Holography, Operators and Finite N

23 Jun 2025, 09:00 → 27 Jun 2025, 16:00 Europe/Stockholm

Motesrum 25 (Albano Building 2)

Pawel Caputa (Stockholm University)

Monday 23 June

09:30 Coffee

1 Loop Space at Finite N and Black Hole Microstates (I)

In these lectures we will introduce basic concepts from invariant theory including Hilbert series, the Hironaka decomposition of the ring of invariants and the Molien-Weyl partition function. Applying these methods to the space of gauge invariant operators ("loops") of multi-matrix models, we are able to completely solve the trace relations exhibiting the algebraic structure of loop space. One of our central results is that the complete loop space can be generated by a finite set of generators of two types. Specifically, for a matrix model with d matrices we show that there are 1+(d-1)N^2 primary invariants that we call gravitons. These generate a Fock space of excitations. There is also a set of secondary invariants which are used at most linearly. The number of secondary invariants grows with the exponential of N^2 providing enough states to explain the entropy of a black hole. Finally, we propose a purely bosonic analog of the recently discovered fortuity mechanism.

Speaker: Robert de Mello Koch (Huzhou University)

12:00 Lunch

Proviant

2 Coulomb branch and integrability

Coulomb branch of the N=4 SYM is a setting where holography (and integrability as it happens) can be applied to a theory with massive particles that have a well-defined S-matrix, form bound states and feature vacuum condensates. I will discuss how condensates (1pt functions) of local operators can be computed using QCD-like sum rules at one loop and to all orders using integrability.

Speaker: Konstantin Zarembo (Nordita)

15:30 Fika/Discussion

Tuesday 24 June

09:30 Coffee

3 Trace Relations, Branes, and Holography (I)

In these lectures, I will discuss a holographic relation between the states of giant graviton branes in anti-de Sitter space and trace relations in the dual gauge theory, and study the implications of this connection for holography at finite N. In a computation of the partition function of BPS states in AdS5 x S5, we will find that the maximal giant is an unstable saddle point and that its states contribute negatively to the partition function. The cohomological treatment of trace relations in N=4 SYM suggests that these D-brane states can be regarded as bulk duals of trace relations. We suggest that the Lefschetz trace formula, which computes bulk observables as an alternating sum of the expectation values in an ensemble of states built on each classical open string vacuum, may be a useful way to formulate the holographic map at finite N. The final lecture will extend these ideas to AdS3 duals of symmetric orbifold CFTs. In this context, we show that a sum over one-loop partition functions of an infinite set of bulk geometries that are asymptotic to AdS3 x S3 x M4 reproduces the integer spectrum of chiral primaries in the symmetric orbifold at finite N.

Speaker: Ji Hoon Lee (ETH Zurich)

12:00 Lunch

Proviant

4 Models of AdS Black Holes from Free CFT Duals

The AdS/CFT correspondence permits the study of bulk duals to free CFTs with gauge symmetries, such as U(N) with large N. In free models CFT correlators are explicitly known. Thermal observables reveal a black hole-like character of massive bulk objects, indicating that classical spacetime emerges by assuming large N in free theories, while strong coupling of the boundary theory is mainly significant quantitatively. I review the origins and the evidence for the above assertions, and preview some new developments. In general, it is important to consider finite but large N to capture non-perturbative gravity effects.

Speaker: Bo Sundborg (OKC, SU)

15:30 Fika/Discussion

Wednesday 25 June

09:30 Coffee

5 Loop Space at Finite N and Black Hole Microstates (II)

In these lectures we will introduce basic concepts from invariant theory including Hilbert series, the Hironaka decomposition of the ring of invariants and the Molien-Weyl partition function. Applying these methods to the space of gauge invariant operators ("loops") of multi-matrix models, we are able to completely solve the trace relations exhibiting the algebraic structure of loop space. One of our central results is that the complete loop space can be generated by a finite set of generators of two types. Specifically, for a matrix model with d matrices we show that there are 1+(d-1)N^2 primary invariants that we call gravitons. These generate a Fock space of excitations. There is also a set of secondary invariants which are used at most linearly. The number of secondary invariants grows with the exponential of N^2 providing enough states to explain the entropy of a black hole. Finally, we propose a purely bosonic analog of the recently discovered fortuity mechanism.

Speaker: Robert de Mello Koch (Huzhou University)

12:00 Lunch

Proviant

6 Trace Relations, Branes, and Holography (II)

In these lectures, I will discuss a holographic relation between the states of giant graviton branes in anti-de Sitter space and trace relations in the dual gauge theory, and study the implications of this connection for holography at finite N. In a computation of the partition function of BPS states in AdS5 x S5, we will find that the maximal giant is an unstable saddle point and that its states contribute negatively to the partition function. The cohomological treatment of trace relations in N=4 SYM suggests that these D-brane states can be regarded as bulk duals of trace relations. We suggest that the Lefschetz trace formula, which computes bulk observables as an alternating sum of the expectation values in an ensemble of states built on each classical open string vacuum, may be a useful way to formulate the holographic map at finite N. The final lecture will extend these ideas to AdS3 duals of symmetric orbifold CFTs. In this context, we show that a sum over one-loop partition functions of an infinite set of bulk geometries that are asymptotic to AdS3 x S3 x M4 reproduces the integer spectrum of chiral primaries in the symmetric orbifold at finite N.

Speaker: Ji Hoon Lee (ETH Zurich)

15:30 Fika/Discussion

18:00 Dinner

TBA

Thursday 26 June

09:30 Coffee

7 Trace Relations, Branes, and Holography (III)

In these lectures, I will discuss a holographic relation between the states of giant graviton branes in anti-de Sitter space and trace relations in the dual gauge theory, and study the implications of this connection for holography at finite N. In a computation of the partition function of BPS states in AdS5 x S5, we will find that the maximal giant is an unstable saddle point and that its states contribute negatively to the partition function. The cohomological treatment of trace relations in N=4 SYM suggests that these D-brane states can be regarded as bulk duals of trace relations. We suggest that the Lefschetz trace formula, which computes bulk observables as an alternating sum of the expectation values in an ensemble of states built on each classical open string vacuum, may be a useful way to formulate the holographic map at finite N. The final lecture will extend these ideas to AdS3 duals of symmetric orbifold CFTs. In this context, we show that a sum over one-loop partition functions of an infinite set of bulk geometries that are asymptotic to AdS3 x S3 x M4 reproduces the integer spectrum of chiral primaries in the symmetric orbifold at finite N.

Speaker: Ji Hoon Lee (ETH Zurich)

12:00 Lunch

Proviant

8 Loop Space at Finite N and Black Hole Microstates (III)

In these lectures we will introduce basic concepts from invariant theory including Hilbert series, the Hironaka decomposition of the ring of invariants and the Molien-Weyl partition function. Applying these methods to the space of gauge invariant operators ("loops") of multi-matrix models, we are able to completely solve the trace relations exhibiting the algebraic structure of loop space. One of our central results is that the complete loop space can be generated by a finite set of generators of two types. Specifically, for a matrix model with d matrices we show that there are 1+(d-1)N^2 primary invariants that we call gravitons. These generate a Fock space of excitations. There is also a set of secondary invariants which are used at most linearly. The number of secondary invariants grows with the exponential of N^2 providing enough states to explain the entropy of a black hole. Finally, we propose a purely bosonic analog of the recently discovered fortuity mechanism.

Speaker: Robert de Mello Koch (Huzhou University)

15:30 Fika/Discussion

Friday 27 June

09:30 Coffee

9 Free Discussion

12:00 Lunch

Proviant