Alberto Za aronineo.phys.wits.ac.za/public/workshop_11/zaffaroni.pdfIntroduction Introduction It is...

Topologically twisted indices and holography

Alberto Zaffaroni

Milano-Bicocca

10th Joburg Worshop4 September 2018

Alberto Zaffaroni (Milano-Bicocca) HoloBH 2018 1 / 35

Introduction

Introduction

It is an old story that you can put supersymmetric theories on many two andfour-dimensional manifolds M with a topological twist [Witten ’88-’91]

We can then define a topologically twisted index as the supersymmetric partitionfunction

ZM×S1 = Tr(−1)F e−βHtwisted ,

which can be computed using localization. Introducing fugacities and magneticfluxes for the flavor symmetries this is a non-trivial function that can be used for

• testing dualities

• counting of states that has been successfully compared with the entropy ofAdS black holes

In this talk, I discuss the index from three to five dimensions with a view toholography.


Introduction

Introduction

There are many interesting solutions in M theory or type II that interpolatesbetween AdS vacua of different dimensions, first studied by Maldacena-Nunez.They can be intepreted as black objects:

I extremal black holes in AdSd+1 with horizon AdS2 ×Md−1

I supersymmetric black strings in AdSd+1 with horizon AdS3 ×Md−2

or, using holography, as an RG flow across dimensions:

I the dual CFTd on Md−1 flows to superconformal QM in the IR

I the dual CFTd on Md−2 flows to a CFT2 in the IR

In most of them the dual CFTd is topologically twisted along M and we can usethe topologically twisted index to extract information about the flow.


Introduction

Based on

S. M. Hosseini-I. Yakoov-AZ; arXiv 1808.06626

S. M. Hosseini-A. Nedelin-AZ; arXiv 1611.09374

F. Benini-K.Hristov-AZ; arXiv 1511.04085 and 1608.07294

see also

F. Benini-AZ; arXiv 1504.03698 and 1605.06120

S. M. Hosseini-AZ; arXiv 1604.03122

S. M. Hosseini-K.Hristov-AZ; arXiv 1803.07588

S. M. Hosseini-K.Hristov-AZ; arXiv 1705.05383

F.Azzurli-N.Bobev-M.Crichigno-V.Min-AZ; arXiv 1707.04257


Introduction

Part I

Three Dimensions


3D and AdS4 black holes

The topological twist

Consider a 3d N = 2 gauge theory on S2 × S1 where susy is preserved by a twiston S2

(∇µ − iARµ)ε ≡ ∂µε = 0 ,

∫S2

FR = 1

[Witten ’88]

The topologically twisted index is a trace over the Hilbert space H of states on asphere in the presence of a magnetic background for the R and the globalsymmetries,

ZS2×S1 = TrH

((−1)F e iJFA

F

e−βH)

where JF is the generator of the global symmetry.

We can easily replace S2 with a Riemann surface Σg.



Localization

Exact quantities in supersymmetric theories with a charge Q2 = 0 can beobtained by a saddle point approximation

Z =

∫e−S =

∫e−S+t{Q,V} =

t�1e−S|class × detfermions

detbosons

∂tZ =

∫{Q, V}e−S+t{Q,V} = 0

Very old idea that has become very concrete recently, with the computation ofpartition functions on spheres and other manifolds supporting supersymmetry.



The partition function

The path integral on S2 × S1 reduces by localization to a matrix modeldepending on few zero modes of the gauge multiplet V = (Aµ, σ, λ, λ

†,D)

• A magnetic flux on S2, m = 12π

∫S2 F in the co-root lattice

• A Wilson line At along S1

• The vacuum expectation value σ of the real scalar

The path integral reduces to an r -dimensional contour integral of a meromorphicform

1

|W |∑m∈Γh

∮C

Zint(u,m) u = At + iσ

[Benini-AZ ’15]



The partition function

• In each sector with gauge flux m we have a meromorphic form

Zint(u,m) = ZclassZ1-loop

ZCSclass = xkm x = e iu

Z chiral1-loop =

∏ρ∈R

[ xρ/2

1− xρ

]ρ(m)−q+1

q = R charge

Z gauge1-loop =

∏α∈G

(1− xα) (i du)r

• Supersymmetric localization selects a particular contour of integration Cand picks some of the residues of the form Zint(u,m).

[Jeffrey-Kirwan residue - similar to Benini,Eager,Hori,Tachikawa ’13; Hori,Kim,Yi ’14]



AdS4 black holesConsider BPS asymptotically AdS4 static black holes

ds2 = −e2U(r)dt2 + e−2U(r)(dr2 + V (r)2ds2

S2

)• vacua of Einstein-Maxwell theories

L =√g(−1

2R +NΛΣF

ΛµνF

Σµν + gij∂Xi∂Xj + · · · )

• characterized by electric and magnetic charges∫S2

FΛ = Vol(S2) pΛ ,

∫S2

∗FΛ = Vol(S2) qΛ ,

• supersymmetry preserved by a topological twist

∇ε = ∂ε+1

4ωabγabε+ iARε = ∂ε

[Cacciatori,Klemm; Gnecchi,Dall’agata; Hristov,Vandoren;Halmagyi;Katmadas]



AdS4 black holes

These extremal static black holes are asymptotic to AdS4 for r � 1 and withhorizon AdS2 × S2 at some r = rh

ds2 ∼dr2

r2+ r2(−dt2 + dθ2 + sin θ2dφ2)) ds2 ∼ −(r − rh)2dt2 +

dr2

(r − rh)2+ (dθ2 + sin θ2dφ2)



Dual Field Theory Perspective

The dimensional reduction on S2 gives a QM with Landau levels

The topologically twisted index is the Witten index of the QM

ZS2×S1 (∆, p) = TrH((−1)F e iJ∆e−βHp

)• magnetic charges p appear in the Hamiltonian and electric charges q are

introduced using chemical potentials ∆

• the entropy is just the Legendre transform: SBH(q, p) = logZ (p,∆)− i∆q



Going back to black holesThere is family of dyonic black holes in AdS4 × S7 with electric and magneticcharges under U(1)4 ⊂ SO(8). The dual field theory is ABJM with gauge groupU(N)× U(N)

Nk

N−k

Ai

Bj

with quartic superpotential

W = A1B1A2B2 − A1B2A2B1

with R and global symmetries

U(1)4 ⊂ SU(2)A × SU(2)B × U(1)B × U(1)R ⊂ SO(8)

.Alberto Zaffaroni (Milano-Bicocca) HoloBH 2018 13 / 35


ABJM twisted index

Luckily enough, the topologically twisted partition function for ABJM

Z susyS2×S1 =

1

(N!)2

∑m,m∈ZN

∫ N∏i=1

dxi2πixi

dxi2πi xi

xkmi

i x−kmi

i ×N∏i 6=j

(1− xi

xj

)(1− xi

xj

)×

×N∏

i,j=1

( √xixjy1

1− xixjy1

)mi−mj−p1+1( √xixjy2

1− xixjy2

)mi−mj−p2+1

( √xjxiy3

1− xjxiy3

)mj−mi−p3+1( √xjxiy4

1− xjxiy4

)mj−mi−p4+1

ya = e i∆a ,∏

a ya = 1 ,∑

pa = 2

can be solved in the large N limit. There is no cancellation between bosons and

fermions and logZ = O(N3/2). [Benini-AZ; Benini-Hristov-AZ]



Black holes in AdS4 × S7

By resumming mi one can reduce the partition function to a sum of residues

Z susyS2×S1 =

∑x∗

R(x∗)

[Nekrasov-Shatashvili; Closset-Kim-Willet]

The poles have the physical interpretation of the Bethe vacua of the 2D theoryobtained by compactification on S1 (with all KK modes)

exp

(i∂W(u)

∂uj

)= 1

where the twisted superpotential W is given (very) schematically by

W =∑i

k(u2i − u2

i ) +∑I

∑ij

Li2(e i(ui−uj+∆I ))

In the large N Limit one Bethe vacuum dominates the partition function!




For the general class of dyonic AdS4 × S7 black holes with electric charges qa andmagnetic charges pa (no rotation) one reproduces indeed the entropy in this way

S(pa, qa) = logZ (∆a, pa)−∑a

i∆aqa

∣∣∣crit

=∑

aipa

∂W∂∆a

− i∆aqa

∣∣∣crit

The function W = 23 iN

3/2√

2∆1∆2∆3∆4 is interesting:

• It is the free energy of ABJM on S3 as a function of a trial R-symmetry

• It is the prepotential F =√X0X1X2X3 of the N = 2 gauged supergravity

obtained by reducing on S7. The formula above is the attractor mechanism

[Benini-Hristov-AZ; Hosseini; AZ]




• Notice that the explicit expression for the entropy of the AdS4 × S7 blackhole is quite complicated. In the case of purely magnetical black holes withjust

p1 = p2 = p3

is given by

S =

√−1 + 6p1 − 6(p1)2 + (−1 + 2p1)3/2

√−1 + 6p1



Generalisations in AdS4

These results are robust and can be generalised to other BH with known dual:

• Universal black hole. Constant scalar black hole with horizon AdS2 × Σg ,g > 1 in minimal gauged supergravity can be embedded in manycompactifications of M theory and type II.

SBH = logZ = (g − 1)FS3

[Azzurli-Bobev-Chicigno-Min-AZ]

• Mass deformed ABJM [Bobev–Min-Pilch]

• BH in massive type IIA truncation on AdS4 × S6. The dual field theory isChern-Simons U(N) with three adjoints and entropy scales like N5/3. Fullmatch of entropy done by

[Hosseini-Hristov-Passias; Benini-Khachatryan-Milan]

• Adding rotation? No conceptual problem, matrix model a mess.



Part II

Four Dimensions


4D and AdS5 black strings

The topologically twisted index

We now consider the index

ZS2×T 2 (∆, τ) = TrH

((−1)FqL0e i

∑a Qa∆a

)

again with a topological twist along S2. q = e2πiτ is the modular parameter ofthe torus. We just obtain an an elliptic generalization of the 3D index

Z chiral1-loop =

∏ρ∈R

[ iη(q)

θ1(x ; q)

]ρ(m)−q+1

q = R charge



Black strings in AdS5

• Consider now black string solutions interpolating between AdS5 andAdS3 × S2. These are RG flows from a CFT4 to an IR CFT2.

• The topologically twisted index becomes the elliptic genus of the horizonCFT

ZS2×T 2 (∆, τ) = TrH

((−1)FqL0e i

∑3a=1 Qa∆a

)

There are examples in AdS5 × S5 corresponding to twisted compactifications ofN = 4 SYM. The central charge of the CFT2 matches between holography andfield theory. [Benini-Bobev]



N = 4 SYM twisted index

Again, the topologically twisted partition function for N = 4 SYM on T 2 can beevaluated using localization

Z =∑

m∈ZN

∫C

N−1∏i=1

dxi2πixi

η(q)2NN∏j 6=i

θ1

(xixj

; q)

iη(q)

3∏a=1

iη(q)

θa

(xixjya; q

)mi−mj−p1+1

too hard in the large N limit. But high temperature τ = iβ → 0 doable and onefinds a Cardy behaviour

logZ =c2D(∆)

β

where c2D(∆) is the trial central charges of the 2d CFT. [Hosseini-Nedelin-AZ; Hong-Liu]



Back to CardyThe black string compactified on a circle with momentum n give rise to a fourdimensional black hole whose entropy is given by Cardy formula

S(pa) =c2D(pa)

β+ nβ ∼

√nc2D(pa)

and matches the entropy of the dimensionally reduced black hole [Hristov].Adding electric charges

S(pa, qa) =c2D(pa,∆a)

β+ nβ − iqa∆a ∼

√g(pa)(n + qaA

−1ab qb)

where Aab t’Hooft anomaly matrix for global symmetries.

The specific dependence on n + qaA−1ab qb

• matches prediction of the attractor mechanism;

• is a consequence of the elliptic modular properties of the elliptic genus(weak Jacobi form of weight zero).



Part III

Five Dimensions


5D and AdS6 - AdS7


In 5D we can compute the partition function on M4 × S1 where M4 is a Kahlermanifold. Things are complicated by instantons. In the toric case we can furtherdeform the theory with an Omega-background (ε1, ε2). We can then follow asuggestion by Nekrasov for an analogous 4D case:

ZM4×S1 (ε1, ε2) =∑pl

∮Cda

χ(M4)∏l=1

ZC2×S1

Nekrasov

(a + ε

(l)1 pl + ε

(l)2 pl+1, ε

(l)1 , ε

(l)2

)[Nekrasov; Bawane,Bershtein,Bonelli,Ronzani,Tanzini ]

where again p are gauge fluxes on M4 and a = At + iβσ. The limit ε1, ε2 → 0 issmooth.


5D and AdS6 - AdS7


The index is obtained by gluing Nekrasov’s partition functions. For the complexprojective space, P2, with toric data:

n1 = (1, 0) , n2 = (0, 1) , n3 = (−1,−1)

P2

l 1 2 3

ε(l)1 ε1 ε2 − ε1 −ε2

ε(l)2 ε2 −ε1 ε1 − ε2

n1, p1

n2, p2

n3, p3


5D and AdS6 - AdS7


For application to holography we can typically neglect instantons. The limitε1, ε2 → 0 then gives

ZM4×S1 =1

|W|∑{pl}

∮C

rk(G)∏i=1

dxi2πixi

e4π2β

g2YM

TrF c(pl ) ∏α∈G

(1− xα

xα/2

)d(α(pl ))

×∏I

∏ρI∈RI

(1− xρI yνI

xρI/2yνI/2

)dhyper(ρI (pl ),νI (tl ))

.

where the multiplicities at exponents are dictated by index theorem. For P2:

cP2 (pl) = (p1 + p2 + p3)2

dP2 (pl) =1

2(p1 + p2 + p3 + 1) (p1 + p2 + p3 + 2)

. . .


5D and AdS6 - AdS7


Due to the quadratic expression for the multiplicities this matrix model is hard tostudy. A natural conjecture is that, similarly to 3D, some important contributionshould come from the critical points of the Seiberg-Witten prepotential

Fpert(a) = −4π2β

g2YM

TrF(a2)− 1

β2

∑α∈G

Li3(xα) +1

β2

∑I

∑ρI∈RI

Li3(xρI yνI )

of the theory compactified on S1 (with all KK modes).


5D and AdS6 - AdS7

The case of S2 × S2 × S1

We studied the large N of the partition function on S2 × S2 × S1 where we canperform a first dimensional reduction on S2 to a 3D theory.

1

|W|∑m,n

∮C

rk(G)∏i=1

dxi2πixi

e8π2β

g2YM

TrF(mn) ∏α∈G

(1− xα

xα/2

)(α(m)+1)(α(n)+1)

×∏I

∏ρI∈RI

(xρI/2yνI/2

1− xρI yνI

)(ρI (m)+νI (s)−1)(ρI (n)+νI (t)−1)

We can interpret as a sum over n of many 3D topologically twisted indices withan effective twisted superpotential W(a, n). With the conjecture that the large Nlimit is dominated by the combined critical points for W(a, n) and F(a) we canfind interesting results.


5D and AdS6 - AdS7

USp(2N) with Nf flavors and an antisimmetric

This theory is conjectured to have a 5d UV fixed point. We find that:

F(∆I ) ∼ FS5 (∆I ) ∼N5/2

√8− Nf

(∆1∆2)3/2

where FS5 (∆) is the free energy on S5 of the corresponding N = 1 theorycomputed by [Pufu-Jafferis ’12], and

logZS2×S2×S1 ∼∑

sI tJ∂2FS5 (∆)

∂∆I∂∆J

in perfect analogy with 3D results.

• These quantities have the right scaling with N and give a prediction for theentropy of AdS6 black holes with horizon AdS2 × S2 × S2 in massive typeIIA.


5D and AdS6 - AdS7

N = 2 SYM in 5D

The maximally supersymmetric YM theory in five dimensions. We find that:

F(∆I ) ∼ FS5 (∆I ) ∼ βg2YMN3 (∆1∆2)2

where FS5 (∆) is the free energy on S5 of the corresponding N = 2 theorycomputed by [Minahan-Nedelin-Zabzine ’12], and

logZS2×S2×S1 ∼∑

sI tJ∂2FS5 (∆)

∂∆I∂∆J

in perfect analogy with 3D results.


5D and AdS6 - AdS7

N = 2 SYM in 5D

The maximally supersymmetric YM theory in five dimensions is conjectured to beequivalent to the (2, 0) theory on a circle. Then

Z 5D SYMS2×S2×S1 = Z

6D (2,0)S2×S2×T 2

and we expect the index to compute the elliptic genus of the twistedcompactification of the (2, 0) theory on S2 × S2. Indeed we find

logZS2×S2×S1 ∼ iπ

12τcr (sI , tJ ,∆I )

where cr is the trial central charge of the CFT2 computed by [Benini-Bobev]. Theholographic dual is a M theory domain wall interpolating between AdS7 andAdS3 × S2 × S2.


5D and AdS6 - AdS7

Conclusions

The main message of this talk is that there is a lot of interesting physics in thetopological twisted indices in particular for applications to AdS BH.

• Better understanding of 5D partition functions and dualities ...

• Black holes in AdS6 ...

Notice however that what we required magnetic charges. The cases of electricallycharged and rotating BH in AdS5 and AdS7 are still open after more than tenyears of attempts.


5D and AdS6 - AdS7

Conclusions

The AdS4 case is essentially understood in the large charge limit.

• it is interesting to investigate 1/N corrections. In particular logN termcomputed numerically by [Liu-PandoZayas-Rathee-Zhao]

• holographic computation of the partition function [Halmagyi,Lal;

Cabo-Bizet,Kol,PandoZayas-Papadimitriou-Rathee]

• localization in supergravity [Hristov,Lodato,Reys]

Huge literature about asymptotically flat BH (remarkable precision tests includinghigher derivatives). The story about AdS BH is just started.


5D and AdS6 - AdS7

Thank you for the attention !


Alberto Za aronineo.phys.wits.ac.za/public/workshop_11/zaffaroni.pdfIntroduction Introduction It is...

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