THE SPECTRAL THEORYOF QUANTUM CURVES

AND TOPOLOGICAL STRINGS

Marcos MariñoUniversity of Geneva

What is a quantum curve?

Algebraic curves are ubiquitous in modern mathematical physics

1) Seiberg-Witten curves of supersymmetric gauge theories

2) Mirror manifolds to toric (non-compact) Calabi-Yaus

3) “Spectral curves” for large N matrix models

4) Spectral curves for integrable systems

5) A-polynomials of knots

0) WKB curve in one-dimensional quantum mechanics

⌃(x, p) = 0

In all these problems, the “classical” theory is described by the periods of the Liouville one-form

p(x)dx

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along the one-cycles of the curve

An important example for this talk is local mirror symmetry

toric Calabi-Yau manifold X

mirror curve

⌃(ex, ep) = 0

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Here, the periods of the Liouville form determine the “prepotential” of topological string theory on X.

AA

BB

mirror map

genus zero GW invariants

t =

I

Ap(x)dx

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@F0

@t=

I

Bp(x)dx

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F0(t) =X

d�1

N0,de�dt

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ex + ep + e�p�x + u = 0

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This looks very much like a WKB approximation to a quantum mechanical problem underlying the curve

Therefore, it is tempting to think that the corrections to the “classical” theory can be obtained by “quantizing” the curve

in an appropriate way

In the case of local mirror symmetry, this leads to the idea that the higher genus free energies of the topological string

Fg(t) =X

d�1

Ng,de�dt

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can be obtained by “quantizing” the mirror curve [ADKMV]

Perturbative quantization schemes

There are two perturbative quantization schemes for curves which produce a series of quantum corrections to

the prepotential.

The first one is based on a direct quantization of the curve, by promoting the canonically conjugate x,p variables to

Heisenberg operators

[x, p] = i~

⌃(x, p = �i~@x) (x, ~) = 0

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This leads to a Schroedinger-type equation

p(x, ~) ⇠ p(x) +X

n�1

pn(x)~2n

One can now use the all-orders WKB method to obtain “quantum” versions of the Liouville form and the periods

t(~) =I

Ap(x, ~)dx

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@F (t, ~)@t

=

I

Bp(x, ~)dx

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This procedure was applied to SW/mirror curves by [Mironov-Morozov, ACDKV]

However, the quantum prepotential obtained in this way

F (t, ~) =X

n�0

Fn(t)~2n

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does not describe the usual topological string free energy. Rather, it gives the so-called “Nekrasov-Shatashvili

free energy”

A different “quantization” scheme is obtained by the topological recursion of Eynard-Orantin

⌃(x, p) ! {Fg(t)}g=0,1,2,···

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This does give the topological string free energies (BKMP conjecture, now a theorem) but there is no obvious

relation to conventional quantization

There has been recent progress in relating topological recursion to some sort of quantization of the curve [Bouchard-

Eynard, Iwaki, Eynard-Garcia Failde, Orantin-Marchal], but their resulting quantum curves have infinitely many quantum corrections and

I will not follow this route

Another shortcoming is that these approaches are perturbative in nature.

We recall that the “total free energy” of topological string theory

F (t, gs) =X

g�0

Fg(t)g2g�2s

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does not define a function, since it is a factorially divergent series

Fg(t) ⇠ (2g)!

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Topological recursion does not give much insight on this problem.

A full quantum realization of topological string theory should give a non-perturbative definition of the theory, in

which the series appears as the asymptotic expansion of a well-defined function.

It turns out that progress along this direction can be made if one looks at the quantum curve as an operator on the

natural Hilbert space L2(R)

This might seem unnatural from the point of view of complex geometry, since the spectral theory of operators is very sensitive to reality and positivity issues. However,

one can “complexify” afterwards, as we will see.

Operators from curves

By quantizing interesting families of curves one obtains well-known operators on , as well as new ones:L2(R)

1) SW curves of Argyres-Douglas theories lead to Schroedinger operators with polynomial potentials

[e.g. Ito-Shu, Grassi-Gu]

p2 + VN (x)� E = 0

2) SW curves of SU(N) theories lead to deformed Schroedinger operators [Grassi-M.M.]

H = p2 + VN (x)

H = 2 cosh p+ VN (x)2 cosh(p) + VN (x)� E = 0

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3) Mirror curves of toric CYs lead to a new family of “Weyl” or exponentiated Heisenberg operators

OP2 = ex + ep + e�x�p

OF0 = ex + ⇠F0e�x + ep + e�p

(1, 0)(1, 0)(�1, 0)(�1, 0)

(0,�1)(0,�1)

(0, 1)(0, 1)

X ! OX

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For simplicity I will consider polytopes with a single inner point

The spectral problem defined by quantum curves is therefore a fascinating generalization of the spectral theory of one-dimensional Schroedinger operators

The main question we want to address is:is there a relation between the spectral theory of these

operators, and the geometric, “quantum” objects associated to the underlying curves?

I will from now on focus on (local) mirror curves

Spectral theory

The operator ⇢X = O�1X

is positive definite and of trace class

Theorem [Grassi-Hatsuda-

M.M., Kashaev-M.M., Laptev-Schimmer-

Takhtakjan]

on L2(R)

e�En , n = 0, 1, · · ·

and all its traces are finite

Tr ⇢`X =X

n�0

e�`En < 1, ` = 1, 2, · · ·

(assuming and some conditions on “mass parameters”)

~ > 0

<latexit sha1_base64="EN39tOkt7fNzP+vAD0EIB5PQMCg=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoF6k6MVjBfsBbSib7aZdutmE3YlQQn+EFw+KePX3ePPfuG1z0NYHA4/3ZpiZFyRSGHTdb6ewtr6xuVXcLu3s7u0flA+PWiZONeNNFstYdwJquBSKN1Gg5J1EcxoFkreD8d3Mbz9xbUSsHnGScD+iQyVCwShaqd0bBVTfuP1yxa26c5BV4uWkAjka/fJXbxCzNOIKmaTGdD03QT+jGgWTfFrqpYYnlI3pkHctVTTixs/m507JmVUGJIy1LYVkrv6eyGhkzCQKbGdEcWSWvZn4n9dNMbzyM6GSFLlii0VhKgnGZPY7GQjNGcqJJZRpYW8lbEQ1ZWgTKtkQvOWXV0nrourVqtcPtUr9No+jCCdwCufgwSXU4R4a0AQGY3iGV3hzEufFeXc+Fq0FJ585hj9wPn8AvgSPMw==</latexit>

Discrete spectrum

Since the operator is of trace class, we can define its Fredholm determinant

which is an entire function of

⌅X() = det (1 + ⇢X) =Y

n�0

�1 + eµ�En

�

= eµ

It can be shown that is identified with the modulus of the CY, therefore this is an entire function on the CY moduli

space

“fermionic” spectral traces

= 1 +1X

N=1

ZX(N, ~)N⌅X()

<latexit sha1_base64="yopYG22f31iypM/iQJsR4hvyjEg=">AAAB9XicbVBNSwMxEJ31s9avqkcvwSLUS9mVgnorevFYwbYL3bVk02wbms2GJKuU0v/hxYMiXv0v3vw3pu0etPXBwOO9GWbmRZIzbVz321lZXVvf2CxsFbd3dvf2SweHLZ1mitAmSXmq/AhrypmgTcMMp75UFCcRp+1oeDP1249UaZaKezOSNExwX7CYEWys9BD4rOujSjDEUuKzbqnsVt0Z0DLxclKGHI1u6SvopSRLqDCEY607nitNOMbKMMLppBhkmkpMhrhPO5YKnFAdjmdXT9CpVXooTpUtYdBM/T0xxonWoySynQk2A73oTcX/vE5m4stwzITMDBVkvijOODIpmkaAekxRYvjIEkwUs7ciMsAKE2ODKtoQvMWXl0nrvOrVqld3tXL9Oo+jAMdwAhXw4ALqcAsNaAIBBc/wCm/Ok/PivDsf89YVJ585gj9wPn8ASzGRwA==</latexit>

<latexit sha1_base64="WlDQHjMqvpFJ+Vnb7Bm8iJDquP0=">AAAB7XicbVBNSwMxEM3Wr1q/qh69BIvgqexKQb0VvXisYD+gXcpsmm1js0lIskJZ+h+8eFDEq//Hm//GtN2Dtj4YeLw3w8y8SHFmrO9/e4W19Y3NreJ2aWd3b/+gfHjUMjLVhDaJ5FJ3IjCUM0GblllOO0pTSCJO29H4dua3n6g2TIoHO1E0TGAoWMwIWCe1emNQCvrlil/158CrJMhJBeVo9MtfvYEkaUKFJRyM6Qa+smEG2jLC6bTUSw1VQMYwpF1HBSTUhNn82ik+c8oAx1K7EhbP1d8TGSTGTJLIdSZgR2bZm4n/ed3UxldhxoRKLRVksShOObYSz17HA6YpsXziCBDN3K2YjEADsS6gkgshWH55lbQuqkGten1fq9Rv8jiK6ASdonMUoEtUR3eogZqIoEf0jF7Rmye9F+/d+1i0Frx85hj9gff5A5oMjys=</latexit>

-150 -100 -50

-0.5

0.5

1.0

1.5

2.0

2.5

zeroes= spectrum �eEn

⌅P2(, 2⇡)

At weak coupling, when , these quantities are related to the NS topological string

~ ! 0

<latexit sha1_base64="E1UCSgMHfjTKzBzgaXOXo9nAFqg=">AAAB/HicbVBNS8NAEJ3Ur1q/oj16WSyCp5JIQb0VvXisYD+gCWWz3TRLN5uwu1FCqX/FiwdFvPpDvPlv3LY5aOuDgcd7M8zMC1LOlHacb6u0tr6xuVXeruzs7u0f2IdHHZVkktA2SXgiewFWlDNB25ppTnuppDgOOO0G45uZ332gUrFE3Os8pX6MR4KFjGBtpIFd9aIAS+RJNoo0ljJ5RM7Arjl1Zw60StyC1KBAa2B/ecOEZDEVmnCsVN91Uu1PsNSMcDqteJmiKSZjPKJ9QwWOqfIn8+On6NQoQxQm0pTQaK7+npjgWKk8DkxnjHWklr2Z+J/Xz3R46U+YSDNNBVksCjOOdIJmSaAhk5RonhuCiWTmVkQiLDHRJq+KCcFdfnmVdM7rbqN+ddeoNa+LOMpwDCdwBi5cQBNuoQVtIJDDM7zCm/VkvVjv1seitWQVM1X4A+vzBzi3lIM=</latexit>

Surprisingly, the strong coupling limit leads to the conventional topological string

~ ! 1

<latexit sha1_base64="uEwE6V57ygv6JB+cqp1jcEkfx0c=">AAACAXicbVDLSsNAFJ34rPUVdSO4GSyCq5JIQd0V3bisYB/QhDKZTpqhk0mYuVFCqBt/xY0LRdz6F+78G6ePhbYeuHA4517uvSdIBdfgON/W0vLK6tp6aaO8ubW9s2vv7bd0kinKmjQRieoERDPBJWsCB8E6qWIkDgRrB8Prsd++Z0rzRN5BnjI/JgPJQ04JGKlnH3pRQBT2FB9EQJRKHrDHZQh5z644VWcCvEjcGamgGRo9+8vrJzSLmQQqiNZd10nBL4gCTgUblb1Ms5TQIRmwrqGSxEz7xeSDET4xSh+HiTIlAU/U3xMFibXO48B0xgQiPe+Nxf+8bgbhhV9wmWbAJJ0uCjOBIcHjOHCfK0ZB5IYQqri5FdOIKELBhFY2IbjzLy+S1lnVrVUvb2uV+tUsjhI6QsfoFLnoHNXRDWqgJqLoET2jV/RmPVkv1rv1MW1dsmYzB+gPrM8fpXKXCw==</latexit>

The strong coupling result is conveniently formulated by using topological string theory in “conifold” coordinates. We recall that, in the moduli space of the CY, there is a conifold point where the

mirror curve becomes singular.

Re(t) ! 1Re(t) ! 1

conifold pointtc = 0

<latexit sha1_base64="GbyF+seeFMd3a+uU2O9o7sirbMg=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoB6EohePFUxbaEPZbDft0s0m7E6EUvobvHhQxKs/yJv/xm2bg7Y+GHi8N8PMvDCVwqDrfjuFtfWNza3idmlnd2//oHx41DRJphn3WSIT3Q6p4VIo7qNAydup5jQOJW+Fo7uZ33ri2ohEPeI45UFMB0pEglG0ko89duP2yhW36s5BVomXkwrkaPTKX91+wrKYK2SSGtPx3BSDCdUomOTTUjczPKVsRAe8Y6miMTfBZH7slJxZpU+iRNtSSObq74kJjY0Zx6HtjCkOzbI3E//zOhlGV8FEqDRDrthiUZRJggmZfU76QnOGcmwJZVrYWwkbUk0Z2nxKNgRv+eVV0ryoerXq9UOtUr/N4yjCCZzCOXhwCXW4hwb4wEDAM7zCm6OcF+fd+Vi0Fpx85hj+wPn8AUb+jls=</latexit>

large radius

We can parametrize the moduli space by a special vanishing coordinate at the conifold, tc

<latexit sha1_base64="vatjQD6dJJFOsJufYTPrHZxcK+0=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoN6KXjxWtB/QhrLZbtqlm03YnQgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUM95ksYx1J6CGS6F4EwVK3kk0p1EgeTsY38789hPXRsTqEScJ9yM6VCIUjKKVHrDP+uWKW3XnIKvEy0kFcjT65a/eIGZpxBUySY3pem6CfkY1Cib5tNRLDU8oG9Mh71qqaMSNn81PnZIzqwxIGGtbCslc/T2R0ciYSRTYzojiyCx7M/E/r5tieOVnQiUpcsUWi8JUEozJ7G8yEJozlBNLKNPC3krYiGrK0KZTsiF4yy+vktZF1atVr+9rlfpNHkcRTuAUzsGDS6jDHTSgCQyG8Ayv8OZI58V5dz4WrQUnnzmGP3A+fwBVQI3a</latexit>

A conjecture

Let us consider the following ’t Hooft limit of the fermionic spectral traces

N ! 1 ~ ! 1

Then,

N

~ = tc

<latexit sha1_base64="HJTnc4JeYSlrOW5DBCutncq6cT8=">AAAB/nicbVDLSsNAFL3xWesrKq7cDBbBVUmkoC6EohtXUsE+oAlhMp20QyeTMDMRSij4K25cKOLW73Dn3zhts9DWAwOHc87l3jlhypnSjvNtLS2vrK6tlzbKm1vbO7v23n5LJZkktEkSnshOiBXlTNCmZprTTiopjkNO2+HwZuK3H6lULBEPepRSP8Z9wSJGsDZSYB/md8hLTAJ5gxDLMbpCOiCBXXGqzhRokbgFqUCBRmB/eb2EZDEVmnCsVNd1Uu3nWGpGOB2XvUzRFJMh7tOuoQLHVPn59PwxOjFKD0WJNE9oNFV/T+Q4VmoUhyYZYz1Q895E/M/rZjq68HMm0kxTQWaLoowjnaBJF6jHJCWajwzBRDJzKyIDLDHRprGyKcGd//IiaZ1V3Vr18r5WqV8XdZTgCI7hFFw4hzrcQgOaQCCHZ3iFN+vJerHerY9ZdMkqZg7gD6zPH+cflNM=</latexit>

log ZX(N, ~) ⇠X

g�0

Fg(tc)~2�2g

<latexit sha1_base64="szLBtrgWm7ruZmiaAiKjCKYejSM=">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</latexit>

~ =1

gsnote that

Corollary: a “volume conjecture”

Explicit calculations show that these spectral traces are very similar to state integrals for hyperbolic three-manifolds (or to

partition functions of 3d susy theories)

~ ! 1

<latexit sha1_base64="a7L3o20/uraKagQuwbUg7dnCw1I=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksgqeSSEG9Fb14rGA/oAlls900SzebsDtRSujFv+LFgyJe/Rne/Ddu2xy09cHA470ZZuYFqeAaHOfbKq2srq1vlDcrW9s7u3v2/kFbJ5mirEUTkahuQDQTXLIWcBCsmypG4kCwTjC6mfqdB6Y0T+Q9jFPmx2QoecgpASP17SMvCojyFB9GQJRKHrHHZQjjvl11as4MeJm4BamiAs2+/eUNEprFTAIVROue66Tg50QBp4JNKl6mWUroiAxZz1BJYqb9fPbABJ8aZYDDRJmSgGfq74mcxFqP48B0xgQivehNxf+8XgbhpZ9zmWbAJJ0vCjOBIcHTNPCAK0ZBjA0hVHFzK6YRUYSCyaxiQnAXX14m7fOaW69d3dWrjesijjI6RifoDLnoAjXQLWqiFqJogp7RK3qznqwX6936mLeWrGLmEP2B9fkDS4qW4Q==</latexit>

V :

<latexit sha1_base64="n3fyFMcsmkl1iCpKUcf34weXd2Q=">AAAB6XicbVDLSgNBEOyNrxhfUY9eBoPgKeyK4OMU9OIxinlAsoTZSW8yZHZ2mZkVwpI/8OJBEa/+kTf/xkmyB00saCiquunuChLBtXHdb6ewsrq2vlHcLG1t7+zulfcPmjpOFcMGi0Ws2gHVKLjEhuFGYDtRSKNAYCsY3U791hMqzWP5aMYJ+hEdSB5yRo2VHprXvXLFrbozkGXi5aQCOeq98le3H7M0QmmYoFp3PDcxfkaV4UzgpNRNNSaUjegAO5ZKGqH2s9mlE3JilT4JY2VLGjJTf09kNNJ6HAW2M6JmqBe9qfif10lNeOlnXCapQcnmi8JUEBOT6dukzxUyI8aWUKa4vZWwIVWUGRtOyYbgLb68TJpnVe+8enV/Xqnd5HEU4QiO4RQ8uIAa3EEdGsAghGd4hTdn5Lw4787HvLXg5DOH8AfO5w8zwo0q</latexit>

value of the Kahler parameter at the conifold

point

It turns out that the volume conjecture for state integrals [Kashaev, Andersen-Kashaev] has a counterpart in this context:

ZX(N, ~) ⇠ exp (�~V )

<latexit sha1_base64="/ix99AE7v7HF25lQKTWaOCMKLo0=">AAACFnicbVDLSgNBEJz1GeNr1aOXwSAkoGFXBPUmevEkCiYGsyHMTnqzQ2YfzPSKIeQrvPgrXjwo4lW8+TdO4h40WtBQVHXT3eWnUmh0nE9ranpmdm6+sFBcXFpeWbXX1us6yRSHGk9koho+0yBFDDUUKKGRKmCRL+Ha752O/OtbUFok8RX2U2hFrBuLQHCGRmrbuzftRvl8h3qhz1SFelpE1IO71JMQYHl3LNM69ZTohlhp2yWn6oxB/xI3JyWS46Jtf3idhGcRxMgl07rpOim2Bkyh4BKGRS/TkDLeY11oGhqzCHRrMH5rSLeN0qFBokzFSMfqz4kBi7TuR77pjBiGetIbif95zQyDw9ZAxGmGEPPvRUEmKSZ0lBHtCAUcZd8QxpUwt1IeMsU4miSLJgR38uW/pL5XdferR5f7peOTPI4C2SRbpExcckCOyRm5IDXCyT15JM/kxXqwnqxX6+27dcrKZzbIL1jvXyiinYo=</latexit>

N fixed

<latexit sha1_base64="a5YWl8jSYzSUbxIphkwEo9n5UVw=">AAAB+nicbVDJSgNBEO2JW4zbRI9eGoPgQcKMBNRb0IsniWAWSELo6dQkTXoWums0YcynePGgiFe/xJt/Y2c5aOKDgsd7VVTV82IpNDrOt5VZWV1b38hu5ra2d3b37Px+TUeJ4lDlkYxUw2MapAihigIlNGIFLPAk1L3B9cSvP4DSIgrvcRRDO2C9UPiCMzRSx87f0tYpbSEMMfXFELrjjl1wis4UdJm4c1Igc1Q69lerG/EkgBC5ZFo3XSfGdsoUCi5hnGslGmLGB6wHTUNDFoBup9PTx/TYKF3qR8pUiHSq/p5IWaD1KPBMZ8Cwrxe9ifif10zQv2inIowThJDPFvmJpBjRSQ60KxRwlCNDGFfC3Ep5nynG0aSVMyG4iy8vk9pZ0S0VL+9KhfLVPI4sOSRH5IS45JyUyQ2pkCrh5JE8k1fyZj1ZL9a79TFrzVjzmQPyB9bnD58fk6A=</latexit>

In order to obtain the traditional Gromov-Witten free energies, one can instead consider the following limit of the

Fredholm determinant

~ ! 1,µ

~ = t fixed

One has, conjecturally, the asymptotic expansion

log⌅X() ⇠X

g�0

Fg(t)~2�2g

(this asymptotics has however oscillatory corrections)

One striking consequence of this picture is the following. The topological string free energies have a branch cut

structure and are singular at the conifold point. However, in the non-perturbative version, the Fredholm determinant is an

entire function with no singularity!

Fg(t)

stringy moduli space

quantum moduli space

Therefore, branch cuts and singularities are artifacts of the asymptotic expansion (i.e. of perturbation theory)

[cf. Maldacena-Moore-Seiberg-Shih]

Although I have emphasized asymptotic results, we have conjectured an exact expression for the Fredholm

determinant of quantum curves, in terms of (refined) BPS invariants of the toric CY. Roughly,

This provides in principle a complete solution of the spectral problem for these “Weyl-type” operators

⌅X() =X

n2Zexp

0

@X

g�0

Fg(t+ 2⇡i~n)~2g�2 +WKB

1

A

<latexit sha1_base64="x+6LkozOJe58XBluFoVgA24Wrxc=">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</latexit>

Note that this spectral problem is different from the one addressed by Nekrasov and Shatashvili: we quantize curves, not integrable systems. The two problems are identical in

curves of genus one with a 5d susy gauge theory realization; for curves of genus g>1 they are different,

albeit not completely unrelated.

However, our solution of the spectral problem gives non-perturbative corrections to the quantization conditions of

NS

Some applications and extensions

In an independent development, it has been found that tau functions of Painleve functions can be written in terms of 4d instanton partition functions by using the same type of Zak

transform [Gamayun-Iorgov-Lissovvy]

This fits in our framework, after taking a “geometric engineering”/4d limit of the appropriate Calabi-Yau. It leads to

a reinterpretation of the tau function as a spectral determinant [Bonelli-Grassi-Tanzini]

SW/4d limit

⌧ =X

n2Zexp

0

@X

g�0

Fg(a+ 2⇡i~n)~2g�2

1

A

<latexit sha1_base64="1/YS1UlM/vZU3lD/D66gRelm1Fc=">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</latexit>

Complexification

So far we have required to be real and positive, but it is clearly interesting to complexify it.

~

<latexit sha1_base64="+kNMvlgaoM4uks8yWlfwXYhxEYk=">AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqexKQb0VvXisYD+gXUo2zbax2WRJskJZ+h+8eFDEq//Hm//GtN2Dtj4YeLw3w8y8MBHcWM/7RoW19Y3NreJ2aWd3b/+gfHjUMirVlDWpEkp3QmKY4JI1LbeCdRLNSBwK1g7HtzO//cS04Uo+2EnCgpgMJY84JdZJrd4oJBr3yxWv6s2BV4mfkwrkaPTLX72BomnMpKWCGNP1vcQGGdGWU8GmpV5qWELomAxZ11FJYmaCbH7tFJ85ZYAjpV1Ji+fq74mMxMZM4tB1xsSOzLI3E//zuqmNroKMyyS1TNLFoigV2Co8ex0PuGbUiokjhGrubsV0RDSh1gVUciH4yy+vktZF1a9Vr+9rlfpNHkcRTuAUzsGHS6jDHTSgCRQe4Rle4Q0p9ILe0ceitYDymWP4A/T5AyCnjts=</latexit>

It turns out that the integral kernel of can be computedexplicitly in many cases [Kashaev-M.M.] . It involves Faddeev’s

quantum dilogarithm

⇢X

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�b(x)

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b2 / ~

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which can be analytically continued to the complex plane. This makes it possible to reformulate topological string partition functions in the language of (double) q-series,

study their resurgent properties in the complex plane, and so on [in progress with Jie Gu].

~

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Open problems

I have presented a precise “duality” between (conventional) topological strings and the spectral theory of trace class operators. This provides a non-perturbative definition of

topological strings (in the spirit of AdS/CFT) and has many implications in both fields.

What is lacking is a more physical understanding of this duality. Formally, it suggests a deep relationship with a 3d

SUSY theory, or with one-dimensional fermions, but this has not been made more concrete

Mathematically, most of our conjectures are unambiguous and well-defined, but probably very hard to prove (it also

seems to be difficult to find mathematicians who know well both sides of the conjectures, i.e. spectral theory and

Gromov-Witten theory!)

Thank you for your attention!