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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!