Supersymmetry on curved spaces and exact partition functions · 2015-11-19 · I’ll review the...

Supersymmetry on curved spacesand

exact partition functions

Giulio Bonelli (SISSA & INFN – Trieste)

November 19, 2015

with A. Bawane, M. Bershtein, M. Ronzani and A. Tanzini

(INFN: ST&FI-GAST-GGS meeting) November 19, 2015 1 / 50

Abstract

I’ll review the technique of equivariant localization of supersymmetric pathintegrals and discuss some properties of space-time manifolds wheresupersymmetry allows the exact evaluation of gauge theories partition functions.As notable examples, we discuss the general structure of partition functions ofN = 2 D = 4 gauge theories on compact (and non compact) toric surfaces andgive some new results about the path integral evaluation for N = 1∗ and N = 2supersymmetric gauge theories on the five sphere.


Introduction

Quantum Field Theory is a tool of paramount importance in the description ofreality at subatomic level (High Energy Particle Physics) as well as at largerscales (Condensed Matter and Statistical Mechanics).

While its Perturbative formulation is very well understood and fruitful, from boththe above viewpoints it is very desirable to be able to understand and keepunder control strong coupling effects.

Perturbative series, when concretely calculable at higher order, are onlyasymptotic ones and can not be resummed as they are (neither in principle) togive a reliable picture at finite coupling.In the last years the exploration of non perturbative aspects of QFTs got boostedat least in two directions:

→ new exact results in supersymmetric gauge theoriesand→ analysis of complex saddles in path integrals [G. Dunne and many others]


Introduction

New exact results in supersymmetric gauge theories

These new results consist of

Exact computation of novel observables in QFTs

New insights into non-perturbative dynamics (RGE flows / dualities)

Connections with other areas of physics and mathematics

and are technically obtained as exact path integral evaluations.→ The computation technique is a very elegant refinement of the non-rinormalisationtheorems in supersymmetric quantum field theories.→ Mathematically, it is an adaptation of the equivariant localization to the path-integralwhich allows, in certain cases, its exact calculation. [much more on this later]

The main theme is supersymmetric quantum field theories on non-trivial backgroundgeometries.


Introduction

A first example is the Witten index:

Z (T D) = TrH(−1)F e−βH

→ Order Parameter for Susy breaking in strongly coupled gauge theories→ up to subtleties, captures NB

E=0 − NFE=0 and is exactly computed

A second example is the Nekrasov partition function for N = 2 theories on R4ε1,ε2

Z(R4ε1,ε2

)=∑

n

qn∫Mn

dµ = e−1

ε1ε2(FSW +O(ε1,ε2))

→ computes the equivariant volumes of instanton moduli spaces→ captures the exact IR physics by the Seiberg-Witten prepotential→ is the generating function of an infinite class of observables? Equivariant vortex counting analog on R2

ε


Introduction

More in general, gauge theories in curved spacetime yield novel observables→ These probe the theory with new scale parameters→ The generating functions are non trivial functions of the parameters of thetheory

Partition function on S1 × SD−1[Romelsberger]

→ measures short representations of susy, not just the vacuum sector→ is a non trivial function of the chemical potentials for global charges→ if the theory flows in the IR to a SCFT, computes the superconformal index

Partition function on SD[Komalgorsky]

→ unified description of [c-, F-, a-]theorem as

Z (SD) = e−F FUV ≥ FIR

→ entanglement entropy of SCFT with spherical entangling surface in R1,D+1

• for N = 2 and D = 4 it is a non trivial computable function of the gaugecoupling [more on this later][Pestun]. Proof of Drukker-Gross conjecture on circularWL in N = 4 and gaussian Matrix Model.


Introduction

New insights into dynamics

check of S-duality→ exact matching of S-dual partition functions→ [Wilson loops]↔ [’t Hooft loops]

dynamics of strongly coupled IR fixed points of renormalization group→ Infrared dualities (Seiberg dualities, Mirror symmetry)→ matching of conformal dimensions of operators in dual phases


Introduction

New connections in math and physAGT correspondence [Alday-Gaiotto-Tachikawa]

ZN=2gauge(S4) = ZToda field th.(Σ2)

3d/3d correspondences [Dimofte-Gaiotto–Gukov]

Zquiver (S3) = ZChern−Simons(M3)

proof of mirror symmetry, exact Kählex potentials [Benini-Cremonesi,Gomis-Lee]

e−K = ZN=(2,2)GLσM (S2) = Zmirror LG

quantum integrable systems and gauge theories [Nekrasov-Okounkov-Shatashvili,

B.-Sciarappa-Vasko-Tanzini]


Introduction

These new results follow from the application of equivariant localization tosupersymmetric path integrals.

This is an extension of the Duistermaat-Heckman theorem to infinite dimensionswhich we are now going to discuss.


Equivariant Localization in a nut-shellConsider a supersymmetric path integral

Z :=

∫D[X ]e−S[X ] (1)

that is it exists a scalar supercharge Q such that the measure in (1) is invariantunder the infinitesimal redefinition δX = QX .Q2 = R is a bosonic symmetry of the theory. We assume it to be compact [atorus action with compact orbits].Then the path integral is supported at the fixed loci of the odd symmetry Q. [Proof:

away from fixed loci the field space can be parameterized by an odd parameter θ describing the Q-flow plus transverse

modes, but the integrand – being Q-invariant – is θ-independent and∫

dθ[const] = 0.]

To evaluate (1) one deforms the action S by a Q-exact term S → S + tQV ,where V is an odd R-invariant fuctional Q2V = RV = 0 (localizing fermion) as

Zt :=

∫D[X ]e−S[X ]−tQV [X ].

This is t-independent [Proof:

− ddt Zt :=

∫D[X ]e−S[X ]−tQV [X ]QV [X ] =

∫D[X ]Q

[e−S[X ]−tQV [X ]V [X ]

]= 0.] and can be evaluated at

large tZ = limt→∞Zt

via semiclassical approximation (as if ~ = 1t ).


Equivariant Localization in a nut-shellNote that

By different choices of V one gets different integral forms of the same object.The semiclassical expansion is well defined if along with a properly chosen V aproper choice of reality conditions on the fields is given.

The path integral then localizes on the space MV of minima of the localizing actionQV = 0 as

Z =

∫MV

dX0e−S[X0][sdet(QV )

(2)X0

]−1/2

There is a common choice of localizing fermion V in supersymmetric theories:distinguish the field coordinate in X = (Φ,Ψ) in bosons and fermions (Grassmanneven and odd fields) and consider V =< Ψ,QΨ > where < ·, · > is a R-invariantscalar product.The localizing action can be expanded in the fermions as

QV =[|QΨ|2

]Ψ=0

+ < Ψ,DΨ > +O(Ψ4)

so that the localization explicitly takes place on the (R-invariant) BPS configurations

MBPS =

Φ| [QΨ]Ψ=0 and Ψ ∈ Ker [D]⊕ coKer [D].

The one loop contribution can be determined by making use of the equivariant indextheorem.


Equivariant Localization in a nut-shellTo compute the one-loop contribution

Find coordinates such that QXe,o = Xo,e, QXo,e = RXe,o [the field space isorganized in multiplets] and the gauge fixing fermion expands asV = XoDXe + . . .

After huge boson-fermion cancelation one stays with (unpaired eigenvalues)

[one − loop] =detcoker(D)Ro

detker(D)Re= EcharR =

∏α

ρα(ε)µα

To compute it: read the weights (ρα, µα) from the equivariant index[indexD] = trker(D)eR − trcoker(D)eR = CcharR =

∑α µαeρα(ε)

The index is computed via Atiyah-Singer index formula

[indexD] = trKerDR − trcoKerDR =∑

P∈fix.pts.

trKerD(P)R − trcoKerD(P)RdetTMBPS (P)(1− R)

The final result typically is in the form

Z(ζ, ε) =

∫MBPS

dm e−S(ζ,m)[one − loop](ε,m) (2)

where ζ are background couplings by which the original action depends and ε are theequivariant parameters in the maximal torus of the global symmetry as R = eε.


Supersymmetric observables & equivariantcohomology

Analogously to the partition function one can evaluate the correlation functions ofsupersymmetric observables, that is the Q-cohomology ring Ker Q/Im Q on theR–invariant fields space.

On the field space therefore susy acts as an R-equivariant differential

The ring of susy observables corresponds to the R-equivariant cohomology ofthe supercharge Q.

Equivariant localization is basically an equivariant contracting homotopy whichreduces the computation of the equivariant Q-cohomolgy

[Ker Q/Im Q]R ∼ H• (MBPS)R

to the moduli space of BPS configurations

The susy path-integral accordingly reduces to the integration over the modulispace and defines the intersection theory of the ring of susy observables.


Supersymmetric gauge theories and integrablesystems

In the general situation the BPS moduli space is stratified by discrete topologicalcharges (e.g. instanton number, vortex number, etc.) asMBPS ∼ ∪nXn.

The topological charge is typically additive in the sense of clustering of finiteaction solutions

Xn × Xn′ → Xn+n′

This makesMBPS an H-space.

The construction is equivariant, so promotes to equivariant cohomology.

By restriction and Kunneth theorem one obtains a graded coproduct on theresummed equivariant cohomology H•T (MBPS) ≡ ⊕nqnH•T (Xn)

promoted to a full Hopf algebra structure: Quantum Integrable system at work!


Rigid supersymmetry and curved spaces

Let’s now look for concrete realizations of the above situation.

Curved spaces where rigid supersymmetry is defined

The program can be realized in different space-time dimensions and withdifferent amounts of supersymmetries

Let’s focus on four and five dimensions


Introduction: 4d N = 1

Let’s start with an appetizer by approaching in a constructive perspective the question:which properties should a (euclidean) four manifold have to host a quantum fieldtheory with rigid N = 1 supersymmetry?In order to answer, let’s study the superalgebra on multiplets.[N.B. This is just 1

2 of the question: the other half is then when these properties arestrong enough to use equivariant localization to compute the supersymmetric pathintegral in a closed form.]


Introduction: 4d N = 1Let’s obtain the supersymmetry algebra with one supercharge (δ2 = 0) parametrizedby a chiral spinor ξα of R-charge +1, as represented on a vector multiplet (in Wessand Bagger notation a gauge field Aαα, gauginos λα and λα, and an auxiliary field D).Supersymmetric variations of the gauge field and the gauginos are fixed by Lorentzcovariance and R-charge conservation to be:

δAαα = ξαλα, δλα = 0, δλα = iξαD + (F +)(αβ)ξβ .

δ2Aαα = 0 and δ2λα = 0 by construction.

δ2λα = iξαδD + [D(ξλ)]+(αβ)ξβ

= iξαδD +∇(αγξβ)λγξβ + (ξ(αDβ)γ λγ)ξβ

=iξαδD +∇(αγξβ)λγξβ + (ξαDβγ λγ)ξβ

[ξ2 = ξαξβεαβ = 0 is used,∇ = covariant derivative containing the spin connection and D = ∇ + A]To get δ2λα = 0, the middle term should align along ξα, so that all the terms canbe compensated by δD. This happens if ∃V such that

∇(αγξβ) = i Vαγξβ + i Vβγξα (3)

and if δD = iDβγ λγξβ − Vβγ λγξβ .

δ2D = 0 identically.



Let’s check the algebra on an anti-chiral multiplet (φ, ψα, F ) to see if there are furtherconditions. Possible Susy variations areδφ = 0, δψα = iξαDααφ, δF = iξαDααψα + ξα[λα, φ] + ξαVααψα

δ2φ = 0 identically.

δ2ψα = iξα(

Dααδφ+ [δAαα, φ])

= iξα(

Dααδφ+ [ξαλα, φ])

= 0 [δφ = 0 and ξ2 = 0]

δ2F =

=iξα[δAαα, ψα] + iξαDαα(iξβDβαφ) + ξα[iξαD + (F +)αβξβ , φ] + ξαVαα(iξβDβαφ)

=− ξα∇ααξβDβαφ− ξαξβDααDβαφ+ ξαξβ [(F +)αβ , φ] + iξαVααξβDβαφ

=− ξα∇ααξβDβαφ+ iξαVααξβDβαφ.

which is vanishing if∇ααξβ = iVααξβ + iWβαξα, (4)

∗ Eq. (4) implies (3) (by symmetrization and setting V = V + W ). Therefore the latteris necessary and sufficient. [One can check that also the linear multiplet closes.]



∇ααξβ = iVααξβ + iWβαξα, (5)

∗∗ Eq.(5) is the generalized Killing spinor equation of [Klare, Tomasiello & Zaffaroni]and [Dumitrescu, Festuccia & Seiberg]. There it is obtained by requiringδ(gravitinos) = 0 in supergravity.In equation (5) V is a background gauge connection for the R-symmetry U(1) bundle.The existence of a solution of equation (4) on a four manifold has someconsequences. It can be shown that [KTZ& DFS] this implies that the almost complexstructure J = ξ ⊗ ξ is integrable and that J is self-dual w.r.t. a compatible hermiteanmetric. [W turns to be a co-closed [d ?W = 0] background 1-form.]∗ Nota Bene: One does not require a Kähler structure.Therefore one concludes that on any hermitean four manifold one can formulate onechiral supersymmetry.


Introduction: generalizations

One can generalize this kind of analysis to

d 6= 4

more supercharges (with different R-charges)

more supersymmetry (more spinorial components that is larger R-symmetrybundle.

The localizing supercharge is effective if δ2 = R is a compact (bosonic) symmetry tobe used in the process of equivariant localization.


Introduction: generalizations

The minimal amount for N = 1 and D = 4 is a linear combination of twosupercharges.

Q = δζ + δζ

Q2 =δζ , δζ

= Lv , where Vαα = ζαζα is a vector on the four manifold.

The four manifold has then to be a T 2 fibration on a Riemann surface.

As first examples, one can have M4 = T 2 × S2 or M4 = S1 × S3.


4d N = 2 theories on C2

D = 4 N = 2 gauge theories have an exactly solvable moduli space of orderparameters while the RG-flow in the coupling constants is still non trivial.The IR effective dynamics of asymptotically free theories can be computed fromsome spectral data (Seiberg-Witten curve & differential) defining an IR integrablesystem. (Toda lattice)The microscopic derivation of the SW solution has been obtained by Nekrasovusing Equivariant Localization.The Ω-background – a specific gravitational coupling – provides a regularizationof the theory on C2, by making the moduli space compact by equivarianceC2 → C2

ε1ε2 (physically, one turns on an attractive gravitational backgroundpotential towards the origin).The instanton part of the Nekrasov partition function ZNek

inst computes theequivariant volume of the ADHM moduli space of instantons and encodes thedata of its equivariant cohomology.Adding the perturbative part is crucial to reproduce the full SW geometry in theε1, ε2 → 0 limit as

ZNekfull = e−

1ε1ε2

[FSW +O(ε)]

The Nekrasov partition function is the building block of the N = 2 D = 4 partitionfunctions on more general geometries.


N = 2 susy gauge theories on four manifoldsThe exact partition function depends in general on all the possible backgroundcouplings that one can turn on by preserving susy up to those removable by asusy transformation.in particular, if the stress-energy tensor of the gauge theory is susy exact, thenthe partition function is independent on the volume moduli/Kähler moduli of thefour manifold, so it is a topological invariant (or better, depends on a reduced setof geometric moduli)long ago (’89) E. Witten proposed that N = 2 D = 4 gauge theories indeedcompute some topological invariants of four manifolds, known as DonaldsonInvariants, and later Moore and Witten proposed a way to make this effective(u-plane integral) in few test casesSome mathematicians (mainly Goettsche, Nakajima and Yoshioka) checked theMW solution and gave some more cases via a very involved technique inalgebric geometrythis proposal got roughly extended by N. Nekrasov to general toric complexsurfaces to explain the computational techniques of GNYour aim (in the 4D case) is to make Nekrasov’s proposal complete and precise,to extend it to equivariant Donaldson invariants and check itthe idea is that indeed the gauge theory, via equivariant localization, is thenatural way to organize those math computations by making them simpler


Let’s study the conditions for the existence of N = 2 D=4 susy on curved manifolds.The computations are much more involved than in the N = 1 case. One can show that

QAµ = iξAσµλA − i ξAσµλA, Qφ = −iξAλA, Qφ = +i ξAλA,

QλA =12σµνξA(Fµν + 8φTµν) + 2σµξADµφ+ σµDµξAφ+ 2iξA[φ, φ] + DABξ

B,

QλA =12σµν ξA(Fµν + 8φTµν) + 2σµξADµφ+ σµDµξAφ− 2i ξA[φ, φ] + DAB ξ

B,

QDAB = −i ξAσmDmλB − i ξBσ

mDmλA + iξAσmDmλB + iξBσ

mDmλA

− 2[φ, ξAλB + ξBλA] + 2[φ, ξAλB + ξBλA].

(6)

(T and T are background tensors) squares to

Q2Aµ = iιV F + iDΦ,

Q2φ = iιV Dφ+ i[Φ, φ] + (w + 2Θ)φ,

Q2φ = iιV Dφ+ i[Φ, φ] + (w − 2Θ)φ,

Q2λA = iιV DλA + i[Φ, λA] + (32

w + Θ)λA +i4

(DρVτ )σρτλA + ΘABλB,

Q2λA = iιV DλA + i[Φ, λA] + (32

w −Θ)λA +i4

(DρVτ )σρτ λA + ΘABλB,

Q2DAB = iιV D(DAB) + i[Φ,DAB] + 2wDAB + ΘACDCB + ΘBCDC

A,

(7)


Introduction: N = 2 D=4 on curved manifoldswhere the parameters

Vµ = 2ξAσµξA, U(1)− isometry

Φ = 2i ξAξAφ+ 2iξAξAφ, gauge transf .

ΘAB = −iξ(AσµDµξB) + iDµξ(Aσ

µξB), SU(2) R − symm.

w = − i2

(ξAσµDµξA + DµξAσµξA), dilatation

Θ = − i4

(ξAσµDµξA − DµξAσµξA) U(1) R − symm. .

(8)

The spinorial parameters satisfy the generalized Killing equations

DµξB + T ρσσρσσµξB −14σµσνDνξB = 0

DµξB + T ρσσρσσµξB −14σµσνDν ξB = 0 (9)

and the auxiliary equations

σµσνDµDνξA + 4DλTµνσµνσλξA = M1ξA,

σµσνDµDν ξA + 4DλTµν σµν σλξA = M2ξA.


N = 2 D=4 on curved manifolds

ξ ∈ Γ(S+ ⊗R⊗LR

)and ξ ∈ Γ

(S− ⊗R† ⊗ L−1

R

), where S± are the spinor

bundles of chirality ±, R is the SU(2) R-symmetry vector bundle and LR is theU(1) R-symmetry line bundle.

The four manifold is subject to the conditions that the above product bundles arewell defined and that a solution to the generalized Killing spinor equations existsand is everywhere well defined.

These conditions differently constrain the space-time four manifold depending onthe choice of R and LR (twisted spinors).

The choice leading to the topologically twisted theory is to set LR = O to be thetrivial line bundle and R = S−.

Therefore, for this choice of the R-symmetry bundles, S+ ⊗ S− ∼ T andS− ⊗ S− ∼ O + T (2,+) with T the tangent bundle and T (2,+) the bundle ofselfdual forms.

The simplest solution is then ξAα = δA

α and ξαA = 0 (Witten’s topological twist)


New solution: N = 2 D=4 on Manifolds with U(1)compact action

If the spacetime manifold admits a compact U(1) action, then one can alwayschoose a metric such that the generating vector field is Killing.

Then one can solve the N = 2 susy conditions with

ξAα = δA

α and ξαA = vµσµαA. (10)

These satisfy the equations with

T = 0 , M1 = M2 = 0 and T = (dλ)− where λ = ?iV ? 1 (11)

The equations reduce to the Killing vector equation for V .


N = 2 D=4 with more natural variablesHaving a smooth solution of the generalized Killing spinor equations, it is thennatural to work out the gauge theory with adapted variables

Φ :=φ− φ ,Φ := 2i ξ2φ+ 2iξ2φ

B+µν := 2(ξ2)2(F +

µν + 8φTµν − 8φSµν)

− (ξAσµν ξB)(ξAσ

κλξB)(Fκλ + 8φTκλ − 8φSκλ)

− 4ξ2V[µD+ν]Φ +

12

(ξ2 + ξ2)(ξAσµν ξB)DAB

η := −i(ξAλA + ξAλA),

Ψµ := i(ξAσµλA − ξAσµλA),

χ+µν := 2ξAσµν ξ

B(ξAλB − ξAλB).

(12)

This change of variables has to be everywhere invertible and well defined. ItsJacobian is given by

J =Jbos

Jferm= 1 , Jbos = Jferm ∼

(ξ2 + ξ2

)4(ξ2)3. (13)

Therefore the change of variables is everywhere well defined if Jbos = Jferm isnever vanishing.In our case we have ξ2 = 1 and ξ2 + ξ2 = 1 + v2 which are nowhere vanishing.


N = 2 D=4 with more natural variables (cohomologicalform)The supersymmetry algebra in terms of the new variables is

QA = Ψ, QΨ = iιV F + DΦ, QΦ = iιV Ψ,

QΦ = η, Qη = i ιV DΦ + i[Φ, Φ],

Qχ+ = B+, QB+ = iLVχ+ + i[Φ, χ+].

(14)

These are the equivariant extension of Witten’s twisted supersymmetry.

In (14) ιV is the contraction with the vector V and LV = DιV + ιV D is thecovariant Lie derivative.

The supercharge (14) manifestly satisfies Q2 = iLV + δgaugeΦ . [There’s still a

consistency condition on the last line, that is the action has to preserve theself-duality of B+ and χ+. This is satisfied iff LV? = ?LV , where ? is the Hodge-?and LV = dιV + ιV d is the Lie derivative. This condition coincides with therequirement that V is an isometry of the four manifold.]

Therefore, we proved that for any four-manifold with a compact U(1) action, once theR-symmetry bundle is properly chosen to fit the equivariant twist, there is a welldefined realization of the corresponding N = 2 equivariant supersymmetry algebra[the twist holds at quantum level! – the Jacobian].


Nekrasov Instanton Partition FunctionConsider N = 2 Yang-Mills theory on C2

ε1,ε2 with gauge group SU(N).

The maximal torus of global symmetries is C2+N? /C?

The BPS moduli space is the moduli space of selfdual configurations F + = 0,that is the solutions of the ADHM constraints (at fixed instanton number k )[B1,B2] + IJ = 0, where Bi ∈ Mat(k × k ,C) and I, J t ∈ Mat(k × N,C), moduloGL(k ,C).

The fixed points are the singular configurations where the instantons are packedat the origin (fixed point of the C2

?)

The partition function reads

Z fullNek (~a, ε1, ε2; Λ) = ZclZ1−loopZinst

where

Zcl = e−~a2ε1ε2 ; Z1−loop = e

∑i<j γΛ(ai−aj ;ε1,ε2) , γ = logΓ2

Zinst =∑

k

Λ2Nk 1k !

εk

(2πiε1ε2)k

∮ k∏s=1

dσs

P(σs)P(σs + ε)

k∏s<t

σ2st (σ

2st − ε2)

(σ2st − ε2

1)(σ2st − ε2

2)(15)

with P(σs) =∏N

j=1(σs − aj ), ε = ε1 + ε2 and Λ a reference scale.


Nekrasov Instanton Partition Function

The link with the equivariant cohmology of the ADHM moduli space becomes cleanerby (carefully) performing the above integrals and obtain

Zinst =∑~Y

Λ2N|~Y |N∏

i,j=1

∏s∈Yi

1E(s) (E(s)− ε)

where E(s) = ai − aj − ε1h(s) + ε2(v(s) + 1) and h(s) and v(s) are the horizontal andvertical relative position of the box s ∈ Yi in the Young diagram Yj .

(I) D-brane realization D(-1)D3 system at the tip of the C2/Z2 quotient: the kD(−1)branes are divided in N groups k =

∑j kj , one for each D3brane. Each

group is in a state labeled by a conjugacy class Yj of the permutations of thekj = |Yj | D(−1)branes.

(II) Via AGT correspondence Zinst is related to Liouville conformal blocks. Thiswill be crucial later on! [The link then is in the fact that Virasoro algebra acts onthe equivariant cohomology of ADHM]


Susy Gauge theory on toric compact surfaces

Toric surfaces are complex four manifolds acted on by C2? with isolated fixed

points.

These can be covered by toric C2 patches glued by C2?-equivariant transition

functions, each patch corresponding to a fixed point.

These are usually presented in terms of a "toric fan" which encodes the fullgeometric data.

Compact toric surfaces are obtained by successive blow-ups (inflating a fixedpoint to a P1) of P2 or of Hirzebruch-Serre Fa surfaces (P1 → Fa → P1 withcharge a; F0 = P1 × P1).

The toric structure implies on Betti numbers that b1 = 0 and b+2 = 1.

Toric surfaces come equipped with C2? invariant Kähler metrics (for example

Fubini-Study for P2 or the sum of round metrics for P1 × P1).


Solving fixed point equtions on toric surfaces

To compute the susy partition function we first solve the fixed points under Q

This corresponds to set [fermions] = 0 and ensure the susy stability asQ[fermions] = 0

From the above algebra, one gets

iιV F + DΦ = 0 and ιV DΦ + [Φ, Φ] = 0 (16)

Choosing the reality condition Φ + Φ† = 0 and studying the integrablityconditions of (16) one gets that all fields are Cartan valued and

(F + Φ)α = F point−likeα + (F + Φ)equivariant

α (17)

where F point−like is the contribution of pointlike instantons at the fixed points ofthe C2

? action (one set for toric patch) and(F + Φ)equivariant,(`) = a + k (`)ω(`) + k (`+1)ω(`+1) where a is a reference Cartanlabel (to be integrated over), ` = 1, . . . , χM labels the toric patches, k’s areinteger flux parameters and ω(`) = ω + H(`) are equivariant extensions (that isiιvω + dH(`) = 0 and H(`)

i (P`) = 0) of the Poincare’ duals of the toric divisors.[For example, for P2 there is a single divisor, but three equivariant extensions]


Evaluation of the path integralthe partition function is supported on vector bundles E such that

µ(E) ≥ µ(G) , where µ(E) ≡∫ω ∧ TrFE

2πrank(E)(18)

for any sub-bundle G, that is on semi-stable vector bundles.

The zero modes for the U(N) theory split in two types: u(1) and su(N) zero modes.

The u(1) zero modes are due to a global u(1) symmetry to gauge fix. This isdone by improving the susy charge and they get eliminated as a perfect quartet.

By correctly treating the gaugino zero modes in the su(N) sector we get preciseinstructions about the integration on the leftover N − 1 Cartan parametersaρ = aα − aβ .

Solving the fixed point equations we bounded the field theory phase to the deepCoulomb branch by declaring Φ and Φ to lay at a generic point in the Cartansubalgebra where the gauge symmetry is completely broken as U(N)→ U(1)N .

⇒ The integral over (a, a) is in CN−1 \ T where T is a tubular neighborhood ofthe hyperplanes set ∆ = aα − aβ = 0 where some extra gauge symmetrycould restore.[For example if N = 2 one gets a single contour integral around the origin in C.]


N = 2 Partition function : explicit formThe complete gauge theory partition function is then given by

Z M,U(N)(q, z ; ε1, ε2)

=∑

k|stable

∮∆

d~a zc1[k ]∏`

ZC2,U(N)full

(q ;~a(`), ε

(`)1 , ε

(`)2

)(19)

where∑k|stable is over fluxes defining equivariant stable vector bundles on M.

∆ is the set of diagonals in the root space.

` = 1, . . . , χ(M) labels the fixed points with weights given by ε(`)1 and ε(`)

2 .

ZC2,U(N)full is the local patch full Nekrasov partition function.

The U(1)N equivariant weights ~a(`) = Φ(`)(P(`))

? Formula (19) was conjectured by Nekrasov and it was proposed as a contourintegral formula for the (equivariant) Donaldson invariants of toric manifolds.?? Formula (19) can be interpreted as a definition of a chiral CFT in d=2, generalizingthe AGT correspondence to compact toric four manifolds. [non-compact ones, namelythe resolution of Hirzebruch-Jung singularities C2/Γ is already known, for exampleC2/Z2 is dual to N = 1 Super Liouville]We matched in detail ? for M = P2 and ?? for M = P1 × P1 which is dual to a chiralversion of Liouville gravity.


Comparison with Donaldson Invariants

If N = 2, then the integral over a12 is a contour integral around the origin.

This prescription agrees and reproduces (in the case c1 odd) with thewall-crossing formulae of Gottsche-Nakajima-Yoshioka.

For manifolds with b+2 = 1 and b−2 = 1, Donaldson invariants are only piece-wise

metric independent: their behavior is described by a chamber structure inH2(M,R) with walls located at H2(M,Z) ∩ H2,−(M,R). [This is encoded in thestability condition] The prototype for this case is P1 × P1.

A common strategy to calculate Donaldson invariants is then given by identifyinga vanishing chamber and then compute the invariants in the other chambers viawall crossing. Our general formulas for N = 2 indeed reproduce the knownresults. [In the non equivariant limit εi → 0 limit, one compares withMoore-Witten u-plane integral.]

Notice that for M = P2 there is a single chamber and the above procedure is notavailable. Moreover, it is neither possible to deform to N = 1 supersymmetrywith mass terms (h(1,1)

(P2) = 1). This makes this case particularly interesting

since it has to be computed directly (and this will be the focus of next talk).


AGT dual interpretationLets go back to our general formula

Z M,U(N)(q, z ; ε1, ε2)

=∑

k|stable

∮∆

d~a zc1[k ]∏`

ZC2,U(N)full

(q ;~a(`), ε

(`)1 , ε

(`)2

)(20)

Have AGT correspondence in mind→ ZC2

inst = ZU(1)FW , and∏

Z1−loop ∼ C3−pt

(20) directly imply that the fixed points of the moduli space of instantons on Mprovide a basis for the representation of L copies of Heisenberg plus WN

algebraeAN(Xp,q) ≡ ⊕L−1

`=0 (H⊕ `WN) (21)with central charge of `WN given by

c` = (N − 1)(

1 + Q2`N(N + 1)

), Q` =

√√√√ ε(`)1

ε(`)2

+

√√√√ ε(`)2

ε(`)1

The overall central charge c =∑` c` coincides with the central charge of the

CFT2 that can be computed from M-theory compactification on M.This extends to the case of compact toric manifolds an analogous constructionvalid for the blow=up of toric local singularities C2/Γp,q (but the compact casetechnically is more difficult because of the integral over the Cartan variables hasto be explicitly performed)


Supersymmetry on squashed S5

One can study the partition function of N = 1∗ susy gauge theories on S5 (andmore in general on 5 dimensional Riemannian manifolds) [Kallen-Zabzine,...]

Notice that S5 is the Hopf S1-fibration over CP2. The round metricds2(S5) = ds2(CP2) + k2, where k = dt + ACP2 and CP2 is equipped with thestandard Fubini-Study structure. The squashed metric is obtained by reducingC3 on the the level set of µ =

∑3i=1 ω

2i |zi |2 and still decomposes according to the

Hopf fibration. This is invariant under a U(1)3 action zi → eiφi zi .

The fixed loci of the V -action are the three S1 at the origins of the three patchesof CP2.

The supersymmetry (already in cohomological form) reads

δA = Ψ δΨ = iV F − Dσ δσ = −iV Ψ

δB = χ δχ = [B, σ] + LV B

where σ is hermitean and (χ,B) are 2-forms in Ker(k ∧+∗) and all in the adjointof the gauge group and the hyper – still in the adjoint –

δΦA = ΨA δΨA = rAB ΦB + iV DΦA + [σ,ΦA]

where rAB rotates the two doublets constrained by the reality condition(

ΦA)∗ = ΩABεΦB .


Supersymmetry on S5

The partition function then localizes on the solitonic configurations given by the“instanton-particles” with world-line the three invariant S1s.

It is given therefore by glueing three copies of the full Nekrasov partition functionon S1 × C2.

The single copy (U(r) theory) is given by

ZS1×C2

U(2),N=1∗(ε1, ε2, β, q,M; a) = ZpertZinst

where

Zpert ∝r∏

α,β=1

Γe(aαβ −M)

Γe(aαβ)and Γe(x) =

∏i,j≥0

(1− e−πiβ(x+iε1+jε2))

and s[x ] ≡ sin(β2 x)

Zinst =∑k≥0

qk∑|~Y |=k

∏α,β

∏s∈Yα

s[eαβ(s)−M]s[eαβ(s)− ε1 − ε2 + M]

s[eαβ(s)]s[eαβ(s)− ε1 − ε2]

where eαβ(s) = aαβ − ε1LYα(s)− ε2(AYβ (s) + 1)


Supersymmetry on S5

The three copies are combined then as

ZS5

U(r),N=1∗(ω,Λ,M) =

∫Rr−1

d~ae−Λ2a2/2ZS5

1−loop(a, ~ω,M)ZS5

inst (Λ, a, ~ω,M)

where Λ2 ≡ 2πω1ω2ω3g2

YM

ZS5

1−loop(a, ~ω,M) =∏α<β

S3(iaαβ)S3(iaαβ + iE)

S3(iaαβ + M)S3(iaαβ −M + iE)

andZS5

inst (Λ, a, ~ω,M) = Z(1)instZ

(2)instZ

(3)inst

whereZ(i)

inst = ZS1×C2

U(r),N=1∗(ε(i)1 , ε

(i)2 , β

(i), q(i),M; a)

and the β(i) = 1ωi

, ε(i)` are the left over ωj , j 6= i and q(i) = e

− 8π3

g2YMωi .


Supersymmetry on S5

If one is able to analyze in detail the analitic properties of the integrand and findsfavourable conditions, then she/he can perform the above integral.That’s what we are aiming to do for the U(2) theory.Let I = Z1−loopZinst , so that

Z S5=

∫R

e−Λ2a2/2I(a)

Actually, via a precise recursion relation and direct analysis, we are able to prove thefollowing points:

The integrand I is an even meromorphic function in a with simple poles only

The poles are in known positions along the imaginary axis in the a space(a = ±ixL)

The integrand I behaves as a constant at large a

then, we can express it as

I(a) = I(∞)

(1 +

∑L

fL(

1a− ixL

− 1a + ixL

))


Supersymmetry on S5

So we can compute the above integral by the formula∫R

e−Λ2a2/2

a− ix= iπsgn(x)eΛ2x2/2Erfc(Λ|x |/

√2)

where Erfc(y) ≡ − 2√π

∫ +∞x e−t2

dt is the complementary error function.Notice that the location of the poles is in a subset of the lattice Z3 at the valuesa = i

∑3i=1 niωi for the vector multiplet and a = i

∑3i=1 niωi + M for the massive hyper.

Moreover the factorI(∞) =

[ZS5

U(1),N=1∗(ωi,Λ,M)]2

where

ZS5

U(1),N=1∗(ωi,Λ,M) =3∏

i=1

ZR4×S1

U(1),N=1∗(ε(i)1 , ε

(i)2 , β

(i), q(i))

is the U(1) gauge theory partition function.


Supersymmetry on S5: N = 2

The formula for the U(2) partition function on S5 is then in the form

ZS5

U(2),N=1∗(ωi,Λ,M) =

=(ZS5

U(1),N=1∗(ωi,Λ))2×

∑ni∈Z3|constraints

fni (ωi,Λ,M)Erfc(

Λ(niωi )/√

2)

where the coefficients fni (ωi,Λ,M) are complicated, but known, functions.These coefficients simplify enormously in the M → 0 limit, that is for the N = 2 theory.In this limit we find the more intelligible formula

ZS5

U(2),N=2(ωi,Λ) =

=(ZS5

U(1),N=2(ωi,Λ))2

Λ

M+

∑ni∈Z3|constraints

e−Λ2n·Q(ωi)·nErfc(

Λ(niωi )/√

2)

where Q(ωi) is a quadratic form which depends on the ωi.


Supersymmetry on S5: N = 2One can add Wilson loop observables to our computation.The equivariant Wilson loop observables are along the three fixed S1s in S5.So we insert in the path integral

W (yi) = TrAd

(3∏

k=1

Peyk

∮S1

k(A+ikσ)

)→ ei

∑k nk yk

and obtain for the generating function the form

WS5

N=2(ωi,Λ|yi) =∑

ni∈Z3|constraints

e−Λ2n·Q(ωi)·n+in·y Erfc(

Λ(niωi )/√

2)

This is in the form of a (generalized) Zwegers mock theta function. [This is generalizedw.r.t. Zwegers case because of the (ωi) dependence in Q, just like Siegel-Naraintheta functions generalize the usual Riemann’s theta function. In the round S5 case(all squashing parameters ωi equal) one removes the generalization because asingle modulus stays, the radius.]Modular properties in the ω’s follows by Poisson resummation and the fact thatf (x) = ex2/2Erfc(x/

√2) is a fixed point under Fourier transform [precisely∫

R+dxf (x) sin(px) =

√π2 f (p)]


Interpretation as regularized Witten index

Since many years there’s a conjecture about the fact that D = 5 susy gaugetheories should capture in the strong coupling the theory of M5-branes on acircle. (I’d say [Douglas])

The radius of the circle is identified with the (dimensional) SYM coupling in 5d.

Therefore, one should identify the N = 2 partition function on S5 with thepartition function of a set of M5-branes on S1 × S5. [or N = (2, 0) An theory onS1 × S5 for n + 1 M5-branes]

Indeed the single M5-brane has been already matched with the 5d partitionfunction on S5 by comparing it with the Witten index of the selfdual tensormultiplet [Lockhart-Vafa]

Nobody (by now) knows much about the multi-M5 brane, but we can try tointerpret our result for 2 M5-branes.

It should be interpreted as the regularized Witten index TrH2

[(−1)F e−βH],

where β is the S1 radius and H the Hamiltonian (plus fugacities of conservedglobal charges) on the Hilbert space H2 of 2 M5-branes.

The two U(1) terms are clearly interpreted as the long distance states (wellseparated M5-branes). The rest should correspond to bound states.


On the regularized Witten index

There are important subtleties in the computation of the Witten index [Cecotti-Girardello,

Imbimbo-Mukhi, Akoury-Comtet, Niemi-Wijewardhana] (’83-’84).

Consider just susy quantum mechanics of a single particle mode on the real linewith superpotential W (X ).

L =12

x2 + iψ†ψ +12

[W ′(x)]2 + ψψ†W ”(x)

and compute the regularized Witten index

∆(β) = Tr(−1)F e−βH =

∫PBCDx Dψe−

∫ β0 L(x,ψ)

The path integral is evaluated by integrating over constant modes, whilefluctations cancel. One stays with

∆(β) =1√2πβ

∫R

dx0 (βW ”(x0)) e−β2 (W ′(x0))2

=1√2π

∫ √βW ′(+∞)

√βW ′(−∞)

dye−y2/2

[Normalization check: harmonic oscillator, W (x) = m2X2/2 and ∆(β) = 1]


Interpretation as regularized Witten index

For a potential with bounded asymptotic values there’s a dependence on β in theform of incomplete Gaussian integrals. (continuum spectrum anomalousbehavior)

Thus one finds

∆(β) =12

[Erfc

(√β

2W ′(+∞)

)− Erfc

(√β

2W ′(−∞)

)]

Therefore the Erfc terms that we find in the dual 2 M5-brane partition funcionhave a possible interpretation as anomalous thresholds terms of an effectivequantum mechanical model.

Notice that this anomalous behavior is usually assumed not to be relevant in theindex computations, but this is done without any justification.

The study of the modular properties of the S5 partition function and itsinterpretation as regularized index are still work in progress.


Open Issues D=4

Compute the partition functions for more general R-symmetry bundle choice(R 6= S±). [Untwisted spinors on P1 × P1 in the paper.]

Extend the equivariant Donaldson recognition to Hirzebruch-Serre surfaces(indeed P1 × P1 would be enough if a blow-up formula is then used)

Extend the residue computation to higher rank→ generalize the Zamolodchikovrecursion relation to W-algebra conformal blocks

Then one can safely use the gauge theory computation to predict equivariantDonaldson invariants at higher rank

Chiral CFT2 defined by a toric diagram: define it off-shell!

Extend the chiral CFT2 definition to non zero fluxes (degenerate field insertions?)

Lift to higher dimensions (6 dim): non-abelian Donaldson Thomas invariants vs.D=6 susy gauge theories


Open Issues D=5

Study the modular properties of the S5 partition function

Interpretation of the S5 partition function as regularized Witten index: what canwe infer from it about the multi M5-brane spectrum?

Study the N = 1∗ deformation and its M-theory interpretation

Obtain higher rank and study sub-leading corrections to the large N limit

Give an interpretation in terms of the theory of M2-branes suspended betweenM5-branes

Compute the 5D partition functions on S1-fibrations over generic compact toriccomplex surfaces


Thank you!


Supersymmetry on curved spaces and exact partition functions · 2015-11-19 · I’ll review the...

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Transcript of Supersymmetry on curved spaces and exact partition functions · 2015-11-19 · I’ll review the...