmpwPjvo/M4KRSsB0lKOjZmw9LmDmvLKIe6m1UGSkyX+BqmIK9T4vcEUyAt73Kzk9yx2RZWZC0j26mQYJH6kSAsvCZVoUMcVt7Ury3a6o34iDHRbXABfga3j1kOEjIlSslDrhBDMYdi7Bs7SgzV46KydOG13AlsF5jPsa8olRaeKmlb1bEwISjxZ/BMIOZGkHDahUiCaZXipv1RHFtY+7h6txgt2tIcL63SCdF4CTYhIeHfaRF7bLwhauqHfM92AaV5VhcHEQdmJ8W4Wf2DTNnctWNPxAvy+yzp5vywJ3/dTfWmbd3azI6QbxlhyPGnKa/l2Ri1eDdDQ4ejfqn50y7KHnqEX6BCl6BidoTfoHE0QRX6jL6gr9Gn6Fv0PfqxDY06Tc5T1EL08w9BYAUS</latexit>

The Weyl algebra is the graded algebra of differential operators on V

<latexit sha1_base64="FbSwyZluNFsdpYaliQbzfOZU6nY=">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</latexit>

W~

V = C[~]hxi, ~∂xi i 2 Ii

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deg xi = deg ~∂xi = 1

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deg ~ = 2

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An Airy structure is a linear map such that L : V → W~

V

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deg 1 condition : ideal condition : [L(V ), L(V )] ⊂ ~W~

V · L(V )

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Li = ~∂xi + O(2)

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slide-6
SLIDE 6
  • I. How 2d topology arises from algebra — Airy structures

An Airy structure is a linear map such that L : V → W~

V

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deg 1 condition : ideal condition : [L(V ), L(V )] ⊂ ~W~

V · L(V )

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Li = ~∂xi + O(2)

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Theorem 1 (Kontsevich, Soibelman 17) There exists a unique with such that F = X

g≥0 n≥1

~g−1 n! Fg,n

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∀i Li · eF = 0

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and F0,1 = 0, F0,2 = 0

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Fg,n ∈ SymnV ∗

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eF

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is called the partition function (uniqueness) (existence)

slide-7
SLIDE 7
  • I. How 2d topology arises from algebra — Partition function

Assume has max. degree 2 Li

<latexit sha1_base64="RF8vRIyMDdOveU9hpNvm+7LCus=">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</latexit>

Li = ~∂xi − X

a,b

1

2Ai a,bxaxb + Bi a,bxa~∂xb + 1 2Ci a,b~2∂xa∂xb

  • − ~Di
<latexit sha1_base64="mNK3UJjq/JgxotBwmV3bxTDOA8=">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</latexit>

Let’s compute e−F Li · eF

<latexit sha1_base64="sbp2HcK4a5GZsbIT1eaCkE2O/Ps=">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</latexit>

[~0 xjxk

2 ]

<latexit sha1_base64="KdUWjh84SjC286A6LM5heIYQ5dg=">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</latexit>

F0,3[i, j, k] − Ai

j,k = 0

<latexit sha1_base64="cYokbvZUSpe0fjE4b9J4QHzgWE=">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</latexit>

= 0

<latexit sha1_base64="4mP7Of2vz8jE/TtbZQoJ6etOf4=">ADCXicZVLbtNAFJ2YVwmvFpZsDNmwsCIbpWq7iFSJDcuCSFsptqp53MRW5mHNjEOrkb8AiS38BjvElq/gK/gFZhIL4XA2c+ace+/4Xl9S8rYNP01iG7dvnP3t794YOHjx4/2T94em5UoynMqOJKXxJsgFcSZrayHC5rDVgQDhdk9Sb4F2vQplLyg72poRB4KatFRbENUjyN06v9UTpON4j/J1lHRqjD2dXB4HfOFG0ESEs5NmaepbUtHNa2ohzaYd4YqDFd4SXMQS59G2XhCbA+15jF8eFq2TdWJC0jT2GuQYJH6kSAkvmci0YLHDbevqup/uqB9CBbovlsDXYNu49xAhU6KUZOo6IcRQzIFNfWOHiaF6yhpLS68VTmBbYr7AvqJUWniqpO1Vx8KEoMSfwTOBmBtBwmlLkQTKsVN/6M4tnDt49rdYpT1pQVeWaUTovEKbEJCwr/TYKx1eXhDC8faHfM9mM5VdRhcHISdGN8W8xObZ4Vz+ZqGH+hXRLb5i21Z4G6U+VvPbIebFTnZIN6So0lHTrK/K3L+epxNxofvJqPT425Z9tBz9BK9Qhk6QqfoLTpDM0RiT6jL+hr9Cn6Fn2PfmxDo0GX8wz1EP38A1zBABM=</latexit>

[~ · 1]

<latexit sha1_base64="qAZWGCN9Ol5P8nW5XMxehDWnzdQ=">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</latexit>

F1,1[i] − Di = 0

<latexit sha1_base64="VCBkWzjAIeTWLWhL0edRWzs9HFo=">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</latexit>

[~g xi2···xin

(n−1)! ]

<latexit sha1_base64="feAEPTD+ghM5U06RlOT4lX+we9I=">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</latexit>

+ 1

2

X

a,b

Ci

a,b

⇣ Fg1,n+1[a, b, i2, . . . , in] + X

ItI0={i2,...,in} h+h0=g

Fh,1+|J|[a, I] Fh0,1+|J0|[b, I0] ⌘◆ = 0

<latexit sha1_base64="49362+8lgurn3jokwuegYO4a0=">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</latexit>

Fg,n[i, i2, . . . , in] − ✓ X

a

X

m=2

Bi

im,aFg,n−1[a, i2, . . . , c

im, . . . , in]

<latexit sha1_base64="XCe/xqDJODoMU3/7ioCobJfemoQ=">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</latexit>

and decompose Fg,n = X

i1,...,in

Fg,n[i1, . . . , in] xi1 · · · xin

<latexit sha1_base64="oPScV1PwjqPSL4eRZQYvK9Hx9c=">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</latexit>

2g − 2 + n ≥ 2

<latexit sha1_base64="MhuxigG5FjreBEGx4DdHL+GZhLk=">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</latexit>
slide-8
SLIDE 8

and such that

  • I. How 2d topology arises from algebra — Partition function

+ 1

2

X

a,b

Ci

a,b

⇣ Fg1,n+1[a, b, i2, . . . , in] + X

ItI0={i2,...,in} h+h0=g

Fh,1+|J|[a, I] Fh0,1+|J0|[b, I0] ⌘◆ = 0

<latexit sha1_base64="49362+8lgurn3jokwuegYO4a0=">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</latexit>

Fg,n[i, i2, . . . , in] − ✓ X

a

X

m=2

Bi

im,aFg,n−1[a, i2, . . . , c

im, . . . , in]

<latexit sha1_base64="XCe/xqDJODoMU3/7ioCobJfemoQ=">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</latexit>

The terms in the bracket are in bijection with

P , → Σ

<latexit sha1_base64="l29wq6G69pbp1wF6EsQJlHGfEgQ=">ADH3icZVLbtNAFJ24PEp4pbBYmPIhoUVxShV20WkSmxYhkfaSrEVzYxv4lHmYc2M01Yj91+Q2MJvsENs+xX8AuPEQjiczZw5947vteXFJwZOxzedoK9O3fv3d9/0H346PGTp72DZ2dGlZrClCqu9AXBjiTMLXMcrgoNGBOJyT1bvaP1+DNkzJz/a6gFTgpWQLRrH10rz3YhImuVIrzZa5xVqryzD5xJYCz3v94WC4Qfg/iRvSRw0m84PO7yRTtBQgLeXYmFk8LGzqsLaMcqi6SWmgwHSFlzADufTN5akjmABve6VdHKeOyaK0IGkVenQTDRIuqRICy8wlWmSwCW3lSuKdrqjfjQMdFvMga/BVmHrIULGRCmZqauIEMxh2zsGzuMDNXjrLQ091rqBLY5gvsK0qlhadK2lZ1LEwdFPmz9kxNzLUg9WlzEdWmVYqb9kdxbOHKx1W7xWjWlhZ4ZWOiMYrsBGpE/6dRpZVLqnf0MJl1Y75EUzjqIeXFgLOzG+rcxPbBanziVrWv9AvziySl5tywJ3/djfWmbV3azIyQbhlhyNGnIS/12Rs7eDeDQ4/Dqnx43y7KPXqLX6A2K0RE6Re/RBE0RTfoK/qGvgdfgh/Bz+DXNjToNDnPUQvB7R/LrgmS</latexit>

Σ − P

<latexit sha1_base64="/fyEHb1qT1lA9QsESPbxavkR7qo=">AD3icZVLbtNAFJ24PEp4tbBkY8iGhYniKlXbRaRKbFiGR9qK2KrmcZNYmYc1Mw6tRv6ISmzhN9ghtnwCX8EvMJNYCIezmTPn3HvH9/qSkhfGDga/OtHOrdt37u7e695/8PDR4739J2dGVZrChCqu9AXBnghYWILy+Gi1IAF4XBOlq+Df74CbQolP9jrEnKB57KYFRbL3M3hdzgeNX8fhyrzfoD9aI/ydpQ3qowfhyv/M7Y4pWAqSlHBszTQelzR3WtqAc6m5WGSgxXeI5TEHOfTOL3BFMgLe9ys6Oc1fIsrIgaR17dDMNEj5RJQSWzGVaMJjhitvalWU73VE/igJ0W1wAX4Gt49ZDhIyIUpKpq4QzEHNvKNHSaG6hGrLF14LXcC2wXmM+wrSqWFp0raVnUsTAhK/Bk8E4i5FiScdiGSYFqluGl/FMcWrnxcvV2MsrY0w0urdEI0XoJNSEj4dxqM1S4Lb2jhWL1lvgPTuKoMg4uDsBXj2J+YtM0dy5b0fAD/aLIOnu+KQvc9VJ/a5l1d70iJ2vEG3I0bMhJ+ndFzg76bB/+HbYOz1ulmUXPUMv0EuUoiN0it6gMZogiT6jL6gr9FN9C36Hv3YhEadJucpaiH6+QeI9AKs</latexit>

stable ∂2,3P ⊂ ˚ Σ

<latexit sha1_base64="2aDguqi1B7yqk5n4i5Zx8i0wU=">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</latexit>

PΣ =

<latexit sha1_base64="EB3B7iLx1QwCkpTeXJbn0dwlak=">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</latexit>

{

<latexit sha1_base64="NxzqCgmvxLJUli1K8p19z8wCs3I=">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</latexit>

∂1P = ∂1Σ

<latexit sha1_base64="RrDx2SWiWUipCU3q2HtEw+WjwKM=">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</latexit>

{

<latexit sha1_base64="NxzqCgmvxLJUli1K8p19z8wCs3I=">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</latexit>

Diff∂

Σ

<latexit sha1_base64="d+n2jETCX9oyJCDCd0XegizRg=">ADIXicZVJLbxMxEHaWVwmvFG5wCeTCYVlq1RtD5EqwYFjeaStlF0i2zubWPFjZXtDK2sl/gsSV/gb3BA3xI/gL2AnK8SGuczn+WY+e8ZDSs6MHQ5/dqJr12/cvLVzu3vn7r37D3q7D8+MqjSFCVc6QuCDXAmYWKZ5XBRasCcDgnyxeBP1+BNkzJd/aqhEzguWQFo9j60Kz32KVa9F+yoqjfpyXWlmE+S9+yucCz3mC4N1xb/3+QNGCAGjud7XZ+p7milQBpKcfGTJNhaTMXVCmHuptWBkpMl3gOU5Bz394icwQT4G2usVR5pgsKwuS1n1v3VSDhA9UCYFlHh6dQ4ErbmtXlu1yR/1wGOh2cAF8Bbuty4iZEyUkrm6jAkxFHPIx76xg9hQPc4rSxc+ljmB7QLzAntFqbTwUEnbUsfChKTY+8CZAMyVIMHbhYgDaZXipv0oji1c+rx6W4zm7VCBl1bpmGi8BuTUPDvNPK8dm4QwuX1vkGzANq8r1b4fAVo5vK/cTmyaZc+mKhg/0qyPr9OlGFrgbJP7UIuvuekWO19bfgMNRA46Tvytytr+XjPYOXo8GJ0fNsuygJ+gZeo4SdIhO0Ct0iaIo/oM/qCvkafom/R9+jHJjXqNDWPUMuiX38Amy0Kkg=</latexit>

{

<latexit sha1_base64="NxzqCgmvxLJUli1K8p19z8wCs3I=">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</latexit>

=

<latexit sha1_base64="4i/V5uY2Z5QXyOQ6lsQqw31FEqs=">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</latexit>

n

[

m=2

<latexit sha1_base64="9bwj6+ANGn8QUorXhlp+N6Cm5F8=">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</latexit>

… with ∂2P = ∂mΣ

<latexit sha1_base64="Jym7v/ZMQiq5LTRqa7/W68JPd9o=">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</latexit>

{

<latexit sha1_base64="NxzqCgmvxLJUli1K8p19z8wCs3I=">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</latexit>

<latexit sha1_base64="kylzgbwHo7SPF9kNhm3wCcOLQw=">ADCXicZVLbtNAFJ2YVwmvFpZsDNmwsKIYpWq7iFSJDcuCSFsptqp5XMdW5mHNjEOrkb8AiS38BjvElq/gK/gFZhIL4XA2c+ace+/4Xl9S8rYyeTXILp1+87de3v3hw8ePnr8ZP/g6blRjaYwp4orfUmwAV5JmNvKcrisNWBOFyQ1ZvgX6xBm0rJD/amhlzgpayKimIbpIw29dX+aDKebBD/T9KOjFCHs6uDwe+MKdoIkJZybMwindQ2d1jbinJoh1ljoMZ0hZewALn0bZS5I5gA73uNLY5zV8m6sSBpG3sMw0SPlIlBJbMZVowKHDbevqup/uqB9CBbovlsDXYNu49xAhM6KUZOo6IcRQzIHNfGOHiaF6xhpLS6/lTmBbYl5gX1EqLTxV0vaqY2FCUOLP4JlAzI0g4bSlSIJpleKm/1EcW7j2ce1uMcr6UoFXVumEaLwCm5CQ8O80GtdFt7QwrF2x3wPpnNVHQYXB2EnxrfF/MQWae5ctqbhB/oVkW32YlsWuBul/tYz2+FmRU42iLfkaNqRk/Tvipy/HqfT8eG76ej0uFuWPfQcvUSvUIqO0Cl6i87QHFUos/oC/oafYq+Rd+jH9vQaNDlPEM9RD/AOiCAQo=</latexit>

{

<latexit sha1_base64="NxzqCgmvxLJUli1K8p19z8wCs3I=">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</latexit>

… with

{

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∂1Σ ∂mΣ Σ − P P

C terms

∂1Σ Σ − P P

Σ

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∂1Σ Σ − P ∂IΣ ∂I0Σ h h0 P

smooth oriented surface, genus , labeled boundaries g

<latexit sha1_base64="iEJ6BI4WQTap1Q9U/CA5m6epZlU=">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</latexit>

Take P

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pair of pants with labeled boundaries n

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= ⇒

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uniquely determined by induction on 2g − 2 + n > 0

<latexit sha1_base64="aWX7Kh1Ifq3NyznLIKpT05WnabI=">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</latexit>

Fg,n

<latexit sha1_base64="/gq7jYwnWF2AqQCrbleWUrjocBc=">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</latexit>

B terms

slide-9
SLIDE 9

and (overdetermined) quadratic relations for i.e. forms a Lie algebra represented by atmost quadratic diff. op

such that and

  • I. How 2d topology arises from algebra — Partition function

The previous argument does not justify that is symmetric in Fg,n[i, i2, . . . , in]

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i ↔ ik

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This is a consequence of the ideal condition. [Li, Lj] = X

a

~f a

i,jLa

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f ∗

∗∗ ∈ C

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(Li)i

<latexit sha1_base64="C5nAvNJXmdt3ecgVctfcetp9Sf8=">ADHicZVLbhMxFHWGVwmvFpZsAtkUaRlqlRtF5EqsWHBoiDSVMqMItzk1jx8j2hFbW/AISW/gNdogt/8BX8AvYyQgx4Wx8fM6913PvXFJyZuxw+KsT3bp95+69vfvdBw8fPX6yf/D0qhKU5hQxZW+ItgAZxImlkOV6UGLAiHKSleB3+6Bm2Ykh/sTQmZwEvJFoxi6Xp4ds5ezVn8/3+cDcoPc/SRrSRw0u5ged32muaCVAWsqxMbNkWNrMYW0Z5VB308pAiWmBlzADufSdrDJHMAHe9iq7OM0ck2VlQdK659FNUj4SJUQWOYu1SKHBa64rV1ZtMd9XNgoNviCvgabN1rPUTImCglc3UdE2Io5pCPfWPHsaF6nFeWryWOYHtCvMF9hWl0sJTJW2rOhYmBMX+DJ4JxNwIEk67EnEwrVLctD+KYwvXPq7eLUbztrTAhVU6JhoXYGMSEv6dRp7XLg1vaOHyesd8D6ZxVRkG1wvCToxvK/cTmyWZc+mah/ot0TW6YtWeCun/hby6y7mxU526C3JSejhpwlf1fk8miQjAbH70b989NmWfbQc/QSHaIEnaBz9AZdoAmiqECf0Rf0NfoUfYu+Rz+2oVGnyXmGWoh+/gHG9QGx</latexit>

For higher order diff. op, we still get a recursion on 2g − 2 + n > 0

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but terms are now in bijection with

Σ0 , → Σ

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{

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{

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Diff∂

Σ

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∂1Σ = ∂1Σ0

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Σ − Σ0

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In the quadratic case, this condition amounts to f k

i,j = Bi j,k − Bj i,k

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(A, B, C, D)

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slide-10
SLIDE 10
  • I. How 2d topology arises from algebra — Comments

The ideal condition is hard to realise : exhibiting Airy structures in not obvious ! The ones we know come from

  • cut and paste relations in 2d geometry
  • conformal field theory (representation theory of VOAs)
  • Lie algebraic techniques (classification for semisimple Lie algebras )

Hadasz, Ruba (19)

In many applications, the interpretation of and as genus and #boundaries g

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n

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  • f a surface is not artificial : “counts” such surfaces

Fg,n

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  • maps (discretized surfaces), and so Feynman expansions of matrix integrals
  • branched covers
  • integrals over , , …

Mg,n

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  • Gromov-Witten invariants (integrals over )

Mg,n(X)

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Indirectly, applications to : knot theory, CFT, integrability, WKB expansions, etc. Mrspin

g,n

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  • branched covers of complex curves (Eynard-Orantin theory)
slide-11
SLIDE 11

II Two examples : 2d TQFT, Virasoro constraints

slide-12
SLIDE 12
  • II. Two examples — The 0th example

V = C

<latexit sha1_base64="EvmOH07yINmozgPISd2et9OIuMA=">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</latexit>

Take L = ~∂x − 1

2x2 + x ~∂x + ~2 2 ∂2 x

  • − ~
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In fact, the equation can be explicitly solved where Bi(y) = y−1/4 exp

  • − 2

3y3/2✓

1 + X

m≥1

6mΓ(m + 1

6)Γ(m + 5 6)

2π y−3m/2 m! ◆

<latexit sha1_base64="xc6AZlw23noknj79GJCxoONONHE=">ADs3icZVLJbtswEJXqLqm7JemxF6W+2KiTWI6d5WAgTQ/tMS2aBTAdg6TGsmBSVEkqjUHwj/pD/Yr+QknZKpkDuLTe7NwhkMKlind6/0OHzUeP3n6bON58XLV6/fbG5tXypRSgoXVDAhrwlWwLIcLnSmGVwXEjAnDK7I4pPXr25Bqkzk3/WygAnHaZ7NMoq1o6abvwySPDrLbHvZiUbR8sbsxvsDi+CuQCRL27tIzySmpm/NgXqwX7feqHjP2k7j5ESJV8aniEUvgRxTZCVcDhDUefMe4zb1PxcXWHNrOA3pY0db0UZFZ1F2R/iYH3FWzhu9UJdPOdLPV2+tVFj0E8Rq0grWdT7fCPygRtOSQa8qwUuO4V+iJwVJnlIFtolJBgekCpzCGPHUDn08MwQRYXSv17HhisrwoNeTURs6aSEIOP6lwzeSJH2MCM1wybU1R1MNdc+VgayTc2C3oG1UK0TIiAiRJ+KuS4imEyco0Nu4rKUVJqOnfcxHCs5jNsMuYC8kdFLmuZcdceaeuO72mPFBLTvyp57zrRS0EU/VLMazhzvnZ+8loUqdmeKGF7BKJF6C7xAf8P40ksQb5GpKbxN4Tv4Faq6Ko9s8T93xcW4mb2DieGINuqX9At8y5RTurtMBMK3Z/NdE2qxU5qSxagaPBGpzE/1bksr8XD/aGXwet0+P1smwE74L3QTuIg6PgNPgSnAcXAQ23w+PwY3jWGDbGDdJIVq6PwnXM26BmDf4XfOs4nw=</latexit>

solves the Airy differential equation

∂2

yBi(y) = y Bi(y)

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Then is the number of terms resulting from the unfolding

  • f the topological recursion, weighted by automorphisms

(~counts pairs of pants decomposition up to diffeo.)

L · eF = 0

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eF = exp ⇣

1 ~(x − x2 2

⌘ Bi 1−2x−~

(2~)2/3

  • <latexit sha1_base64="1N83E/UqcP0VHhK2hd4T9XDVBtE=">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</latexit>

Fg,n = |Gg,n| ∈ Z[ 1

2

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slide-13
SLIDE 13
  • II. Two examples — 2d TQFT

Let be the monoidal category with Bord2

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  • objects : compact 1d smooth oriented manifolds
  • morphisms : cobordisms
  • monoidal structure : disjoint union

Let be the category of finite dim. vector spaces, monoidal structure VectC

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<latexit sha1_base64="l4ZwBiJd6iy+3QcJ5D8q60Dyc4=">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</latexit>

A 2d TQFT is a monoidal functor F : Bord2 → VectC

<latexit sha1_base64="5fti0DaKvfmFDJ06EAp3BCEf45Y=">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</latexit>

It gives - a vector space

F

✓ ◆

= µ F

✓ ◆

= b F

⇣ ⌘

= 1

  • a product

: V ⊗2 → V

<latexit sha1_base64="yTe3mkCdI5bUpW2aZ7oxqUkl8=">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</latexit>
  • a pairing
  • a unit

: V ⊗2 → C

<latexit sha1_base64="MFZkCAbNM+rSg7VQd6pJeu+O2hg=">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</latexit>

(Atiyah) : C → V

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F(S1) = V

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commutative and associative symmetric and compatible :

b

  • µ(a1 ⊗ a2) ⊗ a3
  • = b
  • a1 ⊗ µ(a2 ⊗ a3)
  • <latexit sha1_base64="hOpPvxvL4G0wbuP8PSplHVIWYCg=">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</latexit>
<latexit sha1_base64="FI2vQ3QO8wz4mx3IkX02l+34aPU=">ADF3icZVLbhMxFHWGVwmvtCzZBLJhMYoyKFXbRaRK3bAsiLSVMqPI9txkrPgx2J60lTUfgsQWfoMdYsuSr+AXsJMRYsLZ+Pice6/n3rmk5MzY0ehXJ7pz979B3sPu48eP3n6rLd/cGFUpSlMqeJKXxFsgDMJU8sh6tSAxaEwyVZnQX/cg3aMCU/2NsSMoGXki0YxdZL814v1WxZWPOxYkustbqe9waj4WiD/v8kacgANTif73d+p7milQBpKcfGzJRaTOHtWUQ91NKwMlpiu8hBnIpW+qyBzBHjbq+ziOHNMlpUFSeu+RzfVIOGaKiGwzF2qRQ4LXHFbu7JspzvqR8JAt8UC+Bps3W89RMiEKCVzdRMTYijmkE98Y4exoXqSV5YWXsucwLbAfIF9Ram08FRJ26qOhQlBsT+DZwIxt4KE0xYiDqZVipv2R3Fs4cbH1bvFaN6WFnhlY6JxiuwMQkJ/04jz2uXhje0cHm9Y74H07iqDIPrB2EnxreV+4nNksy5dE3D/QLI+v05bYscDdI/K1l1t3Nipxs0N+So3FDTpK/K3LxZpiMh4fvxoPT42Z9tAL9Aq9Rgk6QqfoLTpHU0TRGn1GX9DX6FP0Lfoe/diGRp0m5zlqIfr5BxVBt0=</latexit>

Frobenius algebra

slide-14
SLIDE 14
  • II. Two examples — 2d TQFT

A 2d TQFT is a monoidal functor F : Bord2 → VectC

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(Atiyah, 81) Theorem (Kontsevich, Soibelman 17) Theorem (Abrams 96) This a 1:1 correspondence between 2d TQFTs and Frobenius algebras We can compute the TQFT functor from the Frobenius algebra using some pair of pants decomposition of the cobordism F

<latexit sha1_base64="PrQiJ+QmL5s0q0eL2LVDQTAngpY=">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</latexit>

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<latexit sha1_base64="QSnJfZv6/jtas8eoDzrn092jKY=">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</latexit>

By the Frobenius algebra axioms, the result is independent of the pair of pants (hence matches the TQFT axioms)

in in in in

: V ⊗n → C

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where and

= ✓ O

{c,c0} glued

b∗

c,c0

O

P = pair of pants

µ∗

P

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µ∗ ∈ (V ∗)⊗3

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b∗ : V ∗ ⊗ V ∗ → C

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slide-15
SLIDE 15
  • II. Two examples — 2d TQFT

Theorem (Kontsevich, Soibelman 17) Lemma 2 (Andersen, B., Chekhov, Orantin 17) Given a 2d TQFT, there is an Airy structure on F(S1) = V

<latexit sha1_base64="Adc1rCav2sYHdx/mMQHU20JetYc=">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</latexit>

whose partition function generate Fg,n = |Gg,n| · F(Σg,n in)

<latexit sha1_base64="7IoLR2W/blzMGKTLBR79EQfyg7o=">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</latexit>

A : V ⊗3 → C

<latexit sha1_base64="l2mlL2gE9KOp2vqfW/D7PKOZKFY=">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</latexit>

B : V ⊗2 → V

<latexit sha1_base64="r1w/NeMVCVhqw3cmpeD9dI8BHWg=">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</latexit>

C : V → V ⊗2

<latexit sha1_base64="dEx3DV3kMsVcz2BN0HPSytSBOPg=">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</latexit>

represents the product when using V

b

' V ∗

<latexit sha1_base64="bQ823s+Id/zHaSM1sgwQO3hRlPY=">ADIHicZVLjtMwFHXDayivDqwQm0A3CEVgzqamUWlkdiwHBDtjNRkKtu5aP6EWynzMiy+BcktvAb7BL+Al+AaeNECln4+N7r32fZCSFdoMhz87wbXrN27e2rvdvXP3v0Hvf2HUy0rRWFCJZPqnGANrBAwMYVhcF4qwJwOCOrV7V+tgalCynemasSUo4XosgLio03zXuPp2HCsVnK0ia64PDeXVjiwunFi3mvPxwMNwj/J3FD+qjB6Xy/8zvJK04CEMZ1noWD0uTWqxMQRm4blJpKDFd4QXMQCx8dcvUEkyAtbXK5EepLURZGRDUhR7dRIGAD1RyjkVmE8UzyHFjLNl2Q631PemANU2LoGtwbiw9RAhYyKlyORlRIimE29oUdRJqcVYZuvS21Nb9wSzHPqOQinsqhWlx1zXTpE/a03XRF9xUp9myaNaNFIy3f4UwYuvZ/bTUaztinHKyNVRBRegYlIHfBvN7LM2c0UFbeZ2xHfgm7UesaKh7Vhx8eXlfmOzeLU2mRN6wH6zREuebpNC8z2Y39ria67WZHjDcItORw15Dj+uyLTl4N4NDh4M+qfHDXLsoeoGfoOYrRITpBr9EpmiCKPqLP6Av6GnwKvgXfgx9b16DTxDxCLQS/gDTKQnt</latexit>

D = F

<latexit sha1_base64="jedZChAv9Tx/juC1K1ClZwX/I=">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</latexit>

<latexit sha1_base64="QSnJfZv6/jtas8eoDzrn092jKY=">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</latexit>

<latexit sha1_base64="QSnJfZv6/jtas8eoDzrn092jKY=">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</latexit>

: V → C

<latexit sha1_base64="4J5iSwrYtmRYQIMA61yAjpVD8E=">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</latexit>

Proved by comparison of TQFT rules with TR The underlying Lie algebra is abelian because the product is symmetric

slide-16
SLIDE 16

and with

  • II. Two examples — Virasoro constraints

The interesting examples of Airy structures have infinite-dimensional V

<latexit sha1_base64="FbSwyZluNFsdpYaliQbzfOZU6nY=">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</latexit>

Take V = zC[ [z2] ]

<latexit sha1_base64="bWXY1oLjOSuzy92/fLQGj6vik4=">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</latexit>

ek = z2k+1 2k + 1

<latexit sha1_base64="z8KXVnYZCamMUmV6xT/Xf7QhQ0Q=">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</latexit>

with basis , and define e∗

k = (2k + 1)dz

z2k+2

<latexit sha1_base64="lPMWx5HpVnL/kt8ns8nf98ZwQ=">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</latexit>

Take Bi

j,k = Res z=0

  • ei · dej · e∗

k · θ

  • =

2k+1 (2i+1)(2j+1) (2j + 1) θk−i−j

<latexit sha1_base64="uQeCq84V+EPlsrK4PLFUv/T1hms=">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</latexit>

Ai

j,k = Res z=0

  • ei · dej · dek · θ
  • = θ−1 δi,j,k,0
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θ = X

s≥−1

θs z2s(dz)−1

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Ci

j,k = Res z=0

  • ei · e∗

j · e∗ k · θ

  • = (2j+1)(2k+1)

2i+1

θk+j+1−i

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Di = θ0

8 δi,0 + θ−1 24 δi,1

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{

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Lemma 3 (Kontsevich, Soibelman 17 ; Andersen, B., Chekhov, Orantin 17 ) Introduce k ∈ N

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These (A,B,C,D) define a quadratic Airy structure based on a Lie algebra isomorphic to spanC(Li)i≥s∗

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[Li, Lj] = (i − j)Li+j

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s∗ = min{s | θs 6= 0}

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slide-17
SLIDE 17

Witten's conjecture (Kontsevich + Dijkgraaf-Verlinde-Verlinde theorem, 91)

  • II. Two examples — Applications

Intersection theory on Mg,n

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moduli space of compact Riemann surfaces

  • f genus g with marked points

C

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p1, . . . , pn

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Mg,n =

<latexit sha1_base64="0dXh61xfkMnp4rWEAlXz3nE+2A0=">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</latexit> <latexit sha1_base64="FI2vQ3QO8wz4mx3IkX02l+34aPU=">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</latexit>

Mg,n

<latexit sha1_base64="n4LEHp2aYFH2Kby4DPDlXOEIc3E=">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</latexit>

Deligne-Mumford compactification by allowing stable (nodal) curves

ψi = c1(T ∗

piC) ∈ H2(Mg,n; Q)

<latexit sha1_base64="FeYnV4MKc+HchfbtSnwzM3cbXA=">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</latexit>

For θ = z−2dz

<latexit sha1_base64="g89aRdW4Gh0TXjBaDQ/yEH+KP4=">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</latexit>

Fg,n[k1, . . . , kn] = ✓ Z

Mg,n n

Y

i=1

ψki

i

n

Y

i=1

(2ki − 1)!!

<latexit sha1_base64="zsLhgkUoIA2IgJYe9wvLMXfQw=">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</latexit>

i.e. Virasoro constraints for - class intersection

ψ

<latexit sha1_base64="DJDqnKpdKySXWBDsk3w/PKC2hEk=">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</latexit>

Wait until Part IV for a geometric explanation

slide-18
SLIDE 18

Mirzakhani's recursion (07)

  • II. Two examples — Applications

Weil-Petersson volumes

moduli space of bordered Riemann surfaces

  • f genus g with n boundaries of lengths

Weil-Petersson volume form

For Wait until Part IV for a geometric explanation (due to Mirzakhani) Mg,n(L) =

<latexit sha1_base64="MYp9/n95sTWo89wu0RuxS8luac=">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</latexit>

L ∈ Rn

+

<latexit sha1_base64="wsjF+zBFfZle36HQi8gpvG9Qphw=">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</latexit>

µWP

<latexit sha1_base64="rSfXw8zclcMSqQW/ow1ukeTzQSw=">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</latexit>

θ = 2π z sin(2πz)dz = X

s≥−1

ζ(2s + 2)(22s+3 − 4)z2s(dz)−1

<latexit sha1_base64="nX389pFEHJGFLFQg+h82w0vdSQ=">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</latexit>

we have Z

Mg,n(L)

dµWP = X

k1,...,kn≥0

Fg,n[k1, . . . , kn]

n

Y

i=1

L2ki

i

2ki!

<latexit sha1_base64="rVSAPsczye1ZgpdFmv1aWwkZM=">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</latexit>
slide-19
SLIDE 19

This amounts to

  • II. Two examples — New Airy structures from old ones

Operations on Airy structures acts by conjugation on U = exp ~

2

X

a,b

φa,b∂xa∂xb

  • <latexit sha1_base64="o2UxMbxP9RzAtb/LoaUh3dtsJzg=">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</latexit>

W~

V

<latexit sha1_base64="yu/hgCimrf1k/ojLOP1ucMqDmk=">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</latexit>

xi → xi + X

a

φi,a ~∂xa

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Lemma 3 still applies when de∗

i → de∗ i +

X

a≥0

φi,a dea

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hence preserves the notion of Airy structure Direct sums of Airy structures are Airy structures

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Lemma 3 has a generalisation to V = zV0[ [z2] ]

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where is a Frobenius algebra V0

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(coupling of the Virasoro example with the 2d TQFT example)

slide-20
SLIDE 20
  • II. Two examples — Abstract loop equations

Back to general . θ

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Let us define the involution and the multidifferentials

ωg,n(z1, . . . , zn) = X

k1,...,kn≥0

Fg,n[k1, . . . , kn]

n

Y

i=1

(2k + 1)dzi z2ki+2

i

<latexit sha1_base64="5Z+KgpH3weZBuRQtiF86IxphBv0=">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</latexit>

2g − 2 + n > 0

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ω0,1(z) = −1 θ

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For any g, n

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abstract loop equations σ(z) = −z

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(B., Eynard, Orantin 13)

ωg,n(z, z2, . . . , zn) + ωg,n(σ(z), z2, . . . , zn)

<latexit sha1_base64="T+YgEtnsDqkHumXJ0w5fqQR40cs=">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</latexit>

holomorphic at z = 0

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ωg1,n+1(z, σ(z), z2, . . . , zn) + X

ItI0={z2,...,zn} h+h0=g

ωh,1+|Z|(z, I)ωh0,1+|I0|(σ(z), I0)

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equivalent to by definition = O

  • z2s∗(dz)2
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n

Y

i=1

e∗

ki(zi)

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ω0,2(z1, z2) = dz1dz2 (z1 − z2)2 + X

a,b≥0

φa,b dea(z1)deb(z2)

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X

i≥s∗

(dz)2 z2i+1 Li · eF = 0

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slide-21
SLIDE 21
  • rthonormal and
  • II. Two examples — Abstract loop equations

More generally, there is a notion of abstract loop equations associated to the data of S

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smooth complex curve

x, y

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meromorphic function on such that

S

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ω0,2

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symmetric bidifferential on double pole with coef. 1 on the diagonal

S2

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dx

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has finitely many zeroes, that are simple and not zeroes of dy

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Frobenius algebra V0 = M

dx(α)=0

C.eα

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µ(eα ⊗ eβ) = δα,βeα

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ω0,1 = ydx

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x = x(α) + z2

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Locally near : α

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local involution σα(z) = −z

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ωg1,n+1(z, σα(z), z2, . . . , zn) + X

ItI0={z2,...,zn} h+h0=g

ωh,1+|Z|(z, I)ωh0,1+|I0|(σα(z), I0) = O

  • y(z)(dz)2
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ωg,n(z, z2, . . . , zn) + ωg,n(σα(z), z2, . . . , zn) = O(dz)

<latexit sha1_base64="V0jZsT2O3cT+5K9XpTUf/Rv4ak=">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</latexit>

∀g, n, α

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slide-22
SLIDE 22
  • II. Two examples — Abstract loop equations

Their set of solutions can be completely described

(B., Shadrin, 15)

and it encodes the partition function of an Airy structure on

ωg1,n+1(z, σα(z), z2, . . . , zn) + X

ItI0={z2,...,zn} h+h0=g

ωh,1+|Z|(z, I)ωh0,1+|I0|(σα(z), I0) = O

  • y(z)(dz)2
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ωg,n(z, z2, . . . , zn) + ωg,n(σα(z), z2, . . . , zn) = O(dz)

<latexit sha1_base64="V0jZsT2O3cT+5K9XpTUf/Rv4ak=">ADYHicZVLbhMxFHUaHiE8msAONgPZJGIUZapUbReRKrFhR0GkLcpEkR83k1H8GNme0Mab+M72CKxpb+AJ4kQk96Nj859q+viTjqbGDwc/aQf3Bw0ePG0+aT589f3HYar+8NCrXFMZUcaWvCTbAUwljm1oO15kGLAiHK7L8UOpXK9AmVfKrvc1gKnAi03lKsfXUrPUtVgISPHNJKIvuOlzPjsKYM2WNh7IXvA8qhtikicCzGPNsgbvr3j3/KPjUjRkL1r1ZqzPoDzYR3AfRDnTQLi5m7dpdzBTNBUhLOTZmEg0yO3VY25RyKJpxbiDdIkTmIBMfHcWU0cwAV7Vcjs/nbpUZrkFSYvARzPWIOE7VUJgyVysBYM5zrktXJZV0x31vU1BV8kF8BXYIqgcRMiIKCWZugkJMRzYCP/sOPQUD1iuaULz02dwHaB+Rz7ilJp4aGStlIdC1OaQr+WmimBuRWkXO1ChKVoleKmeimOLdx4X7FfjLIqNcdLq3RINF6CDUmZ8H83GCtcXJ6hWPFnvgFzE5VWdm4oCT2P5ZzHdsEk2di1e0/EA/ebKI327LAnedyO8qYtHcjMjZJoItOBnuwFn0b0Quj/rRsH/8edg5P90NSwO9Qe9QF0XoBJ2j+gCjRFP9Bv9AfdHfyqN+qH9fbWelDb5bxClai/gtZIR1y</latexit>

∀g, n, α

<latexit sha1_base64="64FlaYRDh4wlaqa9rtoC7VWVyt8=">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</latexit>

There is a unique solution such that

ωg,n(z1, . . . , zn) = X

α

Res

z=α

✓ Z z

α

ω0,2(·, z1) ◆ ωg,n(z, z2, . . . , zn)

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V = zV0[ [z2] ]

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In particular this justifies existence and symmetry of the solution

slide-23
SLIDE 23
  • II. Two examples — Comments

The ideal (here Lie) condition can be checked by direct computation but this looks ad hoc ! There are two conceptual ways to find this Airy structure

  • it can be obtained from free field representation of the Virasoro algebra at c = 1

Other (higher order) Airy structures can be found from the free field rep. VOAs W(glr)

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(Milanov 16 ; B., Bouchard, Chidambaram, Creutzig, Noshchenko 18 ; B., Kramer, Schüler 20)

super-Virasoro

(Bouchard, Ciosmak, Hadasz, Osuga, Ruba, Sulkowski 19)

  • historically, Eynard-Orantin theory preexisted

correspond to higher zeroes of and possibly singular

dx

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S

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correspond to = super-Riemann surface S

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slide-24
SLIDE 24

III Topological expansions in hermitian matrix models

slide-25
SLIDE 25

III Topological expansions in matrix models — Schwinger-Dyson equations Consider the probability measure on the space of hermitian matrices of size

dµ(M) = dM ZN eN Tr V (M)

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N

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M

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V

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: polynomial going to at infinity −∞

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Define the correlators Wn(x1, . . . , xn) = Cumulantµ ✓ Tr 1 x1 − M , . . . , Tr 1 xn − M ◆

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By integration by parts, one can prove

  • r equivalently

µ h Tr

1 x−M

2 − Tr N V 0(M)

x−M

i = 0

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W2(x, x) + W1(x)2 − N V 0(x)W1(x) = −N ⇥ V 0(x)W1(x)]+

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Likewise, for each n ≥ 1

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there is a quadratic functional relation for Wn+1, . . . , W1

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Schwinger-Dyson equations

slide-26
SLIDE 26

(Mhaskar, Saff, Totik,Anderson-Guionnet-Zeitouni …)

III Topological expansions in matrix models — Large N expansion As , the (random) spectral measure of converges to some deterministic (almost surely and in expectation) M

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N → ∞

<latexit sha1_base64="lYm+pO8veVr6enwhbmUanufPKw0=">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</latexit>

λ

<latexit sha1_base64="ZX850Jg2O+A/8TZdbt74TN3Wk=">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</latexit> <latexit sha1_base64="FI2vQ3QO8wz4mx3IkX02l+34aPU=">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</latexit>

W0,1(x) = lim

N→∞ µ

⇥ 1

N Tr 1 x−M

⇤ = Z dλ(ξ) x − ξ

<latexit sha1_base64="YARM6Jhbr76sTuPtZWZQV/8t7gQ=">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</latexit>

exists, holomorphic in

<latexit sha1_base64="FI2vQ3QO8wz4mx3IkX02l+34aPU=">ADF3icZVLbhMxFHWGVwmvtCzZBLJhMYoyKFXbRaRK3bAsiLSVMqPI9txkrPgx2J60lTUfgsQWfoMdYsuSr+AXsJMRYsLZ+Pice6/n3rmk5MzY0ehXJ7pz979B3sPu48eP3n6rLd/cGFUpSlMqeJKXxFsgDMJU8sh6tSAxaEwyVZnQX/cg3aMCU/2NsSMoGXki0YxdZL814v1WxZWPOxYkustbqe9waj4WiD/v8kacgANTif73d+p7milQBpKcfGzJRaTOHtWUQ91NKwMlpiu8hBnIpW+qyBzBHjbq+ziOHNMlpUFSeu+RzfVIOGaKiGwzF2qRQ4LXHFbu7JspzvqR8JAt8UC+Bps3W89RMiEKCVzdRMTYijmkE98Y4exoXqSV5YWXsucwLbAfIF9Ram08FRJ26qOhQlBsT+DZwIxt4KE0xYiDqZVipv2R3Fs4cbH1bvFaN6WFnhlY6JxiuwMQkJ/04jz2uXhje0cHm9Y74H07iqDIPrB2EnxreV+4nNksy5dE3D/QLI+v05bYscDdI/K1l1t3Nipxs0N+So3FDTpK/K3LxZpiMh4fvxoPT42Z9tAL9Aq9Rgk6QqfoLTpHU0TRGn1GX9DX6FP0Lfoe/diGRp0m5zlqIfr5BxVBt0=</latexit>

x ∈ C \ supp λ

<latexit sha1_base64="f46BQbPYam9Lq17nbHwZDJDoxc=">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</latexit>

we have a spectral curve

P(x, y) = y2 − V 0(x)y + Pol(x) = 0

<latexit sha1_base64="FdFoFzBb/nq6TgN/mY6tmJ9TuHM=">ADL3icZVLNjtMwEHbD31L+unDkEuiBrghVsupq2UOlbhwLIh2V2pCZTvTNqp/Itspray8Au+CxBVeA3FBXDnxCjhthMgyF3/zfTNjz3hIzjJtwvB7y7t2/cbNWwe323fu3rv/oHP4cKJloSiMqWRSXRKsgWUCxiYzDC5zBZgTBhdk9arSL9agdCbFO7PNIeF4IbJ5RrFx1KzTG/U2wfbIH/rb98f+C3/yrLc52vrPfRsr7o8kK53v1HDW6Yb9cGf+/yCqQRfVNpodtn7HqaQFB2Eow1pPozA3icXKZJRB2Y4LDTmK7yAKYiFa3aZWIJsKZWmPnLxGYiLwIWvrO2rECAR+o5ByLtHpqCnNcMFPaPG+mW+pGlYFqktgazCl37iIkCGRUqRyExCiKWaQDl1jJ4GmapgWhi4dl1iOzRKzOXYVhVTcQSlMozrmugoK3FlpugJ6y0l1miUPKtFIyXTzUQwb2Li48moxmjapOV4ZqQKi8ApMQKqEf6eRpqWNqzsUt2l5RXwLulZlvjirgS49pK3cSmUWJtvKbVB7pFEmX8ZF8WmO1GzmuIZXu3Imc78/fgdFCDs+jvikyO+9Ggf/Jm0D0f1MtygB6jp6iHInSKztFrNEJjRNFH9Bl9QV+9T94374f3cx/qteqcR6h3q8/1XELyg=</latexit>
  • n which continues analytically to a meromorphic function

S :

<latexit sha1_base64="fFpxiYmRF5qiZQVrWjL0/GxMx8=">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</latexit>

(Tutte 60s, Brezin-Itzykson-Parisi-Zuber 81, …)

Exploiting the Schwinger-Dyson equations and large deviation theory

  • ne can prove the existence of an asymptotic expansion

Wn ∼ X

g≥0

N 2−2g−n Wg,n

<latexit sha1_base64="muy1gUm+17qIM8Dfbv9c8GIZwk8=">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</latexit>

when supp λ = [a, b]

<latexit sha1_base64="tub121CVYn29OzgXA1mnL68yb4=">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</latexit>

(t’Hooft 74, BIPZ 81, Pastur-Shcherbina 01, B. Guionnet 11)

Then, ωg,n(x1, . . . , xn) =

✓ Wg,n(x1, . . . , xn) + δg,0δn,2 (x1 − x2)2 ◆

n

Y

i=1

dxi

<latexit sha1_base64="odpJ5ACoftKxQrzdLoDowp1caf4=">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</latexit>

continues analytically to a meromorphic multidifferential on Sn

<latexit sha1_base64="hNep4UmVz3vAy/lzcCo90pdA=">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</latexit>

(Eynard 04)

with poles at only (for ) dxi = 0

<latexit sha1_base64="4Iw1YZhpZSKmWuQwvn6aXwC0Uj4=">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</latexit>

2g − 2 + n > 0

<latexit sha1_base64="aWX7Kh1Ifq3NyznLIKpT05WnabI=">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</latexit>
slide-27
SLIDE 27

Schwinger-Dyson equations themselves III Topological expansions in matrix models — Large N expansion Inserting in the Schwinger-Dyson equations

(information near , degree 1 condition fails)

Wn ∼ X

g≥0

N 2−2g−n Wg,n

<latexit sha1_base64="muy1gUm+17qIM8Dfbv9c8GIZwk8=">ADN3icZVLNjtMwEHaz/Czlr7scuQR64RCqpOpqdw+VuLCS2IbldqQmU70zSqf4LtlF1ZeQ3eBYkrvAMnbogb4g1w2giRMgf783wzY8/4IwXLtQnDbx1v78bNW7f373Tv3rv/4GHv4PBCy1JRmFDJpLokWAPLBUxMbhcFgowJwymZPWi5qdrUDqX4q25LiDhOBP5IqfYONe8F07nwo91zt1S8rnN/DiD935Y+a/e2aH/3B9mbhFVHEwdGYhq3uHg3Bj/v8gakAfNXY+P+j8ilNJSw7CUIa1nkVhYRKLlckpg6oblxoKTFc4gxmIzDW9TCzBFibK83iJLG5KEoDgla+s26sQMAHKjnHIrWx4ikscMlMZYuinW6pG1kOqu1cAluDqfzWRYSMiZQilVcBIZpiBunYNXYUaKrGaWno0vkSy7FZYrbArqKQijsohWlVx1zXQYHba07XQF9zUu9myYOaNFIy3X4UwauXFy1W4ymbdcCr4xUAVF4BSYgdcK/0jTysb1HYrbtNoh34BuWFnUg/Nrx06Mayt1E5tFibXxmtYf6ATlxPBkWxaY7Ufu1CKr7kYipxvzt+B41IDT6K9ELoaDaDQ4ej3qn50YtlHj9FT9AxF6BidoZfoHE0QR/RZ/QFfU+ed+9H97PbajXaXIeoZ5v/8AM6Q3g=</latexit>

The assumption supp λ = [a, b]

<latexit sha1_base64="tub121CVYn29OzgXA1mnL68yb4=">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</latexit>

xi = ∞

<latexit sha1_base64="oIFfZNJAMCMu1YqSOgWRu6oLkg=">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</latexit>

(information near )

dx = 0

<latexit sha1_base64="OHS5Fh2Re9r0dtpfSF0GFZXozLE=">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</latexit>

and using analytic continuation implies abstract loop equations for (ωg,n)g,n

<latexit sha1_base64="L9iN5dO/w/rnB8ZQWZfZTwIzXFo=">ADG3icZVJLbxMxEHaWVwmvFLhxCeRSpFWURanaHiJV4sKxINJWyq4iP2aTVfxY2d7QYu1PQeIKf4Mb4sqBX8FfwE5WiA1zmc/fNzP2jIeUvDB2NPrViW7cvHX7zt7d7r37Dx4+6u0/Pjeq0hSmVHGlLwk2wAsJU1tYDpelBiwIhwuyeh30izVoUyj53l6XkAm8kEVeUGw9Ne89PUiVgAWeu0Us65dbN+8NRsPRxvr/g6QBA9TY2Xy/8ztlilYCpKUcGzNLRqXNHNa2oBzqbloZKDFd4QXMQC58Y8vMEUyAt7XK5seZK2RZWZC07nvrphokfKBKCyZS7VgkOK29qVZTvdUT+WAnSbXAJfg637rYsImRClJFNXMSGYg5s4hs7jA3VE1ZuvRc5gS2S8xz7CtKpYWHStpWdSxMCIq9D5oJwFwLErxdijiIVilu2o/i2MKVj6t3i1HWpnK8skrHROMV2JiEhH+nwVjt0nCHFo7VO+I7MI2qyjC4fiB2YnxbzE9slmTOpWsaPtAvjazT59uywN0g8aeWHc3K3Kysf4WHI0bcJL8XZHzV8NkPDx8Ox6cHjfLsoeoRfoACXoCJ2iN+gMTRFH9Fn9AV9jT5F36Lv0Y9taNRpcp6glkU/wBRcQed</latexit>

(B., Eynard, Orantin 13)

are not Airy structure constraints/abstract loop equations implies S ' P1

<latexit sha1_base64="XaL4f5v/537hIVtbxHy0jm4NEBg=">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</latexit>

hence automatically

ωg,n(z1, . . . , zn) = X

α

Res

z=α

✓ Z z

α

ω0,2(·, z1) ◆ ωg,n(z, z2, . . . , zn)

<latexit sha1_base64="8N2/weW+j56mIvtNFZQjzLRsaE=">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</latexit>

a

<latexit sha1_base64="408Be2fN6F6Q7i7yKrVG3O/wLwo=">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</latexit>

a

<latexit sha1_base64="408Be2fN6F6Q7i7yKrVG3O/wLwo=">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</latexit>

b

<latexit sha1_base64="0RgdizATZUTa6nUrC+HhKWsKqKg=">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</latexit>

b

<latexit sha1_base64="0RgdizATZUTa6nUrC+HhKWsKqKg=">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</latexit>

S =

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x

− →

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a

<latexit sha1_base64="408Be2fN6F6Q7i7yKrVG3O/wLwo=">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</latexit>

b

<latexit sha1_base64="0RgdizATZUTa6nUrC+HhKWsKqKg=">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</latexit>

(Cauchy formula)

= ⇒

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ωg,n

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computed by topological recursion

(Eynard 05)

slide-28
SLIDE 28

for all , the projection property holds and we have TR III Topological expansions in matrix models — Generalisations The same strategy applies to many other random hermitian matrix models existence of asymptotic expansions

(B., Guionnet, Kozlowski, 15)

Wn ∼ X

g≥0

N 2−2g−n Wg,n

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SD implies abstract loop equations

(B., Eynard, Orantin 13, B. 14)

If p ≥ 3

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Otherwise, it does not and other solutions appear : blobbed TR Blobbed TR appears in random colored tensor models and random spectral triples models

(B. Shadrin 15) (Eynard, Dartois, Bonzom, …) (Azarfar's thesis, …)

dµ(M) = dM ZN exp ✓ X

p≥1 m1,...,mp≥1

N 2−p t(p)

m1,...,mp p

Y

l=1

Tr M ml ◆

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t(p) = 0

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slide-29
SLIDE 29

V From geometric to topological recursion

slide-30
SLIDE 30

An E-valued functorial assignment is the data of for all objects Assume we have a functor IV From geometric to topological recursion — General setting We would like to lift TR to a natural construction associated to surfaces Surf

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Let be the category with

  • objects : compact smooth oriented stable surfaces with labeled boundaries
  • morphisms : isotopy classes of orientation- and label-preserving diffeo.

Let be a category of topological vector spaces V

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E : Surf → V

<latexit sha1_base64="cZkUy9uxQ17xGMpg2r1ezosP3+Y=">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</latexit>

ΩΣ ∈ E(Σ)

<latexit sha1_base64="tpQ1RWU5LGk69GMKQUOxPfn0RKs=">ADJHicZVLbtNAFJ2YVwmvFJYIyZBNkaworlK1XUSqhJDYUR5pK8VWNDO+iUeZhzUzDq1GXvEvSGzhN9ghFmz4BX6BcWIhHM5mzpxz7x3f60sKzowdDn92gmvXb9y8tXO7e+fuvfsPersPz4wqNYUJVzpC4INcCZhYpnlcFowIJwOCfLF7V/vgJtmJLv7VUBqcALyeaMYulWe9J8lrAs9c8o4tBK7ChMnw5d7m9nzW6w8HwzXC/0nckD5qcDrb7fxOMkVLAdJSjo2ZxsPCpg5ryiHqpuUBgpMl3gBU5AL32GeOoIJ8LZX2vlR6pgsSguSVqFHN9Eg4QNVQmCZuUSLDOa45LZyRdFOd9TPh4FuiznwFdgqbD1EyJgoJTN1GRFiKOaQjX1jB5GhepyVluZeS53ANsd8jn1FqbTwVEnbqo6FqYMif9aeqYm5EqQ+bS6i2rRKcdP+KI4tXPq4arsYzdrSHC+t0hHReAk2InXCv9PIsol9RtauKzaMt+CaVxV1IMLa2ErxreV+YlN49S5ZEXrH+i3R1bJ01Z4K4f+1vLrLrFTleI9yQw1FDjuO/K3K2P4hHg4M3o/7JUbMsO+gxeob2UIwO0Ql6hU7RBFH0EX1GX9DX4FPwLfge/NiEBp0m5xFqIfj1B3rCtA=</latexit>

Σ

<latexit sha1_base64="JvtOw0pW3LRCIQPIwjh0ZRWlV8=">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</latexit>

such that, for any we have f : Σ → Σ0

<latexit sha1_base64="KhQ0Y/y18eGXpqsWzrqYrFqZbj8=">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</latexit>

E(f)(ΩΣ) = ΩΣ0

<latexit sha1_base64="tusyUBDAXDryFV/gSoJi7eNp6h8=">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</latexit>

In particular is -invariant ΩΣ

<latexit sha1_base64="4meT/Iw6jBXhi0tUpLyC9eVoh8c=">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</latexit>

Mod∂

Σ := Diff∂ Σ/(Diff∂ Σ)0

<latexit sha1_base64="+Yf5ZQVANu+5yBV54JOtzgiJjvs=">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</latexit>

Geometric recursion constructs such functorial assignments by induction on −χΣ

<latexit sha1_base64="6Al3U2WfngMHpNVWvkldpRz5r0I=">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</latexit>

(Andersen, B., Orantin, 17)

slide-31
SLIDE 31

IV From geometric to topological recursion — Teichmüller setting E(Σ) = C0(TΣ)

<latexit sha1_base64="GRw4SJc9hgqbkyYOtZK3ztGDT8=">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</latexit>

TΣ =

<latexit sha1_base64="68tMyWIQDsHrVjX49ha4ytEmbg=">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</latexit>

{

<latexit sha1_base64="NxzqCgmvxLJUli1K8p19z8wCs3I=">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</latexit>

hyperbolic metrics on such that is geodesic Σ

<latexit sha1_base64="t4nZcNrC/LlUq/+sjdJdFmCy4c=">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</latexit>

∂Σ

<latexit sha1_base64="MH2uXRy6vMBagLCOMEhdnjJe1Y=">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</latexit>

{

<latexit sha1_base64="NxzqCgmvxLJUli1K8p19z8wCs3I=">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</latexit>

(Diff∂

Σ)0

<latexit sha1_base64="h5htlwy1M4BbzfXwnfw2d0Sx/qs=">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</latexit>

MΣ = TΣ/Mod∂

Σ

<latexit sha1_base64="NwO1nE58MlyPcXs3uAIMyPh+eo=">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</latexit>

with topology of convergence on all compacts

Teichmüller space Moduli space E-valued functorial assignments give continuous functions on the moduli space

<latexit sha1_base64="FI2vQ3QO8wz4mx3IkX02l+34aPU=">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</latexit>

such that

PΣ = ✓

n

[

m=2

Pm

Σ

◆ ∪ P∅

Σ

<latexit sha1_base64="h9MAHM+M2IciDLCPB7JO5Uo5YI=">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</latexit>

P∅

Σ =

<latexit sha1_base64="FiMk/UqYhdW0btueqe2M4/H0K8=">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</latexit>

homotopy class of

P , → Σ

<latexit sha1_base64="l29wq6G69pbp1wF6EsQJlHGfEgQ=">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</latexit>

∂1P = ∂1Σ

<latexit sha1_base64="RrDx2SWiWUipCU3q2HtEw+WjwKM=">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</latexit>

∂2P = ∂mΣ

<latexit sha1_base64="Jym7v/ZMQiq5LTRqa7/W68JPd9o=">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</latexit>

Σ − P

<latexit sha1_base64="/fyEHb1qT1lA9QsESPbxavkR7qo=">AD3icZVLbtNAFJ24PEp4tbBkY8iGhYniKlXbRaRKbFiGR9qK2KrmcZNYmYc1Mw6tRv6ISmzhN9ghtnwCX8EvMJNYCIezmTPn3HvH9/qSkhfGDga/OtHOrdt37u7e695/8PDR4739J2dGVZrChCqu9AXBnghYWILy+Gi1IAF4XBOlq+Df74CbQolP9jrEnKB57KYFRbL3M3hdzgeNX8fhyrzfoD9aI/ydpQ3qowfhyv/M7Y4pWAqSlHBszTQelzR3WtqAc6m5WGSgxXeI5TEHOfTOL3BFMgLe9ys6Oc1fIsrIgaR17dDMNEj5RJQSWzGVaMJjhitvalWU73VE/igJ0W1wAX4Gt49ZDhIyIUpKpq4QzEHNvKNHSaG6hGrLF14LXcC2wXmM+wrSqWFp0raVnUsTAhK/Bk8E4i5FiScdiGSYFqluGl/FMcWrnxcvV2MsrY0w0urdEI0XoJNSEj4dxqM1S4Lb2jhWL1lvgPTuKoMg4uDsBXj2J+YtM0dy5b0fAD/aLIOnu+KQvc9VJ/a5l1d70iJ2vEG3I0bMhJ+ndFzg76bB/+HbYOz1ulmUXPUMv0EuUoiN0it6gMZogiT6jL6gr9FN9C36Hv3YhEadJucpaiH6+QeI9AKs</latexit>

stable

{

<latexit sha1_base64="NxzqCgmvxLJUli1K8p19z8wCs3I=">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</latexit>

{

<latexit sha1_base64="NxzqCgmvxLJUli1K8p19z8wCs3I=">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</latexit>

Pm

Σ =

<latexit sha1_base64="FmimKiWuEblsueYLHSDdJh/+oF4=">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</latexit>

{

<latexit sha1_base64="NxzqCgmvxLJUli1K8p19z8wCs3I=">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</latexit>

such that homotopy class of

P , → Σ

<latexit sha1_base64="l29wq6G69pbp1wF6EsQJlHGfEgQ=">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</latexit>

∂1P = ∂1Σ

<latexit sha1_base64="RrDx2SWiWUipCU3q2HtEw+WjwKM=">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</latexit>

Σ − P

<latexit sha1_base64="/fyEHb1qT1lA9QsESPbxavkR7qo=">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</latexit>

stable

{

<latexit sha1_base64="NxzqCgmvxLJUli1K8p19z8wCs3I=">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</latexit>

∂2,3P ⊂ ˚ Σ

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Let us look at Its orbit set is finite and corresponds to the terms in TR PΣ = PΣ/Mod∂

Σ

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slide-32
SLIDE 32

and with symmetric in their last 2 variables Initial data IV From geometric to topological recursion — Teichmüller setting

and note that (boundary lengths)

TP ∼ = R3

+

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A, B, C ∈ C0(R3

+)

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D ∈ C0(TT )SL2(Z)

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pair of pants T =

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P =

<latexit sha1_base64="cLhDXs65XZHQNgiPQTxGqy98Ohk=">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</latexit>

torus with 1 boundary

A, C

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χ = −1

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ΩP = A ~ `(@P)

  • <latexit sha1_base64="XTFsqEr3IDzKwymKJd6vDLn9Eg=">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</latexit>

ΩT = D

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GR construction Disconnected ΩΣ1t···tΣk(σ1, . . . , σk) =

k

Y

i=1

ΩΣi(σi)

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χ ≤ −2

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by induction

ΩΣ() =

n

X

m=2

X

[P ]∈Pm

Σ

B ~ `σ(@P)

  • ΩΣ−P (|Σ−P ) + 1

2

X

[P ]∈P∅

Σ

C ~ `σ(@P)

  • ΩΣ−P (|Σ−P )
<latexit sha1_base64="0yYySRuVzh+KV8+VzW+rQP3XfSA=">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</latexit>

countably many terms, permuted by

<latexit sha1_base64="FI2vQ3QO8wz4mx3IkX02l+34aPU=">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</latexit>

Mod∂

Σ

<latexit sha1_base64="R9IWilcdXI9jnlI375kDEVIvQ=">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</latexit>

∂1Σ ∂mΣ Σ − P P

∂1Σ Σ − P P ∂1Σ Σ − P ∂IΣ ∂I0Σ h h0 P

slide-33
SLIDE 33

IV From geometric to topological recursion — Teichmüller setting and

χ = −1

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ΩP = A ~ `(@P)

  • <latexit sha1_base64="XTFsqEr3IDzKwymKJd6vDLn9Eg=">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</latexit>

ΩT = D

<latexit sha1_base64="gvK+lqwGj0BNV+OjFlxO2qQHi4s=">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</latexit>

Disconnected

ΩΣ1t···tΣk(σ1, . . . , σk) =

k

Y

i=1

ΩΣi(σi)

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χ ≤ −2

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by induction

ΩΣ() =

n

X

m=2

X

[P ]∈Pm

Σ

B ~ `σ(@P)

  • ΩΣ−P (|Σ−P ) + 1

2

X

[P ]∈P∅

Σ

C ~ `σ(@P)

  • ΩΣ−P (|Σ−P )
<latexit sha1_base64="0yYySRuVzh+KV8+VzW+rQP3XfSA=">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</latexit>

Theorem 4 (Andersen, B., Orantin, 17) If A,B,C,D satisfy some decay conditions, then is a well-defined functorial assignment (absolute convergence) ΩΣ

<latexit sha1_base64="4meT/Iw6jBXhi0tUpLyC9eVoh8c=">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</latexit>

V Ωg,n(L) = Z

Mg,n(L)

ΩΣg,ndµWP

<latexit sha1_base64="NYu7nV7Ih+yPTX/C95X/Hwa6d38=">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</latexit>

is a well-defined continuous function of L ∈ Rn

+

<latexit sha1_base64="wsjF+zBFfZle36HQi8gpvG9Qphw=">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</latexit>

and it safisfies topological recursion in the form :

V Ωg,n(L1, . . . , Ln) =

n

X

m=2

Z

R+

d` ` B(L1, Lm, `) V Ωg,n−1(`, L2, . . . , c Lm, . . . , Ln)

<latexit sha1_base64="oYW/8NBR2CwPJt+wsDIl9gyx5o=">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</latexit>

+ 1

2

Z

R2

+

d`d`0 ``0 C(L1, `, `0) ✓ V Ωg1,n+1(`, `0, L2, . . . , Ln) + X

ItI0={L2,...,Ln} h+h0=g

V Ωh,1+|I|(`, I) V Ωh0,1+|I0|(`0, I0) ◆

<latexit sha1_base64="pK4CVRxgD7dYtlDCxG7Q2xXKQAE=">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</latexit>

with base cases and

V Ω0,3(L1, L2, L3) = A(L1, L2, L3)

<latexit sha1_base64="pWAe3dJenLIz8wuZQI1k5JBYCP0=">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</latexit>

V Ω1,1(L) = R

M1,1(L) D dµWP

<latexit sha1_base64="CRxGXg3LvaR3/qkIQ6I+FoG9Hmo=">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</latexit>
slide-34
SLIDE 34

for any and

{

<latexit sha1_base64="NxzqCgmvxLJUli1K8p19z8wCs3I=">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</latexit>

IV From geometric to topological recursion — Examples

with BM(L1, L2, `) = 1 2L1

  • F(L1 + L2 − `) + F(L1 − L2 − `) − F(−L1 + L2 − `) − F(−L1 − L2 − `)
  • <latexit sha1_base64="L3BvB+mN9ZcG9jsSFL1Oby2rSQ=">ADk3icZVLbahsxFJTjXlL3EqelUOiLHL8kZG28xiENxRDSUvqQFqaC2SNkbTH9mKtEjaNEHoD/qD/YpCv6CSbUI3OS8aZkZH0tHQgmfa9Hq/a2v1R4+fPF1/1nj+4uWrjebm63MtS8XgjEku1SUlGngm4MxkhsNloYDklMFnX8K+sU1KJ1J8cPcFjDKyVRk4wR46lx89fR2NpE5fjEue3jcRwdj/tRApzv4CFOJowGzvb94pLaDbd/hJMeBd7G+7gpXEX4yXdqdAdz3Yeu/oijs03xk3271ub1H4IYhXoI1WdTrerP1JUsnKHIRhnGh9FfcKM7JEmYxcI2k1FAQNidTuAIx9QOdjSwlFHhVK83kw8hmoigNCOawr0aiQMBPJvOciDTMKIUJKblxtiq2y3z35GBqpIz4NdgHK4cROmQSilSeRNRqhnhkA79w/YizdQwLQ2beW5kc2JmhE+I7yikyj2UwlS6k1wHU+TXoOkA9G1Ow2pmeREIyX1UtxYuDG+9z9ZiytUhMyN1JFVJE5mIiGDf9PI02dTcIZKrepuyd+B71SZbEIVyDuefyzUj+xq3jk83fNwgf6sAqXtJZtgdu2z1yrIrGIiIHi8JLsD9YgYP4LiLn/W486O59G7QPB6uwrKP3aAtoxjto0P0FZ2iM8TQ39q7Wqu2VX9b/1g/qn9eWtdqz1vUKXqJ/8AzlMi9w=</latexit>

CM(L1, `, `0) = 1 L1

  • F(L1 − ` − `0) − F(−L1 − ` − `0)
  • <latexit sha1_base64="1utfsW/1rPnKJ7+0YC0zjuU7b0=">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</latexit>

F(x) = 2 ln(1 + ex/2)

<latexit sha1_base64="oNxtGS7ue8lYcbphBxAGUs3vsA=">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</latexit>

Take

AM(L1, L2, L3) = 1

<latexit sha1_base64="WkEnJPbFeY+Ds5gIpCIGe/txUqo=">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</latexit>

DM,T () = X

γ=simple closed curve

CM

  • `σ(@T), `σ(), `σ()
  • <latexit sha1_base64="xGBSplYePmMa34+Y81CXOz4S7s=">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</latexit>

Theorem 4 (Mirzakhani, 07) ΩΣ(σ) = 1

<latexit sha1_base64="a4mV75yJ4AGtLZfg9uQsH6JmiY=">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</latexit>

σ ∈ TΣ

<latexit sha1_base64="/hmEVTxf5EVdJTaNyf10rjhEpY=">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</latexit>

Σ

<latexit sha1_base64="JvtOw0pW3LRCIQPIwjh0ZRWlV8=">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</latexit>

As a consequence, satisfies the topological recursion R

Mg,n(L) dµWP

<latexit sha1_base64="wULFrosTvStxF4SbXofjCqWoGCk=">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</latexit>

In fact, the integral operators with kernels B and C preserve the space of even polynomials

A(L1, L2, L3) = X

i,j,k≥0

Ai

j,k ei(L1)ej(L2)ek(L3)

<latexit sha1_base64="JLN1ywyVRVZWGhW1Ccyp3kUwtI=">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</latexit>

Z

R+

d` ` B(L1, L2, `) ek(`) = X

i,j≥0

Bi

j,k ei(L1)ej(L2)

<latexit sha1_base64="+niYOqSYvZHP2s0/eSQ42qQWSBA=">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</latexit>

Z

R2

+

d`d`0 ``0 C(L1, `, `0) ej(`)ek(`0) = X

i0

Ci

j,k ei(L1)

<latexit sha1_base64="mh08PI60lR7VWp6IR4NbZFEDfYc=">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</latexit>

V Ω1,1(L) = X

i≥0

Di ei(L)

<latexit sha1_base64="o2Yp56ZHFV1fTfjTS392I0ImkQ=">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</latexit>

ei(L) = L2i (2i)!

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with the basis yields the Airy structure we’ve seen before …

slide-35
SLIDE 35

IV From geometric to topological recursion — Examples The same thing can be carried on the combinatorial Teichmüller space T comb

Σ

=

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isotopy class of proper embeddings of metric ribbon graphs such that retracts onto Σ

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f(G)

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G

f

, → Σ

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{

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{

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and labels agree In his proof of Witten’s conjecture, Kontsevich constructed a volume form dµK

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Mcomb

Σ

= T comb

Σ

Mod∂

Σ

= [

G ribbon graph type (g,n)

RE(G)

+

Aut G

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  • n the combinatorial Teichmüller space

so that Z

Mcomb

g,n

(L)

dµK = Z

Mg,n

exp ✓1 2

n

X

i=1

L2

i ψi

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and used matrix model techniques to conclude There is an analogue of Mirzakhani’s theorem in the combinatorial case Its integration produces the Virasoro constraint/Airy structure for -intersections

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geometric proof of Witten’s conjecture

ψ

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(Andersen, B., Charbonnier, Giacchetto, Lewanski, Wheeler, to appear)

slide-36
SLIDE 36

Thankyouforyourattention!

  • A. Giacchetto

Airy structures partition function computed by topological recursion Abstract loop equations 2d TQFT

(exact factorization)

geometric recursion

(non local factorization)

Schwinger-Dyson equations matrix models and the like

intersection theory

conformal field theory

integration on moduli space

H∗(Mg,n)

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