RESURGENCE, WKB AND STRINGS

Marcos MariñoUniversity of Geneva

The (exact) WKB method: a little bit of history

Shortly after the discovery of quantum mechanics, it was clear that the one-dimensional Schroedinger equation

� ~22m

00(x) + (V (x)� E) (x) = 0

can be solved in closed form only in very few cases. One needs approximation methods.

One such method was introduced as early as 1926 by Wentzel, Kramers and Brillouin.

One considers the following ansatz

(x, ~) ⇠ 1pp(x, ~)

exp

✓i

~

Z x

p(x0, ~)dx0◆

This defines a “quantum” Liouville one-form

p(x, ~) ⇠ p(x) +X

n�1

pn(x)~2n

The idea of the WKB method is to solve for the wavefunction as an asymptotic expansion in powers of ~

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and solves it with

p(x, ~)dx

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p(x) =p2m(E � V (x))

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The WKB method quickly became a central tool in quantum mechanics.

As a first application, the WKB method explained the Bohr-Sommerfeld quantization condition as the leading approximation

to a more complicated quantization condition, involving corrections in ~

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V (x)

B

E

I

Bp(x, ~)dx =

I

Bp(x)dx+O(~2) = 2⇡

✓k +

1

2

◆

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However, in the period 1930-1970 the understanding of the method was plagued with ambiguities and difficulties.

A particular vexing issue was the “connection problem” relating WKB wavefunctions on the two sides of a turning point.

EE

V (x)V (x)

xxx0x0

Berry-Mount, 1972

The situation was only clarified in 1980-1990 thanks to the work of Voros and Silverstone (building up on previous work

by Dingle). This led to the “exact” WKB method.

The heritage of the previous confusions is that (almost) all standard textbooks and courses on quantum mechanics are

incorrect when it comes to the WKB method!

WKB becomes exact and complex

The reformulation of WKB in the 1980s-1990s was based on two (related) ideas:

1) the right objects to consider are Borel resummations of asymptotic expansions

2) one should extend the Schroedinger equation to the complex realm

This reformulation (at least in its French version) was heavily influenced by Ecalle’s theory of resurgence. Let me now present the basic ingredients of this exact or “resurgent”

WKB method

WKB curve and quantum periods

The starting point of the method is to regard the classical Hamiltonian as defining a complex curve, which I will call the

WKB curve

⌃(x, p) = H(x, p)� E = 0

E

V (x)

�1

�2

�3

�a

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one-cycles of the WKB curve

(2n)!

⇧a(~) =I

�a

p(x, ~)dx ⇠X

n�0

⇧(n)a ~2n

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We can integrate the quantum one-form against the one-cycles of the curve to obtain quantum periods (aka Voros

symbols), which are formal power series in

It is well-known that these series are asymptotic and do not define functions: their coefficients grow as

Can we make sense of them?

~2

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We can think about the different quantum periods as different “sectors” of the theory

b'(⇣) =X

n�0

ann!

⇣nBorel transform

Laplace transform

Borel resummation

The Borel triangle

The Borel method is a systematic (and traditional) way of making sense of factorially divergent formal power series

s(')(z) =

Z 1

0e�⇣ b'(z⇣)d⇣

'(z) =X

n�0

anzn

The Borel transform is analytic at the origin. Very often it can be analytically continued to the complex

plane, displaying singularities (poles, branch cuts).

b'(⇣)

Singularities along the positive real line are obstructions to Borel resummation

Don’t be afraid of Borel singularities: do lateral resummations!

Resurgence

Stokes discontinuity (or Stokes automorphism)

A quantum theory is resurgent if the Stokes discontinuity of the perturbative series in a given sector is a function of the

series in other sectors (and nothing else).

disc↵(') = s+↵(')� s�↵(')<latexit sha1_base64="cuFyE7PCqgoJihqA3TmNVtJ1eao=">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</latexit>

s±↵(')(z) =

Z

C±

e�⇣ b'(z⇣)d⇣C+

↵

C�

�1

�2

In the case of the symmetric double-well in quantum mechanics, we have

s+(⇧1)� s�(⇧1) = �i~ log⇣1 + e�s(⇧2)/~

⌘

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The Borel singularities of the Borel transform of are located at multiples of the instanton action

perturbative

non-perturbative

⇧1

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⇧2

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⇧1

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⇧(0)2

The perturbative sector knows about the non-perturbative sector!

1

~ (s+(⇧1)(~) + s�(⇧1)(~))± tan�1⇣e�

12~ s(⇧2)(~)

⌘= 2⇡

✓k +

1

2

◆

What is the use of quantum periods? One beautiful consequence of the exact WKB method is that exact quantization conditions

(EQC) for the spectrum can be obtained as vanishing conditions for Borel-resummed quantum periods

[Voros, Zinn-Justin]

In the case of the double-well potential, one finds

This requires the exact version of the connection formula due to Voros and Silverstone

Exact quantization conditions

| {z }perturbative

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| {z }instantons

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Complex quantum periods turn out to be crucial in the exact WKB method, as shown by [Balian-Parisi-Voros, Voros] in the case of the

pure quartic oscillator

H = p2 + x

4

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E

�E1/4E1/4

iE1/4

�iE1/4

V (x) = x4

real cycle: only approximate spectrum

The exact spectrum requires the real and the complex cycle:

“complex tunneling”

x4 = E

<latexit sha1_base64="pQhiN9CWVBMlL9jKpS9E7vJD4I0=">AAAB7HicbVBNSwMxEJ3Ur1q/qh69BIvgqexKRS9CUQSPFdy20K4lm2bbYDa7JFmxLP0NXjwo4tUf5M1/Y9ruQVsfDDzem2FmXpAIro3jfKPC0vLK6lpxvbSxubW9U97da+o4VZR5NBaxagdEM8El8ww3grUTxUgUCNYKHq4mfuuRKc1jeWdGCfMjMpA85JQYK3lP97WL61654lSdKfAicXNSgRyNXvmr249pGjFpqCBad1wnMX5GlOFUsHGpm2qWEPpABqxjqSQR0342PXaMj6zSx2GsbEmDp+rviYxEWo+iwHZGxAz1vDcR//M6qQnP/YzLJDVM0tmiMBXYxHjyOe5zxagRI0sIVdzeiumQKEKNzadkQ3DnX14kzZOqW6ue3tYq9cs8jiIcwCEcgwtnUIcbaIAHFDg8wyu8IYle0Dv6mLUWUD6zD3+APn8AIomOQA==</latexit>

Insights from strings and gauge theories

The basic ingredients of the exact WKB method are quantum versions of periods of complex curves. Periods of curves play

an important role in other contexts.

N=2 susy gauge theory

Seiberg-Witten (SW) curve

These periods determine the masses of BPS solitons in the gauge theory

⌃(x, ep) = 0

<latexit sha1_base64="xSgmW0mFpeTBRBigMUb3vTuYdB8=">AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARKkiZkYpuhKIblxXtAzpjyaSZNjTJDElGLGMX/oobF4q49Tfc+Tem7Sy0euDC4Zx7ufeeIGZUacf5snJz8wuLS/nlwsrq2vqGvbnVUFEiManjiEWyFSBFGBWkrqlmpBVLgnjASDMYXIz95h2RikbiRg9j4nPUEzSkGGkjdewd75r2OCrdH8LUkxyS0W18cOZ07KJTdiaAf4mbkSLIUOvYn143wgknQmOGlGq7Tqz9FElNMSOjgpcoEiM8QD3SNlQgTpSfTu4fwX2jdGEYSVNCw4n6cyJFXKkhD0wnR7qvZr2x+J/XTnR46qdUxIkmAk8XhQmDOoLjMGCXSoI1GxqCsKTmVoj7SCKsTWQFE4I7+/Jf0jgqu5Xy8VWlWD3P4siDXbAHSsAFJ6AKLkEN1AEGD+AJvIBX69F6tt6s92lrzspmtsEvWB/fId+U5A==</latexit>

ai =

I

Ai

pdx

<latexit sha1_base64="BtMqWxWDN5+GNYJtDUiFUmlWSKA=">AAACBXicbVDLSsNAFJ34rPUVdamLwSK4KolUdCNU3bisYB/QhDCZTNqhM5MwMxFL6MaNv+LGhSJu/Qd3/o3TNgttPXDhcM693HtPmDKqtON8WwuLS8srq6W18vrG5ta2vbPbUkkmMWnihCWyEyJFGBWkqalmpJNKgnjISDscXI/99j2RiibiTg9T4nPUEzSmGGkjBfYBCugF9BIqdJBfBnQEU5h7ksNoBB8Cu+JUnQngPHELUgEFGoH95UUJzjgRGjOkVNd1Uu3nSGqKGRmVvUyRFOEB6pGuoQJxovx88sUIHhklgnEiTQkNJ+rviRxxpYY8NJ0c6b6a9cbif1430/G5n1ORZpoIPF0UZwzqBI4jgRGVBGs2NARhSc2tEPeRRFib4MomBHf25XnSOqm6terpba1SvyriKIF9cAiOgQvOQB3cgAZoAgwewTN4BW/Wk/VivVsf09YFq5jZA39gff4AO96Xxg==</latexit>

aD,i =

I

Bi

pdx

<latexit sha1_base64="CaKWVZ7rDYvgBttrHCAcfuYF0Xg=">AAACCXicbVDLSsNAFJ3UV62vqEs3g0VwISWRim6EUl24rGAf0IQwmUzaoZNJmJmIJWTrxl9x40IRt/6BO//GaZuFth64cDjnXu69x08Ylcqyvo3S0vLK6lp5vbKxubW9Y+7udWScCkzaOGax6PlIEkY5aSuqGOklgqDIZ6Trj64mfveeCEljfqfGCXEjNOA0pBgpLXkmRF52fULzS+jElCsva3o0hwnMHBHBIIcPnlm1atYUcJHYBamCAi3P/HKCGKcR4QozJGXfthLlZkgoihnJK04qSYLwCA1IX1OOIiLdbPpJDo+0EsAwFrq4glP190SGIinHka87I6SGct6biP95/VSFF25GeZIqwvFsUZgyqGI4iQUGVBCs2FgThAXVt0I8RAJhpcOr6BDs+ZcXSee0ZtdrZ7f1aqNZxFEGB+AQHAMbnIMGuAEt0AYYPIJn8ArejCfjxXg3PmatJaOY2Qd/YHz+AB3PmVc=</latexit>

toric Calabi-Yau manifold X

mirror curve

In this case, the periods determine the prepotential of topological string theory on X, which contains

information about the counting of curves of genus zero on X

aD,i =@F0

@ai

<latexit sha1_base64="X3mIbNNyCd1KZjl1Exsdwx2UPuI=">AAACE3icbVDLSgMxFM34rPU16tJNsAgiUmakohuhqIjLCvYB7TDcSTNtaCYzJBmhDP0HN/6KGxeKuHXjzr8xfSDaeiBwOOeeJPcECWdKO86XNTe/sLi0nFvJr66tb2zaW9s1FaeS0CqJeSwbASjKmaBVzTSnjURSiAJO60HvcujX76lULBZ3up9QL4KOYCEjoI3k24fgZ1dHbIDPs1YCUjPg+Np3cCs2Kfwjgc8Gvl1wis4IeJa4E1JAE1R8+7PVjkkaUaEJB6WarpNoLxteSTgd5FupogmQHnRo01ABEVVeNtppgPeN0sZhLM0RGo/U34kMIqX6UWAmI9BdNe0Nxf+8ZqrDMy9jIkk1FWT8UJhyrGM8LAi3maRE874hQCQzf8WkCxKINjXmTQnu9MqzpHZcdEvFk9tSoXwxqSOHdtEeOkAuOkVldIMqqIoIekBP6AW9Wo/Ws/VmvY9H56xJZgf9gfXxDZ9XnWU=</latexit>

⌃(ex, ep) = 0

<latexit sha1_base64="F7egpDwergs/uFlPbE+zZ1v/Cvg=">AAACB3icbZDLSgMxFIYz9VbrbdSlIMEiVJAyIxXdCEU3LivaC3TGkknTNjTJDElGLEN3bnwVNy4UcesruPNtTNsRtPWHwMd/zuHk/EHEqNKO82Vl5uYXFpeyy7mV1bX1DXtzq6bCWGJSxSELZSNAijAqSFVTzUgjkgTxgJF60L8Y1et3RCoaihs9iIjPUVfQDsVIG6tl73rXtMtRIfEkh2R4e38IfzA6OHNadt4pOmPBWXBTyINUlZb96bVDHHMiNGZIqabrRNpPkNQUMzLMebEiEcJ91CVNgwJxovxkfMcQ7hunDTuhNE9oOHZ/TySIKzXggenkSPfUdG1k/ldrxrpz6idURLEmAk8WdWIGdQhHocA2lQRrNjCAsKTmrxD3kERYm+hyJgR3+uRZqB0V3VLx+KqUL5+ncWTBDtgDBeCCE1AGl6ACqgCDB/AEXsCr9Wg9W2/W+6Q1Y6Uz2+CPrI9vRj2YSg==</latexit>

How do we quantize this classical picture?We can obtain a quantum curve by promoting x, p to

Heisenberg operators

Quantum curves

[x, p] = i~<latexit sha1_base64="ugkZApovRbvoLLdg+/CRuy2cxQQ=">AAACD3icbZDLSsNAFIYn9VbrLerSzWBRXEhJRNCNUHTjsoK9QBPKZDpph85MwsxELCFv4MZXceNCEbdu3fk2TtMI2vrDwMd/zmHO+YOYUaUd58sqLSwuLa+UVytr6xubW/b2TktFicSkiSMWyU6AFGFUkKammpFOLAniASPtYHQ1qbfviFQ0Erd6HBOfo4GgIcVIG6tnH3Y9jvRQhel9dgx/OM78i9STHNIMesMAyZ5ddWpOLjgPbgFVUKjRsz+9foQTToTGDCnVdZ1Y+ymSmmJGsoqXKBIjPEID0jUoECfKT/N7MnhgnD4MI2me0DB3f0+kiCs15oHpzBeerU3M/2rdRIfnfkpFnGgi8PSjMGFQR3ASDuxTSbBmYwMIS2p2hXiIJMLaRFgxIbizJ89D66TmOjX35rRavyziKIM9sA+OgAvOQB1cgwZoAgwewBN4Aa/Wo/VsvVnv09aSVczsgj+yPr4BVu+c0w==</latexit><latexit sha1_base64="ugkZApovRbvoLLdg+/CRuy2cxQQ=">AAACD3icbZDLSsNAFIYn9VbrLerSzWBRXEhJRNCNUHTjsoK9QBPKZDpph85MwsxELCFv4MZXceNCEbdu3fk2TtMI2vrDwMd/zmHO+YOYUaUd58sqLSwuLa+UVytr6xubW/b2TktFicSkiSMWyU6AFGFUkKammpFOLAniASPtYHQ1qbfviFQ0Erd6HBOfo4GgIcVIG6tnH3Y9jvRQhel9dgx/OM78i9STHNIMesMAyZ5ddWpOLjgPbgFVUKjRsz+9foQTToTGDCnVdZ1Y+ymSmmJGsoqXKBIjPEID0jUoECfKT/N7MnhgnD4MI2me0DB3f0+kiCs15oHpzBeerU3M/2rdRIfnfkpFnGgi8PSjMGFQR3ASDuxTSbBmYwMIS2p2hXiIJMLaRFgxIbizJ89D66TmOjX35rRavyziKIM9sA+OgAvOQB1cgwZoAgwewBN4Aa/Wo/VsvVnv09aSVczsgj+yPr4BVu+c0w==</latexit><latexit sha1_base64="ugkZApovRbvoLLdg+/CRuy2cxQQ=">AAACD3icbZDLSsNAFIYn9VbrLerSzWBRXEhJRNCNUHTjsoK9QBPKZDpph85MwsxELCFv4MZXceNCEbdu3fk2TtMI2vrDwMd/zmHO+YOYUaUd58sqLSwuLa+UVytr6xubW/b2TktFicSkiSMWyU6AFGFUkKammpFOLAniASPtYHQ1qbfviFQ0Erd6HBOfo4GgIcVIG6tnH3Y9jvRQhel9dgx/OM78i9STHNIMesMAyZ5ddWpOLjgPbgFVUKjRsz+9foQTToTGDCnVdZ1Y+ymSmmJGsoqXKBIjPEID0jUoECfKT/N7MnhgnD4MI2me0DB3f0+kiCs15oHpzBeerU3M/2rdRIfnfkpFnGgi8PSjMGFQR3ASDuxTSbBmYwMIS2p2hXiIJMLaRFgxIbizJ89D66TmOjX35rRavyziKIM9sA+OgAvOQB1cgwZoAgwewBN4Aa/Wo/VsvVnv09aSVczsgj+yPr4BVu+c0w==</latexit><latexit sha1_base64="ugkZApovRbvoLLdg+/CRuy2cxQQ=">AAACD3icbZDLSsNAFIYn9VbrLerSzWBRXEhJRNCNUHTjsoK9QBPKZDpph85MwsxELCFv4MZXceNCEbdu3fk2TtMI2vrDwMd/zmHO+YOYUaUd58sqLSwuLa+UVytr6xubW/b2TktFicSkiSMWyU6AFGFUkKammpFOLAniASPtYHQ1qbfviFQ0Erd6HBOfo4GgIcVIG6tnH3Y9jvRQhel9dgx/OM78i9STHNIMesMAyZ5ddWpOLjgPbgFVUKjRsz+9foQTToTGDCnVdZ1Y+ymSmmJGsoqXKBIjPEID0jUoECfKT/N7MnhgnD4MI2me0DB3f0+kiCs15oHpzBeerU3M/2rdRIfnfkpFnGgi8PSjMGFQR3ASDuxTSbBmYwMIS2p2hXiIJMLaRFgxIbizJ89D66TmOjX35rRavyziKIM9sA+OgAvOQB1cgwZoAgwewBN4Aa/Wo/VsvVnv09aSVczsgj+yPr4BVu+c0w==</latexit>

⌃(x, ep) = 2⇤2 cosh(p) + x2 � u

<latexit sha1_base64="dvRU/2VJUujh6o9IOpMgltQp3/o=">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</latexit>

Example:SW curve for SU(2),

N=2 SYM

�2⇤2 cosh(p) + x2 � u

�| i = 0

<latexit sha1_base64="lNmDkGO+d7RS/PXzyT7trxrwEP4=">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</latexit>

This gives quantum versions of the periods appearing in gauge theory/topological strings, as in the conventional WKB

method

ai(~) =X

k�0

a(k)i ~2k

<latexit sha1_base64="hd/LrQl5WR5gpOUvbJmkp1dNXlo=">AAACFnicbVDLSsNAFJ34rPUVdelmsAjtwpKUim6EohuXFewDmjRMppN2yOTBzEQoIV/hxl9x40IRt+LOv3GaZqGtBy4czrmXe+9xY0aFNIxvbWV1bX1js7RV3t7Z3dvXDw67Iko4Jh0csYj3XSQIoyHpSCoZ6cecoMBlpOf6NzO/90C4oFF4L6cxsQM0DqlHMZJKcvQz5NCqNXERr11ZIgmc1IfWmEAjg8oZplW/luX2MG34maNXjLqRAy4TsyAVUKDt6F/WKMJJQEKJGRJiYBqxtFPEJcWMZGUrESRG2EdjMlA0RAERdpq/lcFTpYygF3FVoYS5+nsiRYEQ08BVnQGSE7HozcT/vEEivUs7pWGcSBLi+SIvYVBGcJYRHFFOsGRTRRDmVN0K8QRxhKVKsqxCMBdfXibdRt1s1s/vmpXWdRFHCRyDE1AFJrgALXAL2qADMHgEz+AVvGlP2ov2rn3MW1e0YuYI/IH2+QOsBZ53</latexit>

aD,i(~) =X

k�0

a(k)D,i~2k

<latexit sha1_base64="V4OnkHDMfl//HW2iwyf4ZDcyvwc=">AAACHnicbVDLSsNAFJ34rPUVdelmsAgtSElKi26Eoi5cVrAPaNIwmU7bIZMHMxOhhHyJG3/FjQtFBFf6N07TLLT1wMDhnHO5c48bMSqkYXxrK6tr6xubha3i9s7u3r5+cNgRYcwxaeOQhbznIkEYDUhbUslIL+IE+S4jXde7nvndB8IFDYN7OY2I7aNxQEcUI6kkR28gJ7k5o2nZmriIVy4tEftO4kFrTKCRwtwdJGWvkmaRQVLzUkcvGVUjA1wmZk5KIEfL0T+tYYhjnwQSMyRE3zQiaSeIS4oZSYtWLEiEsIfGpK9ogHwi7CQ7L4WnShnCUcjVCyTM1N8TCfKFmPquSvpITsSiNxP/8/qxHF3YCQ2iWJIAzxeNYgZlCGddwSHlBEs2VQRhTtVfIZ4gjrBUjRZVCebiycukU6ua9Wrjrl5qXuV1FMAxOAFlYIJz0AS3oAXaAINH8AxewZv2pL1o79rHPLqi5TNH4A+0rx+iPaGX</latexit>

By using a WKB ansatz for the wavefunction, one obtains again a quantum Liouville one-form

p(x, ~)dx

<latexit sha1_base64="T5DgmqqgHQI/ZWLYh3BiRtI0crQ=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM1iEClISqeiy6MZlBfuAJpTJZNIOnZmEmYm0hIK/4saFIm79Dnf+jdM2C209cOFwzr3ce0+QMKq043xbhZXVtfWN4mZpa3tnd8/eP2ipOJWYNHHMYtkJkCKMCtLUVDPSSSRBPGCkHQxvp377kUhFY/GgxwnxOeoLGlGMtJF69lFSGZ1DbxAgeQYzT3IYTuCoZ5edqjMDXCZuTsogR6Nnf3lhjFNOhMYMKdV1nUT7GZKaYkYmJS9VJEF4iPqka6hAnCg/m50/gadGCWEUS1NCw5n6eyJDXKkxD0wnR3qgFr2p+J/XTXV07WdUJKkmAs8XRSmDOobTLGBIJcGajQ1BWFJzK8QDJBHWJrGSCcFdfHmZtC6qbq16eV8r12/yOIrgGJyACnDBFaiDO9AATYBBBp7BK3iznqwX6936mLcWrHzmEPyB9fkDUe+Ucg==</latexit>

What is the meaning of this quantization?

It turns out that it is related to the “Omega background” for the gauge/string theory, which involves two parameters

✏1, ✏2

<latexit sha1_base64="UDehd3Hp1t13u0z+5jkso091EkM=">AAACAnicbVDLSgMxFM34rPU16krcBIvgQspMqeiy6MZlBfuAzlAy6Z02NJMZkoxQhuLGX3HjQhG3foU7/8a0HUFbDwmcnHMvN/cECWdKO86XtbS8srq2Xtgobm5t7+zae/tNFaeSQoPGPJbtgCjgTEBDM82hnUggUcChFQyvJ37rHqRisbjTowT8iPQFCxkl2khd+9CDRDFuqHuGPXN+npWuXXLKzhR4kbg5KaEc9a796fVimkYgNOVEqY7rJNrPiNSMchgXvVRBQuiQ9KFjqCARKD+brjDGJ0bp4TCW5gqNp+rvjoxESo2iwFRGRA/UvDcR//M6qQ4v/YyJJNUg6GxQmHKsYzzJA/eYBKr5yBBCJTN/xXRAJKHapFY0IbjzKy+SZqXsVsvnt9VS7SqPo4CO0DE6RS66QDV0g+qogSh6QE/oBb1aj9az9Wa9z0qXrLznAP2B9fEN07OWbg==</latexit>

Quantization corresponds to the so-called Nekrasov-Shatashvili (NS) limit

✏1 = ~, ✏2 = 0

<latexit sha1_base64="VYrymid0/u7JVT2p0GYwLxU24lQ=">AAACCnicbZDLSgMxFIYzXmu9jbp0Ey2CCykzpaKbQtGNywr2Ap2hZNLTNjSTGZKMUIau3fgqblwo4tYncOfbmLYjaOshgY//P4fk/EHMmdKO82UtLa+srq3nNvKbW9s7u/befkNFiaRQpxGPZCsgCjgTUNdMc2jFEkgYcGgGw+uJ37wHqVgk7vQoBj8kfcF6jBJtpI595EGsGDfoVrxBQOQZ9sz5EUsVp2MXnKIzLbwIbgYFlFWtY3963YgmIQhNOVGq7Tqx9lMiNaMcxnkvURATOiR9aBsUJATlp9NVxvjEKF3ci6S5QuOp+nsiJaFSozAwnSHRAzXvTcT/vHaie5d+ykScaBB09lAv4VhHeJIL7jIJVPORAUIlM3/FdEAkodqklzchuPMrL0KjVHTLxfPbcqF6lcWRQ4foGJ0iF12gKrpBNVRHFD2gJ/SCXq1H69l6s95nrUtWNnOA/pT18Q1QoJlh</latexit>

Note that we have formulated the correspondence by using WKB quantization of a one-dimensional curve [cf. Mironov-

Morozov]. This might be more fundamental than approaches based on the quantization of a higher-dimensional integrable

system.

The correspondence between the Omega background and quantization is not fully understood. To complicate matters,

we note that the self-dual Omega background,

which gives the conventional genus expansion of topological string, corresponds to a dual quantization [Kallen-M.M.,Grassi-

Hatsuda-M.M., …]

✏1 = �✏2 = gs

<latexit sha1_base64="2SjvNmBM8D4Seqdde10aA62Tprw=">AAACA3icbZDLSsNAFIZPvNZ6i7rTzWAR3FiSUtFNoejGZQV7gTaEyXTaDp1MwsxEKKHgxldx40IRt76EO9/GaRtEW38Y+PjPOZw5fxBzprTjfFlLyyura+u5jfzm1vbOrr2331BRIgmtk4hHshVgRTkTtK6Z5rQVS4rDgNNmMLye1Jv3VCoWiTs9iqkX4r5gPUawNpZvH3ZorBg36FbOfrhU6fvKtwtO0ZkKLYKbQQEy1Xz7s9ONSBJSoQnHSrVdJ9ZeiqVmhNNxvpMoGmMyxH3aNihwSJWXTm8YoxPjdFEvkuYJjabu74kUh0qNwsB0hlgP1HxtYv5Xaye6d+mlTMSJpoLMFvUSjnSEJoGgLpOUaD4ygIlk5q+IDLDERJvY8iYEd/7kRWiUim65eH5bLlSvsjhycATHcAouXEAVbqAGdSDwAE/wAq/Wo/VsvVnvs9YlK5s5gD+yPr4BbqeXZA==</latexit>

gs =1

~

<latexit sha1_base64="cfWZzLo5pqYUcPJoXlfScgv+qZo=">AAAB+3icbVDLSsNAFL2pr1pfsS7dDBbBVUmkohuh6MZlBfuAJoTJdNoOnWTCzEQsIb/ixoUibv0Rd/6N0zYLbT0wcDjnXO6dEyacKe0431ZpbX1jc6u8XdnZ3ds/sA+rHSVSSWibCC5kL8SKchbTtmaa014iKY5CTrvh5Hbmdx+pVEzED3qaUD/Co5gNGcHaSIFdHQXqOnM9YULIG4dY5oFdc+rOHGiVuAWpQYFWYH95A0HSiMaacKxU33US7WdYakY4zSteqmiCyQSPaN/QGEdU+dn89hydGmWAhkKaF2s0V39PZDhSahqFJhlhPVbL3kz8z+unenjlZyxOUk1jslg0TDnSAs2KQAMmKdF8aggmkplbERljiYk2dVVMCe7yl1dJ57zuNuoX941a86aoowzHcAJn4MIlNOEOWtAGAk/wDK/wZuXWi/VufSyiJauYOYI/sD5/AKvblDc=</latexit>

We will not develop this, however, and will restrict ourselves to the original story

This “quantum prepotential” satisfies a version of the holomorphic anomaly equations (HAE) of topological string

theory [Huang-Klemm,Krefl-Walcher]

It follows that the the quantum periods of the WKB method (even in ordinary quantum mechanics) are formal

series of quasi-modular forms on the WKB curve, governed by the HAE [Codesido-M.M.]

The NS correspondence leads to some surprising consequences for the conventional WKB method. It

suggests to define a “quantum prepotential”

aD(~) = @F (a(~), ~)@a(~)

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In some cases the quantum periods can be computed by instanton calculus in the N=2 gauge theory. This expresses them as convergent series in an “instanton counting”

parameter

a(u, ~) =pu

✓1 +

⇤4

u(4u+ ~2) + · · ·◆

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SU(2), N=2 SYM

This can be regarded as a different resummation of the quantum periods. The relation to the standard Borel resummation is non-trivial [Kashani-Poor-Troost, Grassi-Gu-M.M.]

GMN

An important recent development in the interface of WKB/string-gauge theory is the monumental work of Gaiotto-

Moore-Neitzke (GMN) on BPS states in N=2 gauge theories.

It turns out that many ingredients in their theory are related in a precise way to the resurgent WKB method

From WKB to GMN

quantize + WKB gauge/string theory

⌃(x, p) = 0

quantum periods BPS states

h�ai �⇧(0)

a Z(�a)⇧a

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X�a

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WKB GMN

Borel singularities BPS spectrumStokes discontinuities KS morphisms

Voros’ analytic bootstrap

Suppose we have a resurgent quantum theory and we know (1) the Stokes discontinuities of the perturbative

series in all sectors, and (2) their classical limit.

Can we then reconstruct the exact (resummed) series?

The analytic bootstrap is in fact a typical Riemann-Hilbert problem, of the type studied by GMN.

Some of the tools introduced by GMN make it possible to solve old problems in the theory of resurgence. I will focus here on the “analytic bootstrap”, an approach to

quantization proposed by André Voros in 1983.

A solvable example

The analytic bootstrap can be solved with the tools of GMN in an important example: the exact WKB method in QM with

polynomial potentials

E

V (x)

�1

�2

�3

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V (x) = xr+1 �rX

i=1

uixr+1�i

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“minimal”chamber in

moduli space

⇧1 ⇧2

The Stokes discontinuities in this case are given by the Delabaere-Pham formula:

+classical limit

As in GMN, one can solve this Riemann-Hilbert problem in terms of TBA-like equations [Ito-M.M.-Shu]

~ = e�✓ La(✓) = log⇣1 + e�✏a(✓)

⌘

⇧a(~) ⇠ ⇧(0)a , ~ ! 0

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✏a(✓) = ⇧(0)a e✓ �

Z

R

La�1(✓0)

cosh(✓ � ✓0)

d✓0

2⇡�

Z

R

La+1(✓0)

cosh(✓ � ✓0)

d✓0

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✏a(✓) =1

~s(⇧a)(~)

s+(⇧a)� s�(⇧a) = �i~ log⇣1 + e�s(⇧a�1)

⌘� i~ log

⇣1 + e�s(⇧a+1)

⌘

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This provides a “resurgent” derivation and generalization of a conjecture by Gaiotto. It extends the ODE/IM

correspondence of Dorey-Tateo (which was derived for monic potentials) to arbitrary polynomial potentials

As we move in moduli space to different “chambers”, one has to consider additional quantum periods and include them in

theTBA equations. This is the well-known wall-crossing phenomenon.

new (complex!)periods

This picture can be extended to the quantum versions of general SW curves: the resurgent properties of the

corresponding quantum periods can be deduced from the BPS spectrum and its wall-crossing

[Gaiotto, Grassi-Gu-M.M.]

Conclusions and outlook

• WKB is alive and well. Renewed interest in the theory of resurgence, and recent developments in string theory and gauge theory, have provided new insights and fresh solutions of old problems in the theory

• Many open problems! We are still lacking e.g. an exact WKB method for local mirror curves (difference equations).This would be potentially very useful to understand topological strings and BPS states on local Calabi-Yau threefolds

•GMN-like arguments give us the exact resummed quantum periods, but not the quantization conditions. Is there a natural meaning for these in the framework of GMN?

More conceptually, we need a deeper understanding of why many problems in gauge/string theory can be solved by

quantizing the underlying curve

Thank you for your attention!