Sphere Partition Functions, Topology, the Zamolodchikov Metric, and Extremal Correlators Weizmann Institute of Science Efrat Gerchkovitz, Jaume Gomis, ZK [1405.7271] Jaume Gomis, Po-Shen Hsin, ZK, Adam Schwimmer, Nathan Seiberg, Stefan Theisen [1509.08511] Efrat Gerchkovitz, Jaume Gomis, Nafiz Ishtiaque, Avner Karasik, ZK, In Progress
The main purpose of this talk is to re-visit some long-standing problems in SUSY QFT, and to show how modern methods allow to finally solve these problems. I will focus on solving the full chiral ring in (2, 2) theories in d = 2 and N = 2 theories in d = 4. By this I mean an exact computation of all the structure constants as a function of the couplings.
It is important to understand which observables exist in Quantum Field Theory and what is their interpretation. Recently, there has been interest in placing conformal field theories on S d via the stereographic map.
Since this is an angle-preserving transformation, there is a canonical way to implement this compactification of conformal field theories ds 2 = dx i dx i 1 (x 2 + r 2 ) 2 dx i dx i.
Now the theory is free of infrared divergences because space is compact. The UV divergences are the same as in flat space. One can therefore try to compute Z S d [DX ]e S[X ;g ij ], g ij = δ ij 1 (x 2 + r 2 ) 2
Assume the theory has various exactly marginal coupling constants λ i. Then we can compute Is this well defined? Z S d (λ i )
The partition function has various power divergences with the UV cutoff Λ UV of the sort log Z S d = Λ d UV r d + Λ d 2 UV r d 2 +... which correspond to the counter-terms Λ d UV d d x g + Λ d 2 UV d d x gr + Each of these can be multiplied by an arbitrary function of the λ i.
In odd d, we can have a finite piece log Z S d = F (λ i ) and in even d we can have both a log and a finite piece log Z S d = a(λ i ) log(rλ UV ) F (λ i ).
In odd d, the finite piece log Z S d = F (λ i ) is physical! Interesting observable in QFT. Also of great importance in condensed matter physics.
A comment for the advanced: In unitary theories we would like to set the imaginary part of F to zero because of reflection positivity. But actually, this depends on a trivialization of TS 3. Different trivialization give different answers for the imaginary part depending on the framing anomaly coefficient κ g. κ g is useful in the analysis of topological phases of matter.
Can prove that for exactly marginal couplings F (λ i ) = F Measures the total number of degrees of freedom and decreases along renormalization group flows F UV > F IR. Can be mapped to the Vacuum Entanglement Entropy
In even d, log Z S d = a(λ i ) log(rλ UV ) F (λ i ) and we can show that a(λ i ) = a which counts degrees of freedom and decreases along renormalization group flows a UV > a IR. The term F (λ i ) is unphysical. It can be removed by the counter-term d d xf (λ i )E d with E d the Gauss-Bonnet term (which exists only in even dimensions).
It turns out that there are new trace anomalies in the presence of exactly marginal couplings. Consider some general marginal operators in d = 2n CFTs: δs = d d xλ i O i (x). The two-point function O i (x)o j (0) = G ij(λ) x 2d. G ij (λ i ) is the (Zamolodchikov) metric on the conformal manifold {λ i }. This is consistent with conformal Ward identities.
But in momentum space ( ) µ O i (x)o j (0) = G ij (λ)p d 2 log p 2. This logarithm in a CFT signifies a trace anomaly.
( ) µ O i (x)o j (0) = G ij (λ)p d 2 log. The complete trace anomaly is thus given by p 2 T µ µ = c 24π R + G ij(λ) d/2 λ i d/2 λ j. Where the precise action of the derivatives is related to the construction of Paneitz operators and it won t be important for us.
These two trace anomalies are fundamentally different: the usual central charge trace anomaly never manifests itself with logarithms in correlation functions, while the new trace anomaly does come from a logarithm. The former type is called type A and the latter type B.
One can say much more about the general case, but instead I will now discuss SUSY.
Consider a (2, 2) supersymmetric theory in R 2. We have four supercharges Q +, Q, Q +, Q. Suppose the theory is superconformal. Then we have additional four supercharges S +, S, S +, S. In addition we have U(1) V U(1) A R-symmetry.
(2, 2) superconformal field theories often have exactly marginal operators, that are either chiral primaries ( D ± Φ = 0) or twisted chiral primaries ( D + Y = D Y = 0). So we can deform the action by δs = d 2 θλ i Φ i + d 2 θ λ A Y A + c.c. If our SCFT is a sigma model with Calabi-Yau target space, then the chiral and twisted chiral exactly marginal operators describe the complex structure and Kähler deformations of the Calabi-Yau.
So such (2, 2) superconformal field theories are part of some space of conformal theories the conformal manifold M chiral M t.chiral such that both M chiral, and M t.chiral are Kähler spaces with Kähler potential K = K chiral + K t.chiral
Recall that the Kähler potential is not well defined, one has the gauge symmetry This will be important below. K chiral K chiral + F (λ) + F ( λ), K t.chiral K t.chiral + G( λ) + Ḡ( λ).
This space is interesting for various reasons M chiral M t.chiral is the space of massless fields in space-time if we interpret the (2, 2) SCFT as a string worldsheet. K is the Kähler potential of these massless space-time fields. The spaces M chiral, M t.chiral are interchanged by mirror symmetry. Duality symmetries should preserve M chiral M t.chiral. The geometry on the space of CY deformations is interesting to mathematicians.
K is not a holomorphic quantity and so is not accessible via standard techniques that have been utilized in the 90s. So we will use more elaborate ideas.
(2, 2) theories can be placed on S 2 while preserving all the supercharges: m ɛ = γ m η, m ɛ = γ m η, We need to discuss massive subalgebras, namely, subalgebras that only include the isometries. This is because those are the ones controlling the physics of counter-terms.
The two Massive Subalgebras correspond to SU(2 1) A, SU(2 1) B. For example, in SU(2 1) A we retain only SU(2) U(1) V and four supercharges out of the original eight.
Corresponding to these two massive subalgebras, we can define two partition functions, Z S 2 ;A and Z S 2 ;B. The claim is that Z S 2 ;A = r c 3 e K t.chiral Z S 2 ;B = r c 3 e K chiral Notice that this is unlike the non-susy case, in which the finite part of S 2 partition functions is non-universal.
Let us sketch the proof (for SU(2 1) A ). The trace of the EM tensor needs to be supersymmetrized as T µ µ = c 24π R + G i j (λ, λ) µ λ i µ λ j +G AĀ ( λ, λ) µ λ A µ λā + K( λ A, λā). We see that SU(2 1) A fixes the improvement in terms of the Kähler potential for twisted chiral fields. Therefore in the coupling to curved space we have d 2 x grk( λ A, λā).
We see that T µ µ is not well defined under the Kähler gauge transformations and therefore also T ν µ is not well defined. This is unacceptable. A related fact is that the partition function is not a function but a section. Leads to important conclusions: The conformal manifold M has zero Kähler class: i.e. no 2-cycles. K is globally defined. Corollary: The conformal manifold cannot be compact. Classification of singularities?!
The same is true about N = 2 theories in d = 4. There we find Z = r 4a e K(λ, λ)/12 with λ the coefficients of exactly marginal operators δs = λ i d 4 xd 4 θo i + c.c. i
With supersymmetric localization we can now compute these sphere partition functions and thus find the metric on the space of theories. Consider for instance Seiberg-Witten theory with SU(2), N f = 4. Previously there were only perturbative results about this metric but now we can easily compute everything [ ds 2 = dτd τ 3 8(Imτ) 2 135ζ(3) 32π 2 (Imτ) 4 cos(θ)e 8π2 /g 2 3 ] 4π(Imτ) 3/2 + 1575ζ(5) 64π 3 (Imτ) 5 +
One can also obtain explicit forms for the metric on the moduli space of CY s induced by 2d sigma mdoels For example, in a sigma model with only twisted chiral fields, Z S 2 ;B = r c/3 dy 0 e i W (Y 0 )+c.c.
Let us further explain why the claims are correct. Z S 2 ;A = e K t.chiral Z S 2 ;B = e K chiral Z S 4 = e K/12 The partition functions become physical in these theories because the Gauss-Bonnet counter-term cannot be supersymmetrized with a general prefactor. For instance, in SU(2 1) A one finds d 2 θf( λa )R + c.c. = F( λ A )R + c.c. + with F( λ A ) a holomorphic function of the twisted chirals.
These holomorphic counter-terms are in correspondence with Kähler transformations, which are necessary for the formulae to make sense. Z S 2 ;A = e K t.chiral Z S 2 ;B = e K chiral Z S 4 = e K/12
So far we have discussed the metric on the space of theories. A natural generalization is to consider arbitrary extremal correlation functions in N = 2 theories in d = 4. Mathematically one can think of this as a Hermitian vector bundle over M.
Let O I be chiral operators QO I = 0 of dimension I. Extremal correlators are defined to be By extension of the proposal O I1 (x 1 )O I2 (x 2 )...O In (x n )Ō J (w) Z S 4 = e K/12 log Z S 4 = K/12, which allows to compute extremal correlators with I = 2, we claim that all the extremal correlators can be computed by generalized logarithms. There wont be time to explain this, but we note a few results.
It has been suggested long ago that in N = 4 the extremal correlators can be computed at tree-level. Our procedure agrees with this. Our procedure agrees with some kinematical constraints such as the tt equation. It agrees with explicit perturbative computations. One can present results for all the extremal correlators. For example, in Seiberg-Witten theory with gauge group SU(4) Tr(φ 3 )Tr( φ 3 ) = ( g 2 ) 2 ( = YM 30 2295ζ(3) 16π 32π 4 gym 4 + 118575ζ(5) ) 512π 6 gym 6 +
Final Comments Need to explore further N = 2 extremal correlators. Connections with integrability. Work in progress with Gerchkovitz, Gomis, Karasik, Nafiz. In AdS 5, Z S 4 = e K/12 is related to the metric on the space of AdS vacua. Z S 2 ;A,B allows to compute the exact in α metric in 4d supergravity of Type II and heterotic compactifications. Also many connections to mathematics (Gromov-Witten). (4, 4) d = 2 theories? Gaiotto s construction allows to relate our Z S 4 = e K/12 to the metric on punctured Riemann surfaces but one gets in this way a new metric on the moduli space of Riemann surfaces, i.e. not the classical metric. What are the properties of this metric?