# Nordita-Uppsala joint seminars

The meetings are organized interchangeably in Uppsala and Stockholm. When the meetings are held in Uppsala, they take place on Wednesdays, while when they are held in Stockholm, they take place on Mondays, i.e. according to the respective local schedules. The meetings can be combined with a short informal talk by a local speaker. Lunch is usually provided at 12:30.

### (1) Understanding O(N) symmetry for non-integer N via Deligne categories (2) **Localization of 4d N=1 theories on D2×T2**

**Speaker:** (1) Slava Rychkov, (2) Antonio Pittelli

**Department:** (1) Institut des Hautes Études Scientifiques, (2) Uppsala University

**Time:** 2019-12-06 (1) 10:30-11:30, (2) 13:15-14:15

**Location:** Nordita

**Abstract: **(1) In QFT, we often analytically continue in the number of field components, or in the number of spatial dimensions. I always wished to know which rigorous mathematics underlies these computations. What replaces group symmetries and irreducible representations when one goes to non-integer dimensions? It turns out that the appropriate language is provided by symmetric tensor categories introduced by Pierre Deligne in 2004. They have since been studied by mathematicians, and their relevance to QFT and lattice models was pointed out in our recent work 1911.07895. This blackboard talk will expose relevant ideas, without assuming any prior familiarity with categories. (Joint work with Damon Binder.)

(2) We consider 4d N=1 gauge theories with R-symmetry on a hemisphere times a torus. We apply localization techniques to evaluate the exact partition function through a cohomological reformulation of the supersymmetry transformations. Our results represent the natural elliptic lifts of the lower dimensional analogs as well as a field theoretic derivation of the conjectured 4d holomorphic blocks, from which partition functions of compact spaces with diverse topology can be recovered through gluing. We also analyze the different boundary conditions which can naturally be imposed on the chiral multiplets, which turn out to be either Dirichlet or Robin-like. We show that different boundary conditions are related to each other by coupling the bulk to 3d N=1 degrees of freedom on the boundary three-torus, for which we derive explicit 1-loop determinants.

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### (1) Isolated zeros destroy segments of Fermi surface in holographic models with a lattice (2) Higher form symmetries and superfluids

**Speaker:** (1) Alexander Krikun, (2) Diego Hofman

**Department:** (1) Nordita, (2) Amsterdam

**Time:** 2019-10-09 (1) 10:45-12:00, (2) 13:45-15:00

**Location:** Uppsala, (1) Å10134, (2) Å4101

**Abstract: **(1) I will discuss the fermionic response of the holographic model of a strange metal in the non-homogeneous black hole background created by the periodically modulated chemical potential. Apart from the observation of the standard umklapp and the band formation, which is expected to occur in any system with the periodic potential, I will show that in the regimes of strong modulation the lines of zeros of the spectral function get pushed close to the lines of poles indicating the Fermi surface. This proximity destroys the full segments of the latter. The existence of zeros is the generic feature of the holographic fermionic response, which can be related to the alternatively quantized dual CFT. While the effect which we will see, relies substantially on the particularity of the holographic framework, it is quite similar to the Fermi arc formation in the pseudogap phase of the doped cuprates, observed experimentally. Based on: arXiv:1909.09394.

(2) I will describe superfluid hydrodynamics as the hydrodynamic theory of a system with an emergent anomalous higher-form symmetry. The higher-form charge counts the winding planes of the superfluid -- its constitutive relation replaces the Josephson relation of conventional superfluid hydrodynamics. This formulation puts all hydrodynamic equations on equal footing. The anomalous Ward identity can be used as an alternative starting point to prove the existence of a Goldstone boson, without reference to spontaneous symmetry breaking. This provides an alternative characterization of Landau phase transitions in terms of higher-form symmetries and their anomalies instead of how the symmetries are realized. This treatment is more general and, in particular, includes the case of BKT transitions.

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### (1) CFT correlators, string theory scattering amplitudes, and supersymmetric localization (2) Bootstrapping Multiple Scattering Amplitudes

**Speaker:** (1) Silviu Pufu, (2) João Penedones

**Department:** (1) Princeton, (2) EPFL Lausanne

**Time:** 2019-05-08 (1) 10:45-12:00, (2) 13:45-15:00

**Location:** Uppsala, (1) Å4001, (2) Å4101

**Abstract: **(1) In this talk, I will discuss how a combination of supersymmetric localization, Mellin space, and string theory scattering amplitudes can be used to extract new information about the 4-point functions of single trace superconformal primaries in N=4 SYM theory. These computations also provide new precision tests of AdS/CFT beyond the supergravity approximation.

(2) The S-matrix Bootstrap approach can be used to derive bounds on the coupling constants on any Lorentz invariant quantum field theory due to analyticity, crossing symmetry and unitarity. In our previous work, constraints were derived by studying only the 2-to-2 scattering of the lightest particle of the theory. I will describe the extension of our techniques in order to include constraints stemming from 2-to-2 processes which involve the next-to-lightest particle of the theory. We test these techniques on a family of 1+1 dimensional theories inspired by (and including) the three state Potts field theory. We find that the inclusion of more 2-to-2 amplitudes significantly reduces the single amplitude bounds.

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**(1) **Solving q-Virasoro constraints**, (2) **Integrable system constructed from the geometry of a spectral curve

**Speaker:** (1) Rebecca Lodin, (2) Bertrand Eynard

**Department:** (1) Uppsala, (2) IPhT Saclay

**Time:** 2019-03-25 (1) 10:45-11:45, (2) 13:15-14:15

**Location:** Nordita

**Abstract: **(1) The Virasoro constraints – arising from Ward identities – are a key component in understanding the relation between matrix models and conformal field theories; they provide the set of equations constraining the generating function which can then be solved using CFT methods. These Virasoro constraints can be derived either using differential operators or by using the so-called free field representation of the Virasoro algebra. In this talk I will discuss what happens when these constraints are q-deformed. In particular, I will outline how such q-Virasoro constraints can be derived for a large class of deformed eigenvalue matrix models by an elementary trick of inserting certain q-difference operators under the integral. These q-Virasoro constraints can then be solved recursively and they also have applications for gauge theories.

(2) One usual way of defining an integrable system is in terms of a Tau-function obeying Hirota equations.

The Tau-function (example KdV) is usually defined as a function of an infinite set of times t=(t_{0},t_{1},t_{2},t_{3},...).

Here instead we shall define Tau as a function on the moduli space of spectral curves (plane analytic curves with extra structure), and the “times” can be viewed as local coordinates (but not global in general). The tangent space (i.e. the span of all $\partial/\partial t_{k}$, i.e. Hamiltonians) to the moduli space of spectral curves, is isomorphic to the space of meromorphic 1-forms on the curve, and by form-cycle duality is isomorphic to a Lagrangian in the space of cycles. In other words, we reinterpret Hamiltonians as cycles, and the symplectic Poisson structure as the intersection of cycles.

The topological recursion (TR) defines invariants of the spectral curve, and we show how to get a Tau-function from the TR-invariants. This is an efficient method, which gives new insights on integrable systems.

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**(1) **Closed Strings and Moduli in AdS_{3} x S^{3}**, (2) **Light ray operators, event shapes and Einstein gravity

**Speaker:** (1) Olof Ohlsson Sax, (2) Sacha Zhiboedov

**Department:** (1) Nordita, (2) CERN

**Time:** 2018-11-14 (1) 10:15-11:30, (2) 13:45-15:00

**Location:** Uppsala, (1) Å4001, (2) Å80101

**Abstract: **(1) String theory on AdS_{3} x S^{3} x T^{4} has 20 moduli. I will discuss how the perturbative closed string spectrum changes as we move around this moduli space in both the RR and NSNS flux backgrounds. At weak string coupling, only four of the moduli affect the energies. In the RR background the only effect of these moduli is to change the radius of curvature of the background. On the other hand, in the NSNS background, the moduli introduce worldsheet interactions which enable the use of integrability methods to solve the spectral problem. These results show that the worldsheet theory is integrable across the 20 dimensional moduli space.

(2) I will describe recent developments and new results in the understanding of light ray operators in CFTs. One application of these techniques is computation of event shapes that are used to describe collider experiments for generic CFTs. Another application is new constraints on low-energy gravitational theories that admit a UV completion.

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**(1) **Local tree level BCJ numerators for all multiplicity**, (2) Generalized Wilson loops in N=4 SYM and correlators on a line**

**Speaker:** (1) Fei Teng, (2) Arkady Tseytlin

**Department:** (1) Uppsala, (2) Imperial C

**Time:** 2017-12-18 (1) 11:30-12:00, (2) 13:15-14:15

**Location:** Nordita

**Abstract: **(1) we investigate the expansion of tree level multitrace Einstein-Yang-Mills (EYM) amplitudes. First, we propose two types of recursive expansions of tree level EYM amplitudes with an arbitrary number of gluons, gravitons and traces by those amplitudes with fewer traces or/and gravitons. Then we give many support evidence, including proofs using the Cachazo-He-Yuan (CHY) formula and Britto-Cachazo-Feng-Witten (BCFW) recursive relation. As a byproduct, two types of generalized BCJ relations for multitrace EYM are further proposed, which will be useful in the BCFW proof. After one applies the recursive expansions repeatedly, any multitrace EYM amplitudes can be given in the Kleiss-Kuijf (KK) basis of tree level color ordered Yang-Mills (YM) amplitudes. Thus the Bern-Carrasco-Johansson (BCJ) numerators, as the expansion coefficients, for all multitrace EYM amplitudes are naturally constructed.

(2) We will discuss 1d CFT defined by correlators of operators inserted along BPS Wilson line following arXiv:1706.00756 and some ongoing work about generalizations to non-BPS Wilson loops.

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**(1) **Intersecting surface defects, instantons and 2d CFT**, (2) **Yang-Mills origin of gravitational symmetries

**Speaker:** (1) Yiwen Pan, (2) Leron Borsten

**Department:** (1) Uppsala, (2) Dublin

**Time:** 2017-04-10 (1) 11:30-12:00, (2) 13:15-14:15

**Location:** Nordita

**Abstract: **(1) we will introduce and study the intersection of multiple codimension two defects in 4d N=2 (and 5d N=1) gauge theories. These defects can be understood as 4d-2d-0d or 5d-3d-1d coupled systems. On the one hand, they naturally arise in the gauge theoretic operation called Higgsing, and on the other hand, they have 2d CFT interpretation via degeneration of the AGT correspondence. We will compute the partition functions of intersecting defects via the Higgsing procedure. We then discuss their exact relations to Liouville/Toda correlation functions with degenerate insertions. We will also present new Seiberg-like dualities between the two simplest intersecting defect systems.

(2) A recurring theme in the attempts of understanding the quantum theory of gravity is the idea of “gravity as the square of Yang-Mills”. This involves the tensoring of the multiplet content of two super-Yang-Mills theories to obtain the multiplet content of a supergravity theory. A complete understanding of this correspondence requires studying how the gravitational symmetries originate from the corresponding Yang-Mills factors. In this talk I will show how both the global and local symmetries of linear supergravity arise from the corresponding super-Yang-Mills ones through a dictionary between both fields and transformation parameters. Finally, I will comment on how the dictionary seems to restrict the study of dynamics to particular gauge choices.

**(1) **Non-linear partially massless symmetry in an SO(1,5) continuation of conformal gravity**, (2) **The Nekrasov-Shatashvili Limit of the 5D Superconformal Index

**Speaker:** (1) Luis Apolo, (2) Costis Papageorgakis

**Department:** (1) Nordita, (2) Queen Mary

**Time:** 2016-11-23 (1) 11:30-12:00, (2) 13:45-14:45

**Location:** A73121

**Abstract: **(1) Partially massless spin-2 fields in de Sitter space are characterized by a local symmetry that gauges away the helicity-0 mode of an otherwise massive graviton. Unlike massless or massive spin-2 fields, several “no-go” results exist that rule the existence of non-linear, interacting theories of partially massless fields. In this talk I will review some of these no-go results and describe a model that realizes the partially massless symmetry non-linearly.

(2) I will motivate and introduce the Nekrasov-Shatashvili (NS) limit of the five-dimensional N=1 superconformal index. I will then summarise recent results of Cordova and Shao, who related the so-called Schur limit of the four-dimensional N=2 index to the BPS spectrum of 4D theories on the Coulomb branch. Finally, I will describe how the technology of Cordova-Shao can also reproduce the NS index in 5D for simple examples of U(1) theories. In this picture, the 5D instanton solitons are interpreted as additional flavour nodes to an associated 5D BPS quiver.

**(1) Background constraints in the infinite tension limit of the heterotic string, (2) S-matrix bootstrap and QFT in AdS**

**Speaker:** (1) Thales Azevedo, (2) Jonathan Toledo

**Department:** (1) Uppsala, (2) EPFL

**Time:** 2016-10-24 (1) 11:30-12:00, (2) 13:15-14:15

**Location:** Nordita 132:028

**Abstract: **(1) Shortly after Mason & Skinner introduced the so-called Ambitwistor Strings, Berkovits came up with a pure-spinor analogue of the theory, which was later shown to provide the supersymmetric version of the Cachazo-He-Yuan amplitudes. In this talk, after briefly reviewing the infinite tension limit of the pure-spinor heterotic superstring in flat space, I will show how the model can be coupled to a general curved background, and then proceed to show that all the classical constraints on the supergravity and super Yang-Mills background superfields are elegantly obtained from the sole requirement that the BRST charge be nilpotent.

(2) We consider constraints on the S-matrix of any gapped, Lorentz invariant quantum field theory in 1+1 dimensions due to crossing symmetry and unitarity. In this way we establish rigorous bounds on the cubic couplings of a given theory with a fixed mass spectrum. In special cases we identify interesting integrable theories saturating these bounds. Our analytic bounds match precisely with numerical bounds obtained by considering massive QFT in an AdS box and studying the boundary correlators using the technology of the conformal bootstrap. The methods can be generalized to QFT in higher dimensions and I will comment on work underway in 3+1 dimensions. Talk based on 1607.06109 and 1607.06110 with Miguel Paulos, Joao Penedones, Balt van Rees and Pedro Vieira.

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**Supergravity and the cosmological constant**

**Speaker:** Eric Bergshoeff

**Department:** Groningen University

**Time:** 2015-10-28 13:45-15:00

**Location:** Oseenska rummet (Å73121), Ångströmlaboratoriet

**Abstract: **I will discuss under which conditions a cosmological constant can be added to a given supergravity theory. In the case of a positive cosmological constant I will show how the results are related to brane supersymmetry breaking using an anti-D3-brane.