quantum string effective actionsresearch.physics.unc.edu/string/berg.pdfquantum string effective...

Quantum String Effective Actions

Marcus BergKarlstad University, Sweden

UNC Chapel Hill, June 2013

M.B., Haack, Kang ’11M.B., Conlon, Marsh, Witkowski ’12

M.B., Haack, Kang, Sjörs ’13...

• string models in cosmology and phenomenology that seem sensitive to loop corrections

• review of quantum string effective actions of orientifolds with D-branes

• recent work on string one-loop contributions to moduli space metrics

• if there is time: a little math.

Outline

Ultraviolet sensitivity(“effects of high-energy physics make a difference”)

Example: leading-order vanishing due to symmetry, small corrections become important

• approximate shift symmetry in inflationary cosmology

• approximate flavor universality in phenomenology

e.g. Assassi, Baumann, Green, McAllister ’13

Cosmology: axion monodromy

sensitive to nonperturbative corrections to Kähler potential

0 0.02 0.04 0.06 0.08 0.1f

0

0.05

0.1

0.15

0.2

!ns

- 4 -3.5 -3 -2.5 -2 -1.5 -1log10 f

0

0.00025

0.0005

0.00075

0.001

0.00125

0.0015

0.00175

bf

Figure 2: This plot shows the 68% and 95% likelihood contours in the �ns-f , and bf -log10

f

plane, respectively, from the five-year WMAP data on the temperature angular power spec-trum.

5 10 50 100 500 1000

!

!2000

0

2000

4000

6000

8000

!!!"1"C !#2

#$$K2 %

5 10 50 100 500 1000

!

!2000

0

2000

4000

6000

8000

!!!"1"C !#2

#$$K2 %

Figure 3: The left plot shows the angular power spectrum for the best fit point f = 6.67⇥10�4,and �ns = 0.17. The right plot shows the angular power spectrum for the best fit pointtogether with the unbinned WMAP five-year data.

The improvement can be traced to a better fit to the data around the first peak. Wewould like to stress, however, that we do not take this as an indication of oscillations inthe observed angular power spectrum. Similar spikes in the likelihood function occur quitegenerally when fitting an oscillatory model to toy data generated with the conventional powerspectrum without any oscillations, because the oscillations fit some features in the noise. Thepolarization data could provide a cross check, but we find that it is presently not good enoughto do so in a meaningful way.

22

NS5

NS5D4-brane

Figure 1: T-dual, “brane box”, description of this configuration, in which the fractional D3-brane becomes a D4-brane stretched between two NS5-branes on a T-dual circle. Movingin the b direction through multiple periods in closed string moduli space in the originaldescription corresponds to moving one of the NS5-branes around the circle, dragging theD4-brane around with it so as to introduce multiple wrappings.

where we indicated corrections which we will analyze below, suppressing them using sym-metries, warping, and the natural exponential suppression of nonperturbative e↵ects.

In order to obtain 60 e-folds of accelerated expansion, inflation must start at �a

⇠ 11MP

.In addition, the quantum fluctuations of the inflaton must generate a level of scalar curvatureperturbation �R|

60

' 5.4⇥ 10�5, with

�R|N

e

=

s1

12⇡2

V 3

M6

P

V 02

�����N

e

(2.19)

This requiresµ

a

⇠ 6⇥ 10�4MP

(2.20)

Given fa

= �a

/a and the above results, the number of circuits of the fundamental axionperiod (2⇡)2f

a

required for inflation is

Nw

= 11M

P

fa

(2⇡)2

(2.21)

We will compute this number of circuits in each of the specific models below. In the verysimple case with all cycles of the same size, this gives, using (2.15),

Nw

⇠ 11p

6L2

(2⇡)2

(one scale) (2.22)

for b, while the requisite number of circuits for an RR inflaton c is larger by a factor 1/gs

.

2.4 Constraints on Corrections to the Slow-Roll Parameters

Our next task is to ensure that the inflaton potential Vinf

⇡ µ3

a

�a

is the primary term inthe axion potential. All other contributions to the axion potential must make negligible

8

Silverstein, Westphal ’08 +McAllister ’08M.B., Pajer, Sjörs ’09...Pajer, Peloso ’13

2

f✓

are determined by the geometry of the compactifica-tion. We will come back to this point in the next section.We define W (r) such that the vacuum energy densityvanishes at the minimum.

We have performed our computations with W (r) anarbitrary monomial in r and we will give explicit for-mulae for this at the end of this section. In the axionmonodromy string models (about which more in the nextsection), so far only the linear case W (r) = µ3r has beenstudied, where µ is a constant. For the purpose of ex-position, in the following we will discuss details for thequadratic case W (r) = 1

2m2r2 instead, for two reasons.First, it captures a few minor additional complicationswhich are absent in the linear case only. Second, it is thearchetypal potential with an ⌘ problem.

We will soon find out that the whole observable infla-tionary dynamics involves a range d

r

that can be madeparametrically small. If d

r

were arbitrarily small, wecould always expand any W (r) around a minimum, henceobtaining a quadratic potential. In concrete realizations,e.g. the string theory model presented in the next sec-tion, the various parameters are constrained and it isa model-dependent question whether the quadratic ap-proximation is accurate. In any event, the validity of ourmechanism does not rely on W (r) being quadratic.

We would also like to stress that the field space metricis simply dr2 + d✓2, as can be seen from (1), and not theinduced flat space metric in polar coordinates.

We plot the potential as in figure 1. It is clear that thedistance traversed along each “trench” is (almost) thesame at di↵erent heights. However, the periodicity in ✓is not manifest. We therefore also show figure 2, which isthe justification for the title of this work: the potential(2) in polar coordinates looks like a spiral staircase, wherethe periodicity ✓ ! ✓ + 2⇡f

✓

is manifest. However, infigure 2, the induced field space metric is not faithfullyrepresented, unlike in fig. 1 (it is dr2 + d✓2, as statedabove). Indeed, in fig. 2, one can be misled to believethat the spiral trenches at the bottom are much smallerper revolution than those further up, but as we saw in fig.1 they are actually almost the same length at di↵erentheights.

The inflationary dynamics can be intuitively read o↵from either figure. The system starts somewhere up inthe W (r) potential, quickly rolls in the r direction to theclosest trench3 and from there it slowly spirals along thetrench mostly in the ✓ direction. This classical two-fieldslow-roll motion can be captured by an e↵ective single-field potential that we are now going to derive.

3 The initial condition problem here is no worse than in standardchaotic inflation. If the inflaton has some downward initial speed,then it needs to start at some larger r to achieve enough e-foldsof inflation.

Figure 2: The potential in (2) for W (r) = 12m2r2 in cylindri-

cal coordinates, which do not faithfully reproduce the field-space metric as given in (1), contrary to Cartesian coordinatesin figure 1. On the other hand, in polar coordinates, the pe-riodicity in ✓ is apparent and the similarity with Inferno asdescribed by Dante [2] becomes evident.

First it is convenient to rotate our coordinates by✓

✓r

◆=

✓cos ⇠ sin ⇠� sin ⇠ cos ⇠

◆ ✓✓r

◆, (3)

with

sin ⇠ =f

rpf2

r

+ f2✓

, cos ⇠ =f

✓pf2

r

+ f2✓

, (4)

after which the potential becomes

V =12m2

⇣r cos ⇠ + ✓ sin ⇠

⌘2+ ⇤4

1� cos

r

f

�, (5)

where f = fr

f✓

/(f2r

+ f2✓

)1/2. As initial condition, weassume the system starts high up in the quadratic poten-tial, but still at subplanckian values, i.e. f ⌧ rin < M

P

.An interesting regime to consider is then

A. fr

⌧ f✓

⌧MP

,

B. ⇤4 � fm2rin.

It will turn out that useful values of ⇤ range from10�3M

P

to 10�1MP

. Typical values for fr

and f✓

tokeep in mind are 10�3M

P

and 10�1MP

, respectively.The advantage of considering subplanckian f

r

and f✓

isthat we will eventually embed this model in controlledstring models, where superplanckian axion decay con-stants seem elusive [3]. We will assume for the rest ofthis section that these two conditions hold. Notice thatcondition A implies

cos ⇠ ' 1 , sin ⇠ ' fr

f✓

, f ' fr

, (6)

V (r, ✓) = crp + ⇤

4

✓1� cos

✓r

fr� ✓

f✓

◆◆

in KKLT stabilized background Kachru, Kallosh, Linde, Trivedi ’03

Cosmology: axion monodromy

Planck collaboration, ’13

“The models most compatible with the Planck data in the set considered here are the two interesting axion monodromy potentials [McAllister et al] which are motivated by inflationary model building in the context of string theory”

Fun! But: small-field models fit the data better, until there is polarization.

2

f✓

are determined by the geometry of the compactifica-tion. We will come back to this point in the next section.We define W (r) such that the vacuum energy densityvanishes at the minimum.

We have performed our computations with W (r) anarbitrary monomial in r and we will give explicit for-mulae for this at the end of this section. In the axionmonodromy string models (about which more in the nextsection), so far only the linear case W (r) = µ3r has beenstudied, where µ is a constant. For the purpose of ex-position, in the following we will discuss details for thequadratic case W (r) = 1

2m2r2 instead, for two reasons.First, it captures a few minor additional complicationswhich are absent in the linear case only. Second, it is thearchetypal potential with an ⌘ problem.

We will soon find out that the whole observable infla-tionary dynamics involves a range d

r

that can be madeparametrically small. If d

r

were arbitrarily small, wecould always expand any W (r) around a minimum, henceobtaining a quadratic potential. In concrete realizations,e.g. the string theory model presented in the next sec-tion, the various parameters are constrained and it isa model-dependent question whether the quadratic ap-proximation is accurate. In any event, the validity of ourmechanism does not rely on W (r) being quadratic.

We would also like to stress that the field space metricis simply dr2 + d✓2, as can be seen from (1), and not theinduced flat space metric in polar coordinates.

We plot the potential as in figure 1. It is clear that thedistance traversed along each “trench” is (almost) thesame at di↵erent heights. However, the periodicity in ✓is not manifest. We therefore also show figure 2, which isthe justification for the title of this work: the potential(2) in polar coordinates looks like a spiral staircase, wherethe periodicity ✓ ! ✓ + 2⇡f

✓

is manifest. However, infigure 2, the induced field space metric is not faithfullyrepresented, unlike in fig. 1 (it is dr2 + d✓2, as statedabove). Indeed, in fig. 2, one can be misled to believethat the spiral trenches at the bottom are much smallerper revolution than those further up, but as we saw in fig.1 they are actually almost the same length at di↵erentheights.

The inflationary dynamics can be intuitively read o↵from either figure. The system starts somewhere up inthe W (r) potential, quickly rolls in the r direction to theclosest trench3 and from there it slowly spirals along thetrench mostly in the ✓ direction. This classical two-fieldslow-roll motion can be captured by an e↵ective single-field potential that we are now going to derive.

3 The initial condition problem here is no worse than in standardchaotic inflation. If the inflaton has some downward initial speed,then it needs to start at some larger r to achieve enough e-foldsof inflation.

Figure 2: The potential in (2) for W (r) = 12m2r2 in cylindri-

cal coordinates, which do not faithfully reproduce the field-space metric as given in (1), contrary to Cartesian coordinatesin figure 1. On the other hand, in polar coordinates, the pe-riodicity in ✓ is apparent and the similarity with Inferno asdescribed by Dante [2] becomes evident.

First it is convenient to rotate our coordinates by✓

✓r

◆=

✓cos ⇠ sin ⇠� sin ⇠ cos ⇠

◆ ✓✓r

◆, (3)

with

sin ⇠ =f

rpf2

r

+ f2✓

, cos ⇠ =f

✓pf2

r

+ f2✓

, (4)

after which the potential becomes

V =12m2

⇣r cos ⇠ + ✓ sin ⇠

⌘2+ ⇤4

1� cos

r

f

�, (5)

where f = fr

f✓

/(f2r

+ f2✓

)1/2. As initial condition, weassume the system starts high up in the quadratic poten-tial, but still at subplanckian values, i.e. f ⌧ rin < M

P

.An interesting regime to consider is then

A. fr

⌧ f✓

⌧MP

,

B. ⇤4 � fm2rin.

It will turn out that useful values of ⇤ range from10�3M

P

to 10�1MP

. Typical values for fr

and f✓

tokeep in mind are 10�3M

P

and 10�1MP

, respectively.The advantage of considering subplanckian f

r

and f✓

isthat we will eventually embed this model in controlledstring models, where superplanckian axion decay con-stants seem elusive [3]. We will assume for the rest ofthis section that these two conditions hold. Notice thatcondition A implies

cos ⇠ ' 1 , sin ⇠ ' fr

f✓

, f ' fr

, (6)

Silverstein, Westphal ’08 +McAllister ’08M.B., Pajer, Sjörs ’09...Pajer, Peloso ’13

Cosmology: brane inflation

D-brane position scalar is the inflaton (small-field) in KKLT stabilized background

sensitive to loop-induced dependence of stabilizing superpotential on D-brane moduli: “a delicate universe”

���

Dvali, Tye ’98

...

M.B., Haack, Körs ’04

Baumann, Dymarsky, Klebanov, Maldacena, McAllister, Murugan ’06...Baumann, Dymarsky, Klebanov, McAllister, Steinhardt ’07

visiblehidden

Phenomenology: gaugino masses

...but due to further cancellations, soft terms are not sensitive to perturbative corrections to Kähler potential!

Berg, Haack, Pajer ’07

“KKLT v1.1”: the Large Volume Scenario. The soft supersymmetry breaking terms due to flux display leading-order cancellations...

Becker, Becker, Haack, Louis ’02Balasubramanian, Berglund, Conlon, Quevedo ’05

Cicoli, Conlon, Quevedo ’07

called “extended no-scale structure”

Phenomenology: flavor physics

f = fhid + fvis

W = Whid +Wvis

brane models strongly constrain soft terms: “sequestering” supposed to solve supersymmetry flavor problem

���

Aijk = 0 , m2i| = 0)

“clears the way” for anomaly mediation

Randall, Sundrum ’98

(f = �3M2P e

� K3M2

P )

Phenomenology: flavor physics

sequestering in Large Volume Scenariois sensitive to nonperturbative corrections to superpotential:limits on rare decays produce strong constraints on the compactification volume

studies rare decays, e.g. (observed Nov 2012)LHCb collaboration, 1211.2674

Blumenhagen, Conlon et al ’09 M.B., Marsh, McAllister, Pajer ’10M.B. Conlon, Marsh Witkowski ’12

Bs ! µµ

LHCb experiment

Can’t always think “negligible”even when numerically small

at a point in moduli space (i.e. at fixed ).

open and closed string moduli

Le⇥ =1

2�2R�K�i�⇥

⇥µ�i⇥µ�⌅ � 1g2(�)

traF 2

�V (�) + corrections

General D=4, N=1 effective theory of gravity+moduli+gauge fields

�i

String perturbation theory:two expansions

+gs +g2s + . . .hV4i =

V

hV4i =

V

V

quantumz }| {

(semi-)classicalz }| {

E2↵0 ! 0

(semi-)classical, string tree level

dipole charged

quantum, string one-loop

Orbifold: Identify under spatial rotation

Simple models for extra dimensions

••

•

Orientifold: Identify under worldsheet reflection

Z3

unoriented eigenstate

• •

⌦ =

1± ⌦2

makescone

T6/(Z2 � Z2)T6/Z�

6N = 1

Z�6 : (v1, v2, v3) =

�16,�1

2,13

⇥

D3

D7 wraps

D3�1

�2 �3

D7 wraps

U1U2

U3

sample orientifolds:

Simple models for extra dimensions

D7 pointlike

(Z1 = X4 + UX5)

D3

⇥Z1 = e2⇡iv1Z1⇥Z2 = e2⇡iv2Z2

⇥Z3 = e2⇡iv3Z3

sample orientifolds: T6/(Z2 � Z2)

T6/Z�6

N = 1

A first look at corrections: f“N=2 sector”

“partially twisted”“N=1 sector”

“completely twisted”

Z�6 : (v1, v2, v3) =

�16,�1

2,13

⇥

D3

D7 wraps

D3�1

�2 �3

D7 wraps

U1U2

U3

D7 pointlike

(Z1 = X4 + UX5)

D3

⇥Z1 = e2⇡iv1Z1⇥Z2 = e2⇡iv2Z2

⇥Z3 = e2⇡iv3Z3

⇥ ⇥2

A Aa

Singularities of T6/Z6’

orientifold singularity

orbifold singularity

•

Le⇥ =1

2�2R�K�i�⇥

⇥µ�i⇥µ�⌅ � 1g2(�)

traF 2

�V (�) + corrections

• partially twisted: g, K corrections

• completely twisted: g, K corrections

Review: one-loop effective action

correct

threshold corrections

A first look at corrections: f

1

g2ph

(�i)

!1�loop

= � lnM

string

µ+�(�, U)

cf. Cheng, Dong, Duncan, Harvey, Kachru, Wrase ’13

�

Aaµ

Aaµ

ln |z|2 = ln z + ln z = 12Re ln z

U

�D3

D7

f: partially twisted strings(“N=2 sectors”)

Dixon, Kaplunovsky, Louis ‘91...

M.B., Haack, Körs ‘04

�

U

�D3

D7

Aaµ

Aaµ

=a � 1/ND7

Ganor ‘96Plays a role if nonperturbative W:M.B., Haack, Kors ‘04

f corrections

�

U

�D3

D7

Aaµ

Aaµ

Moral: these are string loop corrections,but without them, Wnp doesn’t depend onD3-brane scalar at all. So they are not

“negligible” in any real sense.�

f corrections

�

Aaµ

Aaµ

This gives a new contribution to the inflaton potentialin D-brane inflationary cosmology.

Inflationary cosmology

Baumann, Dymarsky, Klebanov, Maldacena, McAllister, Murugan ’06...

Again, sequestering is one strategy to deal with the flavor problem of gravity-mediated supersymmetry breaking.

Contributions to the effective action like that just discussed can potentially affect sequestering.

Particle phenomenology

Blumenhagen, Conlon et al ’09 M.B., Marsh, McAllister, Pajer ’10

de Alwis ’12M.B. Conlon, Marsh, Witkowski ’12

Aaµ

Aaµ

�j

�k

�i

Tuesday, June 26, 2012

superpotential de-sequesteringZ

d4x

Zd2✓ �new

ij e�aTHQiqj

M.B., Haack, Kang, Sjörs ’13

T A M K

ReT

ReT

K: partially twisted strings

brane at arbitrary position

orientifold plane(fixed locus of )⌦

�

T A M K

ReT

ReT

K: partially twisted strings

brane at arbitrary position

orientifold plane(fixed locus of )⌦

�

TIE Fighter (Star Wars)

M.B., Haack, Kang, Sjörs ’13

K = � ln�(S + S)(T + T )(U + U)

⇥

� ln

⇤1� 1

8�

⇧

i

Ni(⇥i + ⇥i)2

(T + T )(U + U)� 1

128�6

⇧

i

E2(⇥i, U)(S + S)(T + T )

⌅

“integrate” one-loop corrected Kähler metric to get one-loop corrected Kähler potential:

sum over images of E2(�i, U)

K: partially twisted stringsM.B., Haack, Körs, ‘05

E2(�, U) =

X

(n,m) 6=(0,0)

Re(U)

2

|n + mU |4 exp

✓2⇡i

�(n + m ¯U) +

¯�(n + mU)

U +

¯U

◆

K = � ln�(S + S)(T + T )(U + U)

⇥

� ln

⇤1� 1

8�

⇧

i

Ni(⇥i + ⇥i)2

(T + T )(U + U)� 1

128�6

⇧

i

E2(⇥i, U)(S + S)(T + T )

⌅

“integrate” one-loop corrected Kähler metric to get one-loop corrected Kähler potential:

sum over images of E2(�i, U)

K: partially twisted stringsM.B., Haack, Körs, ‘05

E2(�, U) =

X

(n,m) 6=(0,0)

Re(U)

2

|n + mU |4 exp

✓2⇡i

�(n + m ¯U) +

¯�(n + mU)

U +

¯U

◆

puzzle: N=2 supersymmetry requires

?

@�@�E2 ⇠ ln |#1(�, U)|2 + . . .

Efforts by many people:

• moduli-dependent gauge coupling: one-loop order

• Kähler potential: one-loop order

partially twisted strings:summary

K(�, �) = K(S, S, T, T , U, U , �, �)

g(�, �) = g(S, S, T, T , U, U , �, �)

Aaµ

Aaµ

Riemann 1859

(some fine print here)

(“N=1 sectors”)gph: completely twisted strings

angle

Z 1

0d`

#01(v)

#1(v)

= �⇡

2

log

e�2�Ev �(1� v)

�(1 + v)

�

Lüst, Stieberger ’03Akerblom, Blumenhagen, Lüst,

Schmidt-Sommerfeld ’07

D6-branes at angles: Completely twistedstrings give moduli dependence too

v =✓⇡ � ✓0

⇡

K: completely twisted strings

�i

�ı

Bain, M.B. ‘00

Kähler metricof D-brane scalars

K��

(“N=1 sectors”)

K: completely twisted strings

�i

�ı

Bain, M.B. ‘00

Kähler metricof D-brane scalars

K��

state of the art until 2011:extracted field theory anomalous dimensions,

but could not perform integrals

– no explicit result for one-loop Kähler metric even without angles or flux

(“N=1 sectors”)

K: completely twisted strings

M.B., Haack, Kang ’11

(“N=1 sectors”)

�i

�ı labelsspin structures

S =

X

~↵

Zdt Z~↵

|{z}dep. on t

Zd⌫1d⌫2 hV�iV�ıi~↵

| {z }dep. on ⌫’s and t

cf. Anastasopoulos, Antoniadis, Benakli, Goodsell, Vichi ‘11

�i ⌘ Ti�i �Ai

VA ⇠ go

Z

@⌃

d⌧ A↵@⌧X↵+fermions

V� ⇠ go

Z

@⌃

d⌧ �a@�Xa+fermions

along brane

transverse to brane

along and transverse depend on brane

adapt coordinates to brane

Type IIA D-brane moduli

K: completely twisted stringsKähler modulus

Hassan ’03

⇡ 2⇡

(2⇡) = e2i(✓⇡�✓0) (0)

0 ⇡ 2⇡

✓0 ✓⇡ ✓0

| {z }unphysical

unphysicalz }| {

0

work on covering torus with twist of holomorphic fields

Method of images

Bertolini, Billò, Lerda, Morales, Russo ’05contrast:

What string states can run in the loop?

Completely twisted states are localized at intersections of brane images

I = 7

+

+!

UV finiteness

UV

⌧

10

Open string scalars:Finite part vanishes (!)

extend 0-1 integral toentire contour

K1�loop

�

i¯

�

ı = 0v =✓⇡ � ✓0

⇡

twisted worldsheet fermion propagator

Summary: open strings

The moduli space metric of open string moduliin minimally supersymmetric toroidal orientifolds

is not renormalized at the one-loop level.

“String nonrenormalization theorem”

Next, closed strings.

M.B., Haack, Kang ’11M.B., Haack, Kang, Sjörs ’13

K: completely twisted for closed strings – no angles yet!

M.B., Haack, Kang, Sjörs ’13

T A M K

Two-graviton cylinder amplitude

hmn

hmn

VT = eii(@Zi + i(p · ) i)(@Z i + i(p · ) i)eip·X

partition function

worldsheet bosons worldsheet fermions

M.B., Haack, Kang, Sjörs ’13

⇠Z

dt

t4

X

~↵

Z~↵hVT VT i~↵�

⌫

I�(⌫)⌫

identify

fixed line = boundary

IA = 1� ⌫

Worldsheets by identification

I�(⌫)⌫

⌫

Lifting to covering torus

Z

�d2⌫ (f(⌫) + f(I�(⌫)) =

Z

Td2⌫ f(⌫)

K: completely twisted for closed strings – no angles yet!

• lift to covering torus (by previous slide)

• perform torus integral with zeta function regularization

Z

Td2⌫ GF

(1/2,1/2+�)(⌫, ⌧)@⌫GF(1/2,1/2+�)(⌫, ⌧)

• perform spin structure sum

orbifold twist (rational number)

!UV

!UV

UV finiteness

Zeta function regularizationhmn

hmn

lims!0

2(�1)s⇡

s@�

"X

m,n

1((m + �)2 + 2(m + �)n⌧1 + n2|⌧ |2)s

#

plane wave expansion,integrate over positions

K: completely twisted for closed strings

I =Z 1

0

dt

t2GF (�, it/2) =

⇡4

24 sin2 ⇡�� ⇡2

12 0(�)

• use reflection formula for Epstein zeta function

gives derivative of

• finally perform integral over worldsheet modulus

(finite part)

Classical modular form

� · ⌧ =a⌧ + b

c⌧ + d

SL(2, Z)

f(� · ⌧) = (c⌧ + d)wf(⌧)

Classical modular form:reflection formula

(2⇡)�s�(s)'(s) = (�1)k/2(2⇡)s�k�(k � s)'(k � s)

(2⇡)�s�(s)'(s) =Z 1

0(f(iy)� c(0))ys�1dy

f(i/y) = (iy)kf(iy)

cf. Riemann zeta f ! #3

'(s) =1X

n=1

c(n)ns

f(⌧) = c(0) +1X

n=1

c(n)qn

q = e2⇡i⌧

⇡�s�⇣s

2

⌘⇣(s) = ⇡

s�12 �

✓1� s

2

◆⇣(1� s)

Classical modular form

(f |w�)(⌧) = (c⌧ + d)�wf(� · ⌧)

� · ⌧ =a⌧ + b

c⌧ + dSL(2, Z)

(f |w�)(⌧) = f(⌧)

Define Petersson slash operator:

modular form-ness reexpressed as slash invariance:

Automorphic form

P (f, w; ⌧) = 12

X

�2�1\�

f |w�

(f |w�)(⌧) = (c⌧ + d)�wf(� · ⌧)

P (, w; ⌧) = 12

X

(c,d)=1

(c⌧ + d)�we�2⇡i�·⌧

Example: f(⌧) = q�

(holomorphic Eisenstein series for ,not convergent for )

= 0

Poincaré series, “method of images” from “seed” f

�1 =✓

1 n0 1

◆

w 2

(⌧ ! ⌧ + n)

Aaµ

Aaµ

hmn

hmn

holomorphic vertex operatoron D-brane

nonholomorphic vertex operatorin bulk

gauge theory(open strings)

gravity(closed strings)

Automorphic form:nonholomorphic Eisenstein series

⌧2 = Im ⌧

f(⌧) = ⌧ s2seed

e.g. Nakahara’s bookweight zero!

⌧22 (@2

⌧1+ @2

⌧2)⌧ s

2 = s(s� 1)⌧ s2

�z }| {

hyperbolic Laplacian

weight zero!

eigenfunction of �

Im (� · ⌧) = ⌧2|c⌧ + d|2

Es(⌧) =0X

(n,m)

⌧ s2|n+m⌧ |2s

Automorphic form:“generalized”

nonholomorphic Eisenstein series

| {z }mx + ny

z = �x + ⌧y

⌧2 = Im ⌧

z2 = Im z

Es(z, ⌧) =

0X

(n,m)

⌧ s2

|n + m⌧ |2sexp

✓2⇡i

z(n + m⌧)� z(n + m⌧)

⌧ � ⌧

◆

zf(⌧) = ⌧ s

2 exp

✓�2⇡i

z2

⌧2

◆doubly periodic seed:

GF [ ↵� ](z1, z2, ⌧) =

#[↵� ](z12) #0

1(0)#[↵

� ](0) #1(z12), (↵,�) 6= (1/2, 1/2) ,

GB(z1, z2, ⌧) = �↵0

2ln

����#1(z12, ⌧)#0

1(0, ⌧)

����2

+ ↵0 ⇡ (Im z12)2

Im ⌧

z12 = z1 � z22

↵0¯@z@zGB = ��2

(z12) +

1

vol

@zGF = �2(z12)

Worldsheet Green’s functions

bosons:

fermions:

2

↵0¯@z@zGB = ��2

(z12) +

1

vol

Worldsheet Green’s functions(alternative)

! = m + n⌧

p =i

⌧2(n�m⌧)

GB(z, ⌧) =0X

m,n

1|p|2 e2⇡i(pz+pz) =

0X

m,n

⌧22

|n�m⌧ |2 e2⇡i(pz+pz)

generalized nonholomorphic Eisenstein series

plane wave expansion

E1

2

↵0¯@z@zGB = ��2

(z12) +

1

vol

Worldsheet Green’s functions(alternative)

! = m + n⌧

p =i

⌧2(n�m⌧)

GB(z, ⌧) =0X

m,n

1|p|2 e2⇡i(pz+pz) =

0X

m,n

⌧22

|n�m⌧ |2 e2⇡i(pz+pz)

generalized nonholomorphic Eisenstein series

plane wave expansion

E1

resolved: N=2 supersymmetry requires

and it’s true!

@�@�E2 ⇠ ln |#1(�, U)|2 + . . .

Where we left off before:Zeta function regularization

hmn

hmn

lims!0

2(�1)s⇡

s@�

"X

m,n

1((m + �)2 + 2(m + �)n⌧1 + n2|⌧ |2)s

#

brute force:plane wave expansion,integrate over positions

Automorphic form:reflection formula

formally:partition function of twisted boson on torus

e.g. Siegel

formally:torus “Green’s function”but in the twist

0X

m1,m2

⌧ s2|m1 + �1 + ⌧(m2 + �2)|2s ! �(1� s)

�(s)⇡2s�1

0X

m1,m2

⌧1�s2 e2⇡i~m·~�

|m2 + ⌧m1|2�2s.

Integrate worldsheet moduli ( )

State of the art:

Lerche-Nilsson-Schellekens-Warner trick(nice!)

Dixon-Kaplunovsky-Louis unfolding(beautiful!)

⌧

Integrate worldsheet moduli ( )

Brute force (uglier, but works!):

⌧

Z 1

0dy ys�1@� ln#1(�, iy)

reflection in s

twistedholomorphic

Eisenstein seriesGaberdiel, Keller ’09

I(s) =1X

k=1

⇣(2k)�2k�1

Z 1

0dy ys�1(E2k(iy)� 1) .

I(�1) =⇡3

24 sin2(⇡�)� ⇡

12 0(�)

• Selberg-Poincaré series: seed

difficult analytic continuation, not eigenfunction of Laplacian

• keeps T-duality manifest• generalizes Rankin-Selberg-Zagier method

f(⌧) = ⌧ s�w/22 q�

Angelantonj, Florakis, Pioline ’12

AFP approach to tau integrals

AFP approach to tau integrals

• Selberg-Poincaré series: seed

difficult analytic continuation, not eigenfunction of Laplacian

• Niebur-Poincaré series: seed

better analytic continuation, eigenfunction of Laplacian

f(⌧) = ⌧ s�w/22 q�

f(⌧) = Ms,w(�⌧2)e�2⇡i⌧1

Angelantonj, Florakis, Pioline ’12

modifiedWhittaker function

• keeps T-duality manifest• generalizes Rankin-Selberg-Zagier method

• Niebur-Poincaré series: seed

Avoid “unfolding”, keep T-duality manifestgeneralizes Rankin-Selberg-Zagier method

f(⌧) = Ms,w(�⌧2)e�2⇡i⌧1

Angelantonj, Florakis, Pioline ’12

Z

Fdµ ⌧3/2

2 |⌘|6 E2E4(E2E4 � 2E6)�

= �20p

2

dµ =d2⌧

⌧22

AFP approach to tau integrals

Summary

• “string nonrenormalization theorem” for Kähler metric of D-brane moduli

• computed finite constant addition to Kähler metric of closed string moduli

• next set of calculations (angles/flux) can add moduli dependence

M.B., Haack, Kang, Sjörs ’13

• developed/consolidated many techniques for amplitude calculations with D-branes and O-planes

M.B., Haack, Kang ’11

Outlook

• To be sure of Kähler metric, need reduction

• Some of the techniques are brute force: term-by-term integrations, invariances not manifest. AFP approach to tau integrals (Niebur-Poincaré series) leads the way to more general techniques

↵0

Garcia-Etxebarria, Hayashi, Savelli, Shiu ’12Grimm, Savelli, Weissenbacher ’13

Angelantonj, Florakis, Pioline ’12

Outlook: String flux compactifications?

• first step: CFT in AdS with flux

• imaginary antiselfdual fluxes: CFT in pp wave?

• existing anomaly mediation calculations with flux vertex operators neglect backreaction

• interesting to construct better toy models!

e.g. Dolan, Witten ’99

e.g. Conlon, Goodsell, Palti ’10

e.g. Gaberdiel, Green ’02, D’Appollonio, Kiritsis ’02

Thank you!