Modularity of supersymmetric partition functions

We discover a modular property of supersymmetric partition functions of supersymmetric theories with R-symmetry in four dimensions. This modular property is, in a sense, the generalization of the modular invariance of the supersymmetric partition function of two-dimensional supersymmetric theories on a torus i.e. of the elliptic genus. The partition functions in question are on manifolds homeomorphic to the ones obtained by gluing solid tori. Such gluing involves the choice of a large diffeomorphism of the boundary torus, along with the choice of a large gauge transformation for the background flavor symmetry connections, if present. Our modular property is a manifestation of the consistency of the gluing procedure. The modular property is used to rederive a supersymmetric Cardy formula for four dimensional gauge theories that has played a key role in computing the entropy of supersymmetric black holes. To be concrete, we work with four-dimensional $$ \mathcal{N} $$ N = 1 supersymmetric theories but we expect versions of our result to apply more widely to supersymmetric theories in other dimensions.

T $$ \overline{T} $$ deformation in SCFTs and integrable supersymmetric theories

We calculate the $$ \mathcal{S} $$ S -multiplets for two-dimensional Euclidean $$ \mathcal{N} $$ N = (0, 2) and $$ \mathcal{N} $$ N = (2, 2) superconformal field theories under the T$$ \overline{T} $$ T ¯ deformation at leading order of perturbation theory in the deformation coupling. Then, from these $$ \mathcal{N} $$ N = (0, 2) deformed multiplets, we calculate two- and three-point correlators. We show the $$ \mathcal{N} $$ N = (0, 2) chiral ring’s elements do not flow under the T$$ \overline{T} $$ T ¯ deformation. Specializing to integrable supersymmetric seed theories, such as $$ \mathcal{N} $$ N = (2, 2) Landau-Ginzburg models, we use the thermodynamic Bethe ansatz to study the S-matrices and ground state energies. From both an S-matrix perspective and Melzer’s folding prescription, we show that the deformed ground state energy obeys the inviscid Burgers’ equation. Finally, we show that several indices independent of D-term perturbations including the Witten index, Cecotti-Fendley-Intriligator-Vafa index and elliptic genus do not flow under the T$$ \overline{T} $$ T ¯ deformation.

Defects, nested instantons and comet-shaped quivers

We introduce and study a surface defect in four-dimensional gauge theories supporting nested instantons with respect to the parabolic reduction of the gauge group at the defect. This is engineered from a $$\mathrm{{D3/D7}}$$ D 3 / D 7 -branes system on a non-compact Calabi–Yau threefold X. For $$X=T^2\times T^*{{\mathcal {C}}}_{g,k}$$ X = T 2 × T ∗ C g , k , the product of a two torus $$T^2$$ T 2 times the cotangent bundle over a Riemann surface $${{\mathcal {C}}}_{g,k}$$ C g , k with marked points, we propose an effective theory in the limit of small volume of $${\mathcal C}_{g,k}$$ C g , k given as a comet-shaped quiver gauge theory on $$T^2$$ T 2 , the tail of the comet being made of a flag quiver for each marked point and the head describing the degrees of freedom due to the genus g. Mathematically, we obtain for a single $$\mathrm{{D7}}$$ D 7 -brane conjectural explicit formulae for the virtual equivariant elliptic genus of a certain bundle over the moduli space of the nested Hilbert scheme of points on the affine plane. A connection with elliptic cohomology of character varieties and an elliptic version of modified Macdonald polynomials naturally arises.

Topological vertex for 6d SCFTs with ℤ2-twist

We compute the partition function for 6d $$ \mathcal{N} $$ N = 1 SO(2N) gauge theories compactified on a circle with ℤ2 outer automorphism twist. We perform the computation based on 5-brane webs with two O5-planes using topological vertex with two O5-planes. As representative examples, we consider 6d SO(8) and SU(3) gauge theories with ℤ2 twist. We confirm that these partition functions obtained from the topological vertex with O5-planes indeed agree with the elliptic genus computations.

Higher rank K-theoretic Donaldson-Thomas Theory of points

We exploit the critical structure on the Quot scheme $\text {Quot}_{{{\mathbb {A}}}^3}({\mathscr {O}}^{\oplus r}\!,n)$ , in particular the associated symmetric obstruction theory, in order to study rank r K-theoretic Donaldson-Thomas (DT) invariants of the local Calabi-Yau $3$ -fold ${{\mathbb {A}}}^3$ . We compute the associated partition function as a plethystic exponential, proving a conjecture proposed in string theory by Awata-Kanno and Benini-Bonelli-Poggi-Tanzini. A crucial step in the proof is the fact, nontrival if $r>1$ , that the invariants do not depend on the equivariant parameters of the framing torus $({{\mathbb {C}}}^\ast )^r$ . Reducing from K-theoretic to cohomological invariants, we compute the corresponding DT invariants, proving a conjecture of Szabo. Reducing further to enumerative DT invariants, we solve the higher rank DT theory of a pair $(X,F)$ , where F is an equivariant exceptional locally free sheaf on a projective toric $3$ -fold X. As a further refinement of the K-theoretic DT invariants, we formulate a mathematical definition of the chiral elliptic genus studied in physics. This allows us to define elliptic DT invariants of ${{\mathbb {A}}}^3$ in arbitrary rank, which we use to tackle a conjecture of Benini-Bonelli-Poggi-Tanzini.

Quasi-Jacobi forms, elliptic genera and strings in four dimensions

We investigate the interplay between the enumerative geometry of Calabi-Yau fourfolds with fluxes and the modularity of elliptic genera in four-dimensional string theories. We argue that certain contributions to the elliptic genus are given by derivatives of modular or quasi-modular forms, which may encode BPS invariants of Calabi-Yau or non-Calabi-Yau threefolds that are embedded in the given fourfold. As a result, the elliptic genus is only a quasi-Jacobi form, rather than a modular or quasi-modular one in the usual sense. This manifests itself as a holomorphic anomaly of the spectral flow symmetry, and in an elliptic holomorphic anomaly equation that maps between different flux sectors. We support our general considerations by a detailed study of examples, including non-critical strings in four dimensions.For the critical heterotic string, we explain how anomaly cancellation is restored due to the properties of the derivative sector. Essentially, while the modular sector of the elliptic genus takes care of anomaly cancellation involving the universal B-field, the quasi-Jacobi one accounts for additional B-fields that can be present.Thus once again, diverse mathematical ingredients, namely here the algebraic geometry of fourfolds, relative Gromow-Witten theory pertaining to flux backgrounds, and the modular properties of (quasi-)Jacobi forms, conspire in an intriguing manner precisely as required by stringy consistency.

Surface defects on E-string from 5-brane webs

We study 6d E-string theory with defects on a circle. Our basic strategy is to apply the geometric transition to the supersymmetric gauge theories. First, we calculate the partition functions of the 5d SU(3)0 gauge theory with 10 flavors, which is UV-dual to the 5d Sp(2) gauge theory with 10 flavors, based on two different 5-brane web diagrams, and check that two partition functions agree with each other. Then, by utilizing the geometric transition, we find the surface defect partition function for E-string on ℝ4 × T2. We also discuss that our result is consistent with the elliptic genus. Based on the result, we show how the global symmetry is broken by the defects, and discuss that the breaking pattern depends on where/how we insert the defects.

Duality and mock modularity

We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional \mathcal{N} =4𝒩=4 super Yang-Mills theory on \mathbb{CP}^{2}ℂℙ2 for the gauge group SO(3)SO(3) from the path integral of the effective theory on the Coulomb branch. The holomorphic kernel of this equation, which receives contributions only from the instantons, is not modular but ‘mock modular’. The partition function has correct modular properties expected from SS-duality only after including the anomalous nonholomorphic boundary contributions from anti-instantons. Using M-theory duality, we relate this phenomenon to the holomorphic anomaly of the elliptic genus of a two-dimensional noncompact sigma model and compute it independently in two dimensions. The anomaly both in four and in two dimensions can be traced to a topological term in the effective action of six-dimensional (2,0)(2,0) theory on the tensor branch. We consider generalizations to other manifolds and other gauge groups to show that mock modularity is generic and essential for exhibiting duality when the relevant field space is noncompact.