Topological Strings, Resurgence

and Quantum Mechanics

Marcos MariñoUniversity of Geneva

Codesido-M.M., 1612.07687Codesido-M.M.-Schiappa 1712.0260 Alba Grassi-M.M. 1806.01407

In recent years it has become clear that, in some backgrounds, topological string theory is closely related to

quantum-mechanical models and quantum integrable systems.

Introduction

This relation has led to interesting developments, e.g. non-perturbative definitions of some topological string theories.

The quantum systems involved in this correspondence are not always elementary. In this talk, I will discuss

applications of topological string theory to basic models in Quantum Mechanics. This turns out to shed new light on

old problems.

Back to basics: anharmonic oscillators

Arguably, the simplest models in QM are described by Hamiltonians of the form

However, if the degree of the polynomial N is , there is no explicit solution

H = p2 + VN (x)

� 3

degree N polynomial

cubic potential: the basic example in Heisenberg’s seminal paper of 1925

quartic potential: symmetric double-well

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Almost an exact solution: the exact WKB method, i.e. all-orders WKB + Borel-Ecalle resummation

[Voros, Silverstone, Zinn-Justin, Delabaere, Pham, Alvarez…]

(x, ~) = 1pP (x, ~)

exp

i

~

Z x

P (x0, ~)dx0�

All-orders WKB gives a “quantum” Liouville one-form, which is a formal power series in ~2

P (x, ~)dx = p(x)dx+X

n�1

~2npn(x)dx

wavefunction

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H(p, x) = E

Basic building blocks of the exact WKB method: quantum periods and Voros symbols

V� = exp

✓i

~

I

�P (x, ~)dx

◆� one-cycle in

@FNS

@a= i

I

BP (x, ~)dx

Quantum periods define a deformed special geometry [Nekrasov-Shatashvili, Mironov-Morozov, ACDKV]

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A

BBE

quantum period

a =

I

AP (x, ~)dx

quantum free energy

The spectrum of is determined by an “exact” quantization condition (EQC). Typically it is a vanishing

condition involving Voros symbols

A

BB1 + eia/~ + S e�@aFNS/~ = 0

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determines tower of resonancesin the cubic oscillator

1 + eia/~ ⇡ 0<latexit sha1_base64="VnPxLOdaWI6n9A96LU8OxJWbgWw=">AAACDnicbZDLSsNAFIYn9VbrLerSzWApCEJNRNBl0Y3LCvYCTSwn00k7dHJhZiKWkCdw46u4caGIW9fufBunaRba+sPAx3/O4cz5vZgzqSzr2ygtLa+srpXXKxubW9s75u5eW0aJILRFIh6JrgeSchbSlmKK024sKAQepx1vfDWtd+6pkCwKb9Ukpm4Aw5D5jIDSVt+s2cc4dUSAaXaX5sAyDCfOyAORORDHInrAVt+sWnUrF14Eu4AqKtTsm1/OICJJQENFOEjZs61YuSkIxQinWcVJJI2BjGFIexpDCKh00/ycDNe0M8B+JPQLFc7d3xMpBFJOAk93BqBGcr42Nf+r9RLlX7gpC+NE0ZDMFvkJxyrC02zwgAlKFJ9oACKY/ismIxBAlE6wokOw509ehPZp3bbq9s1ZtXFZxFFGB+gQHSEbnaMGukZN1EIEPaJn9IrejCfjxXg3PmatJaOY2Ud/ZHz+AMzjm1I=</latexit><latexit sha1_base64="VnPxLOdaWI6n9A96LU8OxJWbgWw=">AAACDnicbZDLSsNAFIYn9VbrLerSzWApCEJNRNBl0Y3LCvYCTSwn00k7dHJhZiKWkCdw46u4caGIW9fufBunaRba+sPAx3/O4cz5vZgzqSzr2ygtLa+srpXXKxubW9s75u5eW0aJILRFIh6JrgeSchbSlmKK024sKAQepx1vfDWtd+6pkCwKb9Ukpm4Aw5D5jIDSVt+s2cc4dUSAaXaX5sAyDCfOyAORORDHInrAVt+sWnUrF14Eu4AqKtTsm1/OICJJQENFOEjZs61YuSkIxQinWcVJJI2BjGFIexpDCKh00/ycDNe0M8B+JPQLFc7d3xMpBFJOAk93BqBGcr42Nf+r9RLlX7gpC+NE0ZDMFvkJxyrC02zwgAlKFJ9oACKY/ismIxBAlE6wokOw509ehPZp3bbq9s1ZtXFZxFFGB+gQHSEbnaMGukZN1EIEPaJn9IrejCfjxXg3PmatJaOY2Ud/ZHz+AMzjm1I=</latexit><latexit sha1_base64="VnPxLOdaWI6n9A96LU8OxJWbgWw=">AAACDnicbZDLSsNAFIYn9VbrLerSzWApCEJNRNBl0Y3LCvYCTSwn00k7dHJhZiKWkCdw46u4caGIW9fufBunaRba+sPAx3/O4cz5vZgzqSzr2ygtLa+srpXXKxubW9s75u5eW0aJILRFIh6JrgeSchbSlmKK024sKAQepx1vfDWtd+6pkCwKb9Ukpm4Aw5D5jIDSVt+s2cc4dUSAaXaX5sAyDCfOyAORORDHInrAVt+sWnUrF14Eu4AqKtTsm1/OICJJQENFOEjZs61YuSkIxQinWcVJJI2BjGFIexpDCKh00/ycDNe0M8B+JPQLFc7d3xMpBFJOAk93BqBGcr42Nf+r9RLlX7gpC+NE0ZDMFvkJxyrC02zwgAlKFJ9oACKY/ismIxBAlE6wokOw509ehPZp3bbq9s1ZtXFZxFFGB+gQHSEbnaMGukZN1EIEPaJn9IrejCfjxXg3PmatJaOY2Ud/ZHz+AMzjm1I=</latexit><latexit sha1_base64="VnPxLOdaWI6n9A96LU8OxJWbgWw=">AAACDnicbZDLSsNAFIYn9VbrLerSzWApCEJNRNBl0Y3LCvYCTSwn00k7dHJhZiKWkCdw46u4caGIW9fufBunaRba+sPAx3/O4cz5vZgzqSzr2ygtLa+srpXXKxubW9s75u5eW0aJILRFIh6JrgeSchbSlmKK024sKAQepx1vfDWtd+6pkCwKb9Ukpm4Aw5D5jIDSVt+s2cc4dUSAaXaX5sAyDCfOyAORORDHInrAVt+sWnUrF14Eu4AqKtTsm1/OICJJQENFOEjZs61YuSkIxQinWcVJJI2BjGFIexpDCKh00/ycDNe0M8B+JPQLFc7d3xMpBFJOAk93BqBGcr42Nf+r9RLlX7gpC+NE0ZDMFvkJxyrC02zwgAlKFJ9oACKY/ismIxBAlE6wokOw509ehPZp3bbq9s1ZtXFZxFFGB+gQHSEbnaMGukZN1EIEPaJn9IrejCfjxXg3PmatJaOY2Ud/ZHz+AMzjm1I=</latexit>

Perturbatively, we have

Bohr-Sommerfeld

H

I

Ap(x)dx ⇡ 2⇡~

✓n+

1

2

◆

The EQC is only well-defined once a resummation prescription has been chosen for the formal power series.

The coefficients of resummed Voros symbols depend on this prescription (“resurgent” version of Stokes phenomenon)

S = 0, 1<latexit sha1_base64="xTc0dq8Vb+i8r8Ao+gQ2ZQsKH0s=">AAACBHicbVBNS8MwGH7r55xfVY+7BIfgQUYrgl6EoRePE90HbGWkabaFpWlJUmGUHrz4V7x4UMSrP8Kb/8as60E3Hwg8eZ73I3n8mDOlHefbWlpeWV1bL22UN7e2d3btvf2WihJJaJNEPJIdHyvKmaBNzTSnnVhSHPqctv3x9dRvP1CpWCTu9SSmXoiHgg0YwdpIfbuS9vIhqaRBlpoL5uguy7JL58Tt21Wn5uRAi8QtSBUKNPr2Vy+ISBJSoQnHSnVdJ9ZeiqVmhNOs3EsUjTEZ4yHtGipwSJWX5vszdGSUAA0iaY7QKFd/d6Q4VGoS+qYyxHqk5r2p+J/XTfTgwkuZiBNNBZktGiQc6QhNE0EBk5RoPjEEE8nMWxEZYYmJNrmVTQju/JcXSeu05jo19/asWr8q4ihBBQ7hGFw4hzrcQAOaQOARnuEV3qwn68V6tz5mpUtW0XMAf2B9/gC2ypgY</latexit><latexit sha1_base64="xTc0dq8Vb+i8r8Ao+gQ2ZQsKH0s=">AAACBHicbVBNS8MwGH7r55xfVY+7BIfgQUYrgl6EoRePE90HbGWkabaFpWlJUmGUHrz4V7x4UMSrP8Kb/8as60E3Hwg8eZ73I3n8mDOlHefbWlpeWV1bL22UN7e2d3btvf2WihJJaJNEPJIdHyvKmaBNzTSnnVhSHPqctv3x9dRvP1CpWCTu9SSmXoiHgg0YwdpIfbuS9vIhqaRBlpoL5uguy7JL58Tt21Wn5uRAi8QtSBUKNPr2Vy+ISBJSoQnHSnVdJ9ZeiqVmhNOs3EsUjTEZ4yHtGipwSJWX5vszdGSUAA0iaY7QKFd/d6Q4VGoS+qYyxHqk5r2p+J/XTfTgwkuZiBNNBZktGiQc6QhNE0EBk5RoPjEEE8nMWxEZYYmJNrmVTQju/JcXSeu05jo19/asWr8q4ihBBQ7hGFw4hzrcQAOaQOARnuEV3qwn68V6tz5mpUtW0XMAf2B9/gC2ypgY</latexit><latexit sha1_base64="xTc0dq8Vb+i8r8Ao+gQ2ZQsKH0s=">AAACBHicbVBNS8MwGH7r55xfVY+7BIfgQUYrgl6EoRePE90HbGWkabaFpWlJUmGUHrz4V7x4UMSrP8Kb/8as60E3Hwg8eZ73I3n8mDOlHefbWlpeWV1bL22UN7e2d3btvf2WihJJaJNEPJIdHyvKmaBNzTSnnVhSHPqctv3x9dRvP1CpWCTu9SSmXoiHgg0YwdpIfbuS9vIhqaRBlpoL5uguy7JL58Tt21Wn5uRAi8QtSBUKNPr2Vy+ISBJSoQnHSnVdJ9ZeiqVmhNOs3EsUjTEZ4yHtGipwSJWX5vszdGSUAA0iaY7QKFd/d6Q4VGoS+qYyxHqk5r2p+J/XTfTgwkuZiBNNBZktGiQc6QhNE0EBk5RoPjEEE8nMWxEZYYmJNrmVTQju/JcXSeu05jo19/asWr8q4ihBBQ7hGFw4hzrcQAOaQOARnuEV3qwn68V6tz5mpUtW0XMAf2B9/gC2ypgY</latexit><latexit sha1_base64="xTc0dq8Vb+i8r8Ao+gQ2ZQsKH0s=">AAACBHicbVBNS8MwGH7r55xfVY+7BIfgQUYrgl6EoRePE90HbGWkabaFpWlJUmGUHrz4V7x4UMSrP8Kb/8as60E3Hwg8eZ73I3n8mDOlHefbWlpeWV1bL22UN7e2d3btvf2WihJJaJNEPJIdHyvKmaBNzTSnnVhSHPqctv3x9dRvP1CpWCTu9SSmXoiHgg0YwdpIfbuS9vIhqaRBlpoL5uguy7JL58Tt21Wn5uRAi8QtSBUKNPr2Vy+ISBJSoQnHSnVdJ9ZeiqVmhNOs3EsUjTEZ4yHtGipwSJWX5vszdGSUAA0iaY7QKFd/d6Q4VGoS+qYyxHqk5r2p+J/XTfTgwkuZiBNNBZktGiQc6QhNE0EBk5RoPjEEE8nMWxEZYYmJNrmVTQju/JcXSeu05jo19/asWr8q4ihBBQ7hGFw4hzrcQAOaQOARnuEV3qwn68V6tz5mpUtW0XMAf2B9/gC2ypgY</latexit>

1 + eia/~ + S e�@aFNS/~ = 0<latexit sha1_base64="CggBCXqsBJ7BYVOvr2kKgkUBXJw=">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</latexit><latexit sha1_base64="CggBCXqsBJ7BYVOvr2kKgkUBXJw=">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</latexit><latexit sha1_base64="CggBCXqsBJ7BYVOvr2kKgkUBXJw=">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</latexit><latexit sha1_base64="CggBCXqsBJ7BYVOvr2kKgkUBXJw=">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</latexit>

perturbative: how does one calculate Voros symbols?What is their structure?

non-perturbative: can we find EQCs involving actual functions, and not (resummed) formal power series?

Obvious questions:

As we will now see, topological string theory has something interesting to say about both questions

Voros symbolsand the holomorphic anomaly

In topological string theory, we are often interested in the perturbative series of the free energies, . One of the

most powerful methods to obtain them is the BCOV holomorphic anomaly equation (HAE)

-Recursive, with initial data

Fg(t)<latexit sha1_base64="nw7ZQlhJXlSpDiV8/nU1vOXnC+g=">AAAB7XicbVBNS8NAEJ34WetX1aOXxSLUS0lE0GNREI8V7Ae0oWy2m3btJht2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekEhh0HW/nZXVtfWNzcJWcXtnd2+/dHDYNCrVjDeYkkq3A2q4FDFvoEDJ24nmNAokbwWjm6nfeuLaCBU/4DjhfkQHsQgFo2il5m1vUMGzXqnsVt0ZyDLxclKGHPVe6avbVyyNeIxMUmM6npugn1GNgkk+KXZTwxPKRnTAO5bGNOLGz2bXTsipVfokVNpWjGSm/p7IaGTMOApsZ0RxaBa9qfif10kxvPIzEScp8pjNF4WpJKjI9HXSF5ozlGNLKNPC3krYkGrK0AZUtCF4iy8vk+Z51XOr3v1FuXadx1GAYziBCnhwCTW4gzo0gMEjPMMrvDnKeXHenY9564qTzxzBHzifP66ljoc=</latexit><latexit sha1_base64="nw7ZQlhJXlSpDiV8/nU1vOXnC+g=">AAAB7XicbVBNS8NAEJ34WetX1aOXxSLUS0lE0GNREI8V7Ae0oWy2m3btJht2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekEhh0HW/nZXVtfWNzcJWcXtnd2+/dHDYNCrVjDeYkkq3A2q4FDFvoEDJ24nmNAokbwWjm6nfeuLaCBU/4DjhfkQHsQgFo2il5m1vUMGzXqnsVt0ZyDLxclKGHPVe6avbVyyNeIxMUmM6npugn1GNgkk+KXZTwxPKRnTAO5bGNOLGz2bXTsipVfokVNpWjGSm/p7IaGTMOApsZ0RxaBa9qfif10kxvPIzEScp8pjNF4WpJKjI9HXSF5ozlGNLKNPC3krYkGrK0AZUtCF4iy8vk+Z51XOr3v1FuXadx1GAYziBCnhwCTW4gzo0gMEjPMMrvDnKeXHenY9564qTzxzBHzifP66ljoc=</latexit><latexit sha1_base64="nw7ZQlhJXlSpDiV8/nU1vOXnC+g=">AAAB7XicbVBNS8NAEJ34WetX1aOXxSLUS0lE0GNREI8V7Ae0oWy2m3btJht2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekEhh0HW/nZXVtfWNzcJWcXtnd2+/dHDYNCrVjDeYkkq3A2q4FDFvoEDJ24nmNAokbwWjm6nfeuLaCBU/4DjhfkQHsQgFo2il5m1vUMGzXqnsVt0ZyDLxclKGHPVe6avbVyyNeIxMUmM6npugn1GNgkk+KXZTwxPKRnTAO5bGNOLGz2bXTsipVfokVNpWjGSm/p7IaGTMOApsZ0RxaBa9qfif10kxvPIzEScp8pjNF4WpJKjI9HXSF5ozlGNLKNPC3krYkGrK0AZUtCF4iy8vk+Z51XOr3v1FuXadx1GAYziBCnhwCTW4gzo0gMEjPMMrvDnKeXHenY9564qTzxzBHzifP66ljoc=</latexit><latexit sha1_base64="nw7ZQlhJXlSpDiV8/nU1vOXnC+g=">AAAB7XicbVBNS8NAEJ34WetX1aOXxSLUS0lE0GNREI8V7Ae0oWy2m3btJht2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekEhh0HW/nZXVtfWNzcJWcXtnd2+/dHDYNCrVjDeYkkq3A2q4FDFvoEDJ24nmNAokbwWjm6nfeuLaCBU/4DjhfkQHsQgFo2il5m1vUMGzXqnsVt0ZyDLxclKGHPVe6avbVyyNeIxMUmM6npugn1GNgkk+KXZTwxPKRnTAO5bGNOLGz2bXTsipVfokVNpWjGSm/p7IaGTMOApsZ0RxaBa9qfif10kxvPIzEScp8pjNF4WpJKjI9HXSF5ozlGNLKNPC3krYkGrK0AZUtCF4iy8vk+Z51XOr3v1FuXadx1GAYziBCnhwCTW4gzo0gMEjPMMrvDnKeXHenY9564qTzxzBHzifP66ljoc=</latexit>

-Needs boundary conditions (holomorphic ambiguity), but these can be determined for local CY [Huang-Klemm]

-Very efficient if combined with modularity [Huang -Klemm]

-They can be extended to the refined topological string [Huang-Klemm,Krefl-Walcher]

F0(t),<latexit sha1_base64="/NJynpiFAnU7y3jykjYdWPRV6F8=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJUkJKIoMeiIB4r2A9oQ9lsN+3SzSbsToQS+iO8eFDEq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+Oyura+sbm4Wt4vbO7t5+6eCwaeJUM95gsYx1O6CGS6F4AwVK3k40p1EgeSsY3U791hPXRsTqEccJ9yM6UCIUjKKVWnc9t4Jn571S2a26M5Bl4uWkDDnqvdJXtx+zNOIKmaTGdDw3QT+jGgWTfFLspoYnlI3ogHcsVTTixs9m507IqVX6JIy1LYVkpv6eyGhkzDgKbGdEcWgWvan4n9dJMbz2M6GSFLli80VhKgnGZPo76QvNGcqxJZRpYW8lbEg1ZWgTKtoQvMWXl0nzouq5Ve/hsly7yeMowDGcQAU8uIIa3EMdGsBgBM/wCm9O4rw4787HvHXFyWeO4A+czx/D3I6G</latexit><latexit sha1_base64="/NJynpiFAnU7y3jykjYdWPRV6F8=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJUkJKIoMeiIB4r2A9oQ9lsN+3SzSbsToQS+iO8eFDEq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+Oyura+sbm4Wt4vbO7t5+6eCwaeJUM95gsYx1O6CGS6F4AwVK3k40p1EgeSsY3U791hPXRsTqEccJ9yM6UCIUjKKVWnc9t4Jn571S2a26M5Bl4uWkDDnqvdJXtx+zNOIKmaTGdDw3QT+jGgWTfFLspoYnlI3ogHcsVTTixs9m507IqVX6JIy1LYVkpv6eyGhkzDgKbGdEcWgWvan4n9dJMbz2M6GSFLli80VhKgnGZPo76QvNGcqxJZRpYW8lbEg1ZWgTKtoQvMWXl0nzouq5Ve/hsly7yeMowDGcQAU8uIIa3EMdGsBgBM/wCm9O4rw4787HvHXFyWeO4A+czx/D3I6G</latexit><latexit sha1_base64="/NJynpiFAnU7y3jykjYdWPRV6F8=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJUkJKIoMeiIB4r2A9oQ9lsN+3SzSbsToQS+iO8eFDEq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+Oyura+sbm4Wt4vbO7t5+6eCwaeJUM95gsYx1O6CGS6F4AwVK3k40p1EgeSsY3U791hPXRsTqEccJ9yM6UCIUjKKVWnc9t4Jn571S2a26M5Bl4uWkDDnqvdJXtx+zNOIKmaTGdDw3QT+jGgWTfFLspoYnlI3ogHcsVTTixs9m507IqVX6JIy1LYVkpv6eyGhkzDgKbGdEcWgWvan4n9dJMbz2M6GSFLli80VhKgnGZPo76QvNGcqxJZRpYW8lbEg1ZWgTKtoQvMWXl0nzouq5Ve/hsly7yeMowDGcQAU8uIIa3EMdGsBgBM/wCm9O4rw4787HvHXFyWeO4A+czx/D3I6G</latexit><latexit sha1_base64="/NJynpiFAnU7y3jykjYdWPRV6F8=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSJUkJKIoMeiIB4r2A9oQ9lsN+3SzSbsToQS+iO8eFDEq7/Hm//GbZuDtj4YeLw3w8y8IJHCoOt+Oyura+sbm4Wt4vbO7t5+6eCwaeJUM95gsYx1O6CGS6F4AwVK3k40p1EgeSsY3U791hPXRsTqEccJ9yM6UCIUjKKVWnc9t4Jn571S2a26M5Bl4uWkDDnqvdJXtx+zNOIKmaTGdDw3QT+jGgWTfFLspoYnlI3ogHcsVTTixs9m507IqVX6JIy1LYVkpv6eyGhkzDgKbGdEcWgWvan4n9dJMbz2M6GSFLli80VhKgnGZPo76QvNGcqxJZRpYW8lbEg1ZWgTKtoQvMWXl0nzouq5Ve/hsly7yeMowDGcQAU8uIIa3EMdGsBgBM/wCm9O4rw4787HvHXFyWeO4A+czx/D3I6G</latexit>

F1(t)<latexit sha1_base64="U14cRZRdFPIavFRnRAcMew02Ey0=">AAAB7XicbVBNS8NAEJ34WetX1aOXYBHqpSQi6LEoiMcK9gPaUDbbTbt2sxt2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZemAhu0PO+nZXVtfWNzcJWcXtnd2+/dHDYNCrVlDWoEkq3Q2KY4JI1kKNg7UQzEoeCtcLRzdRvPTFtuJIPOE5YEJOB5BGnBK3UvO35FTzrlcpe1ZvBXSZ+TsqQo94rfXX7iqYxk0gFMabjewkGGdHIqWCTYjc1LCF0RAasY6kkMTNBNrt24p5ape9GStuS6M7U3xMZiY0Zx6HtjAkOzaI3Ff/zOilGV0HGZZIik3S+KEqFi8qdvu72uWYUxdgSQjW3t7p0SDShaAMq2hD8xZeXSfO86ntV//6iXLvO4yjAMZxABXy4hBrcQR0aQOERnuEV3hzlvDjvzse8dcXJZ47gD5zPH1wrjlE=</latexit><latexit sha1_base64="U14cRZRdFPIavFRnRAcMew02Ey0=">AAAB7XicbVBNS8NAEJ34WetX1aOXYBHqpSQi6LEoiMcK9gPaUDbbTbt2sxt2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZemAhu0PO+nZXVtfWNzcJWcXtnd2+/dHDYNCrVlDWoEkq3Q2KY4JI1kKNg7UQzEoeCtcLRzdRvPTFtuJIPOE5YEJOB5BGnBK3UvO35FTzrlcpe1ZvBXSZ+TsqQo94rfXX7iqYxk0gFMabjewkGGdHIqWCTYjc1LCF0RAasY6kkMTNBNrt24p5ape9GStuS6M7U3xMZiY0Zx6HtjAkOzaI3Ff/zOilGV0HGZZIik3S+KEqFi8qdvu72uWYUxdgSQjW3t7p0SDShaAMq2hD8xZeXSfO86ntV//6iXLvO4yjAMZxABXy4hBrcQR0aQOERnuEV3hzlvDjvzse8dcXJZ47gD5zPH1wrjlE=</latexit><latexit sha1_base64="U14cRZRdFPIavFRnRAcMew02Ey0=">AAAB7XicbVBNS8NAEJ34WetX1aOXYBHqpSQi6LEoiMcK9gPaUDbbTbt2sxt2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZemAhu0PO+nZXVtfWNzcJWcXtnd2+/dHDYNCrVlDWoEkq3Q2KY4JI1kKNg7UQzEoeCtcLRzdRvPTFtuJIPOE5YEJOB5BGnBK3UvO35FTzrlcpe1ZvBXSZ+TsqQo94rfXX7iqYxk0gFMabjewkGGdHIqWCTYjc1LCF0RAasY6kkMTNBNrt24p5ape9GStuS6M7U3xMZiY0Zx6HtjAkOzaI3Ff/zOilGV0HGZZIik3S+KEqFi8qdvu72uWYUxdgSQjW3t7p0SDShaAMq2hD8xZeXSfO86ntV//6iXLvO4yjAMZxABXy4hBrcQR0aQOERnuEV3hzlvDjvzse8dcXJZ47gD5zPH1wrjlE=</latexit><latexit sha1_base64="U14cRZRdFPIavFRnRAcMew02Ey0=">AAAB7XicbVBNS8NAEJ34WetX1aOXYBHqpSQi6LEoiMcK9gPaUDbbTbt2sxt2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZemAhu0PO+nZXVtfWNzcJWcXtnd2+/dHDYNCrVlDWoEkq3Q2KY4JI1kKNg7UQzEoeCtcLRzdRvPTFtuJIPOE5YEJOB5BGnBK3UvO35FTzrlcpe1ZvBXSZ+TsqQo94rfXX7iqYxk0gFMabjewkGGdHIqWCTYjc1LCF0RAasY6kkMTNBNrt24p5ape9GStuS6M7U3xMZiY0Zx6HtjAkOzaI3Ff/zOilGV0HGZZIik3S+KEqFi8qdvu72uWYUxdgSQjW3t7p0SDShaAMq2hD8xZeXSfO86ntV//6iXLvO4yjAMZxABXy4hBrcQR0aQOERnuEV3hzlvDjvzse8dcXJZ47gD5zPH1wrjlE=</latexit>

Corollary: Voros symbols are formal power series of modular forms on the WKB curve

in genus one:

evidence in genus two: [Fischbach-Klemm-Nega]

Voros symbols for anharmonic oscillators are governed by the holomorphic anomaly equation

(in the NS limit of the refined version)

FNS(⌧, ⌧ , ~) =X

k�0

Fk(⌧, ⌧)~2kcomputed recursively

Claim [Codesido-M.M.]:

The HAE can be also used to calculate exponentially small corrections to this series [Couso et al, Codesido-M.M.-Schiappa]

Why is there such a connection? It turns out that anharmonic oscillators emerge as a scaling limit of SU(N)

Seiberg-Witten (SW) theory near the Argyres-Douglas (AD) point [Grassi-Gu, Grassi-M.M., Ito-Shu]

↵ ! 0<latexit sha1_base64="OksbEv9WfJ3FJ/aebKFvtAI1DN4=">AAAB/XicbVBNS8NAEN3Ur1q/4sfNy2IRPJVEBD0WvXisYFuhCWWy3bRLN7thd6PUUPwrXjwo4tX/4c1/47bNQVsfDDzem2FmXpRypo3nfTulpeWV1bXyemVjc2t7x93da2mZKUKbRHKp7iLQlDNBm4YZTu9SRSGJOG1Hw6uJ376nSjMpbs0opWECfcFiRsBYqeseBMDTAeBAsf7AgFLyAXtdt+rVvCnwIvELUkUFGl33K+hJkiVUGMJB647vpSbMQRlGOB1XgkzTFMgQ+rRjqYCE6jCfXj/Gx1bp4VgqW8Lgqfp7IodE61ES2c4EzEDPexPxP6+TmfgizJlIM0MFmS2KM46NxJMocI8pSgwfWQJEMXsrJgNQQIwNrGJD8OdfXiSt05rv1fybs2r9soijjA7RETpBPjpHdXSNGqiJCHpEz+gVvTlPzovz7nzMWktOMbOP/sD5/AEA/pTq</latexit><latexit sha1_base64="OksbEv9WfJ3FJ/aebKFvtAI1DN4=">AAAB/XicbVBNS8NAEN3Ur1q/4sfNy2IRPJVEBD0WvXisYFuhCWWy3bRLN7thd6PUUPwrXjwo4tX/4c1/47bNQVsfDDzem2FmXpRypo3nfTulpeWV1bXyemVjc2t7x93da2mZKUKbRHKp7iLQlDNBm4YZTu9SRSGJOG1Hw6uJ376nSjMpbs0opWECfcFiRsBYqeseBMDTAeBAsf7AgFLyAXtdt+rVvCnwIvELUkUFGl33K+hJkiVUGMJB647vpSbMQRlGOB1XgkzTFMgQ+rRjqYCE6jCfXj/Gx1bp4VgqW8Lgqfp7IodE61ES2c4EzEDPexPxP6+TmfgizJlIM0MFmS2KM46NxJMocI8pSgwfWQJEMXsrJgNQQIwNrGJD8OdfXiSt05rv1fybs2r9soijjA7RETpBPjpHdXSNGqiJCHpEz+gVvTlPzovz7nzMWktOMbOP/sD5/AEA/pTq</latexit><latexit sha1_base64="OksbEv9WfJ3FJ/aebKFvtAI1DN4=">AAAB/XicbVBNS8NAEN3Ur1q/4sfNy2IRPJVEBD0WvXisYFuhCWWy3bRLN7thd6PUUPwrXjwo4tX/4c1/47bNQVsfDDzem2FmXpRypo3nfTulpeWV1bXyemVjc2t7x93da2mZKUKbRHKp7iLQlDNBm4YZTu9SRSGJOG1Hw6uJ376nSjMpbs0opWECfcFiRsBYqeseBMDTAeBAsf7AgFLyAXtdt+rVvCnwIvELUkUFGl33K+hJkiVUGMJB647vpSbMQRlGOB1XgkzTFMgQ+rRjqYCE6jCfXj/Gx1bp4VgqW8Lgqfp7IodE61ES2c4EzEDPexPxP6+TmfgizJlIM0MFmS2KM46NxJMocI8pSgwfWQJEMXsrJgNQQIwNrGJD8OdfXiSt05rv1fybs2r9soijjA7RETpBPjpHdXSNGqiJCHpEz+gVvTlPzovz7nzMWktOMbOP/sD5/AEA/pTq</latexit><latexit sha1_base64="OksbEv9WfJ3FJ/aebKFvtAI1DN4=">AAAB/XicbVBNS8NAEN3Ur1q/4sfNy2IRPJVEBD0WvXisYFuhCWWy3bRLN7thd6PUUPwrXjwo4tX/4c1/47bNQVsfDDzem2FmXpRypo3nfTulpeWV1bXyemVjc2t7x93da2mZKUKbRHKp7iLQlDNBm4YZTu9SRSGJOG1Hw6uJ376nSjMpbs0opWECfcFiRsBYqeseBMDTAeBAsf7AgFLyAXtdt+rVvCnwIvELUkUFGl33K+hJkiVUGMJB647vpSbMQRlGOB1XgkzTFMgQ+rRjqYCE6jCfXj/Gx1bp4VgqW8Lgqfp7IodE61ES2c4EzEDPexPxP6+TmfgizJlIM0MFmS2KM46NxJMocI8pSgwfWQJEMXsrJgNQQIwNrGJD8OdfXiSt05rv1fybs2r9soijjA7RETpBPjpHdXSNGqiJCHpEz+gVvTlPzovz7nzMWktOMbOP/sD5/AEA/pTq</latexit>

From SW theory to QM

~SW = ↵~QMIn the quantum version:

⇤N�ep + e�p

�+WN (x) = 0

WN (x) = xN +NX

k=2

xN�khk

⇤Np2 + xN +NX

k=2

ehkxN�k = 0

(h2, · · · , hN ) = (0, · · · , 0,�2) + (↵�2eh2, · · · ,↵�NehN )

topologicalstrings

SW theory

anharmonicoscillators

geometric engineering

[Klemm-Lerche-Mayr-Warner-Vafa,

Katz-Klemm-Vafa]

Argyres-Douglasscaling

The holomorphic anomaly equations governing topological strings are inherited in each limiting procedure

The “engineering” of anharmonic oscillators as a limit of SW theory suggests to look more carefully at the quantum SW

curve

Equivalently,

Quantum SW curve: a deformation of QM

We are interested in the actual spectral problem on

L2(R)

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HN | i = �hN | i<latexit sha1_base64="Fpuyu5hjL8x2/IWrKlVDAopZz+w=">AAACE3icbVDLSgMxFM34rPU16tJNsAgiWGZE0I1QdNNVqWAf0ClDJr3ThmYyQ5IRyth/cOOvuHGhiFs37vwb08eith4IHM65l5tzgoQzpR3nx1paXlldW89t5De3tnd27b39uopTSaFGYx7LZkAUcCagppnm0EwkkCjg0Aj6tyO/8QBSsVjc60EC7Yh0BQsZJdpIvn3qRUT3VJiVh37l0UsU8yQRXQ7XZz2/gmcV3y44RWcMvEjcKSmgKaq+/e11YppGIDTlRKmW6yS6nRGpGeUwzHupgoTQPulCy1BBIlDtbJxpiI+N0sFhLM0TGo/V2Y2MREoNosBMjhPMeyPxP6+V6vCqnTGRpBoEnRwKU451jEcF4Q6TQDUfGEKoZOavmPaIJFSbGvOmBHc+8iKpnxddp+jeXRRKN9M6cugQHaET5KJLVEJlVEU1RNETekFv6N16tl6tD+tzMrpkTXcO0B9YX7+8eJ6s</latexit><latexit sha1_base64="Fpuyu5hjL8x2/IWrKlVDAopZz+w=">AAACE3icbVDLSgMxFM34rPU16tJNsAgiWGZE0I1QdNNVqWAf0ClDJr3ThmYyQ5IRyth/cOOvuHGhiFs37vwb08eith4IHM65l5tzgoQzpR3nx1paXlldW89t5De3tnd27b39uopTSaFGYx7LZkAUcCagppnm0EwkkCjg0Aj6tyO/8QBSsVjc60EC7Yh0BQsZJdpIvn3qRUT3VJiVh37l0UsU8yQRXQ7XZz2/gmcV3y44RWcMvEjcKSmgKaq+/e11YppGIDTlRKmW6yS6nRGpGeUwzHupgoTQPulCy1BBIlDtbJxpiI+N0sFhLM0TGo/V2Y2MREoNosBMjhPMeyPxP6+V6vCqnTGRpBoEnRwKU451jEcF4Q6TQDUfGEKoZOavmPaIJFSbGvOmBHc+8iKpnxddp+jeXRRKN9M6cugQHaET5KJLVEJlVEU1RNETekFv6N16tl6tD+tzMrpkTXcO0B9YX7+8eJ6s</latexit><latexit sha1_base64="Fpuyu5hjL8x2/IWrKlVDAopZz+w=">AAACE3icbVDLSgMxFM34rPU16tJNsAgiWGZE0I1QdNNVqWAf0ClDJr3ThmYyQ5IRyth/cOOvuHGhiFs37vwb08eith4IHM65l5tzgoQzpR3nx1paXlldW89t5De3tnd27b39uopTSaFGYx7LZkAUcCagppnm0EwkkCjg0Aj6tyO/8QBSsVjc60EC7Yh0BQsZJdpIvn3qRUT3VJiVh37l0UsU8yQRXQ7XZz2/gmcV3y44RWcMvEjcKSmgKaq+/e11YppGIDTlRKmW6yS6nRGpGeUwzHupgoTQPulCy1BBIlDtbJxpiI+N0sFhLM0TGo/V2Y2MREoNosBMjhPMeyPxP6+V6vCqnTGRpBoEnRwKU451jEcF4Q6TQDUfGEKoZOavmPaIJFSbGvOmBHc+8iKpnxddp+jeXRRKN9M6cugQHaET5KJLVEJlVEU1RNETekFv6N16tl6tD+tzMrpkTXcO0B9YX7+8eJ6s</latexit><latexit sha1_base64="Fpuyu5hjL8x2/IWrKlVDAopZz+w=">AAACE3icbVDLSgMxFM34rPU16tJNsAgiWGZE0I1QdNNVqWAf0ClDJr3ThmYyQ5IRyth/cOOvuHGhiFs37vwb08eith4IHM65l5tzgoQzpR3nx1paXlldW89t5De3tnd27b39uopTSaFGYx7LZkAUcCagppnm0EwkkCjg0Aj6tyO/8QBSsVjc60EC7Yh0BQsZJdpIvn3qRUT3VJiVh37l0UsU8yQRXQ7XZz2/gmcV3y44RWcMvEjcKSmgKaq+/e11YppGIDTlRKmW6yS6nRGpGeUwzHupgoTQPulCy1BBIlDtbJxpiI+N0sFhLM0TGo/V2Y2MREoNosBMjhPMeyPxP6+V6vCqnTGRpBoEnRwKU451jEcF4Q6TQDUfGEKoZOavmPaIJFSbGvOmBHc+8iKpnxddp+jeXRRKN9M6cugQHaET5KJLVEJlVEU1RNETekFv6N16tl6tD+tzMrpkTXcO0B9YX7+8eJ6s</latexit>

�⇤N

�ep + e�p

�+WN (x)

�| i = 0[x, p] = i~

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HN = ⇤N�ep + e�p

�+ x

N +N�1X

k=2

xN�khk

N odd: resonant states

complex eigenvalues

N even: bound states

real eigenvalues �h(n)4 (h2, h3)

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Quantization leads to a discrete family of codimension one submanifolds in SW moduli space

H3 = ⇤3�ep + e�p

�+ x

3 + h2x

H4 = ⇤4�ep + e�p

�+ x

4 + h2x2 + h3x

�h(n)3 (h2)

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A very similar Hamiltonian appears in the Baxter equation of the Toda lattice [Gaudin-Pasquier]. In this case, the boundary

conditions one imposes are much more restrictive: solutions only exist for a discrete set of points in moduli space, which give the eigenvalues of the commuting Hamiltonians of the quantum

Toda lattice.

One can show that these points belong to the submanifolds defined by our quantization problem.

Relation to the Toda lattice

h3

h2

Toda lattice points

It turns out that this deformed version of QM is exactly solvable [Grassi-M.M.]: one can write down exact quantization

conditions for any potential in terms of actual functions (no resummation needed).

Exact quantization conditions

In this problem, the Voros symbols involve the quantum periods of the SW curve:

ai(hk; ~) =I

Ai

�SW(~)

@FNS

@ai(hk; ~) =

I

Bi

�SW(~) i = 1, · · · , N � 1

�SW = pdx

Remarkably, instanton calculus provides a resummation of these formal series in terms of convergent expansions:

The EQC will involve these resummed quantum periods. It is given by the vanishing of a single function in SW moduli

space (codimension one!). What is this function?

Alternatively, one can write TBA-like equations determining these functions [Nekrasov-Shatashvili, Kozlowski-Teschner].

hk(ai; ~) =X

d�0

hk,d(ai; ~)⇤2dN

FNS(ai; ~) =X

d�0

fd(ai; ~)⇤2dN

� =N�1X

i=0

(�1)i�1�i WN · � Weyl orbit

N even

N odd

X

n2WN ·�exp

✓i

~@FNS

@a· n� ⇡a · n

~

◆ Y

↵2�+

⇣2 sinh

⇣⇡a ·↵~

⌘⌘�(n·↵)2

= 0

X

n2WN ·�exp

✓i

~@FNS

@a· n

◆ Y

↵2�+

⇣2 sinh

⇣⇡a ·↵~

⌘⌘�(n·↵)2

= 0

fundamental weights of SU(N)

H4 = ⇤4�ep + e�p

�+ x

4 � 10x2!! !" " !

!"#

!$#

$#

"#

These EQCs can be effectively used to calculate energy levels, in complete agreement with numerical calculations of

the spectrum

⇤ = ~ = 1

Derivation from the TS/ST correspondence

These EQCs are a consequence of the TS/ST correspondence, which relates (conjecturally) the spectral properties of

quantum mirror curves to the BPS invariants of the toric CY [Grassi-Hatsuda-M.M., Codesido-Grassi-M.M.]

topologicalstrings

SW theorygeometric engineering

ep + e�p +NX

k=0

e(N�2k)xHk = 0

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ep + e�p +NX

k=0

xN�khk = 0<latexit sha1_base64="se3+GWXuGRjf+1KdgeoI8xoxiVQ=">AAACG3icbZDLSgMxFIYzXmu9VV26CRZBKC0zRdCNUHTjqlSwF+i0QyZN2zDJTEgyYhnmPdz4Km5cKOJKcOHbmF4Ebf0h8OU/55Cc3xeMKm3bX9bS8srq2npmI7u5tb2zm9vbb6golpjUccQi2fKRIoyGpK6pZqQlJEHcZ6TpB1fjevOOSEWj8FaPBOlwNAhpn2KkjeXlyokrOSRpV8AC/OGkKNKCq2LuJcGFnXar8L6bVItBOvTM3cvl7ZI9EVwEZwZ5MFPNy324vQjHnIQaM6RU27GF7iRIaooZSbNurIhAOEAD0jYYIk5UJ5nslsJj4/RgP5LmhBpO3N8TCeJKjbhvOjnSQzVfG5v/1dqx7p93EhqKWJMQTx/qxwzqCI6Dgj0qCdZsZABhSc1fIR4iibA2cWZNCM78yovQKJccu+TcnOYrl7M4MuAQHIET4IAzUAHXoAbqAIMH8ARewKv1aD1bb9b7tHXJms0cgD+yPr8BbhqgZQ==</latexit><latexit sha1_base64="se3+GWXuGRjf+1KdgeoI8xoxiVQ=">AAACG3icbZDLSgMxFIYzXmu9VV26CRZBKC0zRdCNUHTjqlSwF+i0QyZN2zDJTEgyYhnmPdz4Km5cKOJKcOHbmF4Ebf0h8OU/55Cc3xeMKm3bX9bS8srq2npmI7u5tb2zm9vbb6golpjUccQi2fKRIoyGpK6pZqQlJEHcZ6TpB1fjevOOSEWj8FaPBOlwNAhpn2KkjeXlyokrOSRpV8AC/OGkKNKCq2LuJcGFnXar8L6bVItBOvTM3cvl7ZI9EVwEZwZ5MFPNy324vQjHnIQaM6RU27GF7iRIaooZSbNurIhAOEAD0jYYIk5UJ5nslsJj4/RgP5LmhBpO3N8TCeJKjbhvOjnSQzVfG5v/1dqx7p93EhqKWJMQTx/qxwzqCI6Dgj0qCdZsZABhSc1fIR4iibA2cWZNCM78yovQKJccu+TcnOYrl7M4MuAQHIET4IAzUAHXoAbqAIMH8ARewKv1aD1bb9b7tHXJms0cgD+yPr8BbhqgZQ==</latexit><latexit sha1_base64="se3+GWXuGRjf+1KdgeoI8xoxiVQ=">AAACG3icbZDLSgMxFIYzXmu9VV26CRZBKC0zRdCNUHTjqlSwF+i0QyZN2zDJTEgyYhnmPdz4Km5cKOJKcOHbmF4Ebf0h8OU/55Cc3xeMKm3bX9bS8srq2npmI7u5tb2zm9vbb6golpjUccQi2fKRIoyGpK6pZqQlJEHcZ6TpB1fjevOOSEWj8FaPBOlwNAhpn2KkjeXlyokrOSRpV8AC/OGkKNKCq2LuJcGFnXar8L6bVItBOvTM3cvl7ZI9EVwEZwZ5MFPNy324vQjHnIQaM6RU27GF7iRIaooZSbNurIhAOEAD0jYYIk5UJ5nslsJj4/RgP5LmhBpO3N8TCeJKjbhvOjnSQzVfG5v/1dqx7p93EhqKWJMQTx/qxwzqCI6Dgj0qCdZsZABhSc1fIR4iibA2cWZNCM78yovQKJccu+TcnOYrl7M4MuAQHIET4IAzUAHXoAbqAIMH8ARewKv1aD1bb9b7tHXJms0cgD+yPr8BbhqgZQ==</latexit><latexit sha1_base64="se3+GWXuGRjf+1KdgeoI8xoxiVQ=">AAACG3icbZDLSgMxFIYzXmu9VV26CRZBKC0zRdCNUHTjqlSwF+i0QyZN2zDJTEgyYhnmPdz4Km5cKOJKcOHbmF4Ebf0h8OU/55Cc3xeMKm3bX9bS8srq2npmI7u5tb2zm9vbb6golpjUccQi2fKRIoyGpK6pZqQlJEHcZ6TpB1fjevOOSEWj8FaPBOlwNAhpn2KkjeXlyokrOSRpV8AC/OGkKNKCq2LuJcGFnXar8L6bVItBOvTM3cvl7ZI9EVwEZwZ5MFPNy324vQjHnIQaM6RU27GF7iRIaooZSbNurIhAOEAD0jYYIk5UJ5nslsJj4/RgP5LmhBpO3N8TCeJKjbhvOjnSQzVfG5v/1dqx7p93EhqKWJMQTx/qxwzqCI6Dgj0qCdZsZABhSc1fIR4iibA2cWZNCM78yovQKJccu+TcnOYrl7M4MuAQHIET4IAzUAHXoAbqAIMH8ARewKv1aD1bb9b7tHXJms0cgD+yPr8BbhqgZQ==</latexit>

X

n2WN ·�exp

✓i

~@FNS

@a· n

◆ Y

↵2�+

⇣2 sinh

⇣⇡a ·↵~

⌘⌘�(n·↵)2

= 0

from perturbative WKB for the quantum

mirror curve

from non-perturbative terms/

GV invariants

In contrast, the quantization conditions determining the Toda lattice points [Nekrasov-Shatashvili] only require

perturbative information:

exp

✓i

~@FNS

@a·↵i

◆= �1 i = 1, · · · , N � 1

basic roots of SU(N)

Why Quantum Mechanics is hard

SW theory anharmonicoscillators

Argyres-Douglasscaling

The instanton sum expressing the quantum periods converges very badly near the AD points. So it is not

straightforward to deduce EQCs for QM from our exact result for the quantum SW curve.

We anticipate however that existing EQCs for some QM problems, in terms of integral equations [Dorey-Tateo, Gaiotto],

can be deduced from the TBA form of our EQCs.

Instanton calculus/topological vertex resummations have been crucial in obtaining exact solutions to spectral

problems: they transform asymptotic series into convergent series (in some cases). However, they are tailored for the semiclassical/large radius regions in

moduli spaces.

Are there similar resummations in other regions of moduli space?

quantum mirror curves

quantum SW curves

anharmonicoscillators

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easy

moderately easy

hard

Hierarchy of quantum problems

The 100-year old problem of solving quantum anharmonic oscillators can be addressed in the context of topological

string theory. For example, Voros symbols can be calculated with the holomorphic anomaly equation.

Conclusions

The intermediate stage between topological strings and ordinary quantum mechanics is the quantum SW curve, which

provides a solvable deformation thereof (and a new testing ground for the TS/ST correspondence). Many generalizations

are possible (matter content, gauge groups, …).

Solving QM oscillators amounts in this context to an exact determination of the quantum periods of SW theory near

AD points.

Thank you for your attention!